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Sparsity-certifying Graph Decompositions Ileana Streinu1∗, Louis Theran2 1 Department of Computer Science, Smith College, Northampton, MA. e-mail: streinu@cs.smith.edu 2 Department of Computer Science, University of Massachusetts Amherst. e-mail: theran@cs.umass.edu Abstract. We describe a new algorithm, the (k, `)-pebble game with colors, and use it to obtain a charac- terization of the family of (k, `)-sparse graphs and algorithmic solutions to a family of problems concern- ing tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characteri- zation of arboricity. We also present a new decomposition that certifies sparsity based on the (k, `)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]. 1. Introduction and preliminaries The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a (k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper range of sparse graphs and 0≤ `≤ k the lower range. In this paper, we present efficient algorithms for finding decompositions that certify sparsity in the upper range of `. Our algorithms also apply in the lower range, which was already ad- dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs and graphs admitting the decomposition coincide. Our algorithms are based on a new characterization of sparse graphs, which we call the pebble game with colors. The pebble game with colors is a simple graph construction rule that produces a sparse graph along with a sparsity-certifying decomposition. We define and study a canonical class of pebble game constructions, which correspond to previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide a unifying framework for all the previously known special cases, including Nash-Williams- Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the properties of the augmenting paths used in matroid union and intersection algorithms[5, 6]. Since the sparse graphs in the upper range are not known to be unions or intersections of the matroids for which there are efficient augmenting path algorithms, these do not easily apply in ∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author. 2 Ileana Streinu, Louis Theran Term Meaning Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− ` Block H in G G is sparse, and H is a tight subgraph Component H of G G is sparse and H is a maximal block Map-graph Graph that admits an out-degree-exactly-one orientation (k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs `Tk Union of ` trees, each vertex is in exactly k of them Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′) Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk Table 1. Sparse graph and decomposition terminology used in this paper. the upper range. Pebble game with colors constructions may thus be considered a strengthening of augmenting paths to the upper range of matroidal sparse graphs. 1.1. Sparse graphs A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤ kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges is called tight. For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge. There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of a sparse graph. A component is a maximal block. Table 1 summarizes the sparse graph terminology used in this paper. 1.2. Sparsity-certifying decompositions A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees. Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight graphs. A map-graph is a graph that admits an orientation such that the out-degree of each vertex is exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map- graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible configuration certifying that each color forms a map-graph. Map-graphs may be equivalently defined (see, e.g., [18]) as having exactly one cycle per connected component.1 A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint map-graphs and ` spanning trees. Another characterization of map-graphs, which we will use extensively in this paper, is as the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that the converse holds as well. 1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases of the bicycle matroid or spanning pseudoforests. Sparsity-certifying Graph Decompositions 3 Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a (2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is shown with a certifying orientation. A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2. Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of tree-pieces in any subgraph G′ has size at least `. Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree- pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges. These count as three tree-pieces, even though they come from the same back tree when the whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three gray tree-pieces and one black one. Table 1 contains the decomposition terminology used in this paper. The decomposition problem. We define the decomposition problem for sparse graphs as tak- ing a graph as its input and producing as output, a decomposition that can be used to certify spar- sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions; and the pebble-game-with-colors decomposition, which is defined in the next section. 2. Historical background The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint, 4 Ileana Streinu, Louis Theran Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a single vertex) and one black tree-piece. Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps- and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19]. In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman) graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay [21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1. Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and proper `Tk graphs for the general upper range. We observe that aside from our new pebble- game-with-colors decomposition, all the combinatorial characterizations of the upper range of sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24]. A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick- son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and left as an open problem using the pebble game to find sparsity certifying decompositions. 3. The pebble game with colors Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm for the decomposition problem later in this paper. Since the phrase “with colors” is necessary only for comparison to [12], we will omit it in the rest of the paper when the context is clear. Sparsity-certifying Graph Decompositions 5 We now present the pebble game with colors. The game is played by a single player on a fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the addition and/or orientation of an edge. At any moment of time, the state of the game is captured by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored by the pebbles on them. While playing the pebble game all edges are directed, and we use the notation vw to indicate a directed edge from v to w. We describe the pebble game with colors in terms of its initial configuration and the allowed moves. Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices are shown as black or gray dots. Edges are colored with the color of the pebble on them. Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k. Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume (w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw to E(H) and put the pebble picked up from v on the new edge. Figure 3(a) shows examples of the add-edge move. Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace vw with wv in E(H); put the pebble that was on vw on v; and put p on wv. Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows examples. The convention in these figures, and throughout this paper, is that pebbles on vertices are represented as colored dots, and that edges are shown in the color of the pebble on them. From the definition of the pebble-slide move, it is easy to see that a particular pebble is always either on the vertex where it started or on an edge that has this vertex as the tail. However, when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is sometimes convenient to think of this path reversal sequence as bringing a pebble from the end of the path to the beginning. The output of playing the pebble game is its complete configuration. Output: At the end of the game, we obtain the directed graph H, along with the location and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble game configuration colors the edges. We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game or that H is a pebble-game graph. Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble- game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a pebble-game decomposition. Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges, and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con- 6 Ileana Streinu, Louis Theran (a) (b) (c) Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph. (c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges contain a cycle and do not contribute a piece of tree to the subgraph. Notation Meaning span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)| peb(V ′) Number of pebbles on V ′ ⊂V out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′ pebi(v) Number of pebbles of color ci on v ∈V outi(v) Number of edges vw colored ci for v ∈V Table 2. Pebble game notation used in this paper. nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′ otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub- graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned by G′. The properties of pebble-game decompositions are studied in Section 6, and Theorem 2 shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows this. For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom- position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees. In the following discussion, we use the notation peb(v) for the number of pebbles on v and pebi(v) to indicate the number of pebbles of colors i on v. Table 2 lists the pebble game notation used in this paper. 4. Our Results We describe our results in this section. The rest of the paper provides the proofs. Sparsity-certifying Graph Decompositions 7 Our first result is a strengthening of the pebble games of [12] to include colors. It says that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games discussed in this paper are our pebble game with colors unless noted explicitly. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. Next we consider pebble-game decompositions, showing that they are a generalization of proper `Tk decompositions that extend to the entire matroidal range of sparse graphs. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained by playing the pebble game defined in the previous section. Notice the similarity between the requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a proper `Tk . Our next results show that for any pebble-game graph, we can specialize its pebble game construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these specialized pebble game constructions canonical, and using canonical pebble game construc- tions, we obtain new direct proofs of existing arboricity results. We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo- sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning trees contributes at least one piece of tree to every subgraph. The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper `Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is sufficient to certify sparsity. We now state the main theorem for the upper and lower range. Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. As corollaries, we obtain the existing decomposition results for sparse graphs. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo- rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem. Our last result improves on this, showing that a canonical pebble game construction, and thus 8 Ileana Streinu, Louis Theran a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in O(n2) time and space. These time and space bounds mean that our algorithm can be combined with those of [12] without any change in complexity. 5. Pebble game graphs In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game with colors. Since many of the relevant properties of the pebble game with colors carry over directly from the pebble games of [12], we refer the reader there for the proofs. We begin by establishing some invariants that hold during the execution of the pebble game. Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following invariants are maintained in H: (I1) There are at least ` pebbles on V . [12] (I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12] (I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) For every vertex v ∈V , outi(v)+pebi(v) = 1. (I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with a pebble of color ci or a cycle. Proof. (I1), (I2), and (I3) come directly from [12]. (I4) This invariant clearly holds at the initialization phase of the pebble game with colors. That add-edge and pebble-slide moves preserve (I4) is clear from inspection. (I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of the same color on it. If there is no pebble of that color reachable, then the path must eventually visit some vertex twice. From these invariants, we can show that the pebble game constructible graphs are sparse. Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight. The main step in proving that every sparse graph is a pebble-game graph is the following. Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce the out degree of v by one. Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w. It follows that any sparse graph has a pebble game construction. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. 6. The pebble-game-with-colors decomposition In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We start with the following lemmas about the structure of monochromatic connected components in H, the directed graph maintained during the pebble game. Sparsity-certifying Graph Decompositions 9 Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub- graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for i = 1, . . . ,k. Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex. Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge. Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′. Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and out-edge tail is the root of a tree-piece. Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle. Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′. Since this argument works for any vertex in any color, for each color there is a partitioning of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each pebble and out-edge tail is the root of a monochromatic tree, as desired. Applied to the whole graph Lemma 11 gives us the following. Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of color ci is the root of a (possibly empty) monochromatic tree-piece of color ci. Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′ edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it to the lower range and giving a construction that finds the tree-pieces, showing the connection between the `+1 pebble condition and the hereditary condition on proper `Tk . We conclude our investigation of arbitrary pebble game constructions with a description of the decomposition induced by the pebble game with colors. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub- graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs. For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game decomposition must be sparse. Apply Theorem 1 to complete the proof. Remark: We observe that a pebble-game decomposition for a Laman graph may be read out of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed, pebble game orientations have a natural correspondence with the bipartite matchings used in 10 Ileana Streinu, Louis Theran Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de- composition is a proper `Tk . In the next section, we show that the pebble game can be specialized to correspond to maps- and-trees and proper `Tk decompositions. 7. Canonical Pebble Game Constructions In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves- tigation of decompositions induced by pebble game constructions by studying the case where a minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15 and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that this is always possible, implying that monochromatic map-graphs are created only when we add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that every color is a forest. Every decomposition characterization of tight graphs discussed above follows immediately from the main theorem, giving new proofs of the previous results in a unified framework. In the proof, we will use two specializations of the pebble game moves. The first is a modi- fication of the add-edge move. Canonical add-edge: When performing an add-edge move, cover the new edge with a color that is on both vertices if possible. If not, then take the highest numbered color present. The second is a restriction on which pebble-slide moves we allow. Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a monochromatic cycle. We call a pebble game construction that uses only these moves canonical. In this section we will show that every pebble-game graph has a canonical pebble game construction (Lemma 14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and maps-and-trees decompositions (Theorem 3 and Theorem 4). We begin with a technical lemma that motivates the definition of canonical pebble game constructions. It shows that the situations disallowed by the canonical moves are all the ways for cycles to form in the lowest ` colors. Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created in exactly one of the following ways: (M1) The edge vw is added with an add-edge move. (M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse edge vw. Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7. By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble game construction, since the color of an edge only changes when it is inserted the first time or a new pebble is put on it by a pebble-slide move. Sparsity-certifying Graph Decompositions 11 vw vw Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are labeled according to their role in the definition of the moves. Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves, respectively, in a (2,0)-pebble game construction. We next show that if a graph has a pebble game construction, then it has a canonical peb- ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa- rately. The proof gives two constructions that implement the canonical add-edge and canonical pebble-slide moves. Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc- tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where `′ = min{k, `}. Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If this is not possible, then there are `+1 distinct colors present. Use the highest numbered color to cover the new edge. Remark: We note that in the upper range, there is always a repeated color, so no canonical add-edge moves create cycles in the upper range. The canonical pebble-slide move is defined by a global condition. To prove that we obtain the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma 9 to only canonical moves. The main step is to show that if there is any sequence of moves that reorients a path from v to w, then there is a sequence of canonical moves that does the same thing. Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading to an add-edge move can be replaced with one that has no (M2) steps and allows the same add-edge move. In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added, then it is possible to do this without creating any monochromatic cycles. 12 Ileana Streinu, Louis Theran Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this the shortcut construction by analogy to matroid union and intersection augmenting paths used in previous work on the lower range. Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one application of the shortcut construction reorients a simple path from a vertex w′ to w, and a path from v to w′ is preserved, the shortcut construction can be applied inductively to find the sequence of moves we want. Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle, shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is simple, and the shortcut construction can be applied inductively to it. Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe that it is rooted at w. Now consider the edges reversed in our sequence of moves. As noted above, before we make any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every move before the one that reverses some edge yz; prepend onto what is left a sequence of moves that moves the pebble on w to z in H ′i . Sparsity-certifying Graph Decompositions 13 Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path. The path where the pebbles move is indicated by doubled lines. Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is (M2); (b) avoiding the (M2) and simplifying the path. Since no edges change color in the beginning of the new sequence, we have eliminated the (M2) move. Because our construction does not change any of the edges involved in the remaining tail of the original sequence, the part of the original path that is left in the new sequence will still be a simple path in H, meeting our initial hypothesis. The rest of the lemma follows by induction. Together Lemma 14 and Lemma 15 prove the following. Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction. Using canonical pebble game constructions, we can identify the tight pebble-game graphs with maps-and-trees and `Tk graphs. 14 Ileana Streinu, Louis Theran Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game graph. For the reverse direction, consider a canonical pebble game construction of a tight graph. From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The definition of the canonical add-edge move implies that there must be at least one pebble of each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma 12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding the required ` edge-disjoint spanning trees. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. We next consider the decompositions induced by canonical pebble game constructions when `≥ k +1. Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb- ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show is that a canonical pebble game construction of a tight graph produces a proper `Tk . By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom- position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must in in exactly k of the trees, as required. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . 8. Pebble game algorithms for finding decompositions A naı̈ve implementation of the constructions in the previous section leads to an algorithm re- quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n) applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running time of Θ(n3) for the decomposition problem. In this section, we describe algorithms for the decomposition problem that run in time O(n2). We begin with the overall structure of the algorithm. Algorithm 17 (The canonical pebble game with colors). Input: A graph G. Output: A pebble-game graph H. Method: – Set V (H) = V (G) and place one pebble of each color on the vertices of H. – For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide moves as described by Lemma 15. Sparsity-certifying Graph Decompositions 15 – If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma 14, otherwise discard vw. – Finally, return H, and the locations of the pebbles. Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction found is canonical, the main theorem shows that the coloring of the edges in H gives a maps- and-trees or proper `Tk decomposition. Complexity. We start by observing that the running time of Algorithm 17 is the time taken to process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an edge of G that is added to H. Each of the pebble game moves can be implemented in constant time. What remains is to describe an efficient way to find and move the pebbles. We use the following algorithm as a subroutine of Algorithm 17 to do this. Algorithm 18 (Finding a canonical path to a pebble.). Input: Vertices v and w, and a pebble game configuration on a directed graph H. Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated. Method: – Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and return ‘no.’ – Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use the array c[] to keep track of the colors of pebbles on vertices and edges after we move them and the array s[] to sketch out a canonical path from v to u by finding a successor for each edge. – Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to the next i. Otherwise, check to see if c[vi+1] = c[ei]. – If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge. – Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i = 1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro- matic c[x] path from xq−1 to x using this same procedure. – Finally, slide pebbles along the path from the original endpoints v to u specified by the successor array s[v], s[s[v]], . . . The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut construction. Efficiency comes from the fact that instead of potentially moving the pebble back and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three times: once in the initial depth-first search, and twice while converting the initial path to a canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time spent processing edges in H. Although we have not discussed this explicity, for the algorithm to be efficient we need to maintain components as in [12]. After each accepted edge, the components of H can be updated in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1) time each. 16 Ileana Streinu, Louis Theran Summarizing, we have shown that the canonical pebble game with colors solves the decom- position problem in time O(n2). 9. An important special case: Rigidity in dimension 2 and slider-pinning In this short section we present a new application for the special case of practical importance, k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com- binatorially, we model the bar-slider frameworks as simple graphs together with some loops placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each color. We characterize the minimally rigid bar-slider graphs [20] as graphs that are: 1. (2,3)-sparse for subgraphs containing no loops. 2. (2,0)-tight when loops are included. We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse graphs studied in our paper [14]. The connection with the pebble games in this paper is the following. Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph. Proof. Follows from invariant (I3) of Lemma 7. In [15], we study a special case of slider pinning where every slider is either vertical or horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction. For this axis parallel slider case, the minimally rigid graphs are characterized by: 1. (2,3)-sparse for subgraphs containing no loops. 2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each monochromatic tree spans exactly one loop of its color. This also has an interpretation in terms of colored pebble games. Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)- pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the graph of a minimally pinned axis-parallel bar-slider framework. Proof. Follows from Theorem 4, and Lemma 12. 10. Conclusions and open problems We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and used it to give an efficient algorithm for finding decompositions of sparse graphs into edge- disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the upper range from [12]. We also used the pebble game with colors to describe a new sparsity-certifying decomposi- tion that applies to the entire matroidal range of sparse graphs. Sparsity-certifying Graph Decompositions 17 We defined and studied a class of canonical pebble game constructions that correspond to either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash- Williams arboricity theorem and a unified proof of the previously studied decomposition cer- tificates of sparsity. Canonical pebble game constructions also show the relationship between the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting paths, which do not apply in the upper range. Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2) algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from dense ones. Their technique is based on efficiently finding matroid union augmenting paths, which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch scanning, which finds groups of disjoint augmenting paths. We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester- mann’s algorithm without changing the running time. The data structures used in the implemen- tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those used to support cyclic scanning. The two major open algorithmic problems related to the pebble game are then: Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain an implementable O(n3/2) algorithm for the lower range. Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble game algorithm for the upper range. In particular, it would be of practical importance to find an implementable O(n3/2) algorithm for decompositions into edge-disjoint spanning trees. References 1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965) 4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992) 7. Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompo- sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11 (2007) 9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics 137, 346–365 (1997) 11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 12. Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Cana- dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007) 16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. Journal of the London Mathematical Society 39, 12 (1964) 18. Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992) 19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Mathematics of Operations Research 10(4), 701–708 (1985) 20. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Pro- ceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365– 374. ACM, New York, NY, USA (2008). 21. Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. Journal of Combinato- rial Theory, Series B 26, 95–112 (1984) 22. Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993) 23. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society 142, 221–230 (1961) 24. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introduction and preliminaries Historical background The pebble game with colors Our Results Pebble game graphs The pebble-game-with-colors decomposition Canonical Pebble Game Constructions Pebble game algorithms for finding decompositions An important special case: Rigidity in dimension 2 and slider-pinning Conclusions and open problems
We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,\ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the $(k,\ell)$-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and Westermann and Hendrickson.
Introduction and preliminaries The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a (k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper range of sparse graphs and 0≤ `≤ k the lower range. In this paper, we present efficient algorithms for finding decompositions that certify sparsity in the upper range of `. Our algorithms also apply in the lower range, which was already ad- dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs and graphs admitting the decomposition coincide. Our algorithms are based on a new characterization of sparse graphs, which we call the pebble game with colors. The pebble game with colors is a simple graph construction rule that produces a sparse graph along with a sparsity-certifying decomposition. We define and study a canonical class of pebble game constructions, which correspond to previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide a unifying framework for all the previously known special cases, including Nash-Williams- Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the properties of the augmenting paths used in matroid union and intersection algorithms[5, 6]. Since the sparse graphs in the upper range are not known to be unions or intersections of the matroids for which there are efficient augmenting path algorithms, these do not easily apply in ∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author. 2 Ileana Streinu, Louis Theran Term Meaning Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− ` Block H in G G is sparse, and H is a tight subgraph Component H of G G is sparse and H is a maximal block Map-graph Graph that admits an out-degree-exactly-one orientation (k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs `Tk Union of ` trees, each vertex is in exactly k of them Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′) Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk Table 1. Sparse graph and decomposition terminology used in this paper. the upper range. Pebble game with colors constructions may thus be considered a strengthening of augmenting paths to the upper range of matroidal sparse graphs. 1.1. Sparse graphs A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤ kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges is called tight. For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge. There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of a sparse graph. A component is a maximal block. Table 1 summarizes the sparse graph terminology used in this paper. 1.2. Sparsity-certifying decompositions A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees. Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight graphs. A map-graph is a graph that admits an orientation such that the out-degree of each vertex is exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map- graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible configuration certifying that each color forms a map-graph. Map-graphs may be equivalently defined (see, e.g., [18]) as having exactly one cycle per connected component.1 A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint map-graphs and ` spanning trees. Another characterization of map-graphs, which we will use extensively in this paper, is as the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that the converse holds as well. 1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases of the bicycle matroid or spanning pseudoforests. Sparsity-certifying Graph Decompositions 3 Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a (2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is shown with a certifying orientation. A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2. Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of tree-pieces in any subgraph G′ has size at least `. Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree- pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges. These count as three tree-pieces, even though they come from the same back tree when the whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three gray tree-pieces and one black one. Table 1 contains the decomposition terminology used in this paper. The decomposition problem. We define the decomposition problem for sparse graphs as tak- ing a graph as its input and producing as output, a decomposition that can be used to certify spar- sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions; and the pebble-game-with-colors decomposition, which is defined in the next section. 2. Historical background The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint, 4 Ileana Streinu, Louis Theran Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a single vertex) and one black tree-piece. Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps- and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19]. In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman) graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay [21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1. Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and proper `Tk graphs for the general upper range. We observe that aside from our new pebble- game-with-colors decomposition, all the combinatorial characterizations of the upper range of sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24]. A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick- son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and left as an open problem using the pebble game to find sparsity certifying decompositions. 3. The pebble game with colors Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm for the decomposition problem later in this paper. Since the phrase “with colors” is necessary only for comparison to [12], we will omit it in the rest of the paper when the context is clear. Sparsity-certifying Graph Decompositions 5 We now present the pebble game with colors. The game is played by a single player on a fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the addition and/or orientation of an edge. At any moment of time, the state of the game is captured by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored by the pebbles on them. While playing the pebble game all edges are directed, and we use the notation vw to indicate a directed edge from v to w. We describe the pebble game with colors in terms of its initial configuration and the allowed moves. Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices are shown as black or gray dots. Edges are colored with the color of the pebble on them. Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k. Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume (w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw to E(H) and put the pebble picked up from v on the new edge. Figure 3(a) shows examples of the add-edge move. Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace vw with wv in E(H); put the pebble that was on vw on v; and put p on wv. Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows examples. The convention in these figures, and throughout this paper, is that pebbles on vertices are represented as colored dots, and that edges are shown in the color of the pebble on them. From the definition of the pebble-slide move, it is easy to see that a particular pebble is always either on the vertex where it started or on an edge that has this vertex as the tail. However, when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is sometimes convenient to think of this path reversal sequence as bringing a pebble from the end of the path to the beginning. The output of playing the pebble game is its complete configuration. Output: At the end of the game, we obtain the directed graph H, along with the location and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble game configuration colors the edges. We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game or that H is a pebble-game graph. Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble- game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a pebble-game decomposition. Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges, and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con- 6 Ileana Streinu, Louis Theran (a) (b) (c) Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph. (c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges contain a cycle and do not contribute a piece of tree to the subgraph. Notation Meaning span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)| peb(V ′) Number of pebbles on V ′ ⊂V out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′ pebi(v) Number of pebbles of color ci on v ∈V outi(v) Number of edges vw colored ci for v ∈V Table 2. Pebble game notation used in this paper. nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′ otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub- graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned by G′. The properties of pebble-game decompositions are studied in Section 6, and Theorem 2 shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows this. For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom- position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees. In the following discussion, we use the notation peb(v) for the number of pebbles on v and pebi(v) to indicate the number of pebbles of colors i on v. Table 2 lists the pebble game notation used in this paper. 4. Our Results We describe our results in this section. The rest of the paper provides the proofs. Sparsity-certifying Graph Decompositions 7 Our first result is a strengthening of the pebble games of [12] to include colors. It says that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games discussed in this paper are our pebble game with colors unless noted explicitly. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. Next we consider pebble-game decompositions, showing that they are a generalization of proper `Tk decompositions that extend to the entire matroidal range of sparse graphs. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained by playing the pebble game defined in the previous section. Notice the similarity between the requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a proper `Tk . Our next results show that for any pebble-game graph, we can specialize its pebble game construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these specialized pebble game constructions canonical, and using canonical pebble game construc- tions, we obtain new direct proofs of existing arboricity results. We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo- sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning trees contributes at least one piece of tree to every subgraph. The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper `Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is sufficient to certify sparsity. We now state the main theorem for the upper and lower range. Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. As corollaries, we obtain the existing decomposition results for sparse graphs. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo- rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem. Our last result improves on this, showing that a canonical pebble game construction, and thus 8 Ileana Streinu, Louis Theran a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in O(n2) time and space. These time and space bounds mean that our algorithm can be combined with those of [12] without any change in complexity. 5. Pebble game graphs In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game with colors. Since many of the relevant properties of the pebble game with colors carry over directly from the pebble games of [12], we refer the reader there for the proofs. We begin by establishing some invariants that hold during the execution of the pebble game. Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following invariants are maintained in H: (I1) There are at least ` pebbles on V . [12] (I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12] (I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) For every vertex v ∈V , outi(v)+pebi(v) = 1. (I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with a pebble of color ci or a cycle. Proof. (I1), (I2), and (I3) come directly from [12]. (I4) This invariant clearly holds at the initialization phase of the pebble game with colors. That add-edge and pebble-slide moves preserve (I4) is clear from inspection. (I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of the same color on it. If there is no pebble of that color reachable, then the path must eventually visit some vertex twice. From these invariants, we can show that the pebble game constructible graphs are sparse. Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight. The main step in proving that every sparse graph is a pebble-game graph is the following. Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce the out degree of v by one. Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w. It follows that any sparse graph has a pebble game construction. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. 6. The pebble-game-with-colors decomposition In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We start with the following lemmas about the structure of monochromatic connected components in H, the directed graph maintained during the pebble game. Sparsity-certifying Graph Decompositions 9 Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub- graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for i = 1, . . . ,k. Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex. Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge. Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′. Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and out-edge tail is the root of a tree-piece. Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle. Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′. Since this argument works for any vertex in any color, for each color there is a partitioning of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each pebble and out-edge tail is the root of a monochromatic tree, as desired. Applied to the whole graph Lemma 11 gives us the following. Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of color ci is the root of a (possibly empty) monochromatic tree-piece of color ci. Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′ edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it to the lower range and giving a construction that finds the tree-pieces, showing the connection between the `+1 pebble condition and the hereditary condition on proper `Tk . We conclude our investigation of arbitrary pebble game constructions with a description of the decomposition induced by the pebble game with colors. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub- graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs. For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game decomposition must be sparse. Apply Theorem 1 to complete the proof. Remark: We observe that a pebble-game decomposition for a Laman graph may be read out of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed, pebble game orientations have a natural correspondence with the bipartite matchings used in 10 Ileana Streinu, Louis Theran Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de- composition is a proper `Tk . In the next section, we show that the pebble game can be specialized to correspond to maps- and-trees and proper `Tk decompositions. 7. Canonical Pebble Game Constructions In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves- tigation of decompositions induced by pebble game constructions by studying the case where a minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15 and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that this is always possible, implying that monochromatic map-graphs are created only when we add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that every color is a forest. Every decomposition characterization of tight graphs discussed above follows immediately from the main theorem, giving new proofs of the previous results in a unified framework. In the proof, we will use two specializations of the pebble game moves. The first is a modi- fication of the add-edge move. Canonical add-edge: When performing an add-edge move, cover the new edge with a color that is on both vertices if possible. If not, then take the highest numbered color present. The second is a restriction on which pebble-slide moves we allow. Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a monochromatic cycle. We call a pebble game construction that uses only these moves canonical. In this section we will show that every pebble-game graph has a canonical pebble game construction (Lemma 14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and maps-and-trees decompositions (Theorem 3 and Theorem 4). We begin with a technical lemma that motivates the definition of canonical pebble game constructions. It shows that the situations disallowed by the canonical moves are all the ways for cycles to form in the lowest ` colors. Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created in exactly one of the following ways: (M1) The edge vw is added with an add-edge move. (M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse edge vw. Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7. By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble game construction, since the color of an edge only changes when it is inserted the first time or a new pebble is put on it by a pebble-slide move. Sparsity-certifying Graph Decompositions 11 vw vw Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are labeled according to their role in the definition of the moves. Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves, respectively, in a (2,0)-pebble game construction. We next show that if a graph has a pebble game construction, then it has a canonical peb- ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa- rately. The proof gives two constructions that implement the canonical add-edge and canonical pebble-slide moves. Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc- tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where `′ = min{k, `}. Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If this is not possible, then there are `+1 distinct colors present. Use the highest numbered color to cover the new edge. Remark: We note that in the upper range, there is always a repeated color, so no canonical add-edge moves create cycles in the upper range. The canonical pebble-slide move is defined by a global condition. To prove that we obtain the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma 9 to only canonical moves. The main step is to show that if there is any sequence of moves that reorients a path from v to w, then there is a sequence of canonical moves that does the same thing. Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading to an add-edge move can be replaced with one that has no (M2) steps and allows the same add-edge move. In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added, then it is possible to do this without creating any monochromatic cycles. 12 Ileana Streinu, Louis Theran Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this the shortcut construction by analogy to matroid union and intersection augmenting paths used in previous work on the lower range. Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one application of the shortcut construction reorients a simple path from a vertex w′ to w, and a path from v to w′ is preserved, the shortcut construction can be applied inductively to find the sequence of moves we want. Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle, shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is simple, and the shortcut construction can be applied inductively to it. Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe that it is rooted at w. Now consider the edges reversed in our sequence of moves. As noted above, before we make any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every move before the one that reverses some edge yz; prepend onto what is left a sequence of moves that moves the pebble on w to z in H ′i . Sparsity-certifying Graph Decompositions 13 Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path. The path where the pebbles move is indicated by doubled lines. Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is (M2); (b) avoiding the (M2) and simplifying the path. Since no edges change color in the beginning of the new sequence, we have eliminated the (M2) move. Because our construction does not change any of the edges involved in the remaining tail of the original sequence, the part of the original path that is left in the new sequence will still be a simple path in H, meeting our initial hypothesis. The rest of the lemma follows by induction. Together Lemma 14 and Lemma 15 prove the following. Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction. Using canonical pebble game constructions, we can identify the tight pebble-game graphs with maps-and-trees and `Tk graphs. 14 Ileana Streinu, Louis Theran Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game graph. For the reverse direction, consider a canonical pebble game construction of a tight graph. From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The definition of the canonical add-edge move implies that there must be at least one pebble of each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma 12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding the required ` edge-disjoint spanning trees. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. We next consider the decompositions induced by canonical pebble game constructions when `≥ k +1. Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb- ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show is that a canonical pebble game construction of a tight graph produces a proper `Tk . By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom- position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must in in exactly k of the trees, as required. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . 8. Pebble game algorithms for finding decompositions A naı̈ve implementation of the constructions in the previous section leads to an algorithm re- quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n) applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running time of Θ(n3) for the decomposition problem. In this section, we describe algorithms for the decomposition problem that run in time O(n2). We begin with the overall structure of the algorithm. Algorithm 17 (The canonical pebble game with colors). Input: A graph G. Output: A pebble-game graph H. Method: – Set V (H) = V (G) and place one pebble of each color on the vertices of H. – For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide moves as described by Lemma 15. Sparsity-certifying Graph Decompositions 15 – If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma 14, otherwise discard vw. – Finally, return H, and the locations of the pebbles. Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction found is canonical, the main theorem shows that the coloring of the edges in H gives a maps- and-trees or proper `Tk decomposition. Complexity. We start by observing that the running time of Algorithm 17 is the time taken to process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an edge of G that is added to H. Each of the pebble game moves can be implemented in constant time. What remains is to describe an efficient way to find and move the pebbles. We use the following algorithm as a subroutine of Algorithm 17 to do this. Algorithm 18 (Finding a canonical path to a pebble.). Input: Vertices v and w, and a pebble game configuration on a directed graph H. Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated. Method: – Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and return ‘no.’ – Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use the array c[] to keep track of the colors of pebbles on vertices and edges after we move them and the array s[] to sketch out a canonical path from v to u by finding a successor for each edge. – Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to the next i. Otherwise, check to see if c[vi+1] = c[ei]. – If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge. – Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i = 1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro- matic c[x] path from xq−1 to x using this same procedure. – Finally, slide pebbles along the path from the original endpoints v to u specified by the successor array s[v], s[s[v]], . . . The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut construction. Efficiency comes from the fact that instead of potentially moving the pebble back and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three times: once in the initial depth-first search, and twice while converting the initial path to a canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time spent processing edges in H. Although we have not discussed this explicity, for the algorithm to be efficient we need to maintain components as in [12]. After each accepted edge, the components of H can be updated in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1) time each. 16 Ileana Streinu, Louis Theran Summarizing, we have shown that the canonical pebble game with colors solves the decom- position problem in time O(n2). 9. An important special case: Rigidity in dimension 2 and slider-pinning In this short section we present a new application for the special case of practical importance, k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com- binatorially, we model the bar-slider frameworks as simple graphs together with some loops placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each color. We characterize the minimally rigid bar-slider graphs [20] as graphs that are: 1. (2,3)-sparse for subgraphs containing no loops. 2. (2,0)-tight when loops are included. We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse graphs studied in our paper [14]. The connection with the pebble games in this paper is the following. Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph. Proof. Follows from invariant (I3) of Lemma 7. In [15], we study a special case of slider pinning where every slider is either vertical or horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction. For this axis parallel slider case, the minimally rigid graphs are characterized by: 1. (2,3)-sparse for subgraphs containing no loops. 2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each monochromatic tree spans exactly one loop of its color. This also has an interpretation in terms of colored pebble games. Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)- pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the graph of a minimally pinned axis-parallel bar-slider framework. Proof. Follows from Theorem 4, and Lemma 12. 10. Conclusions and open problems We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and used it to give an efficient algorithm for finding decompositions of sparse graphs into edge- disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the upper range from [12]. We also used the pebble game with colors to describe a new sparsity-certifying decomposi- tion that applies to the entire matroidal range of sparse graphs. Sparsity-certifying Graph Decompositions 17 We defined and studied a class of canonical pebble game constructions that correspond to either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash- Williams arboricity theorem and a unified proof of the previously studied decomposition cer- tificates of sparsity. Canonical pebble game constructions also show the relationship between the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting paths, which do not apply in the upper range. Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2) algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from dense ones. Their technique is based on efficiently finding matroid union augmenting paths, which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch scanning, which finds groups of disjoint augmenting paths. We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester- mann’s algorithm without changing the running time. The data structures used in the implemen- tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those used to support cyclic scanning. The two major open algorithmic problems related to the pebble game are then: Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain an implementable O(n3/2) algorithm for the lower range. Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble game algorithm for the upper range. In particular, it would be of practical importance to find an implementable O(n3/2) algorithm for decompositions into edge-disjoint spanning trees. References 1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965) 4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992) 7. Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompo- sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11 (2007) 9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics 137, 346–365 (1997) 11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 12. Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Cana- dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007) 16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. Journal of the London Mathematical Society 39, 12 (1964) 18. Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992) 19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Mathematics of Operations Research 10(4), 701–708 (1985) 20. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Pro- ceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365– 374. ACM, New York, NY, USA (2008). 21. Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. Journal of Combinato- rial Theory, Series B 26, 95–112 (1984) 22. Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993) 23. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society 142, 221–230 (1961) 24. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introduction and preliminaries Historical background The pebble game with colors Our Results Pebble game graphs The pebble-game-with-colors decomposition Canonical Pebble Game Constructions Pebble game algorithms for finding decompositions An important special case: Rigidity in dimension 2 and slider-pinning Conclusions and open problems
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The evolution of the Earth-Moon system based on the dark fluid model The evolution of the Earth-Moon system based on the dark matter field fluid model Hongjun Pan Department of Chemistry University of North Texas, Denton, Texas 76203, U. S. A. Abstract The evolution of Earth-Moon system is described by the dark matter field fluid model with a non-Newtonian approach proposed in the Meeting of Division of Particle and Field 2004, American Physical Society. The current behavior of the Earth-Moon system agrees with this model very well and the general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago, which is far beyond the Roche’s limit. The result suggests that the tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The average dark matter field fluid constant derived from Earth-Moon system data is 4.39 × 10-22 s-1m-1. This model predicts that the Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. Key Words. dark matter, fluid, evolution, Earth, Moon, Mars 1. Introduction The popularly accepted theory for the formation of the Earth-Moon system is that the Moon was formed from debris of a strong impact by a giant planetesimal with the Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the formation of the Earth-Moon system, it has been evolving at all time scale. It is well known that the Moon is receding from us and both the Earth’s rotation and Moon’s rotation are slowing. The popular theory is that the tidal friction causes all those changes based on the conservation of the angular momentum of the Earth-Moon system. The situation becomes complicated in describing the past evolution of the Earth-Moon system. Because the Moon is moving away from us and the Earth rotation is slowing, this means that the Moon was closer and the Earth rotation was faster in the past. Creationists argue that based on the tidal friction theory, the tidal friction should be stronger and the recessional rate of the Moon should be greater in the past, the distance of the Moon would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence indicates that the recession of the Moon in the past was slower than the present rate, i. e., the recession has been accelerating with time. Therefore, it must be concluded that tidal friction was very much less in the remote past than we would deduce on the basis of present-day observations (Stacey 1977). This was called “geological time scale difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the tidal friction theory (Brush 1983). But we have to consider the case carefully in various aspects. One possible scenario is that the Earth has been undergoing dynamic evolution at all time scale since its inception, the geological and physical conditions (such as the continent positions and drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at remote past could be substantially different from currently, in which the tidal friction could be much less; therefore, the receding rate of the Moon could be slower. Various tidal friction models were proposed in the past to describe the evolution of the Earth- Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present rate of tidal dissipation is anomalously high because the tidal force is close to a resonance in the response function of ocean (Brush 1983). Kagan gave a detailed review about those tidal friction models (Kagan 1997). Those models are based on many assumptions about geological (continental position and drifting) and physical conditions in the past, and many parameters (such as phase lag angle, multi-mode approximation with time dependent frequencies of the resonance modes, etc.) have to be introduced and carefully adjusted to make their predictions close to the geological evidence. However, those assumptions and parameters are still challenged, to certain extent, as concoction. The second possible scenario is that another mechanism could dominate the evolution of the Earth-Moon system and the role of the tidal friction is not significant. In the Meeting of Division of Particle and Field 2004, American Physical Society, University of California at Riverside, the author proposed a dark matter field fluid model (Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with this model very well. This paper will demonstrate that the past evolution of Moon-Earth system can be described by the dark matter field fluid model without any assumptions about past geological and physical conditions. Although the subject of the evolution of the Earth-Moon system has been extensively studied analytically or numerically, to the author’s knowledge, there are no theories similar or equivalent to this model. 2. Invisible matter In modern cosmology, it was proposed that the visible matter in the universe is about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently invisible which is called dark matter and dark energy, such invisible matter has an anti- gravity property to make the universe expanding faster and faster. If the ratio of the matter components of the universe is close to this hypothesis, then, the evolution of the universe should be dominated by the physical mechanism of such invisible matter, such physical mechanism could be far beyond the current Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian physics could reflect only a corner of the iceberg of the greater physics. If the ratio of the matter components of the universe is close to this hypothesis, then, it should be more reasonable to think that such dominant invisible matter spreads in everywhere of the universe (the density of the invisible matter may vary from place to place); in other words, all visible matter objects should be surrounded by such invisible matter and the motion of the visible matter objects should be affected by the invisible matter if there are interactions between the visible matter and the invisible matter. If the ratio of the matter components of the universe is close to this hypothesis, then, the size of the particles of the invisible matter should be very small and below the detection limit of the current technology; otherwise, it would be detected long time ago with such dominant amount. With such invisible matter in mind, we move to the next section to develop the dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible matter (dark matter, dark energy and possible other terms) is called dark matter here. 3. The dark matter field fluid model In this proposed model, it is assumed that: 1. A celestial body rotates and moves in the space, which, for simplicity, is uniformly filled with the dark matter which is in quiescent state relative to the motion of the celestial body. The dark matter possesses a field property and a fluid property; it can interact with the celestial body with its fluid and field properties; therefore, it can have energy exchange with the celestial body, and affect the motion of the celestial body. 2. The fluid property follows the general principle of fluid mechanics. The dark matter field fluid particles may be so small that they can easily permeate into ordinary “baryonic” matter; i. e., ordinary matter objects could be saturated with such dark matter field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the manner of a sponge moving thru water. The nature of the field property of the dark matter field fluid is unknown. It is here assumed that the interaction of the field associated with the dark matter field fluid with the celestial body is proportional to the mass of the celestial body. The dark matter field fluid is assumed to have a repulsive force against the gravitational force towards baryonic matter. The nature and mechanism of such repulsive force is unknown. With the assumptions above, one can study how the dark matter field fluid may influence the motion of a celestial body and compare the results with observations. The common shape of celestial bodies is spherical. According to Stokes's law, a rigid non- permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds number experiences a resistance force F rvF πμ6−= (1) where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual properties of the dark matter field fluid and its permeation into the sphere, the force F may not be proportional to the radius of the sphere. Also, F may be proportional to the mass of the sphere due to the field interaction. Therefore, with the combined effects of both fluid and field, the force exerted on the sphere by the dark matter field fluid is assumed to be of the scaled form (2) mvrF n−−= 16πη where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η is the dark matter field fluid constant, which is equivalent to μ. The direction of the resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force described by Eq. 2 is velocity-dependent and causes negative acceleration. According to Newton's second law of motion, the equation of motion for the sphere is mvr m n−−= 16πη (3) Then (4) )6exp( 10 vtrvv n−−= πη where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a massive gravitational center, there are three forces in the line between the sphere and the gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and (3) the repulsive force of the dark matter field fluid. The drag force in Eq. 3 reduces the orbital velocity and causes the sphere to move inward to the gravitational center. However, if the sum of the centripetal acceleration force and the repulsive force is stronger than the gravitational force, then, the sphere will move outward and recede from the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is sufficiently slow and the repulsive force is small compared to the gravitational force and centripetal acceleration force, then the rate of receding will be accordingly relatively slow. Therefore, the gravitational force and the centripetal acceleration force can be approximately treated in equilibrium at any time. The pseudo equilibrium equation is GMm 2 2 = (5) where G is the gravitational constant, M is the mass of the gravitational center, and R is the radius of the orbit. Inserting v of Eq. 4 into Eq. 5 yields )12exp( 1 R n−= πη (6) (7) )12exp( 10 trRR n−= πη where R = (8) R0 is the initial distance to the gravitational center. Note that R exponentially increases with time. The increase of orbital energy with the receding comes from the repulsive force of dark matter field fluid. The recessional rate of the sphere is dR n−= 112πη (9) The acceleration of the recession is ( Rr Rd n 21 12 −= πη ) . (10) The recessional acceleration is positive and proportional to its distance to the gravitational center, so the recession is faster and faster. According to the mechanics of fluids, for a rigid non-permeable sphere rotating about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere ωπμ 38 rT −= (11) where ω is the angular velocity of the sphere. The direction of the torque in Eq. 11 is opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted on the sphere is ( ) ωπη mrT n 318 −−= (12) The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The torque causes the negative angular acceleration = (13) where I is the moment of inertia of the sphere in the dark matter field fluid ( )21 2 nrmI −= (14) Therefore, the equation of rotation for the sphere in the dark matter field fluid is ωπη d −−= 120 (15) Solving this equation yields (16) )20exp( 10 tr n−−= πηωω where ω0 is the initial angular velocity. One can see that the angular velocity of the sphere exponentially decreases with time and the angular deceleration is proportional to its angular velocity. For the same celestial sphere, combining Eq. 9 and Eq. 15 yields (17) The significance of Eq. 17 is that it contains only observed data without assumptions and undetermined parameters; therefore, it is a critical test for this model. For two different celestial spheres in the same system, combining Eq. 9 and Eq. 15 yields 67.1 1 −=−=⎟⎟ (18) This is another critical test for this model. 4. The current behavior of the Earth-Moon system agrees with the model The Moon-Earth system is the simplest gravitational system. The solar system is complex, the Earth and the Moon experience not only the interaction of the Sun but also interactions of other planets. Let us consider the local Earth-Moon gravitational system as an isolated local gravitational system, i.e., the influence from the Sun and other planets on the rotation and orbital motion of the Moon and on the rotation of the Earth is assumed negligible compared to the forces exerted by the moon and earth on each other. In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data about the Moon and the Earth from references (Dickey et .al., 1994, and Lang, 1992) are listed below for the readers' convenience to verify the calculation because the data may vary slightly with different data sources. Moon: Mean radius: r = 1738.0 km Mass: m = 7.3483 × 1025 gram Rotation period = 27.321661 days Angular velocity of Moon = 2.6617 × 10-6 rad s-1 Mean distance to Earth Rm= 384400 km Mean orbital velocity v = 1.023 km s-1 Orbit eccentricity e = 0.0549 Angular rotation acceleration rate = -25.88 ± 0.5 arcsec century-2 = (-1.255 ± 0.024) × 10-4 rad century-2 = (-1.260 ± 0.024) × 10-23 rad s-2 Receding rate from Earth = 3.82 ± 0.07 cm year-1 = (1.21 ± 0.02) × 10-9 m s-1 Earth: Mean radius: r = 6371.0 km Mass: m = 5.9742 × 1027 gram Rotation period = 23 h 56m 04.098904s = 86164.098904s Angular velocity of rotation = 7.292115 × 10-5 rad s-1 Mean distance to the Sun Rm= 149,597,870.61 km Mean orbital velocity v = 29.78 km s-1 Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2 The Moon's angular rotation acceleration rate and increase in mean distance to the Earth (receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq. 17, the result is 039.054.1 10662.21021.1 1092509.31026.1 (19) The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number 384400 km is assumed to be the distance from the Moon's surface to the Earth's surface. Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord with the model used here. The difference (about 7.8%) between the values of -1.54 and - 1.67 may come from several sources: 1. Moon's orbital is not a perfect circle 2. Moon is not a perfect rigid sphere. 3. The effect from Sun and other planets. 4. Errors in data. 5. Possible other unknown reasons. The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data sets. The third data set can be used to further test the model. If this model correctly describes the situation at hand, it should give consistent results for different motions. The values of n and η calculated from three different data sets are listed below (Note, the mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are used in the calculation). The value of n: n = 0.64 From the Moon's rotation: η = 4.27 × 10-22 s-1 m-1 From the Earth's rotation: η = 4.26 × 10-22 s-1 m-1 From the Moon's recession: η = 4.64 × 10-22 s-1 m-1 One can see that the three values of η are consistent within the range of error in the data. The average value of η: η = (4.39 ± 0.22) × 10-22 s-1 m-1 By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into Eq. 18, the result is 14.053.1 6371000 1738000 1021.11029.7 1092509.3105.5 )64.01( (20) This is also in accord with the model used here. The dragging force exerted on the Moon's orbital motion by the dark matter field fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between the Moon and the Earth ~ 1.90 × 1020 N; and the torque exerted by the dark matter field fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm, respectively. 5. The evolution of Earth-Moon system Sonett et al. found that the length of the terrestrial day 900 million years ago was about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al., 1996). According to the model presented here, back in that time, the length of the day was about 19.2 hours, this agrees very well with Sonett et al.'s result. Another critical aspect of modeling the evolution of the Earth-Moon system is to give a reasonable estimate of the closest distance of the Moon to the Earth when the system was established at 4.5 billion years ago. Based on the dark matter field fluid model, and the above result, the closest distance of the Moon to the Earth was about 259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago, this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion years ago was 250000 km, this is probably the most reasonable number that most astronomers believe and it agrees excellently with the result of this model. The closest distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or 242000 km (Hansen, 1982). According to this model, the length of day of the Earth was about 8 hours at 4.5 billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the length of day of the Earth with the age of the Earth-Moon system described by this model along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One can see that those data fit this model very well in their time range. Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model for past 900 million years. One can see that the model agrees with the geological and fossil data beautifully. The important difference of this model with early models in describing the early evolution of the Earth-Moon system is that this model is based only on current data of the Moon-Earth system and there are no assumptions about the conditions of earlier Earth rotation and continental drifting. Based on this model, the Earth-Moon system has been smoothly evolving to the current position since it was established and the recessional rate of the Moon has been gradually increasing, however, this description does not take it into account that there might be special events happened in the past to cause the suddenly significant changes in the motions of the Earth and the Moon, such as strong impacts by giant asteroids and comets, etc, because those impacts are very common in the universe. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially increases with time. One may then imagine that the recessional rate will quickly become very large. The increase is in fact extremely slow. The Moon's recessional rate will be 3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years. However, whether the Moon's recession will continue or at some time later another mechanism will take over is not known. It should be understood that the tidal friction does affect the evolution of the Earth itself such as the surface crust structure, continental drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s rotation, however, such role is not a dominant mechanism. Unfortunately, there is no data available for the changes of the Earth's orbital motion and all other members of solar system. According to this model and above results, the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km century-1, the length of a year increases about 6.8 ms and the change of the temperature is -1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on the Earth. The length of a year at 1 billion years ago would be 80% of the current length of the year. However, much evidence (growth-bands of corals and shellfish as well as some other evidences) suggest that there has been no apparent change in the length of the year over the billion years and the Earth's orbital motion is more stable than its rotation. This suggests that dark matter field fluid is circulating around Sun with the same direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the Earth's orbital motion experiences very little or no dragging force from the dark matter field fluid. However, this is a conjecture, extensive research has to be conducted to verify if this is the case. 6. Skeptical description of the evolution of the Mars The Moon does not have liquid fluid on its surface, even there is no air, therefore, there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2, which agrees with the model very well. Based on this, one may reasonably think that the Mars’s rotation should be slowing also. The Mars is our nearest neighbor which has attracted human’s great attention since ancient time. The exploration of the Mars has been heating up in recent decades. NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect data and study this mysterious planet. So far there is still not enough data about the history of this planet to describe its evolution. Same as the Earth, the Mars rotates about its central axis and revolves around the Sun, however, the Mars does not have a massive moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to slow its rotation by tidal friction theories. Based on the above result and current the Mars's data, this model predicts that the angular acceleration of the Mars should be about -4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the solar days/Mars year, the vertical dash line marks the current age of the Mars with assumption that the Mars was formed in a similar time period of the Earth formation. Such description was not given before according to the author’s knowledge and is completely skeptical due to lack of reliable data. However, with further expansion of the research and exploration on the Mars, we shall feel confident that the reliable data about the angular rotation acceleration of the Mars will be available in the near future which will provide a vital test for the prediction of this model. There are also other factors which may affect the Mars’s rotation rate such as mass redistribution due to season change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be carefully analyzed. 7. Discussion about the model From the above results, one can see that the current Earth-Moon data and the geological and fossil data agree with the model very well and the past evolution of the Earth-Moon system can be described by the model without introducing any additional parameters; this model reveals the interesting relationship between the rotation and receding (Eq. 17 and Eq. 18) of the same celestial body or different celestial bodies in the same gravitational system, such relationship is not known before. Such success can not be explained by “coincidence” or “luck” because of so many data involved (current Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a “ad hoc” or a wrong model, although the chance for the natural happening of such “coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s data will clarify this; otherwise, a new theory from different approach can be developed to give the same or better description as this model does. It is certain that this model is not perfect and may have defects, further development may be conducted. James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of space, or produce the slightest flaw in its infinite continuity. It extends unbroken from star to star ….” The medium that Maxwell talked about is the aether which was proposed as the carrier of light wave propagation. The Michelson-Morley experiment only proved that the light wave propagation does not depend on such medium and did not reject the existence of the medium in the interstellar space. In fact, the concept of the interstellar medium has been developed dramatically recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field fluid is just a part of such wonderful medium and “precisely” described by Maxwell. 7. Conclusion The evolution of the Earth-Moon system can be described by the dark matter field fluid model with non-Newtonian approach and the current data of the Earth and the Moon fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of day was about 8 hours. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. References S. G. Brush, 1983. L. R. Godfrey (editor), Ghost from the Nineteenth century: Creationist Arguments for a young Earth. Scientists confront creationism. W. W. Norton & Company, New York, London, pp 49. E. Chaisson and S. McMillan. 1993. Astronomy Today, Prentice Hall, Englewood Cliffs, NJ 07632. J. O. Dickey, et al., 1994. Science, 265, 482. D. G. Finch, 1981. Earth, Moon, and Planets, 26(1), 109. K. S. Hansen, 1982. Rev. Geophys. and Space Phys. 20(3), 457. W. K. Hartmann, D. R. Davis, 1975. Icarus, 24, 504. B. A. Kagan, N. B. Maslova, 1994. Earth, Moon and Planets 66, 173. B. A. Kagan, 1997. Prog. Oceanog. 40, 109. E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, and A. Zawistoski, 1999, J. Sediment. Res. 69(6), 1154. K. Lang, 1992. Astrophysical Data: Planets and Stars, Springer-Verlag, New York. H. Pan, 2005. Internat. J. Modern Phys. A, 20(14), 3135. R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104(B8), 17653. C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), Tidal Friction and the Earth’s Rotation. Springer-Verlag, Berlin, pp. 154. L. B. Slichter, 1963. J. Geophys. Res. 68, 14. C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Science, 273, 100. F. D. Stacey, 1977. Physics of the Earth, second edition. John Willey & Sons. J. W. Wells, 1963. Nature, 197, 948. Caption Figure 1, the evolution of Moon’s distance and the length of day of the earth with the age of the Earth-Moon system. Solid lines are calculated according to the dark matter field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al. The dash line marks the current age of the Earth-Moon system. Figure 2, the evolution of Solar days of year with the age of the Earth-Moon system. The solid line is calculated according to dark matter field fluid model. The data are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current age (4.5 billion years). Figure 3, the skeptical description of the evolution of Mars’s length of day and the solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5 billion years). The vertical dash line marks the current age of Mars. Figure 1, Moon's distance and the length of day of Earth change with the age of Earth-Moon system The age of Earth-Moon system (109 years) 0 1 2 3 4 5 Distance Length of day Roche's limit Hansen's result Figure 2, the solar days / year vs. the age of the Earth The age of the Earth (109 years) 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6
The evolution of Earth-Moon system is described by the dark matter field fluid model proposed in the Meeting of Division of Particle and Field 2004, American Physical Society. The current behavior of the Earth-Moon system agrees with this model very well and the general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago, which is far beyond the Roche's limit. The result suggests that the tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The average dark matter field fluid constant derived from Earth-Moon system data is 4.39 x 10^(-22) s^(-1)m^(-1). This model predicts that the Mars's rotation is also slowing with the angular acceleration rate about -4.38 x 10^(-22) rad s^(-2).
Introduction The popularly accepted theory for the formation of the Earth-Moon system is that the Moon was formed from debris of a strong impact by a giant planetesimal with the Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the formation of the Earth-Moon system, it has been evolving at all time scale. It is well known that the Moon is receding from us and both the Earth’s rotation and Moon’s rotation are slowing. The popular theory is that the tidal friction causes all those changes based on the conservation of the angular momentum of the Earth-Moon system. The situation becomes complicated in describing the past evolution of the Earth-Moon system. Because the Moon is moving away from us and the Earth rotation is slowing, this means that the Moon was closer and the Earth rotation was faster in the past. Creationists argue that based on the tidal friction theory, the tidal friction should be stronger and the recessional rate of the Moon should be greater in the past, the distance of the Moon would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence indicates that the recession of the Moon in the past was slower than the present rate, i. e., the recession has been accelerating with time. Therefore, it must be concluded that tidal friction was very much less in the remote past than we would deduce on the basis of present-day observations (Stacey 1977). This was called “geological time scale difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the tidal friction theory (Brush 1983). But we have to consider the case carefully in various aspects. One possible scenario is that the Earth has been undergoing dynamic evolution at all time scale since its inception, the geological and physical conditions (such as the continent positions and drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at remote past could be substantially different from currently, in which the tidal friction could be much less; therefore, the receding rate of the Moon could be slower. Various tidal friction models were proposed in the past to describe the evolution of the Earth- Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present rate of tidal dissipation is anomalously high because the tidal force is close to a resonance in the response function of ocean (Brush 1983). Kagan gave a detailed review about those tidal friction models (Kagan 1997). Those models are based on many assumptions about geological (continental position and drifting) and physical conditions in the past, and many parameters (such as phase lag angle, multi-mode approximation with time dependent frequencies of the resonance modes, etc.) have to be introduced and carefully adjusted to make their predictions close to the geological evidence. However, those assumptions and parameters are still challenged, to certain extent, as concoction. The second possible scenario is that another mechanism could dominate the evolution of the Earth-Moon system and the role of the tidal friction is not significant. In the Meeting of Division of Particle and Field 2004, American Physical Society, University of California at Riverside, the author proposed a dark matter field fluid model (Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with this model very well. This paper will demonstrate that the past evolution of Moon-Earth system can be described by the dark matter field fluid model without any assumptions about past geological and physical conditions. Although the subject of the evolution of the Earth-Moon system has been extensively studied analytically or numerically, to the author’s knowledge, there are no theories similar or equivalent to this model. 2. Invisible matter In modern cosmology, it was proposed that the visible matter in the universe is about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently invisible which is called dark matter and dark energy, such invisible matter has an anti- gravity property to make the universe expanding faster and faster. If the ratio of the matter components of the universe is close to this hypothesis, then, the evolution of the universe should be dominated by the physical mechanism of such invisible matter, such physical mechanism could be far beyond the current Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian physics could reflect only a corner of the iceberg of the greater physics. If the ratio of the matter components of the universe is close to this hypothesis, then, it should be more reasonable to think that such dominant invisible matter spreads in everywhere of the universe (the density of the invisible matter may vary from place to place); in other words, all visible matter objects should be surrounded by such invisible matter and the motion of the visible matter objects should be affected by the invisible matter if there are interactions between the visible matter and the invisible matter. If the ratio of the matter components of the universe is close to this hypothesis, then, the size of the particles of the invisible matter should be very small and below the detection limit of the current technology; otherwise, it would be detected long time ago with such dominant amount. With such invisible matter in mind, we move to the next section to develop the dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible matter (dark matter, dark energy and possible other terms) is called dark matter here. 3. The dark matter field fluid model In this proposed model, it is assumed that: 1. A celestial body rotates and moves in the space, which, for simplicity, is uniformly filled with the dark matter which is in quiescent state relative to the motion of the celestial body. The dark matter possesses a field property and a fluid property; it can interact with the celestial body with its fluid and field properties; therefore, it can have energy exchange with the celestial body, and affect the motion of the celestial body. 2. The fluid property follows the general principle of fluid mechanics. The dark matter field fluid particles may be so small that they can easily permeate into ordinary “baryonic” matter; i. e., ordinary matter objects could be saturated with such dark matter field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the manner of a sponge moving thru water. The nature of the field property of the dark matter field fluid is unknown. It is here assumed that the interaction of the field associated with the dark matter field fluid with the celestial body is proportional to the mass of the celestial body. The dark matter field fluid is assumed to have a repulsive force against the gravitational force towards baryonic matter. The nature and mechanism of such repulsive force is unknown. With the assumptions above, one can study how the dark matter field fluid may influence the motion of a celestial body and compare the results with observations. The common shape of celestial bodies is spherical. According to Stokes's law, a rigid non- permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds number experiences a resistance force F rvF πμ6−= (1) where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual properties of the dark matter field fluid and its permeation into the sphere, the force F may not be proportional to the radius of the sphere. Also, F may be proportional to the mass of the sphere due to the field interaction. Therefore, with the combined effects of both fluid and field, the force exerted on the sphere by the dark matter field fluid is assumed to be of the scaled form (2) mvrF n−−= 16πη where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η is the dark matter field fluid constant, which is equivalent to μ. The direction of the resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force described by Eq. 2 is velocity-dependent and causes negative acceleration. According to Newton's second law of motion, the equation of motion for the sphere is mvr m n−−= 16πη (3) Then (4) )6exp( 10 vtrvv n−−= πη where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a massive gravitational center, there are three forces in the line between the sphere and the gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and (3) the repulsive force of the dark matter field fluid. The drag force in Eq. 3 reduces the orbital velocity and causes the sphere to move inward to the gravitational center. However, if the sum of the centripetal acceleration force and the repulsive force is stronger than the gravitational force, then, the sphere will move outward and recede from the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is sufficiently slow and the repulsive force is small compared to the gravitational force and centripetal acceleration force, then the rate of receding will be accordingly relatively slow. Therefore, the gravitational force and the centripetal acceleration force can be approximately treated in equilibrium at any time. The pseudo equilibrium equation is GMm 2 2 = (5) where G is the gravitational constant, M is the mass of the gravitational center, and R is the radius of the orbit. Inserting v of Eq. 4 into Eq. 5 yields )12exp( 1 R n−= πη (6) (7) )12exp( 10 trRR n−= πη where R = (8) R0 is the initial distance to the gravitational center. Note that R exponentially increases with time. The increase of orbital energy with the receding comes from the repulsive force of dark matter field fluid. The recessional rate of the sphere is dR n−= 112πη (9) The acceleration of the recession is ( Rr Rd n 21 12 −= πη ) . (10) The recessional acceleration is positive and proportional to its distance to the gravitational center, so the recession is faster and faster. According to the mechanics of fluids, for a rigid non-permeable sphere rotating about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere ωπμ 38 rT −= (11) where ω is the angular velocity of the sphere. The direction of the torque in Eq. 11 is opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted on the sphere is ( ) ωπη mrT n 318 −−= (12) The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The torque causes the negative angular acceleration = (13) where I is the moment of inertia of the sphere in the dark matter field fluid ( )21 2 nrmI −= (14) Therefore, the equation of rotation for the sphere in the dark matter field fluid is ωπη d −−= 120 (15) Solving this equation yields (16) )20exp( 10 tr n−−= πηωω where ω0 is the initial angular velocity. One can see that the angular velocity of the sphere exponentially decreases with time and the angular deceleration is proportional to its angular velocity. For the same celestial sphere, combining Eq. 9 and Eq. 15 yields (17) The significance of Eq. 17 is that it contains only observed data without assumptions and undetermined parameters; therefore, it is a critical test for this model. For two different celestial spheres in the same system, combining Eq. 9 and Eq. 15 yields 67.1 1 −=−=⎟⎟ (18) This is another critical test for this model. 4. The current behavior of the Earth-Moon system agrees with the model The Moon-Earth system is the simplest gravitational system. The solar system is complex, the Earth and the Moon experience not only the interaction of the Sun but also interactions of other planets. Let us consider the local Earth-Moon gravitational system as an isolated local gravitational system, i.e., the influence from the Sun and other planets on the rotation and orbital motion of the Moon and on the rotation of the Earth is assumed negligible compared to the forces exerted by the moon and earth on each other. In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data about the Moon and the Earth from references (Dickey et .al., 1994, and Lang, 1992) are listed below for the readers' convenience to verify the calculation because the data may vary slightly with different data sources. Moon: Mean radius: r = 1738.0 km Mass: m = 7.3483 × 1025 gram Rotation period = 27.321661 days Angular velocity of Moon = 2.6617 × 10-6 rad s-1 Mean distance to Earth Rm= 384400 km Mean orbital velocity v = 1.023 km s-1 Orbit eccentricity e = 0.0549 Angular rotation acceleration rate = -25.88 ± 0.5 arcsec century-2 = (-1.255 ± 0.024) × 10-4 rad century-2 = (-1.260 ± 0.024) × 10-23 rad s-2 Receding rate from Earth = 3.82 ± 0.07 cm year-1 = (1.21 ± 0.02) × 10-9 m s-1 Earth: Mean radius: r = 6371.0 km Mass: m = 5.9742 × 1027 gram Rotation period = 23 h 56m 04.098904s = 86164.098904s Angular velocity of rotation = 7.292115 × 10-5 rad s-1 Mean distance to the Sun Rm= 149,597,870.61 km Mean orbital velocity v = 29.78 km s-1 Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2 The Moon's angular rotation acceleration rate and increase in mean distance to the Earth (receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq. 17, the result is 039.054.1 10662.21021.1 1092509.31026.1 (19) The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number 384400 km is assumed to be the distance from the Moon's surface to the Earth's surface. Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord with the model used here. The difference (about 7.8%) between the values of -1.54 and - 1.67 may come from several sources: 1. Moon's orbital is not a perfect circle 2. Moon is not a perfect rigid sphere. 3. The effect from Sun and other planets. 4. Errors in data. 5. Possible other unknown reasons. The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data sets. The third data set can be used to further test the model. If this model correctly describes the situation at hand, it should give consistent results for different motions. The values of n and η calculated from three different data sets are listed below (Note, the mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are used in the calculation). The value of n: n = 0.64 From the Moon's rotation: η = 4.27 × 10-22 s-1 m-1 From the Earth's rotation: η = 4.26 × 10-22 s-1 m-1 From the Moon's recession: η = 4.64 × 10-22 s-1 m-1 One can see that the three values of η are consistent within the range of error in the data. The average value of η: η = (4.39 ± 0.22) × 10-22 s-1 m-1 By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into Eq. 18, the result is 14.053.1 6371000 1738000 1021.11029.7 1092509.3105.5 )64.01( (20) This is also in accord with the model used here. The dragging force exerted on the Moon's orbital motion by the dark matter field fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between the Moon and the Earth ~ 1.90 × 1020 N; and the torque exerted by the dark matter field fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm, respectively. 5. The evolution of Earth-Moon system Sonett et al. found that the length of the terrestrial day 900 million years ago was about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al., 1996). According to the model presented here, back in that time, the length of the day was about 19.2 hours, this agrees very well with Sonett et al.'s result. Another critical aspect of modeling the evolution of the Earth-Moon system is to give a reasonable estimate of the closest distance of the Moon to the Earth when the system was established at 4.5 billion years ago. Based on the dark matter field fluid model, and the above result, the closest distance of the Moon to the Earth was about 259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago, this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion years ago was 250000 km, this is probably the most reasonable number that most astronomers believe and it agrees excellently with the result of this model. The closest distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or 242000 km (Hansen, 1982). According to this model, the length of day of the Earth was about 8 hours at 4.5 billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the length of day of the Earth with the age of the Earth-Moon system described by this model along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One can see that those data fit this model very well in their time range. Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model for past 900 million years. One can see that the model agrees with the geological and fossil data beautifully. The important difference of this model with early models in describing the early evolution of the Earth-Moon system is that this model is based only on current data of the Moon-Earth system and there are no assumptions about the conditions of earlier Earth rotation and continental drifting. Based on this model, the Earth-Moon system has been smoothly evolving to the current position since it was established and the recessional rate of the Moon has been gradually increasing, however, this description does not take it into account that there might be special events happened in the past to cause the suddenly significant changes in the motions of the Earth and the Moon, such as strong impacts by giant asteroids and comets, etc, because those impacts are very common in the universe. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially increases with time. One may then imagine that the recessional rate will quickly become very large. The increase is in fact extremely slow. The Moon's recessional rate will be 3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years. However, whether the Moon's recession will continue or at some time later another mechanism will take over is not known. It should be understood that the tidal friction does affect the evolution of the Earth itself such as the surface crust structure, continental drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s rotation, however, such role is not a dominant mechanism. Unfortunately, there is no data available for the changes of the Earth's orbital motion and all other members of solar system. According to this model and above results, the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km century-1, the length of a year increases about 6.8 ms and the change of the temperature is -1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on the Earth. The length of a year at 1 billion years ago would be 80% of the current length of the year. However, much evidence (growth-bands of corals and shellfish as well as some other evidences) suggest that there has been no apparent change in the length of the year over the billion years and the Earth's orbital motion is more stable than its rotation. This suggests that dark matter field fluid is circulating around Sun with the same direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the Earth's orbital motion experiences very little or no dragging force from the dark matter field fluid. However, this is a conjecture, extensive research has to be conducted to verify if this is the case. 6. Skeptical description of the evolution of the Mars The Moon does not have liquid fluid on its surface, even there is no air, therefore, there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2, which agrees with the model very well. Based on this, one may reasonably think that the Mars’s rotation should be slowing also. The Mars is our nearest neighbor which has attracted human’s great attention since ancient time. The exploration of the Mars has been heating up in recent decades. NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect data and study this mysterious planet. So far there is still not enough data about the history of this planet to describe its evolution. Same as the Earth, the Mars rotates about its central axis and revolves around the Sun, however, the Mars does not have a massive moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to slow its rotation by tidal friction theories. Based on the above result and current the Mars's data, this model predicts that the angular acceleration of the Mars should be about -4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the solar days/Mars year, the vertical dash line marks the current age of the Mars with assumption that the Mars was formed in a similar time period of the Earth formation. Such description was not given before according to the author’s knowledge and is completely skeptical due to lack of reliable data. However, with further expansion of the research and exploration on the Mars, we shall feel confident that the reliable data about the angular rotation acceleration of the Mars will be available in the near future which will provide a vital test for the prediction of this model. There are also other factors which may affect the Mars’s rotation rate such as mass redistribution due to season change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be carefully analyzed. 7. Discussion about the model From the above results, one can see that the current Earth-Moon data and the geological and fossil data agree with the model very well and the past evolution of the Earth-Moon system can be described by the model without introducing any additional parameters; this model reveals the interesting relationship between the rotation and receding (Eq. 17 and Eq. 18) of the same celestial body or different celestial bodies in the same gravitational system, such relationship is not known before. Such success can not be explained by “coincidence” or “luck” because of so many data involved (current Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a “ad hoc” or a wrong model, although the chance for the natural happening of such “coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s data will clarify this; otherwise, a new theory from different approach can be developed to give the same or better description as this model does. It is certain that this model is not perfect and may have defects, further development may be conducted. James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of space, or produce the slightest flaw in its infinite continuity. It extends unbroken from star to star ….” The medium that Maxwell talked about is the aether which was proposed as the carrier of light wave propagation. The Michelson-Morley experiment only proved that the light wave propagation does not depend on such medium and did not reject the existence of the medium in the interstellar space. In fact, the concept of the interstellar medium has been developed dramatically recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field fluid is just a part of such wonderful medium and “precisely” described by Maxwell. 7. Conclusion The evolution of the Earth-Moon system can be described by the dark matter field fluid model with non-Newtonian approach and the current data of the Earth and the Moon fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of day was about 8 hours. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. References S. G. Brush, 1983. L. R. Godfrey (editor), Ghost from the Nineteenth century: Creationist Arguments for a young Earth. Scientists confront creationism. W. W. Norton & Company, New York, London, pp 49. E. Chaisson and S. McMillan. 1993. Astronomy Today, Prentice Hall, Englewood Cliffs, NJ 07632. J. O. Dickey, et al., 1994. Science, 265, 482. D. G. Finch, 1981. Earth, Moon, and Planets, 26(1), 109. K. S. Hansen, 1982. Rev. Geophys. and Space Phys. 20(3), 457. W. K. Hartmann, D. R. Davis, 1975. Icarus, 24, 504. B. A. Kagan, N. B. Maslova, 1994. Earth, Moon and Planets 66, 173. B. A. Kagan, 1997. Prog. Oceanog. 40, 109. E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, and A. Zawistoski, 1999, J. Sediment. Res. 69(6), 1154. K. Lang, 1992. Astrophysical Data: Planets and Stars, Springer-Verlag, New York. H. Pan, 2005. Internat. J. Modern Phys. A, 20(14), 3135. R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104(B8), 17653. C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), Tidal Friction and the Earth’s Rotation. Springer-Verlag, Berlin, pp. 154. L. B. Slichter, 1963. J. Geophys. Res. 68, 14. C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Science, 273, 100. F. D. Stacey, 1977. Physics of the Earth, second edition. John Willey & Sons. J. W. Wells, 1963. Nature, 197, 948. Caption Figure 1, the evolution of Moon’s distance and the length of day of the earth with the age of the Earth-Moon system. Solid lines are calculated according to the dark matter field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al. The dash line marks the current age of the Earth-Moon system. Figure 2, the evolution of Solar days of year with the age of the Earth-Moon system. The solid line is calculated according to dark matter field fluid model. The data are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current age (4.5 billion years). Figure 3, the skeptical description of the evolution of Mars’s length of day and the solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5 billion years). The vertical dash line marks the current age of Mars. Figure 1, Moon's distance and the length of day of Earth change with the age of Earth-Moon system The age of Earth-Moon system (109 years) 0 1 2 3 4 5 Distance Length of day Roche's limit Hansen's result Figure 2, the solar days / year vs. the age of the Earth The age of the Earth (109 years) 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6
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A Determinant of Stirling Cycle Numbers Counts Unlabeled Acyclic Single-Source Automata DAVID CALLAN Department of Statistics University of Wisconsin-Madison 1300 University Ave Madison, WI 53706-1532 callan@stat.wisc.edu March 30, 2007 Abstract We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the deter- minant. 1 Introduction The chief purpose of this paper is to show bijectively that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. Specifically, let Ak(n) denote the kn × kn matrix with (i, j) entry [ ⌊ i−1 ⌊ i−1 ⌋+1+i−j , where is the Stirling cycle number, the number of permutations on [i] with j cycles. For example, A2(5) = 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 10 35 50 24 0 0 0 0 0 0 1 10 35 50 0 0 0 0 0 0 0 1 15 85 0 0 0 0 0 0 0 0 1 15 http://arxiv.org/abs/0704.0004v1 As evident in the example, Ak(n) is formed from k copies of each of rows 2 through n+1 of the Stirling cycle triangle, arranged so that the first nonzero entry in each row is a 1 and, after the first row, this 1 occurs just before the main diagonal; in other words, Ak(n) is a Hessenberg matrix with 1s on the infra-diagonal. We will show Main Theorem. The determinant of Ak(n) is the number of unlabeled acyclic single- source automata with n transient states on a (k + 1)-letter input alphabet. Section 2 reviews basic terminology for automata and recurrence relations to count finite acyclic automata. Section 3 introduces column-marked subdiagonal paths, which play an intermediate role, and a way to code them. Section 4 presents a bijection from these column-marked subdiagonal paths to unlabeled acyclic single-source automata. Fi- nally, Section 5 evaluates detAk(n) using a sign-reversing involution and shows that the determinant counts the codes for column-marked subdiagonal paths. 2 Automata A (complete, deterministic) automaton consists of a set of states and an input alphabet whose letters transform the states among themselves: a letter and a state produce another state (possibly the same one). A finite automaton (finite set of states, finite input alphabet of, say, k letters) can be represented as a k-regular directed multigraph with ordered edges: the vertices represent the states and the first, second, . . . edge from a vertex give the effect of the first, second, . . . alphabet letter on that state. A finite automaton cannot be acyclic in the usual sense of no cycles: pick a vertex and follow any path from it. This path must ultimately hit a previously encountered vertex, thereby creating a cycle. So the term acyclic is used in the looser sense that only one vertex, called the sink, is involved in cycles. This means that all edges from the sink loop back to itself (and may safely be omitted) and all other paths feed into the sink. A non-sink state is called transient. The size of an acyclic automaton is the number of transient states. An acyclic automaton of size n thus has transient states which we label 1, 2, . . . , n and a sink, labeled n + 1. Liskovets [1] uses the inclusion-exclusion principle (more about this below) to obtain the following recurrence relation for the number ak(n) of acyclic automata of size n on a k-letter input alphabet (k ≥ 1): ak(0) = 1; ak(n) = (−1)n−j−1 (j + 1)k(n−j)ak(j), n ≥ 1. A source is a vertex with no incoming edges. A finite acyclic automaton has at least one source because a path traversed backward v1 ← v2 ← v3 ← . . . must have distinct vertices and so cannot continue indefinitely. An automaton is single-source (or initially connected) if it has only one source. Let Bk(n) denote the set of single-source acyclic finite (SAF) automata on a k-letter input alphabet with vertices 1, 2, . . . , n + 1 where 1 is the source and n + 1 is the sink, and set bk(n) = | Bk(n) |. The two-line representation of an automaton in Bk(n) is the 2× kn matrix whose columns list the edges in order. For example, 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2 4 6 6 6 6 6 6 6 3 5 3 2 2 6 is in B3(5) and the source-to-sink paths in B include 1 → 6, 1 → 6, 1 → 6, where the alphabet is {a, b, c}. Proposition 1. The number bk(n) of SAF automata of size n on a k-letter input alphabet (n, k ≥ 1) is given by bk(n) = (−1)n−i (i+ 1)k(n−i)ak(i) Remark This formula is a bit more succinct than the the recurrence in [1, Theorem 3.2]. Proof Consider the setA of acyclic automata with transient vertices [n] = {1, 2, . . . , n} in which 1 is a source. Call 2, 3, . . . , n the interior vertices. For X ⊆ [2, n], let f(X) = # automata in A whose set of interior vertices includes X, g(X) = # automata in A whose set of interior vertices is precisely X. Then f(X) = Y :X⊆Y⊆[2,n] g(Y ) and by Möbius inversion [2] on the lattice of subsets of [2, n], g(X) = Y :X⊆Y⊆[2,n] µ(X, Y )f(Y ) where µ(X, Y ) is the Möbius function for this lattice. Since µ(X, Y ) = (−1)|Y |−|X| if X ⊆ Y , we have in particular that g(∅) = Y⊆[2,n] (−1)| Y |f(Y ). (1) Let | Y | = n − i so that 1 ≤ i ≤ n. When Y consists entirely of sources, the vertices in [n+ 1]\Y and their incident edges form a subautomaton with i transient states; there are ak(i) such. Also, all edges from the n − i vertices comprising Y go directly into [n + 1]\Y : (i + 1)k(n−i) choices. Thus f(Y ) = (i + 1)k(n−i)ak(i). By definition, g(∅) is the number of automata in A for which 1 is the only source, that is, g(∅) = bk(n) and the Proposition now follows from (1). An unlabeled SAF automaton is an equivalence class of SAF automata under relabeling of the interior vertices. Liskovets notes [1] (and we prove below) that Bk(n) has no nontrivial automorphisms, that is, each of the (n− 1)! relabelings of the interior vertices of B ∈ Bk(n) produces a different automaton. So unlabeled SAF automata of size n on a k-letter alphabet are counted by 1 (n−1)! bk(n). The next result establishes a canonical representative in each relabeling class. Proposition 2. Each equivalence class in Bk(n) under relabeling of interior vertices has size (n− 1)! and contains exactly one SAF automaton with the “last occurrences increas- ing” property: the last occurrences of the interior vertices—2, 3, . . . , n—in the bottom row of its two-line representation occur in that order. Proof The first assertion follows from the fact that the interior vertices of an au- tomatonB ∈ bk(n) can be distinguished intrinsically, that is, independent of their labeling. To see this, first mark the source, namely 1, with a mark (new label) v1 and observe that there exists at least one interior vertex whose only incoming edge(s) are from the source (the only currently marked vertex) for otherwise a cycle would be present. For each such interior vertex v, choose the last edge from the marked vertex to v using the built-in ordering of these edges. This determines an order on these vertices; mark them in order v2, v3, . . . , vj (j ≥ 2). If there still remain unmarked interior vertices, at least one of them has incoming edges only from a marked vertex or again a cycle would be present. For each such vertex, use the last incoming edge from a marked vertex, where now edges are arranged in order of initial vertex vi with the built-in order breaking ties, to order and mark these vertices vj+1, vj+2, . . .. Proceed similarly until all interior vertices are marked. For example, for 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2 4 6 6 6 6 6 6 6 3 5 3 2 2 6 v1 = 1 and there is just one interior vertex, namely 4, whose only incoming edge is from the source, and so v2 = 4 and 4 becomes a marked vertex. Now all incoming edges to both 3 and 5 are from marked vertices and the last such edges (built-in order comes into play) are 4 → 5 and 4 → 3 putting vertices 3, 5 in the order 5, 3. So v3 = 5 and v4 = 3. Finally, v5 = 2. This proves the first assertion. By construction of the vs, relabeling each interior vertex i with the subscript of its corresponding v produces an automaton in Bk(n) with the “last occurrences increasing” property and is the only relabeling that does so. The example yields 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 5 2 6 4 3 4 5 5 6 6 6 6 6 6 6 Now let Ck(n) denote the set of canonical SAF automata in Bk(n) representing un- labeled automata; thus | Ck(n) | = (n−1)! bk(n). Henceforth, we identify an unlabeled au- tomaton with its canonical representative. 3 Column-Marked Subdiagonal Paths A subdiagonal (k, n, p)-path is a lattice path of steps E = (1, 0) and N = (0, 1), E for east and N for north, from (0, 0) to (kn, p) that never rise above the line y = 1 x. Let Ck(n, p) denote the set of such paths.For k ≥ 1, it is clear that Ck(n, p) is nonempty only for 0 ≤ p ≤ n and it is known (generalized ballot theorem) that |Ck(n, p) | = kn− kp+ 1 kn+ p+ 1 kn+ p + 1 A path P in Ck(n, n) can be coded by the heights of its E steps above the line y = −1; this gives a a sequence (bi) i=1 subject to the restrictions 1 ≤ b1 ≤ b2 ≤ . . . ≤ bkn and bi ≤ ⌈i/k⌉ for all i. A column-marked subdiagonal (k, n, p)-path is one in which, for each i ∈ [1, kn], one of the lattice squares below the ith E step and above the horizontal line y = −1 is marked, say with a ‘ ∗ ’. Let C k(n, p) denote the set of such marked paths. b b b b b b b b b b b ∗ ∗ ∗ (0,0) (8,4) y = −1 y = 1 A path in C 2(4, 3) A marked path P ∗ in C k(n, n) can be coded by a sequence of pairs (ai, bi) where i=1 is the code for the underlying path P and ai ∈ [1, bi] gives the position of the ∗ in the ith column. The example is coded by (1, 1), (1, 1), (1, 2), (2, 2), (1, 2), (3, 3), (1, 3), (2, 3). An explicit sum for |C k(n, n) | is k(n, n) | = 1≤b1≤b2≤...≤bkn, bi ≤ ⌈i/k⌉ for all i b1b2 . . . bkn, because the summand b1b2 . . . bkn is the number of ways to insert the ‘ ∗ ’s in the underlying path coded by (bi) It is also possible to obtain a recurrence for |C k(n, p) |, and then, using Prop. 1, to show analytically that |C k(n, n) | = | Ck+1(n) |. However, it is much more pleasant to give a bijection and in the next section we will do so. In particular, the number of SAF automata on a 2-letter alphabet is | C2(n) | = |C 1(n, n) | = 1≤b1≤b2≤...≤bn bi ≤ i for all i b1b2 . . . bn = (1, 3, 16, 127, 1363, . . .)n≥1, sequence A082161 in [3]. 4 Bijection from Paths to Automata In this section we exhibit a bijection from C k(n, n) to Ck+1(n). Using the illustrated path as a working example with k = 2 and n = 4, b b b b b b b b b b b ∗ ∗ ∗ (0,0) (8,4) y = −1 y = 1 first construct the top row of a two-line representation consisting of k + 1 each 1s, 2s, . . . ,n s and number them left to right: The last step in the path is necessarily anN step. For the second last, third last,. . .N steps in the path, count the number of steps following it. This gives a sequence i1, i2, . . . , in−1 satisfying 1 ≤ i1 < i2 < . . . < in−1 and ij ≤ (k + 1)j for all j. Circle the positions i1, i2, . . . , in−1 in the two-line representation and then insert (in boldface) 2, 3, . . . , n in the second row in the circled positions: 2 3 4 These will be the last occurrences of 2, 3, . . . , n in the second row. Working from the last column in the path back to the first, fill in the blanks in the second row left to right as follows. Count the number of squares from the ∗ up to the path (including the ∗ square) http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082161 and add this number to the nearest boldface number to the left of the current blank entry (if there are no boldface numbers to the left, add this number to 1) and insert the result in the current blank square. In the example the numbers of squares are 2,3,1,2,1,2,1,1 yielding 2 4 5 3 3 5 4 5 4 5 5 This will fill all blank entries except the last. Note that ∗ s in the bottom row correspond to sink (that is, n+1) labels in the second row. Finally, insert n+1 into the last remaining blank space to give the image automaton: 1 1 1 2 2 2 3 3 3 4 4 4 2 4 5 3 3 5 4 5 4 5 5 5 This process is fully reversible and the map is a bijection. 5 Evaluation of detAk(n) For simplicity, we treat the case k = 1, leaving the generalization to arbitrary k as a not-too-difficult exercise for the interested reader. Write A(n) for A1(n). Thus A(n) = 1≤i,j≤n . From the definition of detA(n) as a sum of signed products, we show that detA(n) is the total weight of certain lists of permutations, each list carrying weight ±1. Then a weight-reversing involution cancels all −1 weights and reduces the problem to counting the surviving lists. These surviving lists are essentially the codes for paths in C 1(n, p), and the Main Theorem follows from §4. To describe the permutations giving a nonzero contribution to detA(n) = σ sgn σ× i=1 ai,σ(i), define the code of a permutation σ on [n] to be the list c = (ci) i=1 with ci = σ(i)−(i−1). Since the (i, j) entry of A(n), , is 0 unless j ≥ i−1, we must have σ(i) ≥ i−1 for all i. It is well known that there are 2n−1 such permutations, corresponding to compositions of n, with codes characterized by the following four conditions: (i) ci ≥ 0 for all i, (ii) c1 ≥ 1, (iii) each ci ≥ 1 is immediately followed by ci − 1 zeros in the list, i=1 ci = n. Let us call such a list a padded composition of n: deleting the zeros is a bijection to ordinary compositions of n. For example, (3, 0, 0, 1, 2, 0) is a padded composition of 6. For a permutation σ with padded composition code c, the nonzero entries in c give the cycle lengths of σ. Hence sgnσ, which is the parity of “n−#cycles in σ”, is given by (−1)#0s in c. We have detA(n) = σ sgn σ i=1 ai,σ(i) = σ sgn σ 2i−σ(i) , and so detA(n) = (−1)#0s in c i+ 1− ci where the sum is restricted to padded compositions c of n with ci ≤ i for all i (A002083) because i+1−ci = 0 unless ci ≤ i. Henceforth, let us write all permutations in standard cycle form whereby the smallest entry occurs first in each cycle and these smallest entries increase left to right. Thus, with dashes separating cycles, 154-2-36 is the standard cycle form of the permutation ( 1 2 3 4 5 65 2 6 1 4 3 ). We define a nonfirst entry to be one that does not start a cycle. Thus the preceding permutation has 3 nonfirst entries: 5,4,6. Note that the number of nonfirst entries is 0 only for the identity permutation. We denote an identity permutation (of any size) by ǫ. By definition of Stirling cycle number, the product in (2) counts lists (πi) i=1 of permu- tations where πi is a permutation on [i+1] with i+1− ci cycles, equivalently, with ci ≤ i nonfirst entries. So define Ln to be the set all lists of permutations π = (πi) i=1 where πi is a permutation on [i + 1], #nonfirst entries in πi is ≤ i, π1 is the transposition (1,2), each nonidentity permutation πi is immediately followed by ci − 1 ǫ’s where ci ≥ 1 is the number of nonfirst entries in πi (so the total number of nonfirst entries is n). Assign a weight to π ∈ Ln by wt(π) = (−1) # ǫ’s in π. Then detA(n) = wt(π). We now define a weight-reversing involution on (most of) Ln. Given π ∈ Ln, scan the list of its component permutations π1 = (1, 2), π2, π3, . . . left to right. Stop at the first one that either (i) has more than one nonfirst entry, or (ii) has only one nonfirst entry, b say, and b > maximum nonfirst entry m of the next permutation in the list. Say πk is the permutation where we stop. http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002083 In case (i) decrement (i.e. decrease by 1) the number of ǫ’s in the list by splitting πk into two nonidentity permutations as follows. Let m be the largest nonfirst entry of πk and let ℓ be its predecessor. Replace πk and its successor in the list (necessarily an ǫ) by the following two permutations: first the transposition (ℓ,m) and second the permutation obtained from πk by erasing m from its cycle and turning it into a singleton. Here are two examples of this case (recall permutations are in standard cycle form and, for clarity, singleton cycles are not shown). i 1 2 3 4 5 6 πi 12 13 23 14-253 ǫ ǫ i 1 2 3 4 5 6 πi 12 13 23 25 14-23 ǫ i 1 2 3 4 5 6 πi 12 23 14 13-24 ǫ 23 i 1 2 3 4 5 6 πi 12 23 14 24 13 23 The reader may readily check that this sends case (i) to case (ii). In case (ii), πk is a transposition (a, b) with b > maximum nonfirst entry m of πk+1. In this case, increment the number of ǫ’s in the list by combining πk and πk+1 into a single permutation followed by an ǫ: in πk+1, b is a singleton; delete this singleton and insert b immediately after a in πk+1 (in the same cycle). The reader may check that this reverses the result in the two examples above and, in general, sends case (ii) to case (i). Since the map alters the number of ǫ’s in the list by 1, it is clearly weight-reversing. The map fails only for lists that both consist entirely of transpositions and have the form (a1, b1), (a2, b2), . . . , (an, bn) with b1 ≤ b2 ≤ . . . ≤ bn. Such lists have weight 1. Hence detA(n) is the number of lists (ai, bi) satisfying 1 ≤ ai < bi ≤ i+ 1 for 1 ≤ i ≤ n, and b1 ≤ b2 ≤ . . . ≤ bn. After subtracting 1 from each bi, these lists code the paths in C 1(n, n) and, using §4, detA(n) = |C 1(n, n) | = | C2(n) |. References [1] Valery A. Liskovets, Exact enumeration of acyclic deterministic au- tomata, Disc. Appl. Math., in press, 2006. Earlier version available at http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html [2] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed., Cambridge University Press, NY, 2001. [3] Neil J. Sloane (founder and maintainer), The On-Line Encyclopedia of Integer Se- quences http://www.research.att.com:80/ njas/sequences/index.html?blank=1 http://www.research.att.com:80/~njas/sequences/index.html?blank=1
We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant.
Introduction The chief purpose of this paper is to show bijectively that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. Specifically, let Ak(n) denote the kn × kn matrix with (i, j) entry [ ⌊ i−1 ⌊ i−1 ⌋+1+i−j , where is the Stirling cycle number, the number of permutations on [i] with j cycles. For example, A2(5) = 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 10 35 50 24 0 0 0 0 0 0 1 10 35 50 0 0 0 0 0 0 0 1 15 85 0 0 0 0 0 0 0 0 1 15 http://arxiv.org/abs/0704.0004v1 As evident in the example, Ak(n) is formed from k copies of each of rows 2 through n+1 of the Stirling cycle triangle, arranged so that the first nonzero entry in each row is a 1 and, after the first row, this 1 occurs just before the main diagonal; in other words, Ak(n) is a Hessenberg matrix with 1s on the infra-diagonal. We will show Main Theorem. The determinant of Ak(n) is the number of unlabeled acyclic single- source automata with n transient states on a (k + 1)-letter input alphabet. Section 2 reviews basic terminology for automata and recurrence relations to count finite acyclic automata. Section 3 introduces column-marked subdiagonal paths, which play an intermediate role, and a way to code them. Section 4 presents a bijection from these column-marked subdiagonal paths to unlabeled acyclic single-source automata. Fi- nally, Section 5 evaluates detAk(n) using a sign-reversing involution and shows that the determinant counts the codes for column-marked subdiagonal paths. 2 Automata A (complete, deterministic) automaton consists of a set of states and an input alphabet whose letters transform the states among themselves: a letter and a state produce another state (possibly the same one). A finite automaton (finite set of states, finite input alphabet of, say, k letters) can be represented as a k-regular directed multigraph with ordered edges: the vertices represent the states and the first, second, . . . edge from a vertex give the effect of the first, second, . . . alphabet letter on that state. A finite automaton cannot be acyclic in the usual sense of no cycles: pick a vertex and follow any path from it. This path must ultimately hit a previously encountered vertex, thereby creating a cycle. So the term acyclic is used in the looser sense that only one vertex, called the sink, is involved in cycles. This means that all edges from the sink loop back to itself (and may safely be omitted) and all other paths feed into the sink. A non-sink state is called transient. The size of an acyclic automaton is the number of transient states. An acyclic automaton of size n thus has transient states which we label 1, 2, . . . , n and a sink, labeled n + 1. Liskovets [1] uses the inclusion-exclusion principle (more about this below) to obtain the following recurrence relation for the number ak(n) of acyclic automata of size n on a k-letter input alphabet (k ≥ 1): ak(0) = 1; ak(n) = (−1)n−j−1 (j + 1)k(n−j)ak(j), n ≥ 1. A source is a vertex with no incoming edges. A finite acyclic automaton has at least one source because a path traversed backward v1 ← v2 ← v3 ← . . . must have distinct vertices and so cannot continue indefinitely. An automaton is single-source (or initially connected) if it has only one source. Let Bk(n) denote the set of single-source acyclic finite (SAF) automata on a k-letter input alphabet with vertices 1, 2, . . . , n + 1 where 1 is the source and n + 1 is the sink, and set bk(n) = | Bk(n) |. The two-line representation of an automaton in Bk(n) is the 2× kn matrix whose columns list the edges in order. For example, 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2 4 6 6 6 6 6 6 6 3 5 3 2 2 6 is in B3(5) and the source-to-sink paths in B include 1 → 6, 1 → 6, 1 → 6, where the alphabet is {a, b, c}. Proposition 1. The number bk(n) of SAF automata of size n on a k-letter input alphabet (n, k ≥ 1) is given by bk(n) = (−1)n−i (i+ 1)k(n−i)ak(i) Remark This formula is a bit more succinct than the the recurrence in [1, Theorem 3.2]. Proof Consider the setA of acyclic automata with transient vertices [n] = {1, 2, . . . , n} in which 1 is a source. Call 2, 3, . . . , n the interior vertices. For X ⊆ [2, n], let f(X) = # automata in A whose set of interior vertices includes X, g(X) = # automata in A whose set of interior vertices is precisely X. Then f(X) = Y :X⊆Y⊆[2,n] g(Y ) and by Möbius inversion [2] on the lattice of subsets of [2, n], g(X) = Y :X⊆Y⊆[2,n] µ(X, Y )f(Y ) where µ(X, Y ) is the Möbius function for this lattice. Since µ(X, Y ) = (−1)|Y |−|X| if X ⊆ Y , we have in particular that g(∅) = Y⊆[2,n] (−1)| Y |f(Y ). (1) Let | Y | = n − i so that 1 ≤ i ≤ n. When Y consists entirely of sources, the vertices in [n+ 1]\Y and their incident edges form a subautomaton with i transient states; there are ak(i) such. Also, all edges from the n − i vertices comprising Y go directly into [n + 1]\Y : (i + 1)k(n−i) choices. Thus f(Y ) = (i + 1)k(n−i)ak(i). By definition, g(∅) is the number of automata in A for which 1 is the only source, that is, g(∅) = bk(n) and the Proposition now follows from (1). An unlabeled SAF automaton is an equivalence class of SAF automata under relabeling of the interior vertices. Liskovets notes [1] (and we prove below) that Bk(n) has no nontrivial automorphisms, that is, each of the (n− 1)! relabelings of the interior vertices of B ∈ Bk(n) produces a different automaton. So unlabeled SAF automata of size n on a k-letter alphabet are counted by 1 (n−1)! bk(n). The next result establishes a canonical representative in each relabeling class. Proposition 2. Each equivalence class in Bk(n) under relabeling of interior vertices has size (n− 1)! and contains exactly one SAF automaton with the “last occurrences increas- ing” property: the last occurrences of the interior vertices—2, 3, . . . , n—in the bottom row of its two-line representation occur in that order. Proof The first assertion follows from the fact that the interior vertices of an au- tomatonB ∈ bk(n) can be distinguished intrinsically, that is, independent of their labeling. To see this, first mark the source, namely 1, with a mark (new label) v1 and observe that there exists at least one interior vertex whose only incoming edge(s) are from the source (the only currently marked vertex) for otherwise a cycle would be present. For each such interior vertex v, choose the last edge from the marked vertex to v using the built-in ordering of these edges. This determines an order on these vertices; mark them in order v2, v3, . . . , vj (j ≥ 2). If there still remain unmarked interior vertices, at least one of them has incoming edges only from a marked vertex or again a cycle would be present. For each such vertex, use the last incoming edge from a marked vertex, where now edges are arranged in order of initial vertex vi with the built-in order breaking ties, to order and mark these vertices vj+1, vj+2, . . .. Proceed similarly until all interior vertices are marked. For example, for 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2 4 6 6 6 6 6 6 6 3 5 3 2 2 6 v1 = 1 and there is just one interior vertex, namely 4, whose only incoming edge is from the source, and so v2 = 4 and 4 becomes a marked vertex. Now all incoming edges to both 3 and 5 are from marked vertices and the last such edges (built-in order comes into play) are 4 → 5 and 4 → 3 putting vertices 3, 5 in the order 5, 3. So v3 = 5 and v4 = 3. Finally, v5 = 2. This proves the first assertion. By construction of the vs, relabeling each interior vertex i with the subscript of its corresponding v produces an automaton in Bk(n) with the “last occurrences increasing” property and is the only relabeling that does so. The example yields 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 5 2 6 4 3 4 5 5 6 6 6 6 6 6 6 Now let Ck(n) denote the set of canonical SAF automata in Bk(n) representing un- labeled automata; thus | Ck(n) | = (n−1)! bk(n). Henceforth, we identify an unlabeled au- tomaton with its canonical representative. 3 Column-Marked Subdiagonal Paths A subdiagonal (k, n, p)-path is a lattice path of steps E = (1, 0) and N = (0, 1), E for east and N for north, from (0, 0) to (kn, p) that never rise above the line y = 1 x. Let Ck(n, p) denote the set of such paths.For k ≥ 1, it is clear that Ck(n, p) is nonempty only for 0 ≤ p ≤ n and it is known (generalized ballot theorem) that |Ck(n, p) | = kn− kp+ 1 kn+ p+ 1 kn+ p + 1 A path P in Ck(n, n) can be coded by the heights of its E steps above the line y = −1; this gives a a sequence (bi) i=1 subject to the restrictions 1 ≤ b1 ≤ b2 ≤ . . . ≤ bkn and bi ≤ ⌈i/k⌉ for all i. A column-marked subdiagonal (k, n, p)-path is one in which, for each i ∈ [1, kn], one of the lattice squares below the ith E step and above the horizontal line y = −1 is marked, say with a ‘ ∗ ’. Let C k(n, p) denote the set of such marked paths. b b b b b b b b b b b ∗ ∗ ∗ (0,0) (8,4) y = −1 y = 1 A path in C 2(4, 3) A marked path P ∗ in C k(n, n) can be coded by a sequence of pairs (ai, bi) where i=1 is the code for the underlying path P and ai ∈ [1, bi] gives the position of the ∗ in the ith column. The example is coded by (1, 1), (1, 1), (1, 2), (2, 2), (1, 2), (3, 3), (1, 3), (2, 3). An explicit sum for |C k(n, n) | is k(n, n) | = 1≤b1≤b2≤...≤bkn, bi ≤ ⌈i/k⌉ for all i b1b2 . . . bkn, because the summand b1b2 . . . bkn is the number of ways to insert the ‘ ∗ ’s in the underlying path coded by (bi) It is also possible to obtain a recurrence for |C k(n, p) |, and then, using Prop. 1, to show analytically that |C k(n, n) | = | Ck+1(n) |. However, it is much more pleasant to give a bijection and in the next section we will do so. In particular, the number of SAF automata on a 2-letter alphabet is | C2(n) | = |C 1(n, n) | = 1≤b1≤b2≤...≤bn bi ≤ i for all i b1b2 . . . bn = (1, 3, 16, 127, 1363, . . .)n≥1, sequence A082161 in [3]. 4 Bijection from Paths to Automata In this section we exhibit a bijection from C k(n, n) to Ck+1(n). Using the illustrated path as a working example with k = 2 and n = 4, b b b b b b b b b b b ∗ ∗ ∗ (0,0) (8,4) y = −1 y = 1 first construct the top row of a two-line representation consisting of k + 1 each 1s, 2s, . . . ,n s and number them left to right: The last step in the path is necessarily anN step. For the second last, third last,. . .N steps in the path, count the number of steps following it. This gives a sequence i1, i2, . . . , in−1 satisfying 1 ≤ i1 < i2 < . . . < in−1 and ij ≤ (k + 1)j for all j. Circle the positions i1, i2, . . . , in−1 in the two-line representation and then insert (in boldface) 2, 3, . . . , n in the second row in the circled positions: 2 3 4 These will be the last occurrences of 2, 3, . . . , n in the second row. Working from the last column in the path back to the first, fill in the blanks in the second row left to right as follows. Count the number of squares from the ∗ up to the path (including the ∗ square) http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082161 and add this number to the nearest boldface number to the left of the current blank entry (if there are no boldface numbers to the left, add this number to 1) and insert the result in the current blank square. In the example the numbers of squares are 2,3,1,2,1,2,1,1 yielding 2 4 5 3 3 5 4 5 4 5 5 This will fill all blank entries except the last. Note that ∗ s in the bottom row correspond to sink (that is, n+1) labels in the second row. Finally, insert n+1 into the last remaining blank space to give the image automaton: 1 1 1 2 2 2 3 3 3 4 4 4 2 4 5 3 3 5 4 5 4 5 5 5 This process is fully reversible and the map is a bijection. 5 Evaluation of detAk(n) For simplicity, we treat the case k = 1, leaving the generalization to arbitrary k as a not-too-difficult exercise for the interested reader. Write A(n) for A1(n). Thus A(n) = 1≤i,j≤n . From the definition of detA(n) as a sum of signed products, we show that detA(n) is the total weight of certain lists of permutations, each list carrying weight ±1. Then a weight-reversing involution cancels all −1 weights and reduces the problem to counting the surviving lists. These surviving lists are essentially the codes for paths in C 1(n, p), and the Main Theorem follows from §4. To describe the permutations giving a nonzero contribution to detA(n) = σ sgn σ× i=1 ai,σ(i), define the code of a permutation σ on [n] to be the list c = (ci) i=1 with ci = σ(i)−(i−1). Since the (i, j) entry of A(n), , is 0 unless j ≥ i−1, we must have σ(i) ≥ i−1 for all i. It is well known that there are 2n−1 such permutations, corresponding to compositions of n, with codes characterized by the following four conditions: (i) ci ≥ 0 for all i, (ii) c1 ≥ 1, (iii) each ci ≥ 1 is immediately followed by ci − 1 zeros in the list, i=1 ci = n. Let us call such a list a padded composition of n: deleting the zeros is a bijection to ordinary compositions of n. For example, (3, 0, 0, 1, 2, 0) is a padded composition of 6. For a permutation σ with padded composition code c, the nonzero entries in c give the cycle lengths of σ. Hence sgnσ, which is the parity of “n−#cycles in σ”, is given by (−1)#0s in c. We have detA(n) = σ sgn σ i=1 ai,σ(i) = σ sgn σ 2i−σ(i) , and so detA(n) = (−1)#0s in c i+ 1− ci where the sum is restricted to padded compositions c of n with ci ≤ i for all i (A002083) because i+1−ci = 0 unless ci ≤ i. Henceforth, let us write all permutations in standard cycle form whereby the smallest entry occurs first in each cycle and these smallest entries increase left to right. Thus, with dashes separating cycles, 154-2-36 is the standard cycle form of the permutation ( 1 2 3 4 5 65 2 6 1 4 3 ). We define a nonfirst entry to be one that does not start a cycle. Thus the preceding permutation has 3 nonfirst entries: 5,4,6. Note that the number of nonfirst entries is 0 only for the identity permutation. We denote an identity permutation (of any size) by ǫ. By definition of Stirling cycle number, the product in (2) counts lists (πi) i=1 of permu- tations where πi is a permutation on [i+1] with i+1− ci cycles, equivalently, with ci ≤ i nonfirst entries. So define Ln to be the set all lists of permutations π = (πi) i=1 where πi is a permutation on [i + 1], #nonfirst entries in πi is ≤ i, π1 is the transposition (1,2), each nonidentity permutation πi is immediately followed by ci − 1 ǫ’s where ci ≥ 1 is the number of nonfirst entries in πi (so the total number of nonfirst entries is n). Assign a weight to π ∈ Ln by wt(π) = (−1) # ǫ’s in π. Then detA(n) = wt(π). We now define a weight-reversing involution on (most of) Ln. Given π ∈ Ln, scan the list of its component permutations π1 = (1, 2), π2, π3, . . . left to right. Stop at the first one that either (i) has more than one nonfirst entry, or (ii) has only one nonfirst entry, b say, and b > maximum nonfirst entry m of the next permutation in the list. Say πk is the permutation where we stop. http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002083 In case (i) decrement (i.e. decrease by 1) the number of ǫ’s in the list by splitting πk into two nonidentity permutations as follows. Let m be the largest nonfirst entry of πk and let ℓ be its predecessor. Replace πk and its successor in the list (necessarily an ǫ) by the following two permutations: first the transposition (ℓ,m) and second the permutation obtained from πk by erasing m from its cycle and turning it into a singleton. Here are two examples of this case (recall permutations are in standard cycle form and, for clarity, singleton cycles are not shown). i 1 2 3 4 5 6 πi 12 13 23 14-253 ǫ ǫ i 1 2 3 4 5 6 πi 12 13 23 25 14-23 ǫ i 1 2 3 4 5 6 πi 12 23 14 13-24 ǫ 23 i 1 2 3 4 5 6 πi 12 23 14 24 13 23 The reader may readily check that this sends case (i) to case (ii). In case (ii), πk is a transposition (a, b) with b > maximum nonfirst entry m of πk+1. In this case, increment the number of ǫ’s in the list by combining πk and πk+1 into a single permutation followed by an ǫ: in πk+1, b is a singleton; delete this singleton and insert b immediately after a in πk+1 (in the same cycle). The reader may check that this reverses the result in the two examples above and, in general, sends case (ii) to case (i). Since the map alters the number of ǫ’s in the list by 1, it is clearly weight-reversing. The map fails only for lists that both consist entirely of transpositions and have the form (a1, b1), (a2, b2), . . . , (an, bn) with b1 ≤ b2 ≤ . . . ≤ bn. Such lists have weight 1. Hence detA(n) is the number of lists (ai, bi) satisfying 1 ≤ ai < bi ≤ i+ 1 for 1 ≤ i ≤ n, and b1 ≤ b2 ≤ . . . ≤ bn. After subtracting 1 from each bi, these lists code the paths in C 1(n, n) and, using §4, detA(n) = |C 1(n, n) | = | C2(n) |. References [1] Valery A. Liskovets, Exact enumeration of acyclic deterministic au- tomata, Disc. Appl. Math., in press, 2006. Earlier version available at http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html [2] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed., Cambridge University Press, NY, 2001. [3] Neil J. Sloane (founder and maintainer), The On-Line Encyclopedia of Integer Se- quences http://www.research.att.com:80/ njas/sequences/index.html?blank=1 http://www.research.att.com:80/~njas/sequences/index.html?blank=1
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FROM DYADIC Λα TO Λα WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY Abstract. In this paper we show how to compute the Λα norm , α ≥ 0, using the dyadic grid. This result is a consequence of the description of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms. Recently, several novel methods for computing the BMO norm of a function f in two dimensions were discussed in [9]. Given its importance, it is also of interest to explore the possibility of computing the norm of a BMO function, or more generally a function in the Lipschitz class Λα, using the dyadic grid in RN . It turns out that the BMO question is closely related to that of approximating functions in the Hardy space H1(RN ) by the Haar system. The approximation in H1(RN ) by affine systems was proved in [2], but this result does not apply to the Haar system. Now, if HA(R) denotes the closure of the Haar system in H1(R), it is not hard to see that the distance d(f,HA) of f ∈ H1(R) to HA is ∼ f(x) dx ∣, see [1]. Thus, neither dyadic atoms suffice to describe the Hardy spaces, nor the evaluation of the norm in BMO can be reduced to a straightforward computation using the dyadic intervals. In this paper we address both of these issues. First, we give a characterization of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms, and then, by a duality argument, we show how to compute the norm in Λα(R N ), α ≥ 0, using the dyadic grid. We begin by introducing some notations. Let J denote a family of cubes Q in RN , and Pd the collection of polynomials in R N of degree less than or equal to d. Given α ≥ 0, Q ∈ J , and a locally integrable function g, let pQ(g) denote the unique polynomial in P[α] such that [g − pQ(g)]χQ has vanishing moments up to order [α]. For a locally square-integrable function g, we consider the maximal function α,J g(x) given by α,J g(x) = sup x∈Q,Q∈J |Q|α/N |g(y)− pQ(g)(y)| 1991 Mathematics Subject Classification. 42B30,42B35. http://arxiv.org/abs/0704.0005v1 2 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY The Lipschitz space Λα,J consists of those functions g such that M α,J g is in L∞, ‖g‖Λα,J = ‖M α,J g‖∞; when the family in question contains all cubes in RN , we simply omit the subscript J . Of course, Λ0 = BMO. Two other families, of dyadic nature, are of interest to us. Intervals in R of the form In,k = [ (k−1)2 n, k2n], where k and n are arbitrary integers, positive, negative or 0, are said to be dyadic. In RN , cubes which are the product of dyadic intervals of the same length, i.e., of the form Qn,k = In,k1 ×· · ·×In,kN , are called dyadic, and the collection of all such cubes is denoted D. There is also the family D0. Let I n,k = [(k− 1)2 n, (k+ 1)2n], where k and n are arbitrary integers. Clearly I ′n,k is dyadic if k is odd, but not if k is even. Now, the collection {I ′n,k : n, k integers} contains all dyadic intervals as well as the shifts [(k − 1)2n + 2n−1, k 2n + 2n−1] of the dyadic intervals by their half length. In RN , put D0 = {Q n,k : Q n,k = I × · · · × I ′n,kN }; Q n,k is called a special cube. Note that D0 contains D properly. Finally, given I ′n,k, let I n,k = [(k − 1)2 n, k2n], and I n,k = [k2 n, (k + 1)2n]. The 2N subcubes of Q′n,k = I × · · · × I ′n,kN of the form I × · · · × I Sj = L or R, 1 ≤ j ≤ N , are called the dyadic subcubes of Q Let Q0 denote the special cube [−1, 1] N . Given α ≥ 0, we construct a family Sα of piecewise polynomial splines in L 2(Q0) that will be useful in characterizing Λα. Let A be the subspace of L 2(Q0) consisting of all functions with vanishing moments up to order [α] which coincide with a polynomial in P[α] on each of the 2 N dyadic subcubes of Q0. A is a finite dimensional subspace of L2(Q0), and, therefore, by the Graham-Schmidt orthogonalization process, say, A has an orthonormal basis in L2(Q0) consisting of functions p1, . . . , pM with vanishing moments up to order [α], which coincide with a polynomial in P[α] on each dyadic subinterval of Q0. Together with each p we also consider all dyadic dilations and integer translations given by pLn,k,α(x) = 2 n(N+α)pL(2nx1 + k1, . . . , 2 nxN + kN ) , 1 ≤ L ≤ M , and let Sα = {p n,k,α : n, k integers, 1 ≤ L ≤ M} . Our first result shows how the dyadic grid can be used to compute the norm in Λα. Theorem A. Let g be a locally square-integrable function and α ≥ 0. Then, g ∈ Λα if, and only if, g ∈ Λα,D and Aα(g) = supp∈Sα ∣〈g, p〉 ∣ < ∞. Moreover, ‖g‖Λα ∼ ‖g‖Λα,D +Aα(g) . Furthermore, it is also true, and the proof is given in Proposition 2.1 be- low, that ‖g‖Λα ∼ ‖g‖Λα,D0 . However, in this simpler formulation, the tree structure of the cubes in D has been lost. FROM DYADIC Λα TO Λα 3 The proof of Theorem A relies on a close investigation of the predual of Λα, namely, the Hardy space H p(RN ) with 0 < p = (α + N)/N ≤ 1. In the process we characterize Hp in terms of simpler subspaces: H , or dyadic Hp, and H , the space generated by the special atoms in Sα. Specifically, we Theorem B. Let 0 < p ≤ 1, and α = N(1/p− 1). We then have Hp = H where the sum is understood in the sense of quasinormed Banach spaces. The paper is organized as follows. In Section 1 we show that individual Hp atoms can be written as a superposition of dyadic and special atoms; this fact may be thought of as an extension of the one-dimensional result of Fridli concerning L∞ 1- atoms, see [5] and [1]. Then, we prove Theorem B. In Section 2 we discuss how to pass from Λα,D, and Λα,D0 , to the Lipschitz space Λα. 1. Characterization of the Hardy spaces Hp We adopt the atomic definition of the Hardy spaces Hp, 0 < p ≤ 1, see [6] and [10]. Recall that a compactly supported function a with [N(1/p− 1)] vanishing moments is an L2 p -atom with defining cube Q if supp(a) ⊆ Q, and |Q|1/p | a(x) |2dx ≤ 1 . The Hardy space Hp(RN ) = Hp consists of those distributions f that can be written as f = λjaj , where the aj ’s are H p atoms, |λj | p < ∞, and the convergence is in the sense of distributions as well as in Hp. Furthermore, ‖f‖Hp ∼ inf |λj | where the infimum is taken over all possible atomic decompositions of f . This last expression has traditionally been called the atomic Hp norm of f . Collections of atoms with special properties can be used to gain a better understanding of the Hardy spaces. Formally, let A be a non-empty subset of L2 p -atoms in the unit ball of Hp. The atomic space H spanned by A consists of those ϕ in Hp of the form λjaj , aj ∈ A , |λj | p < ∞ . It is readily seen that, endowed with the atomic norm ‖ϕ‖Hp = inf |λj | : ϕ = λj aj , aj ∈ A becomes a complete quasinormed space. Clearly, H ⊆ Hp, and, for f ∈ H , ‖f‖Hp ≤ ‖f‖Hp 4 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY Two families are of particular interest to us. When A is the collection of all L2 p -atoms whose defining cube is dyadic, the resulting space is H or dyadic Hp. Now, although ‖f‖Hp ≤ ‖f‖Hp , the two quasinorms are not equivalent on H . Indeed, for p = 1 and N = 1, the functions fn(x) = 2 n[χ[1−2−n,1](x) − χ[1,1+2−n](x)] , satisfy ‖fn‖H1 = 1, but ‖fn‖H1 ∼ |n| tends to infinity with n. Next, when Sα is the family of piecewise polynomial splines constructed above with α = N(1/p − 1), in analogy with the one-dimensional results in [4] and [1], H is referred to as the space generated by special atoms. We are now ready to describe Hp atoms as a superposition of dyadic and special atoms. Lemma 1.1. Let a be an L2 p -atom with defining cube Q, 0 < p ≤ 1, and α = N(1/p − 1). Then a can be written as a linear combination of 2N dyadic atoms ai, each supported in one of the dyadic subcubes of the smallest special cube Qn,k containing Q, and a special atom b in Sα. More precisely, a(x) = i=1 di ai(x) + L=1 cL p −n,−k,α(x), with |di| , |cL| ≤ c. Proof. Suppose first that the defining cube of a is Q0, and let Q1, . . . , Q2N denote the dyadic subcubes of Q0. Furthermore, let {e i , . . . , e i } denote an orthonormal basis of the subspace Ai of L 2(Qi) consisting of polynomials in P[α], 1 ≤ i ≤ 2 N . Put αi(x) = a(x)χQi (x)− 〈aχQi , e j(x) , 1 ≤ i ≤ 2 and observe that 〈αi, e j〉 = 0 for 1 ≤ j ≤ M . Therefore, αi has [α] vanishing moments, is supported in Qi, and ‖αi‖2 ≤ ‖aχQi‖2 + ‖aχQi‖2 ≤ (M + 1) ‖aχQi‖2 . ai(x) = 2N(1/2−1/p) M + 1 αi(x) , 1 ≤ i ≤ N , is an L2 p - dyadic atom. Finally, put b(x) = a(x) − M + 1 2N(1/2−1/p) ai(x) . FROM DYADIC Λα TO Λα 5 Clearly b has [α] vanishing moments, is supported in Q0, coincides with a polynomial in P[α] on each dyadic subcube of Q0, and ‖b‖22 ≤ |〈aχQi , e 2 ≤ M ‖a‖22 . So, b ∈ A, and, consequently, b(x) = L=1 cL p L(x), where |cL| = |〈b, p L〉| ≤ c , 1 ≤ L ≤ M . In the general case, let Q be the defining cube of a, side-length Q = ℓ, and let n and k = (k1, . . . , kN ) be chosen so that 2 n−1 ≤ ℓ < 2n, and Q ⊂ [(k1 − 1)2 n, (k1 + 1)2 n]× · · · × [(kN − 1)2 n, (kN + 1)2 Then, (1/2)N ≤ |Q|/2nN < 1. Now, given x ∈ Q0, let a ′ be the translation and dilation of a given by a′(x) = 2nN/pa(2nx1 − k1, . . . , 2 nxN − kN ) . Clearly, [α] moments of a′ vanish, and ‖a′‖2 = 2 nN/p 2−nN/2‖a‖2 ≤ c |Q| 1/p|Q|−1/2‖a‖2 ≤ c . Thus, a′ is a multiple of an atom with defining cube Q0. By the first part of the proof, a′(x) = i(x) + L(x) , x ∈ Q0 . The support of each a′i is contained in one of the dyadic subcubes of Q0, and, consequently, there is a k such that ai(x) = 2 −nN/pa′i(2 −nx1 − k1, . . . , 2 −nxN − kN ) ai is an L 2p -atom supported in one of the dyadic subcubes of Q. Similarly for the pL’s. Thus, a(x) = di ai(x) + −n,−k,N(1/p−1)(x) , and we have finished. � Theorem B follows readily from Lemma 1.1. Clearly, H →֒ Hp. Conversely, let f = j λj aj be in H p. By Lemma 1.1 each aj can be written as a sum of dyadic and special atoms, and, by distributing the sum, we can write f = fd + fs, with fd in H , fs in H , and ‖fd‖Hp , ‖fs‖Hp |λj | Taking the infimum over the decompositions of f we get ‖f‖Hp c ‖f‖Hp , and H p →֒ H . This completes the proof. 6 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY The meaning of this decomposition is the following. Cubes in D are con- tained in one of the 2N non-overlapping quadrants of RN . To allow for the information carried by a dyadic cube to be transmitted to an adjacent dyadic cube, they must be connected. The pLn,k,α’s channel information across ad- jacent dyadic cubes which would otherwise remain disconnected. The reader will have no difficulty in proving the quantitative version of this observation: Let T be a linear mapping defined on Hp, 0 < p ≤ 1, that assumes values in a quasinormed Banach space X . Then, T is continuous if, and only if, the restrictions of T to H and H are continuous. 2. Characterizations of Λα Theorem A describes how to pass from Λα,D to Λα, and we prove it next. Since (Hp)∗ = Λα and (H )∗ = Λα,D, from Theorem B it follows readily that Λα = Λα,D ∩ (H )∗, so it only remains to show that (H )∗ is characterized by the condition Aα(g) < ∞. First note that if g is a locally square-integrable function with Aα(g) < ∞ and f = j,L cj,L p nj ,kj ,α , since 0 < p ≤ 1, |〈g, f〉| ≤ |cj,L| |〈g, p nj ,kj ,α ≤ Aα(g) |cj,L| and, consequently, taking the infimum over all atomic decompositions of f in , we get g ∈ (H )∗ and ‖g‖(Hp )∗ ≤ Aα(g). To prove the converse we proceed as in [3]. Let Qn = [−2 n, 2n]N . We begin by observing that functions f in L2(Qn) that have vanishing moments up to order [α] and coincide with polynomials of degree [α] on the dyadic subcubes of Qn belong to H ‖f‖Hp ≤ |Qn| 1/p−1/2‖f‖2 . Given ℓ ∈ (H )∗, for a fixed n let us consider the restriction of ℓ to the space of L2 functions f with [α] vanishing moments that are supported in Qn. Since |ℓ(f)| ≤ ‖ℓ‖ ‖f‖Hp ≤ ‖ℓ‖ |Qn| 1/p−1/2‖f‖2 , this restriction is continuous with respect to the norm in L2 and, consequently, it can be extended to a continuous linear functional in L2 and represented as ℓ(f) = f(x) gn(x) dx , FROM DYADIC Λα TO Λα 7 where gn ∈ L 2(Qn) and satisfies ‖gn‖2 ≤ ‖ℓ‖ |Qn| 1/p−1/2. Clearly, gn is uniquely determined in Qn up to a polynomial pn in P[α]. Therefore, gn(x) − pn(x) = gm(x)− pm(x) , a.e. x ∈ Qmin(n,m) . Consequently, if g(x) = gn(x)− pn(x) , x ∈ Qn , g(x) is well defined a.e. and, if f ∈ L2 has [α] vanishing moments and is supported in Qn, we have ℓ(f) = f(x) gn(x) dx f(x) [gn(x)− pn(x)] dx f(x) g(x) dx . Moreover, since each 2nN/ppL(2n ·+k) is an L2 p-atom, 1 ≤ L ≤ M , it readily follows that Aα(g) = sup 1≤L≤M n,k∈Z |〈g, 2−n/ppL(2n ·+k)〉| ≤ ‖ℓ‖ sup ‖pL‖Hp ≤ ‖ℓ‖ , and, consequently, Aα(g) ≤ ‖ℓ‖ , and (H )∗ is the desired space. � The reader will have no difficulty in showing that this result implies the following: Let T be a bounded linear operator from a quasinormed space X into Λα,D. Then, T is bounded from X into Λα if, and only if, Aα(Tx) ≤ c ‖x‖X for every x ∈ X . The process of averaging the translates of dyadic BMO functions leads to BMO, and is an important tool in obtaining results in BMO once they are known to be true in its dyadic counterpart, BMOd, see [7]. It is also known that BMO can be obtained as the intersection of BMOd and one of its shifted counterparts, see [8]. These results motivate our next proposition, which essentially says that g ∈ Λα if, and only if, g ∈ Λα,D and g is in the Lipschitz class obtained from the shifted dyadic grid. Note that the shifts involved in this class are in all directions parallel to the coordinate axis and depend on the side-length of the cube. Proposition 2.1. Λα = Λα,D0 , and ‖g‖Λα ∼ ‖g‖Λα,D0 . Proof. It is obvious that ‖g‖Λα,D0 ≤ ‖g‖Λα . To show the other inequality we invoke Theorem A. Since D ⊂ D0, it suffices to estimate Aα(g), or, equiva- lently, |〈g, p〉| for p ∈ Sα, α = N(1/p − 1). So, pick p = p n,k,α in Sα. The defining cube Q of pLn,k,α is in D0, and, since p n,k,α has [α] vanishing moments, 8 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY 〈pLn,k,α, pQ(g)〉 = 0. Therefore, |〈g, pLn,k,α〉| = |〈g − pQ(g), p n,k,α〉| ≤ ‖pLn,k,α‖2 ‖g − pQ(g)‖L2(Q) ≤ |Q|α/N |Q|1/2‖pLn,k,α‖2 ‖g‖Λα,D0 . Now, a simple change of variables gives |Q|α/N |Q|1/2‖pLn,k,α‖2 ≤ 1, and, con- sequently, also Aα(g) ≤ ‖g‖Λα,D0 . � References [1] W. Abu-Shammala, J.-L. Shiu, and A. Torchinsky, Characterizations of the Hardy space H1 and BMO, preprint. [2] H.-Q. Bui and R. S. Laugesen, Approximation and spanning in the Hardy space, by affine systems, Constr. Approx., to appear. [3] A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distibution, II, Advances in Math., 24 (1977), 101–171. [4] G. S. de Souza, Spaces formed by special atoms, I, Rocky Mountain J. Math. 14 (1984), no. 2, 423–431. [5] S. Fridli, Transition from the dyadic to the real nonperiodic Hardy space, Acta Math. Acad. Paedagog. Niházi (N.S.) 16 (2000), 1–8, (electronic). [6] J. Garćıa-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, Notas de Matemática 116, North Holland, Amsterdam, 1985. [7] J. Garnett and P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. [8] T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003–1006. [9] T. M. Le and L. A. Vese, Image decomposition using total variation and div( BMO)∗, Multiscale Model. Simul. 4, (2005), no. 2, 390–423. [10] A. Torchinsky, Real-variable methods in harmonic analysis, Dover Publications, Inc., Mineola, NY, 2004. Department of Mathematics, Indiana University, Bloomington IN 47405 E-mail address: wabusham@indiana.edu Department of Mathematics, Indiana University, Bloomington IN 47405 E-mail address: torchins@indiana.edu 1. Characterization of the Hardy spaces Hp 2. Characterizations of References
In this paper we show how to compute the $\Lambda_{\alpha}$ norm, $\alpha\ge 0$, using the dyadic grid. This result is a consequence of the description of the Hardy spaces $H^p(R^N)$ in terms of dyadic and special atoms.
FROM DYADIC Λα TO Λα WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY Abstract. In this paper we show how to compute the Λα norm , α ≥ 0, using the dyadic grid. This result is a consequence of the description of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms. Recently, several novel methods for computing the BMO norm of a function f in two dimensions were discussed in [9]. Given its importance, it is also of interest to explore the possibility of computing the norm of a BMO function, or more generally a function in the Lipschitz class Λα, using the dyadic grid in RN . It turns out that the BMO question is closely related to that of approximating functions in the Hardy space H1(RN ) by the Haar system. The approximation in H1(RN ) by affine systems was proved in [2], but this result does not apply to the Haar system. Now, if HA(R) denotes the closure of the Haar system in H1(R), it is not hard to see that the distance d(f,HA) of f ∈ H1(R) to HA is ∼ f(x) dx ∣, see [1]. Thus, neither dyadic atoms suffice to describe the Hardy spaces, nor the evaluation of the norm in BMO can be reduced to a straightforward computation using the dyadic intervals. In this paper we address both of these issues. First, we give a characterization of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms, and then, by a duality argument, we show how to compute the norm in Λα(R N ), α ≥ 0, using the dyadic grid. We begin by introducing some notations. Let J denote a family of cubes Q in RN , and Pd the collection of polynomials in R N of degree less than or equal to d. Given α ≥ 0, Q ∈ J , and a locally integrable function g, let pQ(g) denote the unique polynomial in P[α] such that [g − pQ(g)]χQ has vanishing moments up to order [α]. For a locally square-integrable function g, we consider the maximal function α,J g(x) given by α,J g(x) = sup x∈Q,Q∈J |Q|α/N |g(y)− pQ(g)(y)| 1991 Mathematics Subject Classification. 42B30,42B35. http://arxiv.org/abs/0704.0005v1 2 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY The Lipschitz space Λα,J consists of those functions g such that M α,J g is in L∞, ‖g‖Λα,J = ‖M α,J g‖∞; when the family in question contains all cubes in RN , we simply omit the subscript J . Of course, Λ0 = BMO. Two other families, of dyadic nature, are of interest to us. Intervals in R of the form In,k = [ (k−1)2 n, k2n], where k and n are arbitrary integers, positive, negative or 0, are said to be dyadic. In RN , cubes which are the product of dyadic intervals of the same length, i.e., of the form Qn,k = In,k1 ×· · ·×In,kN , are called dyadic, and the collection of all such cubes is denoted D. There is also the family D0. Let I n,k = [(k− 1)2 n, (k+ 1)2n], where k and n are arbitrary integers. Clearly I ′n,k is dyadic if k is odd, but not if k is even. Now, the collection {I ′n,k : n, k integers} contains all dyadic intervals as well as the shifts [(k − 1)2n + 2n−1, k 2n + 2n−1] of the dyadic intervals by their half length. In RN , put D0 = {Q n,k : Q n,k = I × · · · × I ′n,kN }; Q n,k is called a special cube. Note that D0 contains D properly. Finally, given I ′n,k, let I n,k = [(k − 1)2 n, k2n], and I n,k = [k2 n, (k + 1)2n]. The 2N subcubes of Q′n,k = I × · · · × I ′n,kN of the form I × · · · × I Sj = L or R, 1 ≤ j ≤ N , are called the dyadic subcubes of Q Let Q0 denote the special cube [−1, 1] N . Given α ≥ 0, we construct a family Sα of piecewise polynomial splines in L 2(Q0) that will be useful in characterizing Λα. Let A be the subspace of L 2(Q0) consisting of all functions with vanishing moments up to order [α] which coincide with a polynomial in P[α] on each of the 2 N dyadic subcubes of Q0. A is a finite dimensional subspace of L2(Q0), and, therefore, by the Graham-Schmidt orthogonalization process, say, A has an orthonormal basis in L2(Q0) consisting of functions p1, . . . , pM with vanishing moments up to order [α], which coincide with a polynomial in P[α] on each dyadic subinterval of Q0. Together with each p we also consider all dyadic dilations and integer translations given by pLn,k,α(x) = 2 n(N+α)pL(2nx1 + k1, . . . , 2 nxN + kN ) , 1 ≤ L ≤ M , and let Sα = {p n,k,α : n, k integers, 1 ≤ L ≤ M} . Our first result shows how the dyadic grid can be used to compute the norm in Λα. Theorem A. Let g be a locally square-integrable function and α ≥ 0. Then, g ∈ Λα if, and only if, g ∈ Λα,D and Aα(g) = supp∈Sα ∣〈g, p〉 ∣ < ∞. Moreover, ‖g‖Λα ∼ ‖g‖Λα,D +Aα(g) . Furthermore, it is also true, and the proof is given in Proposition 2.1 be- low, that ‖g‖Λα ∼ ‖g‖Λα,D0 . However, in this simpler formulation, the tree structure of the cubes in D has been lost. FROM DYADIC Λα TO Λα 3 The proof of Theorem A relies on a close investigation of the predual of Λα, namely, the Hardy space H p(RN ) with 0 < p = (α + N)/N ≤ 1. In the process we characterize Hp in terms of simpler subspaces: H , or dyadic Hp, and H , the space generated by the special atoms in Sα. Specifically, we Theorem B. Let 0 < p ≤ 1, and α = N(1/p− 1). We then have Hp = H where the sum is understood in the sense of quasinormed Banach spaces. The paper is organized as follows. In Section 1 we show that individual Hp atoms can be written as a superposition of dyadic and special atoms; this fact may be thought of as an extension of the one-dimensional result of Fridli concerning L∞ 1- atoms, see [5] and [1]. Then, we prove Theorem B. In Section 2 we discuss how to pass from Λα,D, and Λα,D0 , to the Lipschitz space Λα. 1. Characterization of the Hardy spaces Hp We adopt the atomic definition of the Hardy spaces Hp, 0 < p ≤ 1, see [6] and [10]. Recall that a compactly supported function a with [N(1/p− 1)] vanishing moments is an L2 p -atom with defining cube Q if supp(a) ⊆ Q, and |Q|1/p | a(x) |2dx ≤ 1 . The Hardy space Hp(RN ) = Hp consists of those distributions f that can be written as f = λjaj , where the aj ’s are H p atoms, |λj | p < ∞, and the convergence is in the sense of distributions as well as in Hp. Furthermore, ‖f‖Hp ∼ inf |λj | where the infimum is taken over all possible atomic decompositions of f . This last expression has traditionally been called the atomic Hp norm of f . Collections of atoms with special properties can be used to gain a better understanding of the Hardy spaces. Formally, let A be a non-empty subset of L2 p -atoms in the unit ball of Hp. The atomic space H spanned by A consists of those ϕ in Hp of the form λjaj , aj ∈ A , |λj | p < ∞ . It is readily seen that, endowed with the atomic norm ‖ϕ‖Hp = inf |λj | : ϕ = λj aj , aj ∈ A becomes a complete quasinormed space. Clearly, H ⊆ Hp, and, for f ∈ H , ‖f‖Hp ≤ ‖f‖Hp 4 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY Two families are of particular interest to us. When A is the collection of all L2 p -atoms whose defining cube is dyadic, the resulting space is H or dyadic Hp. Now, although ‖f‖Hp ≤ ‖f‖Hp , the two quasinorms are not equivalent on H . Indeed, for p = 1 and N = 1, the functions fn(x) = 2 n[χ[1−2−n,1](x) − χ[1,1+2−n](x)] , satisfy ‖fn‖H1 = 1, but ‖fn‖H1 ∼ |n| tends to infinity with n. Next, when Sα is the family of piecewise polynomial splines constructed above with α = N(1/p − 1), in analogy with the one-dimensional results in [4] and [1], H is referred to as the space generated by special atoms. We are now ready to describe Hp atoms as a superposition of dyadic and special atoms. Lemma 1.1. Let a be an L2 p -atom with defining cube Q, 0 < p ≤ 1, and α = N(1/p − 1). Then a can be written as a linear combination of 2N dyadic atoms ai, each supported in one of the dyadic subcubes of the smallest special cube Qn,k containing Q, and a special atom b in Sα. More precisely, a(x) = i=1 di ai(x) + L=1 cL p −n,−k,α(x), with |di| , |cL| ≤ c. Proof. Suppose first that the defining cube of a is Q0, and let Q1, . . . , Q2N denote the dyadic subcubes of Q0. Furthermore, let {e i , . . . , e i } denote an orthonormal basis of the subspace Ai of L 2(Qi) consisting of polynomials in P[α], 1 ≤ i ≤ 2 N . Put αi(x) = a(x)χQi (x)− 〈aχQi , e j(x) , 1 ≤ i ≤ 2 and observe that 〈αi, e j〉 = 0 for 1 ≤ j ≤ M . Therefore, αi has [α] vanishing moments, is supported in Qi, and ‖αi‖2 ≤ ‖aχQi‖2 + ‖aχQi‖2 ≤ (M + 1) ‖aχQi‖2 . ai(x) = 2N(1/2−1/p) M + 1 αi(x) , 1 ≤ i ≤ N , is an L2 p - dyadic atom. Finally, put b(x) = a(x) − M + 1 2N(1/2−1/p) ai(x) . FROM DYADIC Λα TO Λα 5 Clearly b has [α] vanishing moments, is supported in Q0, coincides with a polynomial in P[α] on each dyadic subcube of Q0, and ‖b‖22 ≤ |〈aχQi , e 2 ≤ M ‖a‖22 . So, b ∈ A, and, consequently, b(x) = L=1 cL p L(x), where |cL| = |〈b, p L〉| ≤ c , 1 ≤ L ≤ M . In the general case, let Q be the defining cube of a, side-length Q = ℓ, and let n and k = (k1, . . . , kN ) be chosen so that 2 n−1 ≤ ℓ < 2n, and Q ⊂ [(k1 − 1)2 n, (k1 + 1)2 n]× · · · × [(kN − 1)2 n, (kN + 1)2 Then, (1/2)N ≤ |Q|/2nN < 1. Now, given x ∈ Q0, let a ′ be the translation and dilation of a given by a′(x) = 2nN/pa(2nx1 − k1, . . . , 2 nxN − kN ) . Clearly, [α] moments of a′ vanish, and ‖a′‖2 = 2 nN/p 2−nN/2‖a‖2 ≤ c |Q| 1/p|Q|−1/2‖a‖2 ≤ c . Thus, a′ is a multiple of an atom with defining cube Q0. By the first part of the proof, a′(x) = i(x) + L(x) , x ∈ Q0 . The support of each a′i is contained in one of the dyadic subcubes of Q0, and, consequently, there is a k such that ai(x) = 2 −nN/pa′i(2 −nx1 − k1, . . . , 2 −nxN − kN ) ai is an L 2p -atom supported in one of the dyadic subcubes of Q. Similarly for the pL’s. Thus, a(x) = di ai(x) + −n,−k,N(1/p−1)(x) , and we have finished. � Theorem B follows readily from Lemma 1.1. Clearly, H →֒ Hp. Conversely, let f = j λj aj be in H p. By Lemma 1.1 each aj can be written as a sum of dyadic and special atoms, and, by distributing the sum, we can write f = fd + fs, with fd in H , fs in H , and ‖fd‖Hp , ‖fs‖Hp |λj | Taking the infimum over the decompositions of f we get ‖f‖Hp c ‖f‖Hp , and H p →֒ H . This completes the proof. 6 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY The meaning of this decomposition is the following. Cubes in D are con- tained in one of the 2N non-overlapping quadrants of RN . To allow for the information carried by a dyadic cube to be transmitted to an adjacent dyadic cube, they must be connected. The pLn,k,α’s channel information across ad- jacent dyadic cubes which would otherwise remain disconnected. The reader will have no difficulty in proving the quantitative version of this observation: Let T be a linear mapping defined on Hp, 0 < p ≤ 1, that assumes values in a quasinormed Banach space X . Then, T is continuous if, and only if, the restrictions of T to H and H are continuous. 2. Characterizations of Λα Theorem A describes how to pass from Λα,D to Λα, and we prove it next. Since (Hp)∗ = Λα and (H )∗ = Λα,D, from Theorem B it follows readily that Λα = Λα,D ∩ (H )∗, so it only remains to show that (H )∗ is characterized by the condition Aα(g) < ∞. First note that if g is a locally square-integrable function with Aα(g) < ∞ and f = j,L cj,L p nj ,kj ,α , since 0 < p ≤ 1, |〈g, f〉| ≤ |cj,L| |〈g, p nj ,kj ,α ≤ Aα(g) |cj,L| and, consequently, taking the infimum over all atomic decompositions of f in , we get g ∈ (H )∗ and ‖g‖(Hp )∗ ≤ Aα(g). To prove the converse we proceed as in [3]. Let Qn = [−2 n, 2n]N . We begin by observing that functions f in L2(Qn) that have vanishing moments up to order [α] and coincide with polynomials of degree [α] on the dyadic subcubes of Qn belong to H ‖f‖Hp ≤ |Qn| 1/p−1/2‖f‖2 . Given ℓ ∈ (H )∗, for a fixed n let us consider the restriction of ℓ to the space of L2 functions f with [α] vanishing moments that are supported in Qn. Since |ℓ(f)| ≤ ‖ℓ‖ ‖f‖Hp ≤ ‖ℓ‖ |Qn| 1/p−1/2‖f‖2 , this restriction is continuous with respect to the norm in L2 and, consequently, it can be extended to a continuous linear functional in L2 and represented as ℓ(f) = f(x) gn(x) dx , FROM DYADIC Λα TO Λα 7 where gn ∈ L 2(Qn) and satisfies ‖gn‖2 ≤ ‖ℓ‖ |Qn| 1/p−1/2. Clearly, gn is uniquely determined in Qn up to a polynomial pn in P[α]. Therefore, gn(x) − pn(x) = gm(x)− pm(x) , a.e. x ∈ Qmin(n,m) . Consequently, if g(x) = gn(x)− pn(x) , x ∈ Qn , g(x) is well defined a.e. and, if f ∈ L2 has [α] vanishing moments and is supported in Qn, we have ℓ(f) = f(x) gn(x) dx f(x) [gn(x)− pn(x)] dx f(x) g(x) dx . Moreover, since each 2nN/ppL(2n ·+k) is an L2 p-atom, 1 ≤ L ≤ M , it readily follows that Aα(g) = sup 1≤L≤M n,k∈Z |〈g, 2−n/ppL(2n ·+k)〉| ≤ ‖ℓ‖ sup ‖pL‖Hp ≤ ‖ℓ‖ , and, consequently, Aα(g) ≤ ‖ℓ‖ , and (H )∗ is the desired space. � The reader will have no difficulty in showing that this result implies the following: Let T be a bounded linear operator from a quasinormed space X into Λα,D. Then, T is bounded from X into Λα if, and only if, Aα(Tx) ≤ c ‖x‖X for every x ∈ X . The process of averaging the translates of dyadic BMO functions leads to BMO, and is an important tool in obtaining results in BMO once they are known to be true in its dyadic counterpart, BMOd, see [7]. It is also known that BMO can be obtained as the intersection of BMOd and one of its shifted counterparts, see [8]. These results motivate our next proposition, which essentially says that g ∈ Λα if, and only if, g ∈ Λα,D and g is in the Lipschitz class obtained from the shifted dyadic grid. Note that the shifts involved in this class are in all directions parallel to the coordinate axis and depend on the side-length of the cube. Proposition 2.1. Λα = Λα,D0 , and ‖g‖Λα ∼ ‖g‖Λα,D0 . Proof. It is obvious that ‖g‖Λα,D0 ≤ ‖g‖Λα . To show the other inequality we invoke Theorem A. Since D ⊂ D0, it suffices to estimate Aα(g), or, equiva- lently, |〈g, p〉| for p ∈ Sα, α = N(1/p − 1). So, pick p = p n,k,α in Sα. The defining cube Q of pLn,k,α is in D0, and, since p n,k,α has [α] vanishing moments, 8 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY 〈pLn,k,α, pQ(g)〉 = 0. Therefore, |〈g, pLn,k,α〉| = |〈g − pQ(g), p n,k,α〉| ≤ ‖pLn,k,α‖2 ‖g − pQ(g)‖L2(Q) ≤ |Q|α/N |Q|1/2‖pLn,k,α‖2 ‖g‖Λα,D0 . Now, a simple change of variables gives |Q|α/N |Q|1/2‖pLn,k,α‖2 ≤ 1, and, con- sequently, also Aα(g) ≤ ‖g‖Λα,D0 . � References [1] W. Abu-Shammala, J.-L. Shiu, and A. Torchinsky, Characterizations of the Hardy space H1 and BMO, preprint. [2] H.-Q. Bui and R. S. Laugesen, Approximation and spanning in the Hardy space, by affine systems, Constr. Approx., to appear. [3] A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distibution, II, Advances in Math., 24 (1977), 101–171. [4] G. S. de Souza, Spaces formed by special atoms, I, Rocky Mountain J. Math. 14 (1984), no. 2, 423–431. [5] S. Fridli, Transition from the dyadic to the real nonperiodic Hardy space, Acta Math. Acad. Paedagog. Niházi (N.S.) 16 (2000), 1–8, (electronic). [6] J. Garćıa-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, Notas de Matemática 116, North Holland, Amsterdam, 1985. [7] J. Garnett and P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. [8] T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003–1006. [9] T. M. Le and L. A. Vese, Image decomposition using total variation and div( BMO)∗, Multiscale Model. Simul. 4, (2005), no. 2, 390–423. [10] A. Torchinsky, Real-variable methods in harmonic analysis, Dover Publications, Inc., Mineola, NY, 2004. Department of Mathematics, Indiana University, Bloomington IN 47405 E-mail address: wabusham@indiana.edu Department of Mathematics, Indiana University, Bloomington IN 47405 E-mail address: torchins@indiana.edu 1. Characterization of the Hardy spaces Hp 2. Characterizations of References
704.001
Polymer Quantum Mechanics and its Continuum Limit Alejandro Corichi,1, 2, 3, ∗ Tatjana Vukašinac,4, † and José A. Zapata1, ‡ Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, Mexico Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F. 04510, Mexico Institute for Gravitational Physics and Geometry, Physics Department, Pennsylvania State University, University Park PA 16802, USA Facultad de Ingenieŕıa Civil, Universidad Michoacana de San Nicolas de Hidalgo, Morelia, Michoacán 58000, Mexico A rather non-standard quantum representation of the canonical commutation relations of quan- tum mechanics systems, known as the polymer representation has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schrödinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schrödinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schrödinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle and a simple cosmological model. PACS numbers: 04.60.Pp, 04.60.Ds, 04.60.Nc 11.10.Gh. I. INTRODUCTION The so-called polymer quantum mechanics, a non- regular and somewhat ‘exotic’ representation of the canonical commutation relations (CCR) [1], has been used to explore both mathematical and physical issues in background independent theories such as quantum grav- ity [2, 3]. A notable example of this type of quantization, when applied to minisuperspace models has given way to what is known as loop quantum cosmology [4, 5]. As in any toy model situation, one hopes to learn about the subtle technical and conceptual issues that are present in full quantum gravity by means of simple, finite di- mensional examples. This formalism is not an exception in this regard. Apart from this motivation coming from physics at the Planck scale, one can independently ask for the relation between the standard continuous repre- sentations and their polymer cousins at the level of math- ematical physics. A deeper understanding of this relation becomes important on its own. The polymer quantization is made of several steps. The first one is to build a representation of the Heisenberg-Weyl algebra on a Kinematical Hilbert space that is “background independent”, and that is sometimes referred to as the polymeric Hilbert space Hpoly. The second and most important part, the implementation of dynamics, deals with the definition of a Hamiltonian (or Hamiltonian constraint) on this space. In the examples ∗Electronic address: corichi@matmor.unam.mx †Electronic address: tatjana@shi.matmor.unam.mx ‡Electronic address: zapata@matmor.unam.mx studied so far, the first part is fairly well understood, yielding the kinematical Hilbert space Hpoly that is, how- ever, non-separable. For the second step, a natural im- plementation of the dynamics has proved to be a bit more difficult, given that a direct definition of the Hamiltonian Ĥ of, say, a particle on a potential on the space Hpoly is not possible since one of the main features of this repre- sentation is that the operators q̂ and p̂ cannot be both simultaneously defined (nor their analogues in theories involving more elaborate variables). Thus, any operator that involves (powers of) the not defined variable has to be regulated by a well defined operator which normally involves introducing some extra structure on the configu- ration (or momentum) space, namely a lattice. However, this new structure that plays the role of a regulator can not be removed when working in Hpoly and one is left with the ambiguity that is present in any regularization. The freedom in choosing it can be sometimes associated with a length scale (the lattice spacing). For ordinary quantum systems such as a simple harmonic oscillator, that has been studied in detail from the polymer view- point, it has been argued that if this length scale is taken to be ‘sufficiently small’, one can arbitrarily approximate standard Schrödinger quantum mechanics [2, 3]. In the case of loop quantum cosmology, the minimum area gap A0 of the full quantum gravity theory imposes such a scale, that is then taken to be fundamental [4]. A natural question is to ask what happens when we change this scale and go to even smaller ‘distances’, that is, when we refine the lattice on which the dynamics of the theory is defined. Can we define consistency con- ditions between these scales? Or even better, can we take the limit and find thus a continuum limit? As it http://arxiv.org/abs/0704.0007v2 mailto:corichi@matmor.unam.mx mailto:tatjana@shi.matmor.unam.mx mailto:zapata@matmor.unam.mx has been shown recently in detail, the answer to both questions is in the affirmative [6]. There, an appropriate notion of scale was defined in such a way that one could define refinements of the theory and pose in a precise fashion the question of the continuum limit of the theory. These results could also be seen as handing a procedure to remove the regulator when working on the appropri- ate space. The purpose of this paper is to further explore different aspects of the relation between the continuum and the polymer representation. In particular in the first part we put forward a novel way of deriving the polymer representation from the ordinary Schrödinger represen- tation as an appropriate limit. In Sec. II we derive two versions of the polymer representation as different lim- its of the Schrödinger theory. In Sec. III we show that these two versions can be seen as different polarizations of the ‘abstract’ polymer representation. These results, to the best of our knowledge, are new and have not been reported elsewhere. In Sec. IV we pose the problem of implementing the dynamics on the polymer representa- tion. In Sec. V we motivate further the question of the continuum limit (i.e. the proper removal of the regulator) and recall the basic constructions of [6]. Several exam- ples are considered in Sec. VI. In particular a simple harmonic oscillator, the polymer free particle and a sim- ple quantum cosmology model are considered. The free particle and the cosmological model represent a general- ization of the results obtained in [6] where only systems with a discrete and non-degenerate spectrum where con- sidered. We end the paper with a discussion in Sec. VII. In order to make the paper self-contained, we will keep the level of rigor in the presentation to that found in the standard theoretical physics literature. II. QUANTIZATION AND POLYMER REPRESENTATION In this section we derive the so called polymer repre- sentation of quantum mechanics starting from a specific reformulation of the ordinary Schrödinger representation. Our starting point will be the simplest of all possible phase spaces, namely Γ = R2 corresponding to a particle living on the real line R. Let us choose coordinates (q, p) thereon. As a first step we shall consider the quantization of this system that leads to the standard quantum theory in the Schrödinger description. A convenient route is to introduce the necessary structure to define the Fock rep- resentation of such system. From this perspective, the passage to the polymeric case becomes clearest. Roughly speaking by a quantization one means a passage from the classical algebraic bracket, the Poisson bracket, {q, p} = 1 (1) to a quantum bracket given by the commutator of the corresponding operators, [ q̂, p̂] = i~ 1̂ (2) These relations, known as the canonical commutation re- lation (CCR) become the most common corner stone of the (kinematics of the) quantum theory; they should be satisfied by the quantum system, when represented on a Hilbert space H. There are alternative points of departure for quantum kinematics. Here we consider the algebra generated by the exponentiated versions of q̂ and p̂ that are denoted U(α) = ei(α q̂)/~ ; V (β) = ei(β p̂)/~ where α and β have dimensions of momentum and length, respectively. The CCR now become U(α) · V (β) = e(−iα β)/~V (β) · U(α) (3) and the rest of the product is U(α1)·U(α2) = U(α1+α2) ; V (β1)·V (β2) = V (β1+β2) The Weyl algebra W is generated by taking finite linear combinations of the generators U(αi) and V (βi) where the product (3) is extended by linearity, (Ai U(αi) +Bi V (βi)) From this perspective, quantization means finding an unitary representation of the Weyl algebra W on a Hilbert space H′ (that could be different from the ordi- nary Schrödinger representation). At first it might look weird to attempt this approach given that we know how to quantize such a simple system; what do we need such a complicated object as W for? It is infinite dimensional, whereas the set S = {1̂, q̂, p̂}, the starting point of the ordinary Dirac quantization, is rather simple. It is in the quantization of field systems that the advantages of the Weyl approach can be fully appreciated, but it is also useful for introducing the polymer quantization and comparing it to the standard quantization. This is the strategy that we follow. A question that one can ask is whether there is any freedom in quantizing the system to obtain the ordinary Schrödinger representation. On a first sight it might seem that there is none given the Stone-Von Neumann unique- ness theorem. Let us review what would be the argument for the standard construction. Let us ask that the repre- sentation we want to build up is of the Schrödinger type, namely, where states are wave functions of configuration space ψ(q). There are two ingredients to the construction of the representation, namely the specification of how the basic operators (q̂, p̂) will act, and the nature of the space of functions that ψ belongs to, that is normally fixed by the choice of inner product on H, or measure µ on R. The standard choice is to select the Hilbert space to be, H = L2(R, dq) the space of square-integrable functions with respect to the Lebesgue measure dq (invariant under constant trans- lations) on R. The operators are then represented as, q̂ · ψ(q) = (q ψ)(q) and p̂ · ψ(q) = −i ~ ∂ ψ(q) (4) Is it possible to find other representations? In order to appreciate this freedom we go to the Weyl algebra and build the quantum theory thereon. The representation of the Weyl algebra that can be called of the ‘Fock type’ involves the definition of an extra structure on the phase space Γ: a complex structure J . That is, a linear map- ping from Γ to itself such that J2 = −1. In 2 dimen- sions, all the freedom in the choice of J is contained in the choice of a parameter d with dimensions of length. It is also convenient to define: k = p/~ that has dimensions of 1/L. We have then, Jd : (q, k) 7→ (−d2 k, q/d2) This object together with the symplectic structure: Ω((q, p); (q′, p′)) = q p′ − p q′ define an inner product on Γ by the formula gd(· ; ·) = Ω(· ; Jd ·) such that: gd((q, p); (q ′, p′)) = q q′ + which is dimension-less and positive definite. Note that with this quantities one can define complex coordinates (ζ, ζ̄) as usual: q + i p ; ζ̄ = q − i d from which one can build the standard Fock representa- tion. Thus, one can alternatively view the introduction of the length parameter d as the quantity needed to de- fine (dimensionless) complex coordinates on the phase space. But what is the relevance of this object (J or d)? The definition of complex coordinates is useful for the construction of the Fock space since from them one can define, in a natural way, creation and annihilation operators. But for the Schrödinger representation we are interested here, it is a bit more subtle. The subtlety is that within this approach one uses the algebraic prop- erties of W to construct the Hilbert space via what is known as the Gel’fand-Naimark-Segal (GNS) construc- tion. This implies that the measure in the Schrödinger representation becomes non trivial and thus the momen- tum operator acquires an extra term in order to render the operator self-adjoint. The representation of the Weyl algebra is then, when acting on functions φ(q) [7]: Û(α) · φ(q) := (eiα q/~ φ)(q) V̂ (β) · φ(q) := e (q−β/2) φ(q − β) The Hilbert space structure is introduced by the defini- tion of an algebraic state (a positive linear functional) ωd : W → C, that must coincide with the expectation value in the Hilbert space taken on a special state ref- ered to as the vacuum: ωd(a) = 〈â〉vac, for all a ∈ W . In our case this specification of J induces such a unique state ωd that yields, 〈Û(α)〉vac = e− d2 α2 ~2 (5) 〈V̂ (β)〉vac = e− d2 (6) Note that the exponents in the vacuum expectation values correspond to the metric constructed out of J : d2 α2 = gd((0, α); (0, α)) and = gd((β, 0); (β, 0)). Wave functions belong to the space L2(R, dµd), where the measure that dictates the inner product in this rep- resentation is given by, dµd = d2 dq In this representation, the vacuum is given by the iden- tity function φ0(q) = 1 that is, just as any plane wave, normalized. Note that for each value of d > 0, the rep- resentation is well defined and continuous in α and β. Note also that there is an equivalence between the q- representation defined by d and the k-representation de- fined by 1/d. How can we recover then the standard representation in which the measure is given by the Lebesgue measure and the operators are represented as in (4)? It is easy to see that there is an isometric isomorphism K that maps the d-representation in Hd to the standard Schrödinger representation in Hschr by: ψ(q) = K · φ(q) = e d1/2π1/4 φ(q) ∈ Hschr = L2(R, dq) Thus we see that all d-representations are unitarily equiv- alent. This was to be expected in view of the Stone-Von Neumann uniqueness result. Note also that the vacuum now becomes ψ0(q) = d1/2π1/4 2 d2 , so even when there is no information about the param- eter d in the representation itself, it is contained in the vacuum state. This procedure for constructing the GNS- Schrödinger representation for quantum mechanics has also been generalized to scalar fields on arbitrary curved space in [8]. Note, however that so far the treatment has all been kinematical, without any knowledge of a Hamil- tonian. For the Simple Harmonic Oscillator of mass m and frequency ω, there is a natural choice compatible with the dynamics given by d = , in which some calculations simplify (for instance for coherent states), but in principle one can use any value of d. Our study will be simplified by focusing on the funda- mental entities in the Hilbert Space Hd , namely those states generated by acting with Û(α) on the vacuum φ0(q) = 1. Let us denote those states by, φα(q) = Û(α) · φ0(q) = ei The inner product between two such states is given by 〈φα, φλ〉d = dµd e ~ = e− (λ−α)2 d2 4 ~2 (7) Note incidentally that, contrary to some common belief, the ‘plane waves’ in this GNS Hilbert space are indeed normalizable. Let us now consider the polymer representation. For that, it is important to note that there are two possible limiting cases for the parameter d: i) The limit 1/d 7→ 0 and ii) The case d 7→ 0. In both cases, we have ex- pressions that become ill defined in the representation or measure, so one needs to be careful. A. The 1/d 7→ 0 case. The first observation is that from the expressions (5) and (6) for the algebraic state ωd, we see that the limiting cases are indeed well defined. In our case we get, ωA := lim1/d→0 ωd such that, ωA(Û(α)) = δα,0 and ωA(V̂ (β)) = 1 (8) From this, we can indeed construct the representation by means of the GNS construction. In order to do that and to show how this is obtained we shall consider several expressions. One has to be careful though, since the limit has to be taken with care. Let us consider the measure on the representation that behaves as: dµd = d2 dq 7→ 1 so the measures tends to an homogeneous measure but whose ‘normalization constant’ goes to zero, so the limit becomes somewhat subtle. We shall return to this point later. Let us now see what happens to the inner product between the fundamental entities in the Hilbert Space Hd given by (7). It is immediate to see that in the 1/d 7→ 0 limit the inner product becomes, 〈φα, φλ〉d 7→ δα,λ (9) with δα,λ being Kronecker’s delta. We see then that the plane waves φα(q) become an orthonormal basis for the new Hilbert space. Therefore, there is a delicate interplay between the two terms that contribute to the measure in order to maintain the normalizability of these functions; we need the measure to become damped (by 1/d) in order to avoid that the plane waves acquire an infinite norm (as happens with the standard Lebesgue measure), but on the other hand the measure, that for any finite value of d is a Gaussian, becomes more and more spread. It is important to note that, in this limit, the operators Û(α) become discontinuous with respect to α, given that for any given α1 and α2 (different), its action on a given basis vector ψλ(q) yields orthogonal vectors. Since the continuity of these operators is one of the hypotesis of the Stone-Von Neumann theorem, the uniqueness result does not apply here. The representation is inequivalent to the standard one. Let us now analyze the other operator, namely the action of the operator V̂ (β) on the basis φα(q): V̂ (β) · φα(q) = e− ~ e(β/d 2+iα/~)q which in the limit 1/d 7→ 0 goes to, V̂ (β) · φα(q) 7→ ei ~ φα(q) that is continuous on β. Thus, in the limit, the operator p̂ = −i~∂q is well defined. Also, note that in this limit the operator p̂ has φα(q) as its eigenstate with eigenvalue given by α: p̂ · φα(q) 7→ αφα(q) To summarize, the resulting theory obtained by taking the limit 1/d 7→ 0 of the ordinary Schrödinger descrip- tion, that we shall call the ‘polymer representation of type A’, has the following features: the operators U(α) are well defined but not continuous in α, so there is no generator (no operator associated to q). The basis vec- tors φα are orthonormal (for α taking values on a contin- uous set) and are eigenvectors of the operator p̂ that is well defined. The resulting Hilbert space HA will be the (A-version of the) polymer representation. Let us now consider the other case, namely, the limit when d 7→ 0. B. The d 7→ 0 case Let us now explore the other limiting case of the Schrödinger/Fock representations labelled by the param- eter d. Just as in the previous case, the limiting algebraic state becomes, ωB := limd→0 ωd such that, ωB(Û(α)) = 1 and ωB(V̂ (β)) = δβ,0 (10) From this positive linear function, one can indeed con- struct the representation using the GNS construction. First let us note that the measure, even when the limit has to be taken with due care, behaves as: dµd = d2 dq 7→ δ(q) dq That is, as Dirac’s delta distribution. It is immediate to see that, in the d 7→ 0 limit, the inner product between the fundamental states φα(q) becomes, 〈φα, φλ〉d 7→ 1 (11) This in fact means that the vector ξ = φα − φλ belongs to the Kernel of the limiting inner product, so one has to mod out by these (and all) zero norm states in order to get the Hilbert space. Let us now analyze the other operator, namely the action of the operator V̂ (β) on the vacuum φ0(q) = 1, which for arbitrary d has the form, φ̃β := V̂ (β) · φ0(q) = e (q−β/2) The inner product between two such states is given by 〈φ̃α, φ̃β〉d = e− (α−β)2 In the limit d → 0, 〈φ̃α, φ̃β〉d → δα,β. We can see then that it is these functions that become the orthonormal, ‘discrete basis’ in the theory. However, the function φ̃β(q) in this limit becomes ill defined. For example, for β > 0, it grows unboundedly for q > β/2, is equal to one if q = β/2 and zero otherwise. In order to overcome these difficulties and make more transparent the resulting the- ory, we shall consider the other form of the representation in which the measure is incorporated into the states (and the resulting Hilbert space is L2(R, dq)). Thus the new state ψβ(q) := K · (V̂ (β) · φ0(q)) = (q−β)2 We can now take the limit and what we get is d 7→0 ψβ(q) := δ 1/2(q, β) where by δ1/2(q, β) we mean something like ‘the square root of the Dirac distribution’. What we really mean is an object that satisfies the following property: δ1/2(q, β) · δ1/2(q, α) = δ(q, β) δβ,α That is, if α = β then it is just the ordinary delta, other- wise it is zero. In a sense these object can be regarded as half-densities that can not be integrated by themselves, but whose product can. We conclude then that the inner product is, 〈ψβ , ψα〉 = dq ψβ(q)ψα(q) = dq δ(q, α) δβ,α = δβ,α which is just what we expected. Note that in this repre- sentation, the vacuum state becomes ψ0(q) := δ 1/2(q, 0), namely, the half-delta with support in the origin. It is important to note that we are arriving in a natural way to states as half-densities, whose squares can be integrated without the need of a nontrivial measure on the configu- ration space. Diffeomorphism invariance arises then in a natural but subtle manner. Note that as the end result we recover the Kronecker delta inner product for the new fundamental states: χβ(q) := δ 1/2(q, β). Thus, in this new B-polymer representation, the Hilbert space HB is the completion with respect to the inner product (13) of the states generated by taking (finite) linear combinations of basis elements of the form χβ : Ψ(q) = bi χβi(q) (14) Let us now introduce an equivalent description of this Hilbert space. Instead of having the basis elements be half-deltas as elements of the Hilbert space where the inner product is given by the ordinary Lebesgue measure dq, we redefine both the basis and the measure. We could consider, instead of a half-delta with support β, a Kronecker delta or characteristic function with support on β: χ′β(q) := δq,β These functions have a similar behavior with respect to the product as the half-deltas, namely: χ′β(q) · χ′α(q) = δβ,α. The main difference is that neither χ ′ nor their squares are integrable with respect to the Lebesgue mea- sure (having zero norm). In order to fix that problem we have to change the measure so that we recover the basic inner product (13) with our new basis. The needed mea- sure turns out to be the discrete counting measure on R. Thus any state in the ‘half density basis’ can be written (using the same expression) in terms of the ‘Kronecker basis’. For more details and further motivation see the next section. Note that in this B-polymer representation, both Û and V̂ have their roles interchanged with that of the A-polymer representation: while U(α) is discontinuous and thus q̂ is not defined in the A-representation, we have that it is V (β) in the B-representation that has this property. In this case, it is the operator p̂ that can not be defined. We see then that given a physical system for which the configuration space has a well defined physi- cal meaning, within the possible representation in which wave-functions are functions of the configuration variable q, the A and B polymer representations are radically dif- ferent and inequivalent. Having said this, it is also true that the A and B representations are equivalent in a different sense, by means of the duality between q and p representations and the d↔ 1/d duality: The A-polymer representation in the “q-representation” is equivalent to the B-polymer representation in the “p-representation”, and conversely. When studying a problem, it is important to decide from the beginning which polymer representation (if any) one should be using (for instance in the q-polarization). This has as a consequence an implication on which variable is naturally “quantized” (even if continuous): p for A and q for B. There could be for instance a physical criteria for this choice. For example a fundamental symmetry could suggest that one representation is more natural than an- other one. This indeed has been recently noted by Chiou in [10], where the Galileo group is investigated and where it is shown that the B representation is better behaved. In the other polarization, namely for wavefunctions of p, the picture gets reversed: q is discrete for the A- representation, while p is for the B-case. Let us end this section by noting that the procedure of obtaining the polymer quantization by means of an appropriate limit of Fock-Schrödinger representations might prove useful in more general settings in field theory or quantum gravity. III. POLYMER QUANTUM MECHANICS: KINEMATICS In previous sections we have derived what we have called the A and B polymer representations (in the q- polarization) as limiting cases of ordinary Fock repre- sentations. In this section, we shall describe, without any reference to the Schrödinger representation, the ‘ab- stract’ polymer representation and then make contact with its two possible realizations, closely related to the A and B cases studied before. What we will see is that one of them (the A case) will correspond to the p-polarization while the other one corresponds to the q−representation, when a choice is made about the physical significance of the variables. We can start by defining abstract kets |µ〉 labelled by a real number µ. These shall belong to the Hilbert space Hpoly. From these states, we define a generic ‘cylinder states’ that correspond to a choice of a finite collection of numbers µi ∈ R with i = 1, 2, . . . , N . Associated to this choice, there are N vectors |µi〉, so we can take a linear combination of them |ψ〉 = ai |µi〉 (15) The polymer inner product between the fundamental kets is given by, 〈ν|µ〉 = δν,µ (16) That is, the kets are orthogonal to each other (when ν 6= µ) and they are normalized (〈µ|µ〉 = 1). Immediately, this implies that, given any two vectors |φ〉 = j=1 bj |νj〉 and |ψ〉 = i=1 ai |µi〉, the inner product between them is given by, 〈φ|ψ〉 = b̄j ai 〈νj |µi〉 = b̄k ak where the sum is over k that labels the intersection points between the set of labels {νj} and {µi}. The Hilbert space Hpoly is the Cauchy completion of finite linear com- bination of the form (15) with respect to the inner prod- uct (16). Hpoly is non-separable. There are two basic operators on this Hilbert space: the ‘label operator’ ε̂: ε̂ |µ〉 := µ |µ〉 and the displacement operator ŝ (λ), ŝ (λ) |µ〉 := |µ+ λ〉 The operator ε̂ is symmetric and the operator(s) ŝ(λ) defines a one-parameter family of unitary operators on Hpoly, where its adjoint is given by ŝ† (λ) = ŝ (−λ). This action is however, discontinuous with respect to λ given that |µ〉 and |µ + λ〉 are always orthogonal, no matter how small is λ. Thus, there is no (Hermitian) operator that could generate ŝ (λ) by exponentiation. So far we have given the abstract characterization of the Hilbert space, but one would like to make contact with concrete realizations as wave functions, or by iden- tifying the abstract operators ε̂ and ŝ with physical op- erators. Suppose we have a system with a configuration space with coordinate given by q, and p denotes its canonical conjugate momenta. Suppose also that for physical rea- sons we decide that the configuration coordinate q will have some “discrete character” (for instance, if it is to be identified with position, one could say that there is an underlying discreteness in position at a small scale). How can we implement such requirements by means of the polymer representation? There are two possibilities, depending on the choice of ‘polarizations’ for the wave- functions, namely whether they will be functions of con- figuration q or momenta p. Let us the divide the discus- sion into two parts. A. Momentum polarization In this polarization, states will be denoted by, ψ(p) = 〈p|ψ〉 where ψµ(p) = 〈p|µ〉 = ei How are then the operators ε̂ and ŝ represented? Note that if we associate the multiplicative operator V̂ (λ) · ψµ(p) = ei ~ = ei (µ+λ) p = ψ(µ+λ)(p) we see then that the operator V̂ (λ) corresponds precisely to the shift operator ŝ (λ). Thus we can also conclude that the operator p̂ does not exist. It is now easy to identify the operator q̂ with: q̂ · ψµ(p) = −i~ ψµ(p) = µ e ~ = µψµ(p) namely, with the abstract operator ε̂. The reason we say that q̂ is discrete is because this operator has as its eigenvalue the label µ of the elementary state ψµ(p), and this label, even when it can take value in a continuum of possible values, is to be understood as a discrete set, given that the states are orthonormal for all values of µ. Given that states are now functions of p, the inner product (16) should be defined by a measure µ on the space on which the wave-functions are defined. In order to know what these two objects are, namely, the quan- tum “configuration” space C and the measure thereon1, we have to make use of the tools available to us from the theory of C∗-algebras. If we consider the operators V̂ (λ), together with their natural product and ∗-relation given by V̂ ∗(λ) = V̂ (−λ), they have the structure of an Abelian C∗-algebra (with unit) A. We know from the representation theory of such objects that A is iso- morphic to the space of continuous functions C0(∆) on a compact space ∆, the spectrum of A. Any representation of A on a Hilbert space as multiplication operator will be on spaces of the form L2(∆, dµ). That is, our quantum configuration space is the spectrum of the algebra, which in our case corresponds to the Bohr compactification Rb of the real line [11]. This space is a compact group and there is a natural probability measure defined on it, the Haar measure µH. Thus, our Hilbert space Hpoly will be isomorphic to the space, Hpoly,p = L2(Rb, dµH) (17) In terms of ‘quasi periodic functions’ generated by ψµ(p), the inner product takes the form 〈ψµ|ψλ〉 := dµH ψµ(p)ψλ(p) := = lim L 7→∞ dpψµ(p)ψλ(p) = δµ,λ (18) note that in the p-polarization, this characterization cor- responds to the ‘A-version’ of the polymer representation of Sec. II (where p and q are interchanged). B. q-polarization Let us now consider the other polarization in which wave functions will depend on the configuration coordinate q: ψ(q) = 〈q|ψ〉 The basic functions, that now will be called ψ̃µ(q), should be, in a sense, the dual of the functions ψµ(p) of the previous subsection. We can try to define them via a ‘Fourier transform’: ψ̃µ(q) := 〈q|µ〉 = 〈q| dµH|p〉〈p|µ〉 which is given by ψ̃µ(q) := dµH〈q|p〉ψµ(p) = dµH e −i p q ~ = δq,µ (19) 1 here we use the standard terminology of ‘configuration space’ to denote the domain of the wave function even when, in this case, it corresponds to the physical momenta p. That is, the basic objects in this representation are Kro- necker deltas. This is precisely what we had found in Sec. II for the B-type representation. How are now the basic operators represented and what is the form of the inner product? Regarding the operators, we expect that they are represented in the opposite manner as in the previous p-polarization case, but that they preserve the same features: p̂ does not exist (the derivative of the Kro- necker delta is ill defined), but its exponentiated version V̂ (λ) does: V̂ (λ) · ψ(q) = ψ(q + λ) and the operator q̂ that now acts as multiplication has as its eigenstates, the functions ψ̃ν(q) = δν,q: q̂ · ψ̃µ(q) := µ ψ̃µ(q) What is now the nature of the quantum configurations space Q? And what is the measure thereon dµq? that defines the inner product we should have: 〈ψ̃µ(q), ψ̃λ(q)〉 = δµ,λ The answer comes from one of the characterizations of the Bohr compactification: we know that it is, in a precise sense, dual to the real line but when equipped with the discrete topology Rd. Furthermore, the measure on Rd will be the ‘counting measure’. In this way we recover the same properties we had for the previous characterization of the polymer Hilbert space. We can thus write: Hpoly,x := L2(Rd, dµc) (20) This completes a precise construction of the B-type poly- mer representation sketched in the previous section. Note that if we had chosen the opposite physical situation, namely that q, the configuration observable, be the quan- tity that does not have a corresponding operator, then we would have had the opposite realization: In the q- polarization we would have had the type-A polymer rep- resentation and the type-B for the p-polarization. As we shall see both scenarios have been considered in the literature. Up to now we have only focused our discussion on the kinematical aspects of the quantization process. Let us now consider in the following section the issue of dynam- ics and recall the approach that had been adopted in the literature, before the issue of the removal of the regulator was reexamined in [6]. IV. POLYMER QUANTUM MECHANICS: DYNAMICS As we have seen the construction of the polymer representation is rather natural and leads to a quan- tum theory with different properties than the usual Schrödinger counterpart such as its non-separability, the non-existence of certain operators and the existence of normalized eigen-vectors that yield a precise value for one of the phase space coordinates. This has been done without any regard for a Hamiltonian that endows the system with a dynamics, energy and so on. First let us consider the simplest case of a particle of mass m in a potential V (q), in which the Hamiltonian H takes the form, p2 + V (q) Suppose furthermore that the potential is given by a non- periodic function, such as a polynomial or a rational func- tion. We can immediately see that a direct implementa- tion of the Hamiltonian is out of our reach, for the simple reason that, as we have seen, in the polymer representa- tion we can either represent q or p, but not both! What has been done so far in the literature? The simplest thing possible: approximate the non-existing term by a well defined function that can be quantized and hope for the best. As we shall see in next sections, there is indeed more that one can do. At this point there is also an important decision to be made: which variable q or p should be regarded as “dis- crete”? Once this choice is made, then it implies that the other variable will not exist: if q is regarded as dis- crete, then p will not exist and we need to approximate the kinetic term p2/2m by something else; if p is to be the discrete quantity, then q will not be defined and then we need to approximate the potential V (q). What hap- pens with a periodic potential? In this case one would be modelling, for instance, a particle on a regular lattice such as a phonon living on a crystal, and then the natural choice is to have q not well defined. Furthermore, the po- tential will be well defined and there is no approximation needed. In the literature both scenarios have been considered. For instance, when considering a quantum mechanical system in [2], the position was chosen to be discrete, so p does not exist, and one is then in the A type for the momentum polarization (or the type B for the q- polarization). With this choice, it is the kinetic term the one that has to be approximated, so once one has done this, then it is immediate to consider any potential that will thus be well defined. On the other hand, when con- sidering loop quantum cosmology (LQC), the standard choice is that the configuration variable is not defined [4]. This choice is made given that LQC is regarded as the symmetric sector of full loop quantum gravity where the connection (that is regarded as the configuration vari- able) can not be promoted to an operator and one can only define its exponentiated version, namely, the holon- omy. In that case, the canonically conjugate variable, closely related to the volume, becomes ‘discrete’, just as in the full theory. This case is however, different from the particle in a potential example. First we could mention that the functional form of the Hamiltonian constraint that implements dynamics has a different structure, but the more important difference lies in that the system is constrained. Let us return to the case of the particle in a po- tential and for definiteness, let us start with the aux- iliary kinematical framework in which: q is discrete, p can not be promoted and thus we have to approximate the kinetic term p̂2/2m. How is this done? The stan- dard prescription is to define, on the configuration space C, a regular ‘graph’ γµ0 . This consists of a numerable set of points, equidistant, and characterized by a pa- rameter µ0 that is the (constant) separation between points. The simplest example would be to consider the set γµ0 = {q ∈ R | q = nµ0 , ∀ n ∈ Z}. This means that the basic kets that will be considered |µn〉 will correspond precisely to labels µn belonging to the graph γµ0 , that is, µn = nµ0. Thus, we shall only consider states of the form, |ψ〉 = bn |µn〉 . (21) This ‘small’ Hilbert space Hγµ0 , the graph Hilbert space, is a subspace of the ‘large’ polymer Hilbert space Hpoly but it is separable. The condition for a state of the form (21) to belong to the Hilbert space Hγµ0 is that the co- efficients bn satisfy: n |bn|2 <∞. Let us now consider the kinetic term p̂2/2m. We have to approximate it by means of trigonometric functions, that can be built out of the functions of the form eiλ p/~. As we have seen in previous sections, these functions can indeed be promoted to operators and act as translation operators on the kets |µ〉. If we want to remain in the graph γ, and not create ‘new points’, then one is con- strained to considering operators that displace the kets by just the right amount. That is, we want the basic shift operator V̂ (λ) to be such that it maps the ket with label |µn〉 to the next ket, namely |µn+1〉. This can in- deed achieved by fixing, once and for all, the value of the allowed parameter λ to be λ = µ0. We have then, V̂ (µ0) · |µn〉 = |µn + µ0〉 = |µn+1〉 which is what we wanted. This basic ‘shift operator’ will be the building block for approximating any (polynomial) function of p. In order to do that we notice that the function p can be approximated by, p ≈ ~ (µ0 p ~ − e−i where the approximation is good for p << ~/µ0. Thus, one can define a regulated operator p̂µ0 that depends on the ‘scale’ µ0 as: p̂µ0 · |µn〉 := [V (µ0) − V (−µ0)] · |µn〉 = (|µn+1〉 − |µn−1〉) (22) In order to regulate the operator p̂2, there are (at least) two possibilities, namely to compose the operator p̂µ0 with itself or to define a new approximation. The oper- ator p̂µ0 · p̂µ0 has the feature that shifts the states two steps in the graph to both sides. There is however an- other operator that only involves shifting once: p̂2µ0 · |νn〉 := [2 − V̂ (µ0) − V̂ (−µ0)] · |νn〉 = (2|νn〉 − |νn+1〉 − |νn−1〉) (23) which corresponds to the approximation p2 ≈ 2~ cos(µ0 p/~)), valid also in the regime p << ~/µ0. With these considerations, one can define the operator Ĥµ0 , the Hamiltonian at scale µ0, that in practice ‘lives’ on the space Hγµ0 as, Ĥµ0 := p̂2µ0 + V̂ (q) , (24) that is a well defined, symmetric operator on Hγµ0 . No- tice that the operator is also defined on Hpoly, but there its physical interpretation is problematic. For example, it turns out that the expectation value of the kinetic term calculated on most states (states which are not tailored to the exact value of the parameter µ0) is zero. Even if one takes a state that gives “reasonable“ expectation values of the µ0-kinetic term and uses it to calculate the expectation value of the kinetic term corresponding to a slight perturbation of the parameter µ0 one would get zero. This problem, and others that arise when working on Hpoly, forces one to assign a physical interpretation to the Hamiltonian Ĥµ0 only when its action is restricted to the subspace Hγµ0 . Let us now explore the form that the Hamiltonian takes in the two possible polarizations. In the q-polarization, the basis, labelled by n is given by the functions χn(q) = δq,µn . That is, the wave functions will only have sup- port on the set γµ0 . Alternatively, one can think of a state as completely characterized by the ‘Fourier coeffi- cients’ an: ψ(q) ↔ an, which is the value that the wave function ψ(q) takes at the point q = µn = nµ0. Thus, the Hamiltonian takes the form of a difference equation when acting on a general state ψ(q). Solving the time independent Schrödinger equation Ĥ · ψ = E ψ amounts to solving the difference equation for the coefficients an. The momentum polarization has a different structure. In this case, the operator p̂2µ0 acts as a multiplication operator, p̂2µ0 · ψ(p) = 1 − cos (µ0 p ψ(p) (25) The operator corresponding to q will be represented as a derivative operator q̂ · ψ(p) := i~ ∂p ψ(p). For a generic potential V (q), it has to be defined by means of spectral theory defined now on a circle. Why on a circle? For the simple reason that by restricting ourselves to a regular graph γµ0 , the functions of p that preserve it (when acting as shift operators) are of the form e(i m µ0 p/~) for m integer. That is, what we have are Fourier modes, labelled by m, of period 2π ~/µ0 in p. Can we pretend then that the phase space variable p is now compactified? The answer is in the affirmative. The inner product on periodic functions ψµ0(p) of p coming from the full Hilbert space Hpoly and given by 〈φ(p)|ψ(p)〉poly = lim L 7→∞ dp φ(p)ψ(p) is precisely equivalent to the inner product on the circle given by the uniform measure 〈φ(p)|ψ(p)〉µ0 = ∫ π~/µ0 −π~/µ0 dp φ(p)ψ(p) with p ∈ (−π~/µ0, π~/µ0). As long as one restricts at- tention to the graph γµ0 , one can work in this separable Hilbert space Hγµ0 of square integrable functions on S Immediately, one can see the limitations of this descrip- tion. If the mechanical system to be quantized is such that its orbits have values of the momenta p that are not small compared with π~/µ0 then the approximation taken will be very poor, and we don’t expect neither the effective classical description nor its quantization to be close to the standard one. If, on the other hand, one is al- ways within the region in which the approximation can be regarded as reliable, then both classical and quantum de- scriptions should approximate the standard description. What does ‘close to the standard description’ exactly mean needs, of course, some further clarification. In particular one is assuming the existence of the usual Schrödinger representation in which the system has a be- havior that is also consistent with observations. If this is the case, the natural question is: How can we approxi- mate such description from the polymer picture? Is there a fine enough graph γµ0 that will approximate the system in such a way that all observations are indistinguishable? Or even better, can we define a procedure, that involves a refinement of the graph γµ0 such that one recovers the standard picture? It could also happen that a continuum limit can be de- fined but does not coincide with the ‘expected one’. But there might be also physical systems for which there is no standard description, or it just does not make sense. Can in those cases the polymer representation, if it ex- ists, provide the correct physical description of the sys- tem under consideration? For instance, if there exists a physical limitation to the minimum scale set by µ0, as could be the case for a quantum theory of gravity, then the polymer description would provide a true physical bound on the value of certain quantities, such as p in our example. This could be the case for loop quantum cosmology, where there is a minimum value for physical volume (coming from the full theory), and phase space points near the ‘singularity’ lie at the region where the approximation induced by the scale µ0 departs from the standard classical description. If in that case the poly- mer quantum system is regarded as more fundamental than the classical system (or its standard Wheeler-De Witt quantization), then one would interpret this dis- crepancies in the behavior as a signal of the breakdown of classical description (or its ‘naive’ quantization). In the next section we present a method to remove the regulator µ0 which was introduced as an intermedi- ate step to construct the dynamics. More precisely, we shall consider the construction of a continuum limit of the polymer description by means of a renormalization procedure. V. THE CONTINUUM LIMIT This section has two parts. In the first one we motivate the need for a precise notion of the continuum limit of the polymeric representation, explaining why the most direct, and naive approach does not work. In the sec- ond part, we shall present the main ideas and results of the paper [6], where the Hamiltonian and the physical Hilbert space in polymer quantum mechanics are con- structed as a continuum limit of effective theories, follow- ing Wilson’s renormalization group ideas. The resulting physical Hilbert space turns out to be unitarily isomor- phic to the ordinary Hs = L2(R, dq) of the Schrödinger theory. Before describing the results of [6] we should discuss the precise meaning of reaching a theory in the contin- uum. Let us for concreteness consider the B-type repre- sentation in the q-polarization. That is, states are func- tions of q and the orthonormal basis χµ(q) is given by characteristic functions with support on q = µ. Let us now suppose we have a Schrödinger state Ψ(q) ∈ Hs = L2(R, dq). What is the relation between Ψ(q) and a state in Hpoly,x? We are also interested in the opposite ques- tion, that is, we would like to know if there is a preferred state in Hs that is approximated by an arbitrary state ψ(q) in Hpoly,x. The first obvious observation is that a Schödinger state Ψ(q) does not belong to Hpoly,x since it would have an infinite norm. To see that note that even when the would-be state can be formally expanded in the χµ basis as, Ψ(q) = Ψ(µ) χµ(q) where the sum is over the parameter µ ∈ R. Its associ- ated norm in Hpoly,x would be: |Ψ(q)|2poly = |Ψ(µ)|2 → ∞ which blows up. Note that in order to define a mapping P : Hs → Hpoly,x, there is a huge ambiguity since the values of the function Ψ(q) are needed in order to expand the polymer wave function. Thus we can only define a mapping in a dense subset D of Hs where the values of the functions are well defined (recall that in Hs the value of functions at a given point has no meaning since states are equivalence classes of functions). We could for instance ask that the mapping be defined for representatives of the equivalence classes in Hs that are piecewise continuous. From now on, when we refer to an element of the space Hs we shall be refereeing to one of those representatives. Notice then that an element of Hs does define an element of Cyl∗γ , the dual to the space Cylγ , that is, the space of cylinder functions with support on the (finite) lattice γ = {µ1, µ2, . . . , µN}, in the following way: Ψ(q) : Cylγ −→ C such that Ψ(q)[ψ(q)] = (Ψ|ψ〉 := Ψ(µ) 〈χµ| ψi χµi〉polyγ Ψ(µi)ψi < ∞ (26) Note that this mapping could be seen as consisting of two parts: First, a projection Pγ : Cyl ∗ → Cylγ such that Pγ(Ψ) = Ψγ(q) := i Ψ(µi)χµi(q) ∈ Cylγ . The state Ψγ is sometimes refereed to as the ‘shadow of Ψ(q) on the lattice γ’. The second step is then to take the inner product between the shadow Ψγ(q) and the state ψ(q) with respect to the polymer inner product 〈Ψγ |ψ〉polyγ . Now this inner product is well defined. Notice that for any given lattice γ the corresponding projector Pγ can be intuitively interpreted as some kind of ‘coarse graining map’ from the continuum to the lattice γ. In terms of functions of q the projection is replacing a continuous function defined on R with a function over the lattice γ ⊂ R which is a discrete set simply by restricting Ψ to γ. The finer the lattice the more points that we have on the curve. As we shall see in the second part of this section, there is indeed a precise notion of coarse graining that implements this intuitive idea in a concrete fashion. In particular, we shall need to replace the lattice γ with a decomposition of the real line in intervals (having the lattice points as end points). Let us now consider a system in the polymer represen- tation in which a particular lattice γ0 was chosen, say with points of the form {qk ∈ R |qk = ka0 , ∀ k ∈ Z}, namely a uniform lattice with spacing equal to a0. In this case, any Schrödinger wave function (of the type that we consider) will have a unique shadow on the lattice γ0. If we refine the lattice γ 7→ γn by dividing each interval in 2n new intervals of length an = a0/2 n we have new shad- ows that have more and more points on the curve. Intu- itively, by refining infinitely the graph we would recover the original function Ψ(q). Even when at each finite step the corresponding shadow has a finite norm in the poly- mer Hilbert space, the norm grows unboundedly and the limit can not be taken, precisely because we can not em- bed Hs into Hpoly. Suppose now that we are interested in the reverse process, namely starting from a polymer theory on a lattice and asking for the ‘continuum wave function’ that is best approximated by a wave function over a graph. Suppose furthermore that we want to con- sider the limit of the graph becoming finer. In order to give precise answers to these (and other) questions we need to introduce some new technology that will allow us to overcome these apparent difficulties. In the remaining of this section we shall recall these constructions for the benefit of the reader. Details can be found in [6] (which is an application of the general formalism discussed in [9]). The starting point in this construction is the concept of a scale C, which allows us to define the effective the- ories and the concept of continuum limit. In our case a scale is a decomposition of the real line in the union of closed-open intervals, that cover the whole line and do not intersect. Intuitively, we are shifting the emphasis from the lattice points to the intervals defined by the same points with the objective of approximating con- tinuous functions defined on R with functions that are constant on the intervals defined by the lattice. To be precise, we define an embedding, for each scale Cn from Hpoly to Hs by means of a step function: Ψ(man) χman(q) → Ψ(man) χαm(q) ∈ Hs with χαn(q) a characteristic function on the interval αm = [man, (m + 1)an). Thus, the shadows (living on the lattice) were just an intermediate step in the con- struction of the approximating function; this function is piece-wise constant and can be written as a linear com- bination of step functions with the coefficients provided by the shadows. The challenge now is to define in an appropriate sense how one can approximate all the aspects of the theory by means of this constant by pieces functions. Then the strategy is that, for any given scale, one can define an effective theory by approximating the kinetic operator by a combination of the translation operators that shift between the vertices of the given decomposition, in other words by a periodic function in p. As a result one has a set of effective theories at given scales which are mutually related by coarse graining maps. This framework was developed in [6]. For the convenience of the reader we briefly recall part of that framework. Let us denote the kinematic polymer Hilbert space at the scale Cn as HCn , and its basis elements as eαi,Cn , where αi = [ian, (i + 1)an) ∈ Cn. By construction this basis is orthonormal. The basis elements in the dual Hilbert space H∗Cn are denoted by ωαi,Cn ; they are also orthonormal. The states ωαi,Cn have a simple action on Cyl, ωαi,Cn(δx0,q) = χαi,Cn(x0). That is, if x0 is in the interval αi of Cn the result is one and it is zero if it is not there. Given any m ≤ n, we define d∗m,n : H∗Cn → H as the ‘coarse graining’ map between the dual Hilbert spaces, that sends the part of the elements of the dual basis to zero while keeping the information of the rest: d∗m,n(ωαi,Cn) = ωβj ,Cm if i = j2 n−m, in the opposite case d∗m,n(ωαi,Cn) = 0. At every scale the corresponding effective theory is given by the hamiltonian Hn. These Hamiltonians will be treated as quadratic forms, hn : HCn → R, given by hn(ψ) = λ (ψ,Hnψ) , (27) where λ2Cn is a normalizaton factor. We will see later that this rescaling of the inner product is necessary in order to guarantee the convergence of the renormalized theory. The completely renormalized theory at this scale is obtained as hrenm := lim d⋆m,nhn. (28) and the renormalized Hamiltonians are compatible with each other, in the sense that d⋆m,nh n = h In order to analyze the conditions for the convergence in (28) let us express the Hamiltonian in terms of its eigen-covectors end eigenvalues. We will work with effec- tive Hamiltonians that have a purely discrete spectrum (labelled by ν) Hn · Ψν,Cn = Eν,Cn Ψν,Cn . We shall also introduce, as an intermediate step, a cut-off in the energy levels. The origin of this cut-off is in the approximation of the Hamiltonian of our system at a given scale with a Hamiltonian of a periodic system in a regime of small energies, as we explained earlier. Thus, we can write hνcut−offm = νcut−off Eν,CmΨν,Cm ⊗ Ψν,Cm , (29) where the eigen covectors Ψν,Cm are normalized accord- ing to the inner product rescaled by 1 , and the cut- off can vary up to a scale dependent bound, νcut−off ≤ νmax(Cm). The Hilbert space of covectors together with such inner product will be called H⋆renCm . In the presence of a cut-off, the convergence of the microscopically corrected Hamiltonians, equation (28) is equivalent to the existence of the following two limits. The first one is the convergence of the energy levels, Eν,Cn = E ν . (30) Second is the existence of the completely renormalized eigen covectors, d⋆m,n Ψν,Cn = Ψ ∈ H⋆renCm ⊂ Cyl ⋆ . (31) We clarify that the existence of the above limit means that Ψrenν,Cm(δx0,q) is well defined for any δx0,q ∈ Cyl. No- tice that this point-wise convergence, if it can take place at all, will require the tuning of the normalization factors λ2Cn . Now we turn to the question of the continuum limit of the renormalized covectors. First we can ask for the existence of the limit Ψrenν,Cn(δx0,q) (32) for any δx0,q ∈ Cyl. When this limits exists there is a natural action of the eigen covectors in the continuum limit. Below we consider another notion of the continuum limit of the renormalized eigen covectors. When the completely renormalized eigen covectors exist, they form a collection that is d⋆-compatible, d⋆m,nΨ = Ψrenν,Cm . A sequence of d ⋆-compatible nor- malizable covectors define an element of , which is the projective limit of the renormalized spaces of covec- H⋆renCn . (33) The inner product in this space is defined by ({ΨCn}, {ΦCn})renR := lim (ΨCn ,ΦCn) The natural inclusion of C∞0 in is by an antilinear map which assigns to any Ψ ∈ C∞0 the d⋆-compatible collection ΨshadCn := ωαiΨ̄(L(αi)) ∈ H⋆renCn ⊂ Cyl ΨshadCn will be called the shadow of Ψ at scale Cn and acts in Cyl as a piecewise constant function. Clearly other types of test functions like Schwartz functions are also naturally included in . In this context a shadow is a state of the effective theory that approximates a state in the continuum theory. Since the inner product in is degenerate, the physical Hilbert space is defined as H⋆phys := / ker(·, ·)ren Hphys := H⋆⋆phys The nature of the physical Hilbert space, whether it is isomorphic to the Schrödinger Hilber space, Hs, or not, is determined by the normalization factors λ2Cn which can be obtained from the conditions asking for compatibil- ity of the dynamics of the effective theories at different scales. The dynamics of the system under consideration selects the continuum limit. Let us now return to the definition of the Hamilto- nian in the continuum limit. First consider the contin- uum limit of the Hamiltonian (with cut-off) in the sense of its point-wise convergence as a quadratic form. It turns out that if the limit of equation (32) exists for all the eigencovectors allowed by the cut-off, we have νcut−off ren : Hpoly,x → R defined by νcut−off ren (δx0,q) := lim hνcut−off renn ([δx0,q]Cn). (34) This Hamiltonian quadratic form in the continuum can be coarse grained to any scale and, as can be ex- pected, it yields the completely renormalized Hamilto- nian quadratic forms at that scale. However, this is not a completely satisfactory continuum limit because we can not remove the auxiliary cut-off νcut−off . If we tried, as we include more and more eigencovectors in the Hamilto- nian the calculations done at a given scale would diverge and doing them in the continuum is just as divergent. Below we explore a more successful path. We can use the renormalized inner product to induce an action of the cut–off Hamiltonians on νcut−off ren ({ΨCn}) := lim hνcut−off renn ((ΨCn , ·)renCn ), where we have used the fact that (ΨCn , ·)renCn ∈ HCn . The existence of this limit is trivial because the renormalized Hamiltonians are finite sums and the limit exists term by term. These cut-off Hamiltonians descend to the physical Hilbert space νcut−off ren ([{ΨCn}]) := h νcut−off ren ({ΨCn}) for any representative {ΨCn} ∈ [{ΨCn}] ∈ H⋆phys. Finally we can address the issue of removal of the cut- off. The Hamiltonian hren → R is defined by the limit := lim νcut−off→∞ νcut−off ren when the limit exists. Its corresponding Hermitian form in Hphys is defined whenever the above limit exists. This concludes our presentation of the main results of [6]. Let us now consider several examples of systems for which the continuum limit can be investigated. VI. EXAMPLES In this section we shall develop several examples of systems that have been treated with the polymer quanti- zation. These examples are simple quantum mechanical systems, such as the simple harmonic oscillator and the free particle, as well as a quantum cosmological model known as loop quantum cosmology. A. The Simple Harmonic Oscillator In this part, let us consider the example of a Simple Har- monic Oscillator (SHO) with parameters m and ω, clas- sically described by the following Hamiltonian mω2 x2. Recall that from these parameters one can define a length scale D = ~/mω. In the standard treatment one uses this scale to define a complex structure JD (and an in- ner product from it), as we have described in detail that uniquely selects the standard Schrödinger representation. At scale Cn we have an effective Hamiltonian for the Simple Harmonic Oscillator (SHO) given by HCn = 1 − cos anp mω2x2 . (35) If we interchange position and momentum, this Hamilto- nian is exactly that of a pendulum of mass m, length l and subject to a constant gravitational field g: ĤCn = − +mgl(1 − cos θ) where those quantities are related to our system by, mω an , g = , θ = That is, we are approximating, for each scale Cn the SHO by a pendulum. There is, however, an important difference. From our knowledge of the pendulum system, we know that the quantum system will have a spectrum for the energy that has two different asymptotic behav- iors, the SHO for low energies and the planar rotor in the higher end, corresponding to oscillating and rotating solutions respectively2. As we refine our scale and both the length of the pendulum and the height of the periodic potential increase, we expect to have an increasing num- ber of oscillating states (for a given pendulum system, there is only a finite number of such states). Thus, it is justified to consider the cut-off in the energy eigenval- ues, as discussed in the last section, given that we only expect a finite number of states of the pendulum to ap- proximate SHO eigenstates. With these consideration in mind, the relevant question is whether the conditions for the continuum limit to exist are satisfied. This question has been answered in the affirmative in [6]. What was shown there was that the eigen-values and eigen func- tions of the discrete systems, which represent a discrete and non-degenerate set, approximate those of the contin- uum, namely, of the standard harmonic oscillator when the inner product is renormalized by a factor λ2Cn = 1/2 This convergence implies that the continuum limit exists as we understand it. Let us now consider the simplest possible system, a free particle, that has nevertheless the particular feature that the spectrum of the energy is con- tinuous. 2 Note that both types of solutions are, in the phase space, closed. This is the reason behind the purely discrete spectrum. The distinction we are making is between those solutions inside the separatrix, that we call oscillating, and those that are above it that we call rotating. B. Free Polymer Particle In the limit ω → 0, the Hamiltonian of the Simple Harmonic oscillator (35) goes to the Hamiltonian of a free particle and the corresponding time independent Schrödinger equation, in the p−polarization, is given by (1 − cos anp ) − ECn ψ̃(p) = 0 where we now have that p ∈ S1, with p ∈ (−π~ Thus, we have ECn = 1 − cos ≤ ECn,max ≡ 2 . (36) At each scale the energy of the particle we can describe is bounded from above and the bound depends on the scale. Note that in this case the spectrum is continu- ous, which implies that the ordinary eigenfunctions of the Hilbert are not normalizable. This imposes an upper bound in the value that the energy of the particle can have, in addition to the bound in the momentum due to its “compactification”. Let us first look for eigen-solutions to the time inde- pendent Schrödinger equation, that is, for energy eigen- states. In the case of the ordinary free particle, these correspond to constant momentum plane waves of the form e±( ) and such that the ordinary dispersion re- lation p2/2m = E is satisfied. These plane waves are not square integrable and do not belong to the ordinary Hilbert space of the Schrödinger theory but they are still useful for extracting information about the system. For the polymer free particle we have, ψ̃Cn(p) = c1δ(p− PCn) + c2δ(p+ PCn) where PCn is a solution of the previous equation consid- ering a fixed value of ECn . That is, PCn = P (ECn) = arccos 1 − ma The inverse Fourier transform yields, in the ‘x represen- tation’, ψCn(xj) = ∫ π~/an −π~/an ψ̃(p) e p j dp = ixjPCn /~ + c2e −ixjPCn /~ .(37) with xj = an j for j ∈ Z. Note that the eigenfunctions are still delta functions (in the p representation) and thus not (square) normalizable with respect to the polymer inner product, that in the p polarization is just given by the ordinary Haar measure on S1, and there is no quantization of the momentum (its spectrum is still truly continuous). Let us now consider the time dependent Schrödinger equation, i~ ∂t Ψ̃(p, t) = Ĥ · Ψ̃(p, t). Which now takes the form, Ψ̃(p, t) = (1 − cos (an p/~)) Ψ̃(p, t) that has as its solution, Ψ̃(p, t) = e− (1−cos (an p/~)) t ψ̃(p) = e(−iECn /~) t ψ̃(p) for any initial function ψ̃(p), where ECn satisfy the dis- persion relation (36). The wave function Ψ(xj , t), the xj-representation of the wave function, can be obtained for any given time t by Fourier transforming with (37) the wave function Ψ̃(p, t). In order to check out the convergence of the micro- scopically corrected Hamiltonians we should analyze the convergence of the energy levels and of the proper cov- ectors. In the limit n → ∞, ECn → E = p2/2m so we can be certain that the eigen-values for the energy converge (when fixing the value of p). Let us write the proper covector as ΨCn = (ψCn , ·)renCn ∈ H . Then we can bring microscopic corrections to scale Cm and look for convergence of such corrections ΨrenCm = lim d⋆m,nΨCn . It is easy to see that given any basis vector eαi ∈ HCm the following limit ΨrenCm(eαi,Cm) = limCn→∞ ΨCn(dn,m(eαi,Cm)) exists and is equal to ΨshadCm (eαi,Cm) = [d ⋆ΨSchr](eαi,Cm) = Ψ Schr(iam) where ΨshadCm is calculated using the free particle Hamilto- nian in the Schrödinger representation. This expression defines the completely renormalized proper covector at the scale Cm. C. Polymer Quantum Cosmology In this section we shall present a version of quantum cosmology that we call polymer quantum cosmology. The idea behind this name is that the main input in the quan- tization of the corresponding mini-superspace model is the use of a polymer representation as here understood. Another important input is the choice of fundamental variables to be used and the definition of the Hamiltonian constraint. Different research groups have made differ- ent choices. We shall take here a simple model that has received much attention recently, namely an isotropic, homogeneous FRW cosmology with k = 0 and coupled to a massless scalar field ϕ. As we shall see, a proper treatment of the continuum limit of this system requires new tools under development that are beyond the scope of this work. We will thus restrict ourselves to the intro- duction of the system and the problems that need to be solved. The system to be quantized corresponds to the phase space of cosmological spacetimes that are homogeneous and isotropic and for which the homogeneous spatial slices have a flat intrinsic geometry (k = 0 condition). The only matter content is a mass-less scalar field ϕ. In this case the spacetime geometry is given by metrics of the form: ds2 = −dt2 + a2(t) (dx2 + dy2 + dz2) where the function a(t) carries all the information and degrees of freedom of the gravity part. In terms of the coordinates (a, pa, ϕ, pϕ) for the phase space Γ of the the- ory, all the dynamics is captured in the Hamiltonian con- straint C := −3 + 8πG 2|a|3 The first step is to define the constraint on the kine- matical Hilbert space to find physical states and then a physical inner product to construct the physical Hilbert space. First note that one can rewrite the equation as: p2a a 2 = 8πG If, as is normally done, one chooses ϕ to act as an in- ternal time, the right hand side would be promoted, in the quantum theory, to a second derivative. The left hand side is, furthermore, symmetric in a and pa. At this point we have the freedom in choosing the variable that will be quantized and the variable that will not be well defined in the polymer representation. The standard choice is that pa is not well defined and thus, a and any geometrical quantity derived from it, is quantized. Fur- thermore, we have the choice of polarization on the wave function. In this respect the standard choice is to select the a-polarization, in which a acts as multiplication and the approximation of pa, namely sin(λ pa)/λ acts as a difference operator on wave functions of a. For details of this particular choice see [5]. Here we shall adopt the op- posite polarization, that is, we shall have wave functions Ψ(pa, ϕ). Just as we did in the previous cases, in order to gain intuition about the behavior of the polymer quantized theory, it is convenient to look at the equivalent prob- lem in the classical theory, namely the classical system we would get be approximating the non-well defined ob- servable (pa in our present case) by a well defined object (made of trigonometric functions). Let us for simplicity choose to replace pa 7→ sin(λ pa)/λ. With this choice we get an effective classical Hamiltonian constraint that depends on λ: Cλ := − sin(λ pa) λ2|a| + 8πG 2|a|3 We can now compute effective equations of motion by means of the equations: Ḟ := {F, Cλ}, for any observable F ∈ C∞(Γ), and where we are using the effective (first order) action: dτ(pa ȧ+ pϕ ϕ̇−N Cλ) with the choice N = 1. The first thing to notice is that the quantity pϕ is a constant of the motion, given that the variable ϕ is cyclic. The second observation is that ϕ̇ = 8πG has the same sign as pϕ and never vanishes. Thus ϕ can be used as a (n internal) time variable. The next observation is that the equation for , namely the effective Friedman equation, will have a zero for a non-zero value of a given by λ2p2ϕ. This is the value at which there will be bounce if the trajectory started with a large value of a and was con- tracting. Note that the ‘size’ of the universe when the bounce occurs depends on both the constant pϕ (that dictates the matter density) and the value of the lattice size λ. Here it is important to stress that for any value of pϕ (that uniquely fixes the trajectory in the (a, pa) plane), there will be a bounce. In the original description in terms of Einstein’s equations (without the approxima- tion that depends on λ), there in no such bounce. If ȧ < 0 initially, it will remain negative and the universe collapses, reaching the singularity in a finite proper time. What happens within the effective description if we re- fine the lattice and go from λ to λn := λ/2 n? The only thing that changes, for the same classical orbit labelled by pϕ, is that the bounce occurs at a ‘later time’ and for a smaller value of a∗ but the qualitative picture remains the same. This is the main difference with the systems considered before. In those cases, one could have classical trajecto- ries that remained, for a given choice of parameter λ, within the region where sin(λp)/λ is a good approxima- tion to p. Of course there were also classical trajectories that were outside this region but we could then refine the lattice and find a new value λ′ for which the new clas- sical trajectory is well approximated. In the case of the polymer cosmology, this is never the case: Every classical trajectory will pass from a region where the approxima- tion is good to a region where it is not; this is precisely where the ‘quantum corrections’ kick in and the universes bounces. Given that in the classical description, the ‘original’ and the ‘corrected’ descriptions are so different we expect that, upon quantization, the corresponding quantum the- ories, namely the polymeric and the Wheeler-DeWitt will be related in a non-trivial way (if at all). In this case, with the choice of polarization and for a particular factor ordering we have, sin(λpa) · Ψ(pa, ϕ) = 0 as the Polymer Wheeler-DeWitt equation. In order to approach the problem of the continuum limit of this quantum theory, we have to realize that the task is now somewhat different than before. This is so given that the system is now a constrained system with a constraint operator rather than a regular non-singular system with an ordinary Hamiltonian evolution. Fortu- nately for the system under consideration, the fact that the variable ϕ can be regarded as an internal time allows us to interpret the quantum constraint as a generalized Klein-Gordon equation of the form Ψ = Θλ · Ψ where the operator Θλ is ‘time independent’. This al- lows us to split the space of solutions into ‘positive and negative frequency’, introduce a physical inner product on the positive frequency solutions of this equation and a set of physical observables in terms of which to de- scribe the system. That is, one reduces in practice the system to one very similar to the Schrödinger case by taking the positive square root of the previous equation: Θλ · Ψ. The question we are interested is whether the continuum limit of these theories (labelled by λ) exists and whether it corresponds to the Wheeler- DeWitt theory. A complete treatment of this problem lies, unfortunately, outside the scope of this work and will be reported elsewhere [12]. VII. DISCUSSION Let us summarize our results. In the first part of the article we showed that the polymer representation of the canonical commutation relations can be obtained as the limiting case of the ordinary Fock-Schrödinger represen- tation in terms of the algebraic state that defines the representation. These limiting cases can also be inter- preted in terms of the naturally defined coherent states associated to each representation labelled by the param- eter d, when they become infinitely ‘squeezed’. The two possible limits of squeezing lead to two different polymer descriptions that can nevertheless be identified, as we have also shown, with the two possible polarizations for an abstract polymer representation. This resulting the- ory has, however, very different behavior as the standard one: The Hilbert space is non-separable, the representa- tion is unitarily inequivalent to the Schrödinger one, and natural operators such as p̂ are no longer well defined. This particular limiting construction of the polymer the- ory can shed some light for more complicated systems such as field theories and gravity. In the regular treatments of dynamics within the poly- mer representation, one needs to introduce some extra structure, such as a lattice on configuration space, to con- struct a Hamiltonian and implement the dynamics for the system via a regularization procedure. How does this re- sulting theory compare to the original continuum theory one had from the beginning? Can one hope to remove the regulator in the polymer description? As they stand there is no direct relation or mapping from the polymer to a continuum theory (in case there is one defined). As we have shown, one can indeed construct in a systematic fashion such relation by means of some appropriate no- tions related to the definition of a scale, closely related to the lattice one had to introduce in the regularization. With this important shift in perspective, and an appro- priate renormalization of the polymer inner product at each scale one can, subject to some consistency condi- tions, define a procedure to remove the regulator, and arrive to a Hamiltonian and a Hilbert space. As we have seen, for some simple examples such as a free particle and the harmonic oscillator one indeed recovers the Schrödinger description back. For other sys- tems, such as quantum cosmological models, the answer is not as clear, since the structure of the space of classi- cal solutions is such that the ‘effective description’ intro- duced by the polymer regularization at different scales is qualitatively different from the original dynamics. A proper treatment of these class of systems is underway and will be reported elsewhere [12]. Perhaps the most important lesson that we have learned here is that there indeed exists a rich inter- play between the polymer description and the ordinary Schrödinger representation. The full structure of such re- lation still needs to be unravelled. We can only hope that a full understanding of these issues will shed some light in the ultimate goal of treating the quantum dynamics of background independent field systems such as general relativity. Acknowledgments We thank A. Ashtekar, G. Hossain, T. Pawlowski and P. Singh for discussions. This work was in part supported by CONACyT U47857-F and 40035-F grants, by NSF PHY04-56913, by the Eberly Research Funds of Penn State, by the AMC-FUMEC exchange program and by funds of the CIC-Universidad Michoacana de San Nicolás de Hidalgo. [1] R. Beaume, J. Manuceau, A. Pellet and M. Sirugue, “Translation Invariant States In Quantum Mechanics,” Commun. Math. Phys. 38, 29 (1974); W. E. Thirring and H. Narnhofer, “Covariant QED without indefinite met- ric,” Rev. Math. Phys. 4, 197 (1992); F. Acerbi, G. Mor- chio and F. 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A rather non-standard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schroedinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schroedinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schroedinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle and a simple cosmological model.
Polymer Quantum Mechanics and its Continuum Limit Alejandro Corichi,1, 2, 3, ∗ Tatjana Vukašinac,4, † and José A. Zapata1, ‡ Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, Mexico Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F. 04510, Mexico Institute for Gravitational Physics and Geometry, Physics Department, Pennsylvania State University, University Park PA 16802, USA Facultad de Ingenieŕıa Civil, Universidad Michoacana de San Nicolas de Hidalgo, Morelia, Michoacán 58000, Mexico A rather non-standard quantum representation of the canonical commutation relations of quan- tum mechanics systems, known as the polymer representation has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schrödinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schrödinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schrödinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle and a simple cosmological model. PACS numbers: 04.60.Pp, 04.60.Ds, 04.60.Nc 11.10.Gh. I. INTRODUCTION The so-called polymer quantum mechanics, a non- regular and somewhat ‘exotic’ representation of the canonical commutation relations (CCR) [1], has been used to explore both mathematical and physical issues in background independent theories such as quantum grav- ity [2, 3]. A notable example of this type of quantization, when applied to minisuperspace models has given way to what is known as loop quantum cosmology [4, 5]. As in any toy model situation, one hopes to learn about the subtle technical and conceptual issues that are present in full quantum gravity by means of simple, finite di- mensional examples. This formalism is not an exception in this regard. Apart from this motivation coming from physics at the Planck scale, one can independently ask for the relation between the standard continuous repre- sentations and their polymer cousins at the level of math- ematical physics. A deeper understanding of this relation becomes important on its own. The polymer quantization is made of several steps. The first one is to build a representation of the Heisenberg-Weyl algebra on a Kinematical Hilbert space that is “background independent”, and that is sometimes referred to as the polymeric Hilbert space Hpoly. The second and most important part, the implementation of dynamics, deals with the definition of a Hamiltonian (or Hamiltonian constraint) on this space. In the examples ∗Electronic address: corichi@matmor.unam.mx †Electronic address: tatjana@shi.matmor.unam.mx ‡Electronic address: zapata@matmor.unam.mx studied so far, the first part is fairly well understood, yielding the kinematical Hilbert space Hpoly that is, how- ever, non-separable. For the second step, a natural im- plementation of the dynamics has proved to be a bit more difficult, given that a direct definition of the Hamiltonian Ĥ of, say, a particle on a potential on the space Hpoly is not possible since one of the main features of this repre- sentation is that the operators q̂ and p̂ cannot be both simultaneously defined (nor their analogues in theories involving more elaborate variables). Thus, any operator that involves (powers of) the not defined variable has to be regulated by a well defined operator which normally involves introducing some extra structure on the configu- ration (or momentum) space, namely a lattice. However, this new structure that plays the role of a regulator can not be removed when working in Hpoly and one is left with the ambiguity that is present in any regularization. The freedom in choosing it can be sometimes associated with a length scale (the lattice spacing). For ordinary quantum systems such as a simple harmonic oscillator, that has been studied in detail from the polymer view- point, it has been argued that if this length scale is taken to be ‘sufficiently small’, one can arbitrarily approximate standard Schrödinger quantum mechanics [2, 3]. In the case of loop quantum cosmology, the minimum area gap A0 of the full quantum gravity theory imposes such a scale, that is then taken to be fundamental [4]. A natural question is to ask what happens when we change this scale and go to even smaller ‘distances’, that is, when we refine the lattice on which the dynamics of the theory is defined. Can we define consistency con- ditions between these scales? Or even better, can we take the limit and find thus a continuum limit? As it http://arxiv.org/abs/0704.0007v2 mailto:corichi@matmor.unam.mx mailto:tatjana@shi.matmor.unam.mx mailto:zapata@matmor.unam.mx has been shown recently in detail, the answer to both questions is in the affirmative [6]. There, an appropriate notion of scale was defined in such a way that one could define refinements of the theory and pose in a precise fashion the question of the continuum limit of the theory. These results could also be seen as handing a procedure to remove the regulator when working on the appropri- ate space. The purpose of this paper is to further explore different aspects of the relation between the continuum and the polymer representation. In particular in the first part we put forward a novel way of deriving the polymer representation from the ordinary Schrödinger represen- tation as an appropriate limit. In Sec. II we derive two versions of the polymer representation as different lim- its of the Schrödinger theory. In Sec. III we show that these two versions can be seen as different polarizations of the ‘abstract’ polymer representation. These results, to the best of our knowledge, are new and have not been reported elsewhere. In Sec. IV we pose the problem of implementing the dynamics on the polymer representa- tion. In Sec. V we motivate further the question of the continuum limit (i.e. the proper removal of the regulator) and recall the basic constructions of [6]. Several exam- ples are considered in Sec. VI. In particular a simple harmonic oscillator, the polymer free particle and a sim- ple quantum cosmology model are considered. The free particle and the cosmological model represent a general- ization of the results obtained in [6] where only systems with a discrete and non-degenerate spectrum where con- sidered. We end the paper with a discussion in Sec. VII. In order to make the paper self-contained, we will keep the level of rigor in the presentation to that found in the standard theoretical physics literature. II. QUANTIZATION AND POLYMER REPRESENTATION In this section we derive the so called polymer repre- sentation of quantum mechanics starting from a specific reformulation of the ordinary Schrödinger representation. Our starting point will be the simplest of all possible phase spaces, namely Γ = R2 corresponding to a particle living on the real line R. Let us choose coordinates (q, p) thereon. As a first step we shall consider the quantization of this system that leads to the standard quantum theory in the Schrödinger description. A convenient route is to introduce the necessary structure to define the Fock rep- resentation of such system. From this perspective, the passage to the polymeric case becomes clearest. Roughly speaking by a quantization one means a passage from the classical algebraic bracket, the Poisson bracket, {q, p} = 1 (1) to a quantum bracket given by the commutator of the corresponding operators, [ q̂, p̂] = i~ 1̂ (2) These relations, known as the canonical commutation re- lation (CCR) become the most common corner stone of the (kinematics of the) quantum theory; they should be satisfied by the quantum system, when represented on a Hilbert space H. There are alternative points of departure for quantum kinematics. Here we consider the algebra generated by the exponentiated versions of q̂ and p̂ that are denoted U(α) = ei(α q̂)/~ ; V (β) = ei(β p̂)/~ where α and β have dimensions of momentum and length, respectively. The CCR now become U(α) · V (β) = e(−iα β)/~V (β) · U(α) (3) and the rest of the product is U(α1)·U(α2) = U(α1+α2) ; V (β1)·V (β2) = V (β1+β2) The Weyl algebra W is generated by taking finite linear combinations of the generators U(αi) and V (βi) where the product (3) is extended by linearity, (Ai U(αi) +Bi V (βi)) From this perspective, quantization means finding an unitary representation of the Weyl algebra W on a Hilbert space H′ (that could be different from the ordi- nary Schrödinger representation). At first it might look weird to attempt this approach given that we know how to quantize such a simple system; what do we need such a complicated object as W for? It is infinite dimensional, whereas the set S = {1̂, q̂, p̂}, the starting point of the ordinary Dirac quantization, is rather simple. It is in the quantization of field systems that the advantages of the Weyl approach can be fully appreciated, but it is also useful for introducing the polymer quantization and comparing it to the standard quantization. This is the strategy that we follow. A question that one can ask is whether there is any freedom in quantizing the system to obtain the ordinary Schrödinger representation. On a first sight it might seem that there is none given the Stone-Von Neumann unique- ness theorem. Let us review what would be the argument for the standard construction. Let us ask that the repre- sentation we want to build up is of the Schrödinger type, namely, where states are wave functions of configuration space ψ(q). There are two ingredients to the construction of the representation, namely the specification of how the basic operators (q̂, p̂) will act, and the nature of the space of functions that ψ belongs to, that is normally fixed by the choice of inner product on H, or measure µ on R. The standard choice is to select the Hilbert space to be, H = L2(R, dq) the space of square-integrable functions with respect to the Lebesgue measure dq (invariant under constant trans- lations) on R. The operators are then represented as, q̂ · ψ(q) = (q ψ)(q) and p̂ · ψ(q) = −i ~ ∂ ψ(q) (4) Is it possible to find other representations? In order to appreciate this freedom we go to the Weyl algebra and build the quantum theory thereon. The representation of the Weyl algebra that can be called of the ‘Fock type’ involves the definition of an extra structure on the phase space Γ: a complex structure J . That is, a linear map- ping from Γ to itself such that J2 = −1. In 2 dimen- sions, all the freedom in the choice of J is contained in the choice of a parameter d with dimensions of length. It is also convenient to define: k = p/~ that has dimensions of 1/L. We have then, Jd : (q, k) 7→ (−d2 k, q/d2) This object together with the symplectic structure: Ω((q, p); (q′, p′)) = q p′ − p q′ define an inner product on Γ by the formula gd(· ; ·) = Ω(· ; Jd ·) such that: gd((q, p); (q ′, p′)) = q q′ + which is dimension-less and positive definite. Note that with this quantities one can define complex coordinates (ζ, ζ̄) as usual: q + i p ; ζ̄ = q − i d from which one can build the standard Fock representa- tion. Thus, one can alternatively view the introduction of the length parameter d as the quantity needed to de- fine (dimensionless) complex coordinates on the phase space. But what is the relevance of this object (J or d)? The definition of complex coordinates is useful for the construction of the Fock space since from them one can define, in a natural way, creation and annihilation operators. But for the Schrödinger representation we are interested here, it is a bit more subtle. The subtlety is that within this approach one uses the algebraic prop- erties of W to construct the Hilbert space via what is known as the Gel’fand-Naimark-Segal (GNS) construc- tion. This implies that the measure in the Schrödinger representation becomes non trivial and thus the momen- tum operator acquires an extra term in order to render the operator self-adjoint. The representation of the Weyl algebra is then, when acting on functions φ(q) [7]: Û(α) · φ(q) := (eiα q/~ φ)(q) V̂ (β) · φ(q) := e (q−β/2) φ(q − β) The Hilbert space structure is introduced by the defini- tion of an algebraic state (a positive linear functional) ωd : W → C, that must coincide with the expectation value in the Hilbert space taken on a special state ref- ered to as the vacuum: ωd(a) = 〈â〉vac, for all a ∈ W . In our case this specification of J induces such a unique state ωd that yields, 〈Û(α)〉vac = e− d2 α2 ~2 (5) 〈V̂ (β)〉vac = e− d2 (6) Note that the exponents in the vacuum expectation values correspond to the metric constructed out of J : d2 α2 = gd((0, α); (0, α)) and = gd((β, 0); (β, 0)). Wave functions belong to the space L2(R, dµd), where the measure that dictates the inner product in this rep- resentation is given by, dµd = d2 dq In this representation, the vacuum is given by the iden- tity function φ0(q) = 1 that is, just as any plane wave, normalized. Note that for each value of d > 0, the rep- resentation is well defined and continuous in α and β. Note also that there is an equivalence between the q- representation defined by d and the k-representation de- fined by 1/d. How can we recover then the standard representation in which the measure is given by the Lebesgue measure and the operators are represented as in (4)? It is easy to see that there is an isometric isomorphism K that maps the d-representation in Hd to the standard Schrödinger representation in Hschr by: ψ(q) = K · φ(q) = e d1/2π1/4 φ(q) ∈ Hschr = L2(R, dq) Thus we see that all d-representations are unitarily equiv- alent. This was to be expected in view of the Stone-Von Neumann uniqueness result. Note also that the vacuum now becomes ψ0(q) = d1/2π1/4 2 d2 , so even when there is no information about the param- eter d in the representation itself, it is contained in the vacuum state. This procedure for constructing the GNS- Schrödinger representation for quantum mechanics has also been generalized to scalar fields on arbitrary curved space in [8]. Note, however that so far the treatment has all been kinematical, without any knowledge of a Hamil- tonian. For the Simple Harmonic Oscillator of mass m and frequency ω, there is a natural choice compatible with the dynamics given by d = , in which some calculations simplify (for instance for coherent states), but in principle one can use any value of d. Our study will be simplified by focusing on the funda- mental entities in the Hilbert Space Hd , namely those states generated by acting with Û(α) on the vacuum φ0(q) = 1. Let us denote those states by, φα(q) = Û(α) · φ0(q) = ei The inner product between two such states is given by 〈φα, φλ〉d = dµd e ~ = e− (λ−α)2 d2 4 ~2 (7) Note incidentally that, contrary to some common belief, the ‘plane waves’ in this GNS Hilbert space are indeed normalizable. Let us now consider the polymer representation. For that, it is important to note that there are two possible limiting cases for the parameter d: i) The limit 1/d 7→ 0 and ii) The case d 7→ 0. In both cases, we have ex- pressions that become ill defined in the representation or measure, so one needs to be careful. A. The 1/d 7→ 0 case. The first observation is that from the expressions (5) and (6) for the algebraic state ωd, we see that the limiting cases are indeed well defined. In our case we get, ωA := lim1/d→0 ωd such that, ωA(Û(α)) = δα,0 and ωA(V̂ (β)) = 1 (8) From this, we can indeed construct the representation by means of the GNS construction. In order to do that and to show how this is obtained we shall consider several expressions. One has to be careful though, since the limit has to be taken with care. Let us consider the measure on the representation that behaves as: dµd = d2 dq 7→ 1 so the measures tends to an homogeneous measure but whose ‘normalization constant’ goes to zero, so the limit becomes somewhat subtle. We shall return to this point later. Let us now see what happens to the inner product between the fundamental entities in the Hilbert Space Hd given by (7). It is immediate to see that in the 1/d 7→ 0 limit the inner product becomes, 〈φα, φλ〉d 7→ δα,λ (9) with δα,λ being Kronecker’s delta. We see then that the plane waves φα(q) become an orthonormal basis for the new Hilbert space. Therefore, there is a delicate interplay between the two terms that contribute to the measure in order to maintain the normalizability of these functions; we need the measure to become damped (by 1/d) in order to avoid that the plane waves acquire an infinite norm (as happens with the standard Lebesgue measure), but on the other hand the measure, that for any finite value of d is a Gaussian, becomes more and more spread. It is important to note that, in this limit, the operators Û(α) become discontinuous with respect to α, given that for any given α1 and α2 (different), its action on a given basis vector ψλ(q) yields orthogonal vectors. Since the continuity of these operators is one of the hypotesis of the Stone-Von Neumann theorem, the uniqueness result does not apply here. The representation is inequivalent to the standard one. Let us now analyze the other operator, namely the action of the operator V̂ (β) on the basis φα(q): V̂ (β) · φα(q) = e− ~ e(β/d 2+iα/~)q which in the limit 1/d 7→ 0 goes to, V̂ (β) · φα(q) 7→ ei ~ φα(q) that is continuous on β. Thus, in the limit, the operator p̂ = −i~∂q is well defined. Also, note that in this limit the operator p̂ has φα(q) as its eigenstate with eigenvalue given by α: p̂ · φα(q) 7→ αφα(q) To summarize, the resulting theory obtained by taking the limit 1/d 7→ 0 of the ordinary Schrödinger descrip- tion, that we shall call the ‘polymer representation of type A’, has the following features: the operators U(α) are well defined but not continuous in α, so there is no generator (no operator associated to q). The basis vec- tors φα are orthonormal (for α taking values on a contin- uous set) and are eigenvectors of the operator p̂ that is well defined. The resulting Hilbert space HA will be the (A-version of the) polymer representation. Let us now consider the other case, namely, the limit when d 7→ 0. B. The d 7→ 0 case Let us now explore the other limiting case of the Schrödinger/Fock representations labelled by the param- eter d. Just as in the previous case, the limiting algebraic state becomes, ωB := limd→0 ωd such that, ωB(Û(α)) = 1 and ωB(V̂ (β)) = δβ,0 (10) From this positive linear function, one can indeed con- struct the representation using the GNS construction. First let us note that the measure, even when the limit has to be taken with due care, behaves as: dµd = d2 dq 7→ δ(q) dq That is, as Dirac’s delta distribution. It is immediate to see that, in the d 7→ 0 limit, the inner product between the fundamental states φα(q) becomes, 〈φα, φλ〉d 7→ 1 (11) This in fact means that the vector ξ = φα − φλ belongs to the Kernel of the limiting inner product, so one has to mod out by these (and all) zero norm states in order to get the Hilbert space. Let us now analyze the other operator, namely the action of the operator V̂ (β) on the vacuum φ0(q) = 1, which for arbitrary d has the form, φ̃β := V̂ (β) · φ0(q) = e (q−β/2) The inner product between two such states is given by 〈φ̃α, φ̃β〉d = e− (α−β)2 In the limit d → 0, 〈φ̃α, φ̃β〉d → δα,β. We can see then that it is these functions that become the orthonormal, ‘discrete basis’ in the theory. However, the function φ̃β(q) in this limit becomes ill defined. For example, for β > 0, it grows unboundedly for q > β/2, is equal to one if q = β/2 and zero otherwise. In order to overcome these difficulties and make more transparent the resulting the- ory, we shall consider the other form of the representation in which the measure is incorporated into the states (and the resulting Hilbert space is L2(R, dq)). Thus the new state ψβ(q) := K · (V̂ (β) · φ0(q)) = (q−β)2 We can now take the limit and what we get is d 7→0 ψβ(q) := δ 1/2(q, β) where by δ1/2(q, β) we mean something like ‘the square root of the Dirac distribution’. What we really mean is an object that satisfies the following property: δ1/2(q, β) · δ1/2(q, α) = δ(q, β) δβ,α That is, if α = β then it is just the ordinary delta, other- wise it is zero. In a sense these object can be regarded as half-densities that can not be integrated by themselves, but whose product can. We conclude then that the inner product is, 〈ψβ , ψα〉 = dq ψβ(q)ψα(q) = dq δ(q, α) δβ,α = δβ,α which is just what we expected. Note that in this repre- sentation, the vacuum state becomes ψ0(q) := δ 1/2(q, 0), namely, the half-delta with support in the origin. It is important to note that we are arriving in a natural way to states as half-densities, whose squares can be integrated without the need of a nontrivial measure on the configu- ration space. Diffeomorphism invariance arises then in a natural but subtle manner. Note that as the end result we recover the Kronecker delta inner product for the new fundamental states: χβ(q) := δ 1/2(q, β). Thus, in this new B-polymer representation, the Hilbert space HB is the completion with respect to the inner product (13) of the states generated by taking (finite) linear combinations of basis elements of the form χβ : Ψ(q) = bi χβi(q) (14) Let us now introduce an equivalent description of this Hilbert space. Instead of having the basis elements be half-deltas as elements of the Hilbert space where the inner product is given by the ordinary Lebesgue measure dq, we redefine both the basis and the measure. We could consider, instead of a half-delta with support β, a Kronecker delta or characteristic function with support on β: χ′β(q) := δq,β These functions have a similar behavior with respect to the product as the half-deltas, namely: χ′β(q) · χ′α(q) = δβ,α. The main difference is that neither χ ′ nor their squares are integrable with respect to the Lebesgue mea- sure (having zero norm). In order to fix that problem we have to change the measure so that we recover the basic inner product (13) with our new basis. The needed mea- sure turns out to be the discrete counting measure on R. Thus any state in the ‘half density basis’ can be written (using the same expression) in terms of the ‘Kronecker basis’. For more details and further motivation see the next section. Note that in this B-polymer representation, both Û and V̂ have their roles interchanged with that of the A-polymer representation: while U(α) is discontinuous and thus q̂ is not defined in the A-representation, we have that it is V (β) in the B-representation that has this property. In this case, it is the operator p̂ that can not be defined. We see then that given a physical system for which the configuration space has a well defined physi- cal meaning, within the possible representation in which wave-functions are functions of the configuration variable q, the A and B polymer representations are radically dif- ferent and inequivalent. Having said this, it is also true that the A and B representations are equivalent in a different sense, by means of the duality between q and p representations and the d↔ 1/d duality: The A-polymer representation in the “q-representation” is equivalent to the B-polymer representation in the “p-representation”, and conversely. When studying a problem, it is important to decide from the beginning which polymer representation (if any) one should be using (for instance in the q-polarization). This has as a consequence an implication on which variable is naturally “quantized” (even if continuous): p for A and q for B. There could be for instance a physical criteria for this choice. For example a fundamental symmetry could suggest that one representation is more natural than an- other one. This indeed has been recently noted by Chiou in [10], where the Galileo group is investigated and where it is shown that the B representation is better behaved. In the other polarization, namely for wavefunctions of p, the picture gets reversed: q is discrete for the A- representation, while p is for the B-case. Let us end this section by noting that the procedure of obtaining the polymer quantization by means of an appropriate limit of Fock-Schrödinger representations might prove useful in more general settings in field theory or quantum gravity. III. POLYMER QUANTUM MECHANICS: KINEMATICS In previous sections we have derived what we have called the A and B polymer representations (in the q- polarization) as limiting cases of ordinary Fock repre- sentations. In this section, we shall describe, without any reference to the Schrödinger representation, the ‘ab- stract’ polymer representation and then make contact with its two possible realizations, closely related to the A and B cases studied before. What we will see is that one of them (the A case) will correspond to the p-polarization while the other one corresponds to the q−representation, when a choice is made about the physical significance of the variables. We can start by defining abstract kets |µ〉 labelled by a real number µ. These shall belong to the Hilbert space Hpoly. From these states, we define a generic ‘cylinder states’ that correspond to a choice of a finite collection of numbers µi ∈ R with i = 1, 2, . . . , N . Associated to this choice, there are N vectors |µi〉, so we can take a linear combination of them |ψ〉 = ai |µi〉 (15) The polymer inner product between the fundamental kets is given by, 〈ν|µ〉 = δν,µ (16) That is, the kets are orthogonal to each other (when ν 6= µ) and they are normalized (〈µ|µ〉 = 1). Immediately, this implies that, given any two vectors |φ〉 = j=1 bj |νj〉 and |ψ〉 = i=1 ai |µi〉, the inner product between them is given by, 〈φ|ψ〉 = b̄j ai 〈νj |µi〉 = b̄k ak where the sum is over k that labels the intersection points between the set of labels {νj} and {µi}. The Hilbert space Hpoly is the Cauchy completion of finite linear com- bination of the form (15) with respect to the inner prod- uct (16). Hpoly is non-separable. There are two basic operators on this Hilbert space: the ‘label operator’ ε̂: ε̂ |µ〉 := µ |µ〉 and the displacement operator ŝ (λ), ŝ (λ) |µ〉 := |µ+ λ〉 The operator ε̂ is symmetric and the operator(s) ŝ(λ) defines a one-parameter family of unitary operators on Hpoly, where its adjoint is given by ŝ† (λ) = ŝ (−λ). This action is however, discontinuous with respect to λ given that |µ〉 and |µ + λ〉 are always orthogonal, no matter how small is λ. Thus, there is no (Hermitian) operator that could generate ŝ (λ) by exponentiation. So far we have given the abstract characterization of the Hilbert space, but one would like to make contact with concrete realizations as wave functions, or by iden- tifying the abstract operators ε̂ and ŝ with physical op- erators. Suppose we have a system with a configuration space with coordinate given by q, and p denotes its canonical conjugate momenta. Suppose also that for physical rea- sons we decide that the configuration coordinate q will have some “discrete character” (for instance, if it is to be identified with position, one could say that there is an underlying discreteness in position at a small scale). How can we implement such requirements by means of the polymer representation? There are two possibilities, depending on the choice of ‘polarizations’ for the wave- functions, namely whether they will be functions of con- figuration q or momenta p. Let us the divide the discus- sion into two parts. A. Momentum polarization In this polarization, states will be denoted by, ψ(p) = 〈p|ψ〉 where ψµ(p) = 〈p|µ〉 = ei How are then the operators ε̂ and ŝ represented? Note that if we associate the multiplicative operator V̂ (λ) · ψµ(p) = ei ~ = ei (µ+λ) p = ψ(µ+λ)(p) we see then that the operator V̂ (λ) corresponds precisely to the shift operator ŝ (λ). Thus we can also conclude that the operator p̂ does not exist. It is now easy to identify the operator q̂ with: q̂ · ψµ(p) = −i~ ψµ(p) = µ e ~ = µψµ(p) namely, with the abstract operator ε̂. The reason we say that q̂ is discrete is because this operator has as its eigenvalue the label µ of the elementary state ψµ(p), and this label, even when it can take value in a continuum of possible values, is to be understood as a discrete set, given that the states are orthonormal for all values of µ. Given that states are now functions of p, the inner product (16) should be defined by a measure µ on the space on which the wave-functions are defined. In order to know what these two objects are, namely, the quan- tum “configuration” space C and the measure thereon1, we have to make use of the tools available to us from the theory of C∗-algebras. If we consider the operators V̂ (λ), together with their natural product and ∗-relation given by V̂ ∗(λ) = V̂ (−λ), they have the structure of an Abelian C∗-algebra (with unit) A. We know from the representation theory of such objects that A is iso- morphic to the space of continuous functions C0(∆) on a compact space ∆, the spectrum of A. Any representation of A on a Hilbert space as multiplication operator will be on spaces of the form L2(∆, dµ). That is, our quantum configuration space is the spectrum of the algebra, which in our case corresponds to the Bohr compactification Rb of the real line [11]. This space is a compact group and there is a natural probability measure defined on it, the Haar measure µH. Thus, our Hilbert space Hpoly will be isomorphic to the space, Hpoly,p = L2(Rb, dµH) (17) In terms of ‘quasi periodic functions’ generated by ψµ(p), the inner product takes the form 〈ψµ|ψλ〉 := dµH ψµ(p)ψλ(p) := = lim L 7→∞ dpψµ(p)ψλ(p) = δµ,λ (18) note that in the p-polarization, this characterization cor- responds to the ‘A-version’ of the polymer representation of Sec. II (where p and q are interchanged). B. q-polarization Let us now consider the other polarization in which wave functions will depend on the configuration coordinate q: ψ(q) = 〈q|ψ〉 The basic functions, that now will be called ψ̃µ(q), should be, in a sense, the dual of the functions ψµ(p) of the previous subsection. We can try to define them via a ‘Fourier transform’: ψ̃µ(q) := 〈q|µ〉 = 〈q| dµH|p〉〈p|µ〉 which is given by ψ̃µ(q) := dµH〈q|p〉ψµ(p) = dµH e −i p q ~ = δq,µ (19) 1 here we use the standard terminology of ‘configuration space’ to denote the domain of the wave function even when, in this case, it corresponds to the physical momenta p. That is, the basic objects in this representation are Kro- necker deltas. This is precisely what we had found in Sec. II for the B-type representation. How are now the basic operators represented and what is the form of the inner product? Regarding the operators, we expect that they are represented in the opposite manner as in the previous p-polarization case, but that they preserve the same features: p̂ does not exist (the derivative of the Kro- necker delta is ill defined), but its exponentiated version V̂ (λ) does: V̂ (λ) · ψ(q) = ψ(q + λ) and the operator q̂ that now acts as multiplication has as its eigenstates, the functions ψ̃ν(q) = δν,q: q̂ · ψ̃µ(q) := µ ψ̃µ(q) What is now the nature of the quantum configurations space Q? And what is the measure thereon dµq? that defines the inner product we should have: 〈ψ̃µ(q), ψ̃λ(q)〉 = δµ,λ The answer comes from one of the characterizations of the Bohr compactification: we know that it is, in a precise sense, dual to the real line but when equipped with the discrete topology Rd. Furthermore, the measure on Rd will be the ‘counting measure’. In this way we recover the same properties we had for the previous characterization of the polymer Hilbert space. We can thus write: Hpoly,x := L2(Rd, dµc) (20) This completes a precise construction of the B-type poly- mer representation sketched in the previous section. Note that if we had chosen the opposite physical situation, namely that q, the configuration observable, be the quan- tity that does not have a corresponding operator, then we would have had the opposite realization: In the q- polarization we would have had the type-A polymer rep- resentation and the type-B for the p-polarization. As we shall see both scenarios have been considered in the literature. Up to now we have only focused our discussion on the kinematical aspects of the quantization process. Let us now consider in the following section the issue of dynam- ics and recall the approach that had been adopted in the literature, before the issue of the removal of the regulator was reexamined in [6]. IV. POLYMER QUANTUM MECHANICS: DYNAMICS As we have seen the construction of the polymer representation is rather natural and leads to a quan- tum theory with different properties than the usual Schrödinger counterpart such as its non-separability, the non-existence of certain operators and the existence of normalized eigen-vectors that yield a precise value for one of the phase space coordinates. This has been done without any regard for a Hamiltonian that endows the system with a dynamics, energy and so on. First let us consider the simplest case of a particle of mass m in a potential V (q), in which the Hamiltonian H takes the form, p2 + V (q) Suppose furthermore that the potential is given by a non- periodic function, such as a polynomial or a rational func- tion. We can immediately see that a direct implementa- tion of the Hamiltonian is out of our reach, for the simple reason that, as we have seen, in the polymer representa- tion we can either represent q or p, but not both! What has been done so far in the literature? The simplest thing possible: approximate the non-existing term by a well defined function that can be quantized and hope for the best. As we shall see in next sections, there is indeed more that one can do. At this point there is also an important decision to be made: which variable q or p should be regarded as “dis- crete”? Once this choice is made, then it implies that the other variable will not exist: if q is regarded as dis- crete, then p will not exist and we need to approximate the kinetic term p2/2m by something else; if p is to be the discrete quantity, then q will not be defined and then we need to approximate the potential V (q). What hap- pens with a periodic potential? In this case one would be modelling, for instance, a particle on a regular lattice such as a phonon living on a crystal, and then the natural choice is to have q not well defined. Furthermore, the po- tential will be well defined and there is no approximation needed. In the literature both scenarios have been considered. For instance, when considering a quantum mechanical system in [2], the position was chosen to be discrete, so p does not exist, and one is then in the A type for the momentum polarization (or the type B for the q- polarization). With this choice, it is the kinetic term the one that has to be approximated, so once one has done this, then it is immediate to consider any potential that will thus be well defined. On the other hand, when con- sidering loop quantum cosmology (LQC), the standard choice is that the configuration variable is not defined [4]. This choice is made given that LQC is regarded as the symmetric sector of full loop quantum gravity where the connection (that is regarded as the configuration vari- able) can not be promoted to an operator and one can only define its exponentiated version, namely, the holon- omy. In that case, the canonically conjugate variable, closely related to the volume, becomes ‘discrete’, just as in the full theory. This case is however, different from the particle in a potential example. First we could mention that the functional form of the Hamiltonian constraint that implements dynamics has a different structure, but the more important difference lies in that the system is constrained. Let us return to the case of the particle in a po- tential and for definiteness, let us start with the aux- iliary kinematical framework in which: q is discrete, p can not be promoted and thus we have to approximate the kinetic term p̂2/2m. How is this done? The stan- dard prescription is to define, on the configuration space C, a regular ‘graph’ γµ0 . This consists of a numerable set of points, equidistant, and characterized by a pa- rameter µ0 that is the (constant) separation between points. The simplest example would be to consider the set γµ0 = {q ∈ R | q = nµ0 , ∀ n ∈ Z}. This means that the basic kets that will be considered |µn〉 will correspond precisely to labels µn belonging to the graph γµ0 , that is, µn = nµ0. Thus, we shall only consider states of the form, |ψ〉 = bn |µn〉 . (21) This ‘small’ Hilbert space Hγµ0 , the graph Hilbert space, is a subspace of the ‘large’ polymer Hilbert space Hpoly but it is separable. The condition for a state of the form (21) to belong to the Hilbert space Hγµ0 is that the co- efficients bn satisfy: n |bn|2 <∞. Let us now consider the kinetic term p̂2/2m. We have to approximate it by means of trigonometric functions, that can be built out of the functions of the form eiλ p/~. As we have seen in previous sections, these functions can indeed be promoted to operators and act as translation operators on the kets |µ〉. If we want to remain in the graph γ, and not create ‘new points’, then one is con- strained to considering operators that displace the kets by just the right amount. That is, we want the basic shift operator V̂ (λ) to be such that it maps the ket with label |µn〉 to the next ket, namely |µn+1〉. This can in- deed achieved by fixing, once and for all, the value of the allowed parameter λ to be λ = µ0. We have then, V̂ (µ0) · |µn〉 = |µn + µ0〉 = |µn+1〉 which is what we wanted. This basic ‘shift operator’ will be the building block for approximating any (polynomial) function of p. In order to do that we notice that the function p can be approximated by, p ≈ ~ (µ0 p ~ − e−i where the approximation is good for p << ~/µ0. Thus, one can define a regulated operator p̂µ0 that depends on the ‘scale’ µ0 as: p̂µ0 · |µn〉 := [V (µ0) − V (−µ0)] · |µn〉 = (|µn+1〉 − |µn−1〉) (22) In order to regulate the operator p̂2, there are (at least) two possibilities, namely to compose the operator p̂µ0 with itself or to define a new approximation. The oper- ator p̂µ0 · p̂µ0 has the feature that shifts the states two steps in the graph to both sides. There is however an- other operator that only involves shifting once: p̂2µ0 · |νn〉 := [2 − V̂ (µ0) − V̂ (−µ0)] · |νn〉 = (2|νn〉 − |νn+1〉 − |νn−1〉) (23) which corresponds to the approximation p2 ≈ 2~ cos(µ0 p/~)), valid also in the regime p << ~/µ0. With these considerations, one can define the operator Ĥµ0 , the Hamiltonian at scale µ0, that in practice ‘lives’ on the space Hγµ0 as, Ĥµ0 := p̂2µ0 + V̂ (q) , (24) that is a well defined, symmetric operator on Hγµ0 . No- tice that the operator is also defined on Hpoly, but there its physical interpretation is problematic. For example, it turns out that the expectation value of the kinetic term calculated on most states (states which are not tailored to the exact value of the parameter µ0) is zero. Even if one takes a state that gives “reasonable“ expectation values of the µ0-kinetic term and uses it to calculate the expectation value of the kinetic term corresponding to a slight perturbation of the parameter µ0 one would get zero. This problem, and others that arise when working on Hpoly, forces one to assign a physical interpretation to the Hamiltonian Ĥµ0 only when its action is restricted to the subspace Hγµ0 . Let us now explore the form that the Hamiltonian takes in the two possible polarizations. In the q-polarization, the basis, labelled by n is given by the functions χn(q) = δq,µn . That is, the wave functions will only have sup- port on the set γµ0 . Alternatively, one can think of a state as completely characterized by the ‘Fourier coeffi- cients’ an: ψ(q) ↔ an, which is the value that the wave function ψ(q) takes at the point q = µn = nµ0. Thus, the Hamiltonian takes the form of a difference equation when acting on a general state ψ(q). Solving the time independent Schrödinger equation Ĥ · ψ = E ψ amounts to solving the difference equation for the coefficients an. The momentum polarization has a different structure. In this case, the operator p̂2µ0 acts as a multiplication operator, p̂2µ0 · ψ(p) = 1 − cos (µ0 p ψ(p) (25) The operator corresponding to q will be represented as a derivative operator q̂ · ψ(p) := i~ ∂p ψ(p). For a generic potential V (q), it has to be defined by means of spectral theory defined now on a circle. Why on a circle? For the simple reason that by restricting ourselves to a regular graph γµ0 , the functions of p that preserve it (when acting as shift operators) are of the form e(i m µ0 p/~) for m integer. That is, what we have are Fourier modes, labelled by m, of period 2π ~/µ0 in p. Can we pretend then that the phase space variable p is now compactified? The answer is in the affirmative. The inner product on periodic functions ψµ0(p) of p coming from the full Hilbert space Hpoly and given by 〈φ(p)|ψ(p)〉poly = lim L 7→∞ dp φ(p)ψ(p) is precisely equivalent to the inner product on the circle given by the uniform measure 〈φ(p)|ψ(p)〉µ0 = ∫ π~/µ0 −π~/µ0 dp φ(p)ψ(p) with p ∈ (−π~/µ0, π~/µ0). As long as one restricts at- tention to the graph γµ0 , one can work in this separable Hilbert space Hγµ0 of square integrable functions on S Immediately, one can see the limitations of this descrip- tion. If the mechanical system to be quantized is such that its orbits have values of the momenta p that are not small compared with π~/µ0 then the approximation taken will be very poor, and we don’t expect neither the effective classical description nor its quantization to be close to the standard one. If, on the other hand, one is al- ways within the region in which the approximation can be regarded as reliable, then both classical and quantum de- scriptions should approximate the standard description. What does ‘close to the standard description’ exactly mean needs, of course, some further clarification. In particular one is assuming the existence of the usual Schrödinger representation in which the system has a be- havior that is also consistent with observations. If this is the case, the natural question is: How can we approxi- mate such description from the polymer picture? Is there a fine enough graph γµ0 that will approximate the system in such a way that all observations are indistinguishable? Or even better, can we define a procedure, that involves a refinement of the graph γµ0 such that one recovers the standard picture? It could also happen that a continuum limit can be de- fined but does not coincide with the ‘expected one’. But there might be also physical systems for which there is no standard description, or it just does not make sense. Can in those cases the polymer representation, if it ex- ists, provide the correct physical description of the sys- tem under consideration? For instance, if there exists a physical limitation to the minimum scale set by µ0, as could be the case for a quantum theory of gravity, then the polymer description would provide a true physical bound on the value of certain quantities, such as p in our example. This could be the case for loop quantum cosmology, where there is a minimum value for physical volume (coming from the full theory), and phase space points near the ‘singularity’ lie at the region where the approximation induced by the scale µ0 departs from the standard classical description. If in that case the poly- mer quantum system is regarded as more fundamental than the classical system (or its standard Wheeler-De Witt quantization), then one would interpret this dis- crepancies in the behavior as a signal of the breakdown of classical description (or its ‘naive’ quantization). In the next section we present a method to remove the regulator µ0 which was introduced as an intermedi- ate step to construct the dynamics. More precisely, we shall consider the construction of a continuum limit of the polymer description by means of a renormalization procedure. V. THE CONTINUUM LIMIT This section has two parts. In the first one we motivate the need for a precise notion of the continuum limit of the polymeric representation, explaining why the most direct, and naive approach does not work. In the sec- ond part, we shall present the main ideas and results of the paper [6], where the Hamiltonian and the physical Hilbert space in polymer quantum mechanics are con- structed as a continuum limit of effective theories, follow- ing Wilson’s renormalization group ideas. The resulting physical Hilbert space turns out to be unitarily isomor- phic to the ordinary Hs = L2(R, dq) of the Schrödinger theory. Before describing the results of [6] we should discuss the precise meaning of reaching a theory in the contin- uum. Let us for concreteness consider the B-type repre- sentation in the q-polarization. That is, states are func- tions of q and the orthonormal basis χµ(q) is given by characteristic functions with support on q = µ. Let us now suppose we have a Schrödinger state Ψ(q) ∈ Hs = L2(R, dq). What is the relation between Ψ(q) and a state in Hpoly,x? We are also interested in the opposite ques- tion, that is, we would like to know if there is a preferred state in Hs that is approximated by an arbitrary state ψ(q) in Hpoly,x. The first obvious observation is that a Schödinger state Ψ(q) does not belong to Hpoly,x since it would have an infinite norm. To see that note that even when the would-be state can be formally expanded in the χµ basis as, Ψ(q) = Ψ(µ) χµ(q) where the sum is over the parameter µ ∈ R. Its associ- ated norm in Hpoly,x would be: |Ψ(q)|2poly = |Ψ(µ)|2 → ∞ which blows up. Note that in order to define a mapping P : Hs → Hpoly,x, there is a huge ambiguity since the values of the function Ψ(q) are needed in order to expand the polymer wave function. Thus we can only define a mapping in a dense subset D of Hs where the values of the functions are well defined (recall that in Hs the value of functions at a given point has no meaning since states are equivalence classes of functions). We could for instance ask that the mapping be defined for representatives of the equivalence classes in Hs that are piecewise continuous. From now on, when we refer to an element of the space Hs we shall be refereeing to one of those representatives. Notice then that an element of Hs does define an element of Cyl∗γ , the dual to the space Cylγ , that is, the space of cylinder functions with support on the (finite) lattice γ = {µ1, µ2, . . . , µN}, in the following way: Ψ(q) : Cylγ −→ C such that Ψ(q)[ψ(q)] = (Ψ|ψ〉 := Ψ(µ) 〈χµ| ψi χµi〉polyγ Ψ(µi)ψi < ∞ (26) Note that this mapping could be seen as consisting of two parts: First, a projection Pγ : Cyl ∗ → Cylγ such that Pγ(Ψ) = Ψγ(q) := i Ψ(µi)χµi(q) ∈ Cylγ . The state Ψγ is sometimes refereed to as the ‘shadow of Ψ(q) on the lattice γ’. The second step is then to take the inner product between the shadow Ψγ(q) and the state ψ(q) with respect to the polymer inner product 〈Ψγ |ψ〉polyγ . Now this inner product is well defined. Notice that for any given lattice γ the corresponding projector Pγ can be intuitively interpreted as some kind of ‘coarse graining map’ from the continuum to the lattice γ. In terms of functions of q the projection is replacing a continuous function defined on R with a function over the lattice γ ⊂ R which is a discrete set simply by restricting Ψ to γ. The finer the lattice the more points that we have on the curve. As we shall see in the second part of this section, there is indeed a precise notion of coarse graining that implements this intuitive idea in a concrete fashion. In particular, we shall need to replace the lattice γ with a decomposition of the real line in intervals (having the lattice points as end points). Let us now consider a system in the polymer represen- tation in which a particular lattice γ0 was chosen, say with points of the form {qk ∈ R |qk = ka0 , ∀ k ∈ Z}, namely a uniform lattice with spacing equal to a0. In this case, any Schrödinger wave function (of the type that we consider) will have a unique shadow on the lattice γ0. If we refine the lattice γ 7→ γn by dividing each interval in 2n new intervals of length an = a0/2 n we have new shad- ows that have more and more points on the curve. Intu- itively, by refining infinitely the graph we would recover the original function Ψ(q). Even when at each finite step the corresponding shadow has a finite norm in the poly- mer Hilbert space, the norm grows unboundedly and the limit can not be taken, precisely because we can not em- bed Hs into Hpoly. Suppose now that we are interested in the reverse process, namely starting from a polymer theory on a lattice and asking for the ‘continuum wave function’ that is best approximated by a wave function over a graph. Suppose furthermore that we want to con- sider the limit of the graph becoming finer. In order to give precise answers to these (and other) questions we need to introduce some new technology that will allow us to overcome these apparent difficulties. In the remaining of this section we shall recall these constructions for the benefit of the reader. Details can be found in [6] (which is an application of the general formalism discussed in [9]). The starting point in this construction is the concept of a scale C, which allows us to define the effective the- ories and the concept of continuum limit. In our case a scale is a decomposition of the real line in the union of closed-open intervals, that cover the whole line and do not intersect. Intuitively, we are shifting the emphasis from the lattice points to the intervals defined by the same points with the objective of approximating con- tinuous functions defined on R with functions that are constant on the intervals defined by the lattice. To be precise, we define an embedding, for each scale Cn from Hpoly to Hs by means of a step function: Ψ(man) χman(q) → Ψ(man) χαm(q) ∈ Hs with χαn(q) a characteristic function on the interval αm = [man, (m + 1)an). Thus, the shadows (living on the lattice) were just an intermediate step in the con- struction of the approximating function; this function is piece-wise constant and can be written as a linear com- bination of step functions with the coefficients provided by the shadows. The challenge now is to define in an appropriate sense how one can approximate all the aspects of the theory by means of this constant by pieces functions. Then the strategy is that, for any given scale, one can define an effective theory by approximating the kinetic operator by a combination of the translation operators that shift between the vertices of the given decomposition, in other words by a periodic function in p. As a result one has a set of effective theories at given scales which are mutually related by coarse graining maps. This framework was developed in [6]. For the convenience of the reader we briefly recall part of that framework. Let us denote the kinematic polymer Hilbert space at the scale Cn as HCn , and its basis elements as eαi,Cn , where αi = [ian, (i + 1)an) ∈ Cn. By construction this basis is orthonormal. The basis elements in the dual Hilbert space H∗Cn are denoted by ωαi,Cn ; they are also orthonormal. The states ωαi,Cn have a simple action on Cyl, ωαi,Cn(δx0,q) = χαi,Cn(x0). That is, if x0 is in the interval αi of Cn the result is one and it is zero if it is not there. Given any m ≤ n, we define d∗m,n : H∗Cn → H as the ‘coarse graining’ map between the dual Hilbert spaces, that sends the part of the elements of the dual basis to zero while keeping the information of the rest: d∗m,n(ωαi,Cn) = ωβj ,Cm if i = j2 n−m, in the opposite case d∗m,n(ωαi,Cn) = 0. At every scale the corresponding effective theory is given by the hamiltonian Hn. These Hamiltonians will be treated as quadratic forms, hn : HCn → R, given by hn(ψ) = λ (ψ,Hnψ) , (27) where λ2Cn is a normalizaton factor. We will see later that this rescaling of the inner product is necessary in order to guarantee the convergence of the renormalized theory. The completely renormalized theory at this scale is obtained as hrenm := lim d⋆m,nhn. (28) and the renormalized Hamiltonians are compatible with each other, in the sense that d⋆m,nh n = h In order to analyze the conditions for the convergence in (28) let us express the Hamiltonian in terms of its eigen-covectors end eigenvalues. We will work with effec- tive Hamiltonians that have a purely discrete spectrum (labelled by ν) Hn · Ψν,Cn = Eν,Cn Ψν,Cn . We shall also introduce, as an intermediate step, a cut-off in the energy levels. The origin of this cut-off is in the approximation of the Hamiltonian of our system at a given scale with a Hamiltonian of a periodic system in a regime of small energies, as we explained earlier. Thus, we can write hνcut−offm = νcut−off Eν,CmΨν,Cm ⊗ Ψν,Cm , (29) where the eigen covectors Ψν,Cm are normalized accord- ing to the inner product rescaled by 1 , and the cut- off can vary up to a scale dependent bound, νcut−off ≤ νmax(Cm). The Hilbert space of covectors together with such inner product will be called H⋆renCm . In the presence of a cut-off, the convergence of the microscopically corrected Hamiltonians, equation (28) is equivalent to the existence of the following two limits. The first one is the convergence of the energy levels, Eν,Cn = E ν . (30) Second is the existence of the completely renormalized eigen covectors, d⋆m,n Ψν,Cn = Ψ ∈ H⋆renCm ⊂ Cyl ⋆ . (31) We clarify that the existence of the above limit means that Ψrenν,Cm(δx0,q) is well defined for any δx0,q ∈ Cyl. No- tice that this point-wise convergence, if it can take place at all, will require the tuning of the normalization factors λ2Cn . Now we turn to the question of the continuum limit of the renormalized covectors. First we can ask for the existence of the limit Ψrenν,Cn(δx0,q) (32) for any δx0,q ∈ Cyl. When this limits exists there is a natural action of the eigen covectors in the continuum limit. Below we consider another notion of the continuum limit of the renormalized eigen covectors. When the completely renormalized eigen covectors exist, they form a collection that is d⋆-compatible, d⋆m,nΨ = Ψrenν,Cm . A sequence of d ⋆-compatible nor- malizable covectors define an element of , which is the projective limit of the renormalized spaces of covec- H⋆renCn . (33) The inner product in this space is defined by ({ΨCn}, {ΦCn})renR := lim (ΨCn ,ΦCn) The natural inclusion of C∞0 in is by an antilinear map which assigns to any Ψ ∈ C∞0 the d⋆-compatible collection ΨshadCn := ωαiΨ̄(L(αi)) ∈ H⋆renCn ⊂ Cyl ΨshadCn will be called the shadow of Ψ at scale Cn and acts in Cyl as a piecewise constant function. Clearly other types of test functions like Schwartz functions are also naturally included in . In this context a shadow is a state of the effective theory that approximates a state in the continuum theory. Since the inner product in is degenerate, the physical Hilbert space is defined as H⋆phys := / ker(·, ·)ren Hphys := H⋆⋆phys The nature of the physical Hilbert space, whether it is isomorphic to the Schrödinger Hilber space, Hs, or not, is determined by the normalization factors λ2Cn which can be obtained from the conditions asking for compatibil- ity of the dynamics of the effective theories at different scales. The dynamics of the system under consideration selects the continuum limit. Let us now return to the definition of the Hamilto- nian in the continuum limit. First consider the contin- uum limit of the Hamiltonian (with cut-off) in the sense of its point-wise convergence as a quadratic form. It turns out that if the limit of equation (32) exists for all the eigencovectors allowed by the cut-off, we have νcut−off ren : Hpoly,x → R defined by νcut−off ren (δx0,q) := lim hνcut−off renn ([δx0,q]Cn). (34) This Hamiltonian quadratic form in the continuum can be coarse grained to any scale and, as can be ex- pected, it yields the completely renormalized Hamilto- nian quadratic forms at that scale. However, this is not a completely satisfactory continuum limit because we can not remove the auxiliary cut-off νcut−off . If we tried, as we include more and more eigencovectors in the Hamilto- nian the calculations done at a given scale would diverge and doing them in the continuum is just as divergent. Below we explore a more successful path. We can use the renormalized inner product to induce an action of the cut–off Hamiltonians on νcut−off ren ({ΨCn}) := lim hνcut−off renn ((ΨCn , ·)renCn ), where we have used the fact that (ΨCn , ·)renCn ∈ HCn . The existence of this limit is trivial because the renormalized Hamiltonians are finite sums and the limit exists term by term. These cut-off Hamiltonians descend to the physical Hilbert space νcut−off ren ([{ΨCn}]) := h νcut−off ren ({ΨCn}) for any representative {ΨCn} ∈ [{ΨCn}] ∈ H⋆phys. Finally we can address the issue of removal of the cut- off. The Hamiltonian hren → R is defined by the limit := lim νcut−off→∞ νcut−off ren when the limit exists. Its corresponding Hermitian form in Hphys is defined whenever the above limit exists. This concludes our presentation of the main results of [6]. Let us now consider several examples of systems for which the continuum limit can be investigated. VI. EXAMPLES In this section we shall develop several examples of systems that have been treated with the polymer quanti- zation. These examples are simple quantum mechanical systems, such as the simple harmonic oscillator and the free particle, as well as a quantum cosmological model known as loop quantum cosmology. A. The Simple Harmonic Oscillator In this part, let us consider the example of a Simple Har- monic Oscillator (SHO) with parameters m and ω, clas- sically described by the following Hamiltonian mω2 x2. Recall that from these parameters one can define a length scale D = ~/mω. In the standard treatment one uses this scale to define a complex structure JD (and an in- ner product from it), as we have described in detail that uniquely selects the standard Schrödinger representation. At scale Cn we have an effective Hamiltonian for the Simple Harmonic Oscillator (SHO) given by HCn = 1 − cos anp mω2x2 . (35) If we interchange position and momentum, this Hamilto- nian is exactly that of a pendulum of mass m, length l and subject to a constant gravitational field g: ĤCn = − +mgl(1 − cos θ) where those quantities are related to our system by, mω an , g = , θ = That is, we are approximating, for each scale Cn the SHO by a pendulum. There is, however, an important difference. From our knowledge of the pendulum system, we know that the quantum system will have a spectrum for the energy that has two different asymptotic behav- iors, the SHO for low energies and the planar rotor in the higher end, corresponding to oscillating and rotating solutions respectively2. As we refine our scale and both the length of the pendulum and the height of the periodic potential increase, we expect to have an increasing num- ber of oscillating states (for a given pendulum system, there is only a finite number of such states). Thus, it is justified to consider the cut-off in the energy eigenval- ues, as discussed in the last section, given that we only expect a finite number of states of the pendulum to ap- proximate SHO eigenstates. With these consideration in mind, the relevant question is whether the conditions for the continuum limit to exist are satisfied. This question has been answered in the affirmative in [6]. What was shown there was that the eigen-values and eigen func- tions of the discrete systems, which represent a discrete and non-degenerate set, approximate those of the contin- uum, namely, of the standard harmonic oscillator when the inner product is renormalized by a factor λ2Cn = 1/2 This convergence implies that the continuum limit exists as we understand it. Let us now consider the simplest possible system, a free particle, that has nevertheless the particular feature that the spectrum of the energy is con- tinuous. 2 Note that both types of solutions are, in the phase space, closed. This is the reason behind the purely discrete spectrum. The distinction we are making is between those solutions inside the separatrix, that we call oscillating, and those that are above it that we call rotating. B. Free Polymer Particle In the limit ω → 0, the Hamiltonian of the Simple Harmonic oscillator (35) goes to the Hamiltonian of a free particle and the corresponding time independent Schrödinger equation, in the p−polarization, is given by (1 − cos anp ) − ECn ψ̃(p) = 0 where we now have that p ∈ S1, with p ∈ (−π~ Thus, we have ECn = 1 − cos ≤ ECn,max ≡ 2 . (36) At each scale the energy of the particle we can describe is bounded from above and the bound depends on the scale. Note that in this case the spectrum is continu- ous, which implies that the ordinary eigenfunctions of the Hilbert are not normalizable. This imposes an upper bound in the value that the energy of the particle can have, in addition to the bound in the momentum due to its “compactification”. Let us first look for eigen-solutions to the time inde- pendent Schrödinger equation, that is, for energy eigen- states. In the case of the ordinary free particle, these correspond to constant momentum plane waves of the form e±( ) and such that the ordinary dispersion re- lation p2/2m = E is satisfied. These plane waves are not square integrable and do not belong to the ordinary Hilbert space of the Schrödinger theory but they are still useful for extracting information about the system. For the polymer free particle we have, ψ̃Cn(p) = c1δ(p− PCn) + c2δ(p+ PCn) where PCn is a solution of the previous equation consid- ering a fixed value of ECn . That is, PCn = P (ECn) = arccos 1 − ma The inverse Fourier transform yields, in the ‘x represen- tation’, ψCn(xj) = ∫ π~/an −π~/an ψ̃(p) e p j dp = ixjPCn /~ + c2e −ixjPCn /~ .(37) with xj = an j for j ∈ Z. Note that the eigenfunctions are still delta functions (in the p representation) and thus not (square) normalizable with respect to the polymer inner product, that in the p polarization is just given by the ordinary Haar measure on S1, and there is no quantization of the momentum (its spectrum is still truly continuous). Let us now consider the time dependent Schrödinger equation, i~ ∂t Ψ̃(p, t) = Ĥ · Ψ̃(p, t). Which now takes the form, Ψ̃(p, t) = (1 − cos (an p/~)) Ψ̃(p, t) that has as its solution, Ψ̃(p, t) = e− (1−cos (an p/~)) t ψ̃(p) = e(−iECn /~) t ψ̃(p) for any initial function ψ̃(p), where ECn satisfy the dis- persion relation (36). The wave function Ψ(xj , t), the xj-representation of the wave function, can be obtained for any given time t by Fourier transforming with (37) the wave function Ψ̃(p, t). In order to check out the convergence of the micro- scopically corrected Hamiltonians we should analyze the convergence of the energy levels and of the proper cov- ectors. In the limit n → ∞, ECn → E = p2/2m so we can be certain that the eigen-values for the energy converge (when fixing the value of p). Let us write the proper covector as ΨCn = (ψCn , ·)renCn ∈ H . Then we can bring microscopic corrections to scale Cm and look for convergence of such corrections ΨrenCm = lim d⋆m,nΨCn . It is easy to see that given any basis vector eαi ∈ HCm the following limit ΨrenCm(eαi,Cm) = limCn→∞ ΨCn(dn,m(eαi,Cm)) exists and is equal to ΨshadCm (eαi,Cm) = [d ⋆ΨSchr](eαi,Cm) = Ψ Schr(iam) where ΨshadCm is calculated using the free particle Hamilto- nian in the Schrödinger representation. This expression defines the completely renormalized proper covector at the scale Cm. C. Polymer Quantum Cosmology In this section we shall present a version of quantum cosmology that we call polymer quantum cosmology. The idea behind this name is that the main input in the quan- tization of the corresponding mini-superspace model is the use of a polymer representation as here understood. Another important input is the choice of fundamental variables to be used and the definition of the Hamiltonian constraint. Different research groups have made differ- ent choices. We shall take here a simple model that has received much attention recently, namely an isotropic, homogeneous FRW cosmology with k = 0 and coupled to a massless scalar field ϕ. As we shall see, a proper treatment of the continuum limit of this system requires new tools under development that are beyond the scope of this work. We will thus restrict ourselves to the intro- duction of the system and the problems that need to be solved. The system to be quantized corresponds to the phase space of cosmological spacetimes that are homogeneous and isotropic and for which the homogeneous spatial slices have a flat intrinsic geometry (k = 0 condition). The only matter content is a mass-less scalar field ϕ. In this case the spacetime geometry is given by metrics of the form: ds2 = −dt2 + a2(t) (dx2 + dy2 + dz2) where the function a(t) carries all the information and degrees of freedom of the gravity part. In terms of the coordinates (a, pa, ϕ, pϕ) for the phase space Γ of the the- ory, all the dynamics is captured in the Hamiltonian con- straint C := −3 + 8πG 2|a|3 The first step is to define the constraint on the kine- matical Hilbert space to find physical states and then a physical inner product to construct the physical Hilbert space. First note that one can rewrite the equation as: p2a a 2 = 8πG If, as is normally done, one chooses ϕ to act as an in- ternal time, the right hand side would be promoted, in the quantum theory, to a second derivative. The left hand side is, furthermore, symmetric in a and pa. At this point we have the freedom in choosing the variable that will be quantized and the variable that will not be well defined in the polymer representation. The standard choice is that pa is not well defined and thus, a and any geometrical quantity derived from it, is quantized. Fur- thermore, we have the choice of polarization on the wave function. In this respect the standard choice is to select the a-polarization, in which a acts as multiplication and the approximation of pa, namely sin(λ pa)/λ acts as a difference operator on wave functions of a. For details of this particular choice see [5]. Here we shall adopt the op- posite polarization, that is, we shall have wave functions Ψ(pa, ϕ). Just as we did in the previous cases, in order to gain intuition about the behavior of the polymer quantized theory, it is convenient to look at the equivalent prob- lem in the classical theory, namely the classical system we would get be approximating the non-well defined ob- servable (pa in our present case) by a well defined object (made of trigonometric functions). Let us for simplicity choose to replace pa 7→ sin(λ pa)/λ. With this choice we get an effective classical Hamiltonian constraint that depends on λ: Cλ := − sin(λ pa) λ2|a| + 8πG 2|a|3 We can now compute effective equations of motion by means of the equations: Ḟ := {F, Cλ}, for any observable F ∈ C∞(Γ), and where we are using the effective (first order) action: dτ(pa ȧ+ pϕ ϕ̇−N Cλ) with the choice N = 1. The first thing to notice is that the quantity pϕ is a constant of the motion, given that the variable ϕ is cyclic. The second observation is that ϕ̇ = 8πG has the same sign as pϕ and never vanishes. Thus ϕ can be used as a (n internal) time variable. The next observation is that the equation for , namely the effective Friedman equation, will have a zero for a non-zero value of a given by λ2p2ϕ. This is the value at which there will be bounce if the trajectory started with a large value of a and was con- tracting. Note that the ‘size’ of the universe when the bounce occurs depends on both the constant pϕ (that dictates the matter density) and the value of the lattice size λ. Here it is important to stress that for any value of pϕ (that uniquely fixes the trajectory in the (a, pa) plane), there will be a bounce. In the original description in terms of Einstein’s equations (without the approxima- tion that depends on λ), there in no such bounce. If ȧ < 0 initially, it will remain negative and the universe collapses, reaching the singularity in a finite proper time. What happens within the effective description if we re- fine the lattice and go from λ to λn := λ/2 n? The only thing that changes, for the same classical orbit labelled by pϕ, is that the bounce occurs at a ‘later time’ and for a smaller value of a∗ but the qualitative picture remains the same. This is the main difference with the systems considered before. In those cases, one could have classical trajecto- ries that remained, for a given choice of parameter λ, within the region where sin(λp)/λ is a good approxima- tion to p. Of course there were also classical trajectories that were outside this region but we could then refine the lattice and find a new value λ′ for which the new clas- sical trajectory is well approximated. In the case of the polymer cosmology, this is never the case: Every classical trajectory will pass from a region where the approxima- tion is good to a region where it is not; this is precisely where the ‘quantum corrections’ kick in and the universes bounces. Given that in the classical description, the ‘original’ and the ‘corrected’ descriptions are so different we expect that, upon quantization, the corresponding quantum the- ories, namely the polymeric and the Wheeler-DeWitt will be related in a non-trivial way (if at all). In this case, with the choice of polarization and for a particular factor ordering we have, sin(λpa) · Ψ(pa, ϕ) = 0 as the Polymer Wheeler-DeWitt equation. In order to approach the problem of the continuum limit of this quantum theory, we have to realize that the task is now somewhat different than before. This is so given that the system is now a constrained system with a constraint operator rather than a regular non-singular system with an ordinary Hamiltonian evolution. Fortu- nately for the system under consideration, the fact that the variable ϕ can be regarded as an internal time allows us to interpret the quantum constraint as a generalized Klein-Gordon equation of the form Ψ = Θλ · Ψ where the operator Θλ is ‘time independent’. This al- lows us to split the space of solutions into ‘positive and negative frequency’, introduce a physical inner product on the positive frequency solutions of this equation and a set of physical observables in terms of which to de- scribe the system. That is, one reduces in practice the system to one very similar to the Schrödinger case by taking the positive square root of the previous equation: Θλ · Ψ. The question we are interested is whether the continuum limit of these theories (labelled by λ) exists and whether it corresponds to the Wheeler- DeWitt theory. A complete treatment of this problem lies, unfortunately, outside the scope of this work and will be reported elsewhere [12]. VII. DISCUSSION Let us summarize our results. In the first part of the article we showed that the polymer representation of the canonical commutation relations can be obtained as the limiting case of the ordinary Fock-Schrödinger represen- tation in terms of the algebraic state that defines the representation. These limiting cases can also be inter- preted in terms of the naturally defined coherent states associated to each representation labelled by the param- eter d, when they become infinitely ‘squeezed’. The two possible limits of squeezing lead to two different polymer descriptions that can nevertheless be identified, as we have also shown, with the two possible polarizations for an abstract polymer representation. This resulting the- ory has, however, very different behavior as the standard one: The Hilbert space is non-separable, the representa- tion is unitarily inequivalent to the Schrödinger one, and natural operators such as p̂ are no longer well defined. This particular limiting construction of the polymer the- ory can shed some light for more complicated systems such as field theories and gravity. In the regular treatments of dynamics within the poly- mer representation, one needs to introduce some extra structure, such as a lattice on configuration space, to con- struct a Hamiltonian and implement the dynamics for the system via a regularization procedure. How does this re- sulting theory compare to the original continuum theory one had from the beginning? Can one hope to remove the regulator in the polymer description? As they stand there is no direct relation or mapping from the polymer to a continuum theory (in case there is one defined). As we have shown, one can indeed construct in a systematic fashion such relation by means of some appropriate no- tions related to the definition of a scale, closely related to the lattice one had to introduce in the regularization. With this important shift in perspective, and an appro- priate renormalization of the polymer inner product at each scale one can, subject to some consistency condi- tions, define a procedure to remove the regulator, and arrive to a Hamiltonian and a Hilbert space. As we have seen, for some simple examples such as a free particle and the harmonic oscillator one indeed recovers the Schrödinger description back. For other sys- tems, such as quantum cosmological models, the answer is not as clear, since the structure of the space of classi- cal solutions is such that the ‘effective description’ intro- duced by the polymer regularization at different scales is qualitatively different from the original dynamics. A proper treatment of these class of systems is underway and will be reported elsewhere [12]. Perhaps the most important lesson that we have learned here is that there indeed exists a rich inter- play between the polymer description and the ordinary Schrödinger representation. The full structure of such re- lation still needs to be unravelled. We can only hope that a full understanding of these issues will shed some light in the ultimate goal of treating the quantum dynamics of background independent field systems such as general relativity. Acknowledgments We thank A. Ashtekar, G. Hossain, T. Pawlowski and P. Singh for discussions. This work was in part supported by CONACyT U47857-F and 40035-F grants, by NSF PHY04-56913, by the Eberly Research Funds of Penn State, by the AMC-FUMEC exchange program and by funds of the CIC-Universidad Michoacana de San Nicolás de Hidalgo. [1] R. Beaume, J. Manuceau, A. Pellet and M. Sirugue, “Translation Invariant States In Quantum Mechanics,” Commun. Math. Phys. 38, 29 (1974); W. E. Thirring and H. Narnhofer, “Covariant QED without indefinite met- ric,” Rev. Math. Phys. 4, 197 (1992); F. Acerbi, G. Mor- chio and F. Strocchi, “Infrared singular fields and non- regular representations of canonical commutation rela- tion algebras”, J. Math. Phys. 34, 899 (1993); F. Cav- allaro, G. Morchio and F. Strocchi, “A generalization of the Stone-von Neumann theorem to non-regular repre- sentations of the CCR-algebra”, Lett. Math. Phys. 47 307 (1999); H. Halvorson, “Complementarity of Repre- sentations in quantum mechanics”, Studies in History and Philosophy of Modern Physics 35 45 (2004). [2] A. Ashtekar, S. Fairhurst and J.L. Willis, “Quantum gravity, shadow states, and quantum mechanics”, Class. Quant. Grav. 20 1031 (2003) [arXiv:gr-qc/0207106]. [3] K. Fredenhagen and F. Reszewski, “Polymer state ap- proximations of Schrödinger wave functions”, Class. Quant. Grav. 23 6577 (2006) [arXiv:gr-qc/0606090]. [4] M. Bojowald, “Loop quantum cosmology”, Living Rev. Rel. 8, 11 (2005) [arXiv:gr-qc/0601085]; A. Ashtekar, M. Bojowald and J. Lewandowski, “Mathematical struc- ture of loop quantum cosmology”, Adv. Theor. Math. Phys. 7 233 (2003) [arXiv:gr-qc/0304074]; A. Ashtekar, T. Pawlowski and P. Singh, “Quantum nature of the big bang: Improved dynamics” Phys. Rev. D 74 084003 (2006) [arXiv:gr-qc/0607039] [5] V. Husain and O. Winkler, “Semiclassical states for quantum cosmology” Phys. Rev. D 75 024014 (2007) [arXiv:gr-qc/0607097]; V. Husain V and O. Winkler, “On singularity resolution in quantum gravity”, Phys. Rev. D 69 084016 (2004). [arXiv:gr-qc/0312094]. [6] A. Corichi, T. Vukasinac and J.A. Zapata. “Hamil- tonian and physical Hilbert space in polymer quan- tum mechanics”, Class. Quant. Grav. 24 1495 (2007) [arXiv:gr-qc/0610072] [7] A. Corichi and J. Cortez, “Canonical quantization from an algebraic perspective” (preprint) [8] A. Corichi, J. Cortez and H. Quevedo, “Schrödinger and Fock Representations for a Field Theory on Curved Spacetime”, Annals Phys. (NY) 313 446 (2004) [arXiv:hep-th/0202070]. [9] E. Manrique, R. Oeckl, A. Weber and J.A. Zapata, “Loop quantization as a continuum limit” Class. Quant. Grav. 23 3393 (2006) [arXiv:hep-th/0511222]; E. Manrique, R. Oeckl, A. Weber and J.A. Zapata, “Effective theo- ries and continuum limit for canonical loop quantization” (preprint) [10] D.W. Chiou, “Galileo symmetries in polymer particle representation”, Class. Quant. Grav. 24, 2603 (2007) [arXiv:gr-qc/0612155]. [11] W. Rudin, Fourier analysis on groups, (Interscience, New York, 1962) [12] A. Ashtekar, A. Corichi, P. Singh, “Contrasting LQC and WDW using an exactly soluble model” (preprint); A. Corichi, T. Vukasinac, and J.A. 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704.001
Numerical solution of shock and ramp compression for general material properties Damian C. Swift∗ Materials Science and Technology Division, Lawrence Livermore National Laboratory, 7000, East Avenue, Livermore, CA 94550, U.S.A. (Dated: March 7, 2007; revised April 8, 2008 and July 1, 2008 – LA-UR-07-2051) Abstract A general formulation was developed to represent material models for applications in dynamic loading. Numerical methods were devised to calculate response to shock and ramp compression, and ramp decompression, generalizing previous solutions for scalar equations of state. The numerical methods were found to be flexible and robust, and matched analytic results to a high accuracy. The basic ramp and shock solution methods were coupled to solve for composite deformation paths, such as shock-induced impacts, and shock interactions with a planar interface between different materials. These calculations capture much of the physics of typical material dynamics experiments, without requiring spatially-resolving simulations. Example calculations were made of loading histories in metals, illustrating the effects of plastic work on the temperatures induced in quasi-isentropic and shock-release experiments, and the effect of a phase transition. PACS numbers: 62.50.+p, 47.40.-x, 62.20.-x, 46.35.+z Keywords: material dynamics, shock, isentrope, adiabat, numerical solution, constitutive behavior ∗Electronic address: damian.swift@physics.org http://arxiv.org/abs/0704.0008v3 mailto:damian.swift@physics.org I. INTRODUCTION The continuum representation of matter is widely used for material dynamics in sci- ence and engineering. Spatially-resolved continuum dynamics simulations are the most widespread and familiar, solving the initial value problem by discretizing the spatial domain and integrating the dynamical equations forward in time to predict the motion and defor- mation of components of the system. This type of simulation is used, for instance, to study hypervelocity impact problems such as the vulnerability of armor to projectiles [1, 2], the performance of satellite debris shields [3], and the impact of meteorites with planets, notably the formation of the moon [4]. The problem can be divided into the dynamical equations of the continuum, the state field of the components s(~r), and the inherent properties of the materials. Given the local material state s, the material properties allow the stress τ to be determined. Given the stress field τ(~r) and mass density field ρ(~r), the dynamical equations describe the fields of acceleration, compression, and thermodynamic work done on the materials. The equations of continuum dynamics describe the behavior of a dynamically deforming system of arbitrary complexity. Particular, simpler deformation paths can be described more compactly by different sets of equations, and solved by different techniques than those used for continuum dynamics in general. Simpler deformation paths occur often in experiments designed to develop and calibrate models of material properties. These paths can be regarded as different ways of interrogating the material properties. The principal examples in material dynamics are shock and ramp compression [5, 6]. Typical experiments are designed to induce such loading histories and measure or infer the properties of the material in these states before they are destroyed by release from the edges or by reflected waves. The development of the field of material dynamics was driven by applications in the physics of hypervelocity impact and high explosive systems, including nuclear weapons [7]. In the regimes of interest, typically components with dimensions ranging from millime- ters to meters and pressures from 1GPa to 1TPa, material behavior is dominated by the scalar equation of state (EOS): the relationship between pressure, compression (or mass density), and internal energy. Other components of stress (specifically shear stresses) are much smaller, and chemical explosives react promptly so can be treated by simple mod- els of complete detonation. EOS were developed as fits to experimental data, particularly to series of shock states and to isothermal compression measurements [8]. It is relatively straightforward to construct shock and ramp compression states from an EOS algebraically or numerically depending on the EOS, and to fit an EOS to these measurements. More recently, applications and scientific interest have grown to include a wider range of pressures and time scales, such as laser-driven inertial confinement fusion [9], and experiments are designed to measure other aspects than the EOS, such as the kinetics of phase changes, con- stitutive behavior describing shear stresses, incomplete chemical reactions, and the effects of microstructure, including grain orientation and porosity. Theoretical techniques have also evolved to predict the EOS with ∼1% accuracy [10] and elastic contributions to shear stress with slightly poorer accuracy [11]. A general convention for representing material states is described, and numerical methods are reported for calculating shock and ramp compression states from general representations of material properties. II. CONCEPTUAL STRUCTURE FOR MATERIAL PROPERTIES The desired structure for the description of the material state and properties under dy- namic loading was developed to be as general as possible with respect to the types of material or models to be represented in the same framework, and designed to give the greatest amount of commonality between spatially-resolved simulations and calculations of shock and ramp compressions. In condensed matter on sub-microsecond time scales, heat conduction is often too slow to have a significant effect on the response of the material, and is ignored here. The equations of non-relativistic continuum dynamics are, in Lagrangian form, i.e. along characteristics moving with the local material velocity ~u(~r), Dρ(~r, t) = −ρ(~r, t)div~u(~r, t) (1) D~u(~r, t) ρ(~r, t) div τ(~r, t) (2) De(~r, t) = ||τ(~r, t)grad~u(~r, t)|| (3) where ρ is the mass density and e the specific internal energy. Changes in e can be related to changes in the temperature T through the heat capacity. The inherent properties of each material in the problem are described by its constitutive relation or equation of state τ(s). As well as experiencing compression and work from mechanical deformation, the local material state s(~r, t) can evolve through internal processes such as plastic flow. In general, Ds(~r, t) ≡ ṡ[s(~r, t), U(~r, t)] : U ≡ grad ~u(~r, t) (4) which can also include the equations for ∂ρ/∂t and ∂e/∂t. Thus the material properties must describe at a minimum τ(s) and ṡ[s(~r, t), U(~r, t)] for each material. If they also describe T (s), the conductivity, and ṡ(ė), then heat conduction can be treated. Other functions may be needed for particular numerical methods in continuum dynamics, such as the need for wave speeds (e.g. the longitudinal sound speed), which are needed for time step control in explicit time integration. Internally, within the material properties models, it is desirable to re-use software as much as possible, and other functions of the state are therefore desirable to allow models to be constructed in a modular and hierarchical way. Arithmetic manipulations must be performed on the state during numerical integration, and these can be encoded neatly using operator overloading, so the operator of the appropriate type is invoked automatically without having to include ‘if-then-else’ structures for each operator as is the case in non- object-oriented programming languages such as Fortran-77. For instance, if ṡ is calculated in a forward-time numerical method then changes of state are calculated using numerical evolution equations such as s(t+ δt) = s(t) + δtṡ. (5) Thus for a general state s and its time derivative ṡ, which has an equivalent set of compo- nents, it is necessary to multiply a state by a real number and to add two states together. For a specific software implementation, other operations may be needed, for example to create, copy, or destroy a new instance of a state. The attraction of this approach is that, by choosing a reasonably general form for the constitutive relation and associated operations, it is possible to separate the continuum dynamics part of the problem from the inherent behavior of the material. The relations describing the properties of different types of material can be encapsulated in a library form where the continuum dynamics program need know nothing about the relations for any spe- cific type of material, and vice versa. The continuum dynamics programs and the material properties relations can be developed and maintained independently of each other, provided that the interface remains the same (Table I). This is an efficient way to make complicated material models available for simulations of different types, including Lagrangian and Eule- rian hydrocodes operating on different numbers of dimensions, and calculations of specific loading or heating histories such as shock and ramp loading discussed below. Software in- terfaces have been developed in the past for scalar EOS with a single structure for the state [12], but object-oriented techniques make it practical to extend the concept to much more complicated states, to combinations of models, and to alternative types of model selected when the program is run, without having to find a single super-set state encompassing all possible states as special cases. A very wide range of types of material behavior can be represented with this formalism. At the highest level, different types of behavior are characterized by different structures for the state s (Table II). For each type of state, different specific models can be defined, such as perfect gas, polytropic and Grüneisen EOS. For each specific model, different materials are represented by choosing different values for the parameters in the model, and different local material states are represented through different values for the components of s. In the jargon of object-oriented programming, the ability to define an object whose precise type is undetermined until the program is run is known as polymorphism. For our application, polymorphism is used at several levels in the hierarchy of objects, from the overall type of a material (such as ‘one represented by a pressure-density-energy EOS’ or ‘one represented by a deviatoric stress model’) through the type of relation used to describe the properties of that material type (such as perfect gas, polytropic, or Grüneisen for a pressure-density-energy EOS, or Steinberg-Guinan [13] or Preston-Tonks-Wallace [14] for a deviatoric stress model), to the type of general mathematical function used to represent some of these relations (such as a polynomial or a tabular representation of γ(ρ) in a polytropic EOS) (Table III). States or models may be defined by extending or combining other states or models – this can be implemented using the object-oriented programming concept of inheritance. Thus deviatoric stress models can be defined as an extension to any pressure-density-energy EOS (they are usually written assuming a specific type, such as Steinberg’s cubic Grüneisen form), homo- geneous mixtures can be defined as combinations of any pressure-density-temperature EOS, and heterogeneous mixtures can be defined as combinations of materials each represented by any type of material model. Trial implementations have been made as libraries in the C++ and Java programming languages [15]. The external interface to the material properties was general at the level of representing a generic material type and state. The type of state and model were then selected when programs using the material properties library were run. In C++, objects which were polymorphic at run time had to be represented as pointers, requiring additional software constructions to allocate and free up physical memory associated with each object. It was possible to include general re-usable functions as polymorphic objects when defining models: real functions of one real parameter could be polynomials, transcendentals, tabular with different interpolation schemes, piecewise definitions over different regions of the one dimensional line, sums, products, etc; again defined specifically at run time. Object-oriented polymorphism and inheritance were thus very powerful techniques for increasing software re-use, making the software more compact and more reliable through the greater use of functions which had already been tested. Given conceptual and software structures designed to represent general material proper- ties suitable for use in spatially-resolved continuum dynamics simulations, we now consider the use of these generic material models for calculating idealized loading paths. III. IDEALIZED ONE-DIMENSIONAL LOADING Experiments to investigate the response of materials to dynamic loading, and to calibrate parameters in models of their behavior, are usually designed to apply as simple a loading history as is consistent with the transient state of interest. The simplest canonical types of loading history are shock and ramp [5, 6]. Methods of solution are presented for calculating the result of shock and ramp loading for materials described by generalized material models discussed in the previous section. Such direct solution removes the need to use a time- and space-resolved continuum dynamics simulation, allowing states to be calculated with far greater efficiency and without the need to consider and make allowance for attributes of resolved simulations such as the finite numerical resolution and the effect of numerical and artificial viscosities. A. Ramp compression Ramp compression is taken here to mean compression or decompression. If the material is represented by an inviscid scalar EOS, i.e. ignoring dissipative processes and non-scalar effects from elastic strain, ramp compression follows an isentrope. This is no longer true when dissipative processes such as plastic heating occur. The term ‘quasi-isentropic’ is sometimes used in this context, particularly for shockless compression; here we prefer to refer to the thermodynamic trajectories as adiabats since this is a more appropriate term: no heat is exchanged with the surroundings on the time scales of interest. For adiabatic compression, the state evolves according to the second law of thermody- namics, de = T dS − p dv (6) where T is the temperature and S the specific entropy. Thus ė = T Ṡ − p v̇ = T Ṡ − pdiv~u , (7) or for a more general material whose stress tensor is more complicated than a scalar pressure, de = T dS + τn dv ⇒ ė = T Ṡ + τndiv~u where τn is the component of stress normal to the direction of deformation. The velocity gradient was expressed through a compression factor η ≡ ρ′/ρ and a strain rate ǫ̇. In all ramp experiments used in the development and calibration of accurate material models, the strain has been applied uniaxially. More general strain paths, for instance isotropic or including a shear component, can be treated by the same formalism, and that the working rate is then a full inner product of the stress and strain tensors. The acceleration or deceleration of the material normal to the wave as it is compressed or expanded adiabatically is , (9) from which it can be deduced that where cl is the longitudinal wave speed. As with continuum dynamics, internal evolution of the material state can be calculated simultaneously with the continuum equations, or operator split and calculated periodically at constant compression [16]. The results are the same to second order in the compression increment. Operator-splitting allows calculations to be performed without an explicit en- tropy, if the continuum equations are integrated isentropically and dissipative processes are captured by internal evolution at constant compression. Operator-splitting is desirable when internal evolution can produce highly nonlinear changes, such as reaction from solid to gas: rapid changes in state and properties can make numerical schemes unstable. Operator-splitting is also desirable when the integration time step for internal evolution is much shorter than the continuum dynamics time step. Neither of these considerations is very important for ramp compression without spatial res- olution, but operator-splitting was used as an option in the ramp compression calculations for consistency with continuum dynamics simulations. The ramp compression equations were integrated using forward-time Runge-Kutta nu- merical schemes of second order. The fourth order scheme is a trivial extension. The sequence of operations to calculate an increment of ramp compression is as follows: 1. Time increment: δt = − | ln η| 2. Predictor: s(t + δt/2) = s(t) + ṡm(s(t), ǫ̇) (12) 3. Corrector: s(t+ δt) = s(t) + δtṡm(s(t+ δt/2), ǫ̇) (13) 4. Internal evolution: s(t+ δt) → s(t+ δt) + ∫ t+δt ṡi(s(t ′), ǫ̇) dt′ (14) where ṡm is the model-dependent state evolution from applied strain, and ṡi is internal evolution at constant compression. The independent variable for integration is specific volume v or mass density ρ; for numerical integration finite steps are taken in ρ and v. The step size ∆ρ can be controlled so that the numerical error during integration remains within chosen limits. A tabular adiabat can be calculated by integrating over a range of v or ρ, but when simulating experimental scenarios the upper limit for integration is usually that one of the other thermodynamic quantities reaches a certain value, for example that the normal component of stress reaches zero, which is the case on release from a high pressure state at a free surface. Specific end conditions were found by monitoring the quantity of interest until bracketed by a finite integration step, then bisecting until the stop condition was satisfied to a chosen accuracy. During bisection, each trial calculation was performed as an integration from the first side of the bracket by the trial compression. B. Shock compression Shock compression is the solution of a Riemann problem for the dynamics of a jump in compression moving with constant speed and with a constant thickness. The Rankine- Hugoniot (RH) equations [5] describing the shock compression of matter are derived in the continuum approximation, where the shock is a formal discontinuity in the continuum fields. In reality, matter is composed of atoms, and shocks have a finite width governed by the kinetics of dissipative processes – at a fundamental level, matter does not distinguish between shock compression and ramp compression with a high strain rate – but the RH equations apply as long as the width of the region of matter where unresolved processes occur is constant. Compared with the isentropic states induced by ramp compression in a material represented by an EOS, a shock always increases the entropy and hence the temperature. With dissipative processes included, the distinction between a ramp and a shock may become blurred. The RH equations express the conservation of mass, momentum, and energy across a moving discontinuity in state. They are usually expressed in terms of the pressure, but are readily generalized for materials supporting shear stresses by using the component of stress normal to the shock (i.e., parallel with the direction of propagation of the shock), τn: u2s = −v τn − τn0 v0 − v , (15) ∆up = −(τn − τn0)(v0 − v), (16) e = e0 − (τn + τn0)(v0 − v), (17) where us is the speed of the shock wave with respect to the material, ∆up is the change in material speed normal to the shock wave (i.e., parallel to its direction of propagation), and subscript 0 refers to the initial state. The RH relations can be applied to general material models if a time scale or strain rate is imposed, and an orientation chosen for the material with respect to the shock. Shock compression in continuum dynamics is almost always uniaxial. The RH equations involve only the initial and final states in the material. If a material has properties that depend on the deformation path – such as plastic flow or viscosity – then physically the detailed shock structure may make a difference [17]. This is a limitation of discontinuous shocks in continuum dynamics: it may be addressed as discussed above by including dissipative processes and considering ramp compression, if the dissipative pro- cesses can be represented adequately in the continuum approximation. Spatially-resolved simulations with numerical differentiation to obtain spatial derivatives and forward time differencing are usually not capable of representing shock discontinuities directly, and an artificial viscosity is used to smear shock compression over a few spatial cells [18]. The trajectory followed by the material in thermodynamic space is a smooth adiabat with dissi- pative heating supplied by the artificial viscosity. If plastic work is also included during this adiabatic compression, the overall heating for a given compression is greater than from the RH equations. To be consistent, plastic flow should be neglected while the artificial viscosity is non-zero. This localized disabling of physical processes, particularly time-dependent ones, during the passage of the unphysically smeared shock was previously found necessary for numerically stable simulations of detonation waves by reactive flow [19]. Detonation waves are reactive shock waves. Steady planar detonation (the Chapman- Jouguet state [20]) may be calculated using the RH relations, by imposing the condition that the material state behind the shock is fully reacted. Several numerical methods have been used to solve the RH equations for materials repre- sented by an EOS only [21, 22]. The general RH equations may be solved numerically for a given shock compression ∆ρ by varying the specific internal energy e until the normal stress from the material model equals that from the RH energy equation, Eq. 17. The shock and particle speeds are then calculated from Eqs 15 and 16. This numerical method is particu- larly convenient for EOS of the form p(ρ, e), as e may be varied directly. Solutions may still be found for general material models using ṡ(ė), by which the energy may be varied until the solution is found. Numerically, the solution was found by bracketing and bisection: 1. For given compression ∆ρ, take the low-energy end for bracketing as a nearby state s− (e.g. the previous state, of lower compression, on the Hugoniot), compressed adia- batically (to state s̃), and cooled so the specific internal energy is e(s−). 2. Bracket the desired state: apply successively larger heating increments ∆e to s̃, evolv- ing each trial state internally, until τn(s) from the material model exceeds τn(e − e0) from Eq. 17. 3. Bisect in ∆e, evolving each trial state internally, until τn(s) equals τn(e − e0) to the desired accuracy. As with ramp compression, the independent variable for solution was mass density ρ, and finite steps ∆ρ were taken. Each shock state was calculated independently of the rest, so numerical errors did not accumulate along the shock Hugoniot. The accuracy of the solution was independent of ∆ρ. A tabular Hugoniot can be calculated by solving over a range of ρ, but again when simulating experimental scenarios it is usually more useful to calculate the shock state where one of the other thermodynamic quantities reaches a certain value, often that up and τn match the values from another, simultaneous shock calculation for another material – the situation in impact and shock transmission problems, discussed below. Specific stop conditions were found by monitoring the quantity of interest until bracketed by a finite solution step, then bisecting until the stop condition was satisfied to a chosen accuracy. During bisection, each trial calculation was performed as a shock from the initial conditions to the trial shock compression. C. Accuracy: application to air The accuracy of these numerical schemes was tested by comparing with shock and ramp compression of a material represented by a perfect gas EOS, p = (γ − 1)ρe. (18) The numerical solution requires a value to be chosen for every parameter in the material model, here γ. Air was chosen as an example material, with γ = 1.4. Air at standard tem- perature and pressure has approximately ρ = 10−3 g/cm3 and e = 0.25MJ/kg. Isentropes for the perfect gas EOS have the form pρ−γ = constant, (19) and shock Hugoniots have the form p = (γ − 1) 2e0ρ0ρ+ p0(ρ− ρ0) (γ + 1)ρ0 − (γ − 1)ρ . (20) The numerical solutions reproduced the principal isentrope and Hugoniot to 10−3% and 0.1% respectively, for a compression increment of 1% along the isentrope and a solution tolerance of 10−6GPa for each shock state (Fig. 1). Over most of the range, the error in the Hugoniot was 0.02% or less, only approaching 0.1% near the maximum shock compression. IV. COMPLEX BEHAVIOR OF CONDENSED MATTER The ability to calculate shock and ramp loci in state space, i.e. as a function of vary- ing loading conditions, is particularly convenient for investigating complex aspects of the response of condensed matter to dynamic loading. Each locus can be obtained by a single series of shock or ramp solutions, rather than having to perform a series of time- and space- resolved continuum dynamics simulations, varying the initial or boundary conditions and reducing the solution. We consider the calculation of temperature in the scalar EOS, the effect of material strength and the effect of phase changes. A. Temperature The continuum dynamics equations can be closed using a mechanical EOS relating stress to mass density, strain, and internal energy. For a scalar EOS, the ideal form to close the continuum equations is p(ρ, e), with s = {ρ, e} the natural choice for the primitive state fields. However, the temperature is needed as a parameter in physical descriptions of many contributions to the constitutive response, including plastic flow, phase transitions, and chemical reactions. Here, we discuss the calculation of temperature in different forms of the scalar EOS. 1. Density-temperature equations of state If the scalar EOS is constructed from its underlying physical contributions for continuum dynamics, it may take the form e(ρ, T ), from which p(ρ, T ) can be calculated using the second law of thermodynamics [10]. An example is the ‘SESAME’ form of EOS, based on interpolated tabular relations for {p, e}(ρ, T ) [23]. A pair of relations {p, e}(ρ, T ) can be used as a mechanical EOS by eliminating T , which is equivalent to inverting e(ρ, T ) to find T (ρ, e), then substituting in p(ρ, T ). For a general e(ρ, T ) relation, for example for the SESAME EOS, the inverse can be calculated numerically as required, along an isochore. In this way, a {p, e}(ρ, T ) can be used as a p(ρ, e) EOS. Alternatively, the same p(ρ, T ) relation can be used directly with a primitive state field including temperature instead of energy: s = {ρ, T}. The evolution of the state under mechanical work then involves the calculation of Ṫ (ė), i.e. the reciprocal of the specific heat capacity, which is a derivative of e(ρ, T ). As this calculation does not require e(ρ, T ) to be inverted, it is computationally more efficient to use {p, e}(ρ, T ) EOS with a temperature- based, rather than energy-based, state. The main disadvantage is that it is more difficult to ensure exact energy conservation as the continuum dynamics equations are integrated in time, but any departure from exact conservation is at the level of accuracy of the algorithm used to integrate the heat capacity. Both structures of EOS have been implemented for material property calculations. Taking a SESAME type EOS, thermodynamic loci were calculated with {ρ, e} or {ρ, T} primitive states, for comparison (Fig. 2). For a monotonic EOS, the results were indistinguishable within differences from forward or reverse interpolation of the tabular relations. When the EOS, or the effective surface using a given order of interpolating function, was non- monotonic, the results varied greatly because of non-uniqueness when eliminating T for the {ρ, e} primitive state. 2. Temperature model for mechanical equations of state Mechanical EOS are often available as empirical, algebraic relations p(ρ, e), derived from shock data. Temperature can be calculated without altering the mechanical EOS by adding a relation T (ρ, e). While this relation could take any form in principle, one can also follow the logic of the Grüneisen EOS, in which the pressure is defined in terms of its deviation ∆p(ρ, e − er) from a reference curve {pr, er}(ρ). Thus temperatures can be calculated by reference to a compression curve along which the temperature and specific internal energy are known, {Tr, er}(ρ), and a specific heat capacity defined as a function of density cv(ρ). In the calculations, this augmented EOS was represented as a ‘mechanical-thermal’ form comprising any p(ρ, e) EOS plus the reference curves – an example of software inheritance and polymorphism. One natural reference curve for temperature is the cold curve, Tr = 0K. The cold curve can be estimated from the principal isentrope e(ρ)|s0 using the estimated density variation of the Grüneisen parameter: er(ρ) = e(ρ)|s0 − T0cpe a(1−ρ0/ρ) )γ0−a [24]. In this work, the principal isentrope was calculated in tabular form from the mechanical EOS, using the ramp compression algorithm described above. Empirical EOS are calibrated using experimental data. Shock and adiabatic compression measurements on strong materials inevitably include elastic-plastic contributions as well as the scalar EOS itself. If the elastic-plastic contributions are not taken into account self- consistently, the EOS may implicitly include contributions from the strength. A unique scalar EOS can be constructed to reproduce the normal stress as a function of compression for any unique loading path: shock or adiabat, for a constant or smoothly-varying strain rate. Such an EOS would not generally predict the response to other loading histories. The EOS and constitutive properties for the materials considered here were constructed self- consistently from shock data – this does not mean the models are accurate for other loading paths, as neither the EOS nor the strength model includes all the physical terms that real materials exhibit. This does not in any case matter for the purposes of demonstrating the properties of the numerical schemes. This mechanical-thermal procedure was applied to Al using a Grüneisen EOS fitted to the same shock data used to calculate the {p, e}(ρ, T ) EOS discussed above [24]. Temperatures were in good agreement (Fig. 2). The mechanical-thermal calculations required a similar computational effort to the tabular {p, e}(ρ, T ) EOS with a {ρ, T} primitive states (and were thus much more efficient than the tabular EOS with {ρ, e} states), and described the EOS far more compactly. B. Strength For dynamic compressions to o(10GPa) and above, on microsecond time scales, the flow stress of solids is often treated as a correction or small perturbation to the scalar EOS. However, the flow stress has been observed to be much higher on nanosecond time scales [25], and interactions between elastic and plastic waves may have a significant effect on the compression and wave propagation. The Rankine-Hugoniot equations should be solved self-consistently with strength included. 1. Preferred representation of isotropic strength There is an inconsistency in the standard continuum dynamics treatment of scalar (pres- sure) and tensor (stress) response. The scalar EOS expresses the pressure p(ρ, e) as the dependent quantity, which is the most convenient form for use in the continuum equations. Standard practice is to use sub-Hookean elasticity (hypoelastic form) [16] (Table II), in which the state parameters include the stress deviator σ, evolved by integration σ̇ = G(s)ǫ̇ (22) where G is the shear modulus and ǫ̇ the strain rate deviator. Thus the isotropic and devia- toric contributions to stress are not treated in an equivalent way: the pressure is calculated from a local state involving a strain-like parameter (mass density), whereas the stress de- viator evolves with the time-derivative of strain. This inconsistency causes problems along complicated loading paths because G varies strongly with compression: if a material is sub- jected to a shear strain ǫ, then isotropic compression (increasing the shear modulus from G to G′, leaving ǫ unchanged), then shear unloading to isotropic stress, the true unloading strain is −ǫ, whereas the hypoelastic calculation would require a strain of −ǫG/G′. Using Be and the Steinberg-Guinan strength model as an example of the difference between hy- poelastic and hyperelastic calculations, consider an initial strain to a flow stress of 0.3GPa followed by isothermal, isotropic compression to 100GPa,. the strain to unload to a state of isotropic stress is 0.20% (hyperelastic) and 0.09% (hypoelastic). The discrepancy arises because the hypoelastic model does not increase the deviatoric stress under compression at constant deviatoric strain. The stress can be considered as a direct response of the material to the instantaneous state of elastic strain: σ(ǫ, T ). This relation can be predicted directly with electronic structure calculations of the stress tensor in a solid for a given compression and elastic strain state [11], and is a direct generalization of the scalar equation of state. A more consistent representation of the state parameters is to use the strain deviator ǫ rather than σ, and to calculate σ from scratch when required using σ = G(s)ǫ (23) – a hyperelastic formulation. The state parameters are then {ρ, e, ǫ, ǫ̃p}. The different formulations give different answers when deviatoric strain is accumulated at different compressions, in which case the hyperelastic formulation is correct. If the shear modulus varies with strain deviator – i.e., for nonlinear elasticity – then the definition of G(ǫ) must be adjusted to give the same stress for a given strain. Many isotropic strength models use scalar measures of the strain and stress to parame- terize work hardening and to apply a yield model of flow stress: fǫ||ǫ2||, σ̃ = fσ||σ2||. (24) Inconsistent conventions for equivalent scalar measures have been used by different workers. In the present work, the common shock physics convention was used that the flow stress component of τn is Y where Y is the flow stress. For consistency with published speeds and amplitudes for elastic waves, fǫ = fσ = , in contrast to other values previously used for lower-rate deformation [26]. In principle, the values of fǫ and fσ do not matter as long as the strength parameters were calibrated using the same values then used in any simulations. 2. Beryllium The flow stress measured from laser-driven shock experiments on Be crystals a few tens of micrometers thick is, at around 5-9GPa [25], much greater than the 0.3-1.3GPa mea- sured on microsecond time scales. A time-dependent crystal plasticity model for Be is being developed, and the behavior under dynamic loading depends on the detailed time depen- dence of plasticity. Calculations were performed with the Steinberg-Guinan strength model developed for microsecond scale data [24], and, for the purposes of rough comparison, with elastic-perfectly plastic response with a flow stress of 10GPa. The elastic-perfectly plastic model neglected pressure- and work- hardening. Calculations were made of the principal adiabat and shock Hugoniot, and of a release adiabat from a state on the principal Hugoniot. Calculations were made with and without strength. Considering the state trajectories in stress-volume space, it is interesting to note that heating from plastic flow may push the adiabat above the Hugoniot, because of the greater heating obtained by integrating along the adiabat compared with jumping from the initial to the final state on the Hugoniot (Fig. 3). Even with an elastic-perfectly plastic strength model, the with-strength curves do not lie exactly 2 Y above the strengthless curves, because heating from plastic flow contributes an increasing amount of internal energy to the EOS as compression increases. An important characteristic for the seeding of instabilities by microstructural variations in shock response is the shock stress at which an elastic wave does not run ahead of the shock. In Be with the high flow stress of nanosecond response, the relation between shock and particle speeds is significantly different from the relation for low flow stress (Fig. 4). For low flow stress, the elastic wave travels at 13.2 km/s. A plastic shock travels faster than this for pressures greater than 110GPa, independent of the constitutive model. The speed of a plastic shock following the initial elastic wave is similar to the low strength case, because the material is already at its flow stress, but the speed of a single plastic shock is appreciably higher. For compression to a given normal stress, the temperature is significantly higher with plastic flow included. The additional heating is particularly striking on the principal adi- abat: the temperature departs significantly from the principal isentrope. Thus ramp-wave compression of strong materials may lead to significant levels of heating, contrary to com- mon assumptions of small temperature increases [27]. Plastic flow is largely irreversible, so heating occurs on unloading as well as loading. Thus, on adiabatic release from a shock- compressed state, additional heating occurs compared with the no-strength case. These levels of heating are important as shock or release melting may occur at a significantly lower shock pressure than would be expected ignoring the effect of strength. (Fig. 5.) C. Phase changes An important property of condensed matter is phase changes, including solid-solid poly- morphism and solid-liquid. An equilibrium phase diagram can be represented as a single overall EOS surface as before. Multiple, competing phases with kinetics for each phase trans- formation can be represented conveniently using the structure described above for general material properties, for example by describing the local state as a set of volume fractions fi of each possible simple-EOS phase, with transition rates and equilibration among them. This model is described in more detail elsewhere [19]. However, it is interesting to investi- gate the robustness of the numerical scheme for calculating shock Hugoniots when the EOS has the discontinuities in value and gradient associated with phase changes. The EOS of molten metal, and the solid-liquid phase transition, can be represented to a reasonable approximation as an adjustment to the EOS of the solid: ptwo-phase(ρ, e) = psolid(ρ, ẽ) (25) where e : T (ρ, e) < Tm(ρ) e−∆h̃m : ∆h̃m ≡ cv(ρ, e) [T (ρ, e)− Tm(ρ)] < ∆hm e−∆hm : otherwise and ∆hm is the specific latent heat of fusion. Taking the EOS and a modified Lindemann melting curve for Al [24], and using ∆hm = 0.397MJ/kg, the shock Hugoniot algorithm was found to operate stably across the phase transition (Fig. 6). V. COMPOSITE LOADING PATHS Given methods to calculate shock and adiabatic loading paths from arbitrary initial states, a considerable variety of experimental scenarios can be treated from the interaction of loading or unloading waves with interfaces between different materials, in planar geometry for uniaxial compression. The key physical constraint is that, if two dissimilar materials are to remain in contact after an interaction such as an impact or the passage of a shock, the normal stress τn and particle speed up in both materials must be equal on either side of the interface. The change in particle speed and stress normal to the waves were calculated above for compression waves running in the direction of increasing spatial ordinate (left to right). Across an interface, the sense is reversed for the material at the left. Thus a projectile impacting a stationary target to the right is decelerated from its initial speed by the shock induced by impact. The general problem at an interface can be analyzed by considering the states at the instant of first contact – on impact, or when a shock traveling through a sandwich of ma- terials first reaches the interface. The initial states are {ul, sl; ur, sr}. The final states are {uj, s l; uj, r r} where uj is the joint particle speed, τn(s l) = τn(s r), and s i is connected to si by either a shock or an adiabat, starting at the appropriate initial velocity and stress, and with orientation given by the side of the system each material occurs on. Each type of wave is considered in turn, looking for an intersection in the up − τn plane. Examples of these wave interactions are the impact of a projectile with a stationary target (Fig. 7), release of a shock state at a free surface or a material (e.g. a window) of lower shock impedance (hence reflecting a release wave into the shocked material – Fig. 8), reshocking at a surface with a material of higher shock impedance (Fig. 8), or tension induced as materials try to separate in opposite directions when joined by a bonded interface (Fig. 9). Each of these scenarios may occur in turn following the impact of a projectile with a target: if the target is layered then a shock is transmitted across each interface with a release or a reshock reflected back, depending on the materials; release ultimately occurs at the rear of the projectile and the far end of the target, and the oppositely-moving release waves subject the projectile and target to tensile stresses when they interact (Fig. 10). As an illustration of combining shock and ramp loading calculations, consider the problem of an Al projectile, initially traveling at 3.6 km/s, impacting a stationary, composite target comprising a Mo sample and a LiF release window [28, 29]. The shock and release states were calculated using published material properties [24]. The initial shock state was calculated to have a normal stress of 63.9GPa. On reaching the LiF, the shock was calculated to transmit at 27.1GPa, reflecting as a release in the Mo. These stresses match the continuum dynamics simulation to within 0.1GPa in the Mo and 0.3GPa in the LiF, using the same material properties (Fig. 11). The associated wave and particle speeds match to a similar accuracy; wave speeds are much more difficult to extract from the continuum dynamics simulation. An extension of this analysis can be used to calculate the interaction of oblique shocks with an interface [30]. VI. CONCLUSIONS A general formulation was developed to represent material models for applications in dynamic loading, suitable for software implementation in object-oriented programming lan- guages. Numerical methods were devised to calculate the response of matter represented by the general material models to shock and ramp compression, and ramp decompression, by direct evaluation of the thermodynamic pathways for these compressions rather than spatially-resolved simulations. This approach is a generalization of earlier work on solutions for materials represented by a scalar equation of state. The numerical methods were found to be flexible and robust: capable of application to materials with very different properties. The numerical solutions matched analytic results to a high accuracy. Care was needed with the interpretation of some types of physical response, such as plas- tic flow, when applied to deformation at high strain rates. The underlying time-dependence of processes occurring during deformation should be taken into account. The actual history of loading and heating experienced by material during the passage of a shock may influence the final state – this history is not captured in the continuum approximation to material dynamics, where shocks are treated as discontinuities. Thus care is also needed in spa- tially resolved simulations when shocks are modeled using artificial viscosity to smear them unphysically over a finite thickness. Calculations were shown to demonstrate the operation of the algorithms for shock and ramp compression with material models representative of complex solids including strength and phase transformations. The basic ramp and shock solution methods were coupled to solve for composite defor- mation paths, such as shock-induced impacts, and shock interactions with a planar interface between different materials. Such calculations capture much of the physics of typical ma- terial dynamics experiments, without requiring spatially-resolving simulations. The results of direct solution of the relevant shock and ramp loading conditions were compared with hydrocode simulations, showing complete consistency. Acknowledgments Ian Gray introduced the author to the concept of multi-model material properties soft- ware. Lee Markland developed a prototype Hugoniot-calculating computer program for equations of state while working for the author as an undergraduate summer student. Evolutionary work on material properties libraries was supported by the U.K. Atomic Weapons Establishment, Fluid Gravity Engineering Ltd, andWessex Scientific and Technical Services Ltd. Refinements to the technique and applications to the problems described were undertaken at Los Alamos National Laboratory (LANL) and Lawrence Livermore National Laboratory (LLNL). The work was performed partially in support of, and funded by, the National Nuclear Se- curity Agency’s Inertial Confinement Fusion program at LANL (managed by Steven Batha), and LLNL’s Laboratory-Directed Research and Development project 06-SI-004 (Principal Investigator: Hector Lorenzana). The work was performed under the auspices of the U.S. Department of Energy under contracts W-7405-ENG-36, DE-AC52-06NA25396, and DE- AC52-07NA27344. References [1] J.K. Dienes, J.M. Walsh, in R. Kinslow (Ed), “High-Velocity Impact Phenomena” (Academic Press, New York, 1970). [2] D.J. Benson, Comp. Mech. 15, 6, pp 558-571 (1995). [3] J.W. Gehring, Jr, in R. Kinslow (Ed), “High-Velocity Impact Phenomena” (Academic Press, New York, 1970). [4] R.M. Canup, E. Asphaug, Nature 412, pp 708-712 (2001). [5] For a recent review and introduction, see e.g. M.R. Boslough and J.R. Asay, in J.R. Asay, M. Shahinpoor (Eds), “High-Pressure Shock Compression of Solids” (Springer-Verlag, New York, 1992). [6] For example, C.A. Hall, J.R. Asay, M.D. Knudson, W.A. Stygar, R.B. Spielman, T.D. Pointon, D.B. Reisman, A. Toor, and R.C. Cauble, Rev. Sci. Instrum. 72, 3587 (2001). [7] M.A. Meyers, “Dynamic Behavior of Materials” (Wiley, New York, 1994). [8] G. McQueen, S.P. March, J.W. Taylor, J.N. Fritz, W.J. Carter, in R. Kinslow (Ed), “High- Velocity Impact Phenomena” (Academic Press, New York, 1970). [9] J.D. Lindl, “Inertial Confinement Fusion” (Springer-Verlag, New York, 1998). [10] D.C. Swift, G.J. Ackland, A. Hauer, G.A. Kyrala, Phys. Rev. B 64, 214107 (2001). [11] J.P. Poirier, G.D. Price, Phys. of the Earth and Planetary Interiors 110, pp 147-56 (1999). [12] I.N. Gray, P.C. Thompson, B.J. Parker, D.C. Swift, J.R. Maw, A. Giles and others (AWE Aldermaston), unpublished. [13] D.J. Steinberg, S.G. Cochran, M.W. Guinan, J. Appl. Phys. 51, 1498 (1980). [14] D.L. Preston, D.L. Tonks, and D.C. Wallace, J. Appl. Phys. 93, 211 (2003). [15] A version of the software, including representative parts of the material model library and the algorithms for calculating the ramp adiabat and shock Hugoniot, is available as a supplemen- tary file provided with the preprint of this manuscript, arXiv:0704.0008. Software support, and versions with additional models, are available commercially from Wessex Scientific and Technical Services Ltd (http://wxres.com). [16] D. Benson, Computer Methods in Appl. Mechanics and Eng. 99, 235 (1992). http://arxiv.org/abs/0704.0008 http://wxres.com [17] J.L. Ding, J. Mech. and Phys. of Solids 54, pp 237-265 (2006). [18] J. von Neumann, R.D. Richtmyer, J. Appl. Phys. 21, 3, pp 232-237 (1950). [19] R.M. Mulford, D.C. Swift, in preparation. [20] W. Fickett, W.C. Davis, “Detonation” (University of California Press, Berkeley, 1979). [21] R. Menikoff, B.J. Plohr, Rev. Mod. Phys. 61, pp 75-130 (1989). [22] A. Majda, Mem. Amer. Math. Soc., 41, 275 (1983). [23] K.S. Holian (Ed.), T-4 Handbook of Material Property Data Bases, Vol 1c: Equations of State, Los Alamos National Laboratory report LA-10160-MS (1984). [24] D.J. Steinberg, Equation of State and Strength Properties of Selected Materials, Lawrence Livermore National Laboratory report UCRL-MA-106439 change 1 (1996). [25] D.C. Swift, T.E. Tierney, S.-N. Luo, D.L. Paisley, G.A. Kyrala, A. Hauer, S.R. Greenfield, A.C. Koskelo, K.J. McClellan, H.E. Lorenzana, D. Kalantar, B.A. Remington, P. Peralta, E. Loomis, Phys.Plasmas 12, 056308 (2005). [26] R. Hill, “The Mathematical Theory of Plasticity” (Clarendon Press, Oxford, 1950). [27] C.A. Hall, Phys. Plasmas 7, 5, pp 2069-2075 (2000). [28] D.C. Swift, A. Seifter, D.B. Holtkamp, and D.A. Clark, Phys. Rev. B 76, 054122 (2007). [29] A. Seifter and D.C. Swift, Phys. Rev. B 77, 134104 (2008). [30] E. Loomis, D.C. Swift, J. Appl. Phys. 103, 023518 (2008). TABLE I: Interface to material models required for explicit forward-time continuum dynamics simulations. purpose interface calls program set-up read/write material data continuum dynamics equations stress(state) time step control sound speed(state) evolution of state (deformation) d(state)/dt(state,grad ~u) evolution of state (heating) d(state)/dt(state,ė) internal evolution of state d(state)/dt manipulation of states create and delete add states multiply state by a scalar check for self-consistency Parentheses in the interface calls denote functions, e.g. “stress(state)” for “stress as a function of the instantaneous, local state.” The evolution functions are shown in the operator-split structure that is most robust for explicit, forward-time numerical solutions and can also be used for calculations of the shock Hugoniot and ramp compression. Checks for self-consistency include that mass density is positive, volume or mass fractions of components of a mixture add up to one, TABLE II: Examples of types of material model, distinguished by different structures in the state vector. model state vector effect of mechanical strain s ṡm(s, gradu) mechanical equation of state ρ, e −ρdiv~u,−pdiv~u/ρ thermal equation of state ρ, T −ρdiv~u,−pdiv~u/ρcv heterogeneous mixture {ρ, e, fv}i {−ρdiv~u,−pdiv~u/ρ, 0}i homogeneous mixture ρ, T, {fm}i {−ρdiv~u,−pdiv~u/ρcv , 0i traditional deviatoric strength ρ, e, σ, ǫ̃p −ρdiv~u, −pdiv~u+fp||σǫ̇p|| , Gǫ̇e, fǫ||ǫ̇ The symbols are ρ: mass density; e: specific internal energy, T : temperature, fv: volume fraction, fm: mass fraction, σ: stress deviator, fp: fraction of plastic work converted to heat, gradup: plastic part of velocity gradient, G: shear modulus, ǫ̇e,p: elastic and plastic parts of strain rate deviator, ǫ̃p: scalar equivalent plastic strain, fǫ: factor in effective strain magnitude. Reacting solid explosives can be represented as heterogeneous mixtures, one component being the reacted products; reaction, a process of internal evolution, transfers material from unreacted to reacted components. Gas-phase reaction can be represented as a homogeneous mixture, reactions transferring mass between components representing different types of molecule. Symmetric tensors such as the stress deviator are represented more compactly by their 6 unique upper triangular components, e.g. using Voigt notation. TABLE III: Outline hierarchy of material models, illustrating the use of polymorphism (in the object-oriented programming sense). material (or state) type model type mechanical equation of state polytropic, Grüneisen, energy-based Jones-Wilkins-Lee, (ρ, T ) table, etc thermal equation of state temperature-based Jones-Wilkins- Lee, quasiharmonic, (ρ, T ) table, reactive equation of state modified polytropic, reactive Jones- Wilkins-Lee spall Cochran-Banner deviatoric stress elastic-plastic, Steinberg-Guinan, Steinberg-Lund, Preston-Tonks- Wallace, etc homogeneous mixture mixing and reaction models heterogeneous mixture equilibration and reaction models Continuum dynamics programs can refer to material properties as an abstract ‘material type’ with an abstract material state. The actual type of a material (e.g. mechanical equation of state), the specific model type (e.g. polytropic), and the state of material of that type are all handled transparently by the object-oriented software structure. The reactive equation of state has an additional state parameter λ, and the software operations are defined by extending those of the mechanical equation of state. Spalling materials can be represented by a solid state plus a void fraction fv, with operations defined by extending those of the solid material. Homogeneous mixtures are defined as a set of thermal equations of state, and the state is the set of states and mass fractions for each. Heterogeneous mixtures are defined as a set of ‘pure’ material properties of any type, and the state is the set of states for each component plus its volume fraction. 0.0001 0.001 0.01 0.001 0.01 mass density (g/cm3) isentrope Hugoniot 0.0001 0.001 0.01 0.001 0.01 mass density (g/cm3) isentrope Hugoniot FIG. 1: Principal isentrope and shock Hugoniot for air (perfect gas): numerical calculations for general material models, compared with analytic solutions. 0 1000 2000 3000 4000 5000 temperature (K) solid: Grueneisen dashed: SESAME 3716 FIG. 2: Shock Hugoniot for Al in pressure-temperature space, for different representations of the equation of state. 0.7 0.75 0.8 0.85 0.9 0.95 1 volume compression each pair of lines: upper is Hugoniot, lower is adiabat FIG. 3: Principal adiabat and shock Hugoniot for Be in normal stress-compression space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with Y = 10GPa (dotted). 0 20 40 60 80 100 120 140 normal stress (GPa) elastic wave plastic shock FIG. 4: Principal adiabat and shock Hugoniot for Be in shock speed-normal stress space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with Y = 10GPa (dotted). 0 1000 2000 3000 4000 5000 temperature (K) principal adiabat principal Hugoniot release adiabat FIG. 5: Principal adiabat, shock Hugoniot, and release adiabat for Be in normal stress-temperature space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with Y = 10GPa (dotted). 0 1000 2000 3000 4000 5000 temperature (K) melt locus solid Hugoniot FIG. 6: Demonstration of shock Hugoniot solution across a phase boundary: shock-melting of Al, for different initial porosities. initial state particle speed initial state of projectile principal Hugoniot of target principal Hugoniot of projectile shock state: intersection of target FIG. 7: Wave interactions for the impact of a flat projectile moving from left to right with a stationary target. Dashed arrows are a guide to the sequence of states. For a projectile moving from right to left, the construction is the mirror image reflected in the normal stress axis. states particle speed secondary Hugoniot of target initial shock state in target principal Hugoniot: high impedance window low impedance window target release isentrope target release at free surface window release FIG. 8: Wave interactions for the release of a shocked state (shock moving from left to right) into a stationary ‘window’ material to its right. The release state depends whether the window has a higher or lower shock impedance than the shocked material. Dashed arrows are a guide to the sequence of states. For a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis. projectile release in projectile and target final tensile state in projectile and target particle speed target release target release projectile release initial shock state FIG. 9: Wave interactions for the release of a shocked state by tension induced as materials try to separate in opposite directions when joined by a bonded interface. Material damage, spall, and separation are neglected: the construction shows the maximum tensile stress possible. For general material properties, e.g. if plastic flow is included, the state of maximum tensile stress is not just the negative of the initial shock state. Dashed arrows are a guide to the sequence of states. The graph shows the initial state after an impact by a projectile moving from right to left; for a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis. tension ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� target impact shocks transmitted shock; reflected wave free surface release release interactions: FIG. 10: Schematic of uniaxial wave interactions induced by the impact of a flat projectile with a composite target. 0 5 10 15 20 position (mm) LiFAl Mo reflected transmitted release shock original shock state FIG. 11: Hydrocode simulation of Al projectile at 3.6 km/s impacting a Mo target with a LiF release window, 1.1µs after impact. Structures on the waves are elastic precursors. List of figures 1. Principal isentrope and shock Hugoniot for air (perfect gas): numerical calculations for general material models, compared with analytic solutions. 2. Shock Hugoniot for Al in pressure-temperature space, for different representations of the equation of state. 3. Principal adiabat and shock Hugoniot for Be in normal stress-compression space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic- perfectly plastic with Y = 10GPa (dotted). 4. Principal adiabat and shock Hugoniot for Be in shock speed-normal stress space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic- perfectly plastic with Y = 10GPa (dotted). 5. Principal adiabat, shock Hugoniot, and release adiabat for Be in normal stress- temperature space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with Y = 10GPa (dotted). 6. Demonstration of shock Hugoniot solution across a phase boundary: shock-melting of Al, for different initial porosities. 7. Wave interactions for the impact of a flat projectile moving from left to right with a stationary target. Dashed arrows are a guide to the sequence of states. For a projectile moving from right to left, the construction is the mirror image reflected in the normal stress axis. 8. Wave interactions for the release of a shocked state (shock moving from left to right) into a stationary ‘window’ material to its right. The release state depends whether the window has a higher or lower shock impedance than the shocked material. Dashed arrows are a guide to the sequence of states. For a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis. 9. Wave interactions for the release of a shocked state by tension induced as materials try to separate in opposite directions when joined by a bonded interface. Material damage, spall, and separation are neglected: the construction shows the maximum tensile stress possible. For general material properties, e.g. if plastic flow is included, the state of maximum tensile stress is not just the negative of the initial shock state. Dashed arrows are a guide to the sequence of states. The graph shows the initial state after an impact by a projectile moving from right to left; for a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis. 10. Schematic of uniaxial wave interactions induced by the impact of a flat projectile with a composite target. 11. Hydrocode simulation of Al projectile at 3.6 km/s impacting a Mo target with a LiF release window, 1.1µs after impact. Structures on the waves are elastic precursors. Introduction Conceptual structure for material properties Idealized one-dimensional loading Ramp compression Shock compression Accuracy: application to air Complex behavior of condensed matter Temperature Density-temperature equations of state Temperature model for mechanical equations of state Strength Preferred representation of isotropic strength Beryllium Phase changes Composite loading paths Conclusions Acknowledgments References References List of figures
A general formulation was developed to represent material models for applications in dynamic loading. Numerical methods were devised to calculate response to shock and ramp compression, and ramp decompression, generalizing previous solutions for scalar equations of state. The numerical methods were found to be flexible and robust, and matched analytic results to a high accuracy. The basic ramp and shock solution methods were coupled to solve for composite deformation paths, such as shock-induced impacts, and shock interactions with a planar interface between different materials. These calculations capture much of the physics of typical material dynamics experiments, without requiring spatially-resolving simulations. Example calculations were made of loading histories in metals, illustrating the effects of plastic work on the temperatures induced in quasi-isentropic and shock-release experiments, and the effect of a phase transition.
Introduction Conceptual structure for material properties Idealized one-dimensional loading Ramp compression Shock compression Accuracy: application to air Complex behavior of condensed matter Temperature Density-temperature equations of state Temperature model for mechanical equations of state Strength Preferred representation of isotropic strength Beryllium Phase changes Composite loading paths Conclusions Acknowledgments References References List of figures
704.001
Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov Mathematics Department San Francisco State University San Francisco, CA 94132 sergei@sfsu.edu May 8, 2006 Abstract Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković’s and Winkler’s rela- tions play an important role in the theory of partial cubes. These struc- tures are employed in the paper to characterize bipartite graphs and par- tial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated. Key words: Hypercube, partial cube, semicube 1 Introduction A hypercube H(X) on a set X is a graph which vertices are the finite subsets of X ; two vertices are joined by an edge if they differ by a singleton. A partial cube is a graph that can be isometrically embedded into a hypercube. There are three general graph-theoretical structures that play a prominent role in the theory of partial cubes; namely, semicubes, Djoković’s relation θ, and Winkler’s relation Θ. We use these structures, in particular, to characterize bi- partite graphs and partial cubes. The characterization problem for partial cubes was considered as an important one and many characterizations are known. We list contributions in the chronological order: Djoković [9] (1973), Avis [2] (1981), Winkler [20] (1984), Roth and Winkler [18] (1986), Chepoi [6, 7] (1988 and 1994). In the paper, we present new proofs for the results of Djoković [9], Winkler [20], and Chepoi [6], and obtain two more characterizations of partial cubes. http://arxiv.org/abs/0704.0010v1 The paper is also concerned with some ways of constructing new partial cubes from old ones. Properties of subcubes, the Cartesian product of partial cubes, and expansion and contraction of a partial cube are investigated. We introduce a construction based on pasting two graphs together and show how new partial cubes can be obtained from old ones by pasting them together. The paper is organized as follows. Hypercubes and partial cubes are introduced in Section 2 together with two basic examples of infinite partial cubes. Vertex sets of partial cubes are described in terms of well graded families of finite sets. In Section 3 we introduce the concepts of a semicube, Djoković’s θ and Win- kler’s Θ relations, and establish some of their properties. Bipartite graphs and partial cubes are characterized by means of these structures. One more charac- terization of partial cubes is obtained in Section 4, where so-called fundamental sets in a graph are introduced. The rest of the paper is devoted to constructions: subcubes and the Carte- sian product (Section 6), pasting (Section 7), and expansions and contractions (Section 8). We show that these constructions produce new partial cubes from old ones. Isometric and lattice dimensions of new partial cubes are calculated. These dimensions are introduced in Section 5. Few words about conventions used in the paper are in order. The sum (disjoint union) A+B of two sets A and B is the union ({1} ×A) ∪ ({2} ×B). All graphs in the paper are simple undirected graphs. In the notation G = (V,E), the symbol V stands for the set of vertices of the graph G and E stands for its set of edges. By abuse of language, we often write ab for an edge in a graph; if this is the case, ab is an unordered pair of distinct vertices. We denote 〈U〉 the graph induced by the set of vertices U ⊆ V . If G is a connected graph, then dG(a, b) stands for the distance between two vertices a and b of the graph G. Wherever it is clear from the context which graph is under consideration, we drop the subscript G in dG(a, b). A subgraph H ⊆ G is an isometric subgraph if dH(a, b) = dG(a, b) for all vertices a and b of H ; it is convex if any shortest path in G between vertices of H belongs to H . 2 Hypercubes and partial cubes Let X be a set. We denote Pf (X) the set of all finite subsets of X . Definition 2.1. A graph H(X) has the set Pf (X) as the set of its vertices; a pair of vertices PQ is an edge of H(X) if the symmetric difference P∆Q is a singleton. The graph H(X) is called the hypercube on X [9]. If X is a finite set of cardinality n, then the graph H(X) is the n-cube Qn. The dimension of the hypercube H(X) is the cardinality of the set X . The shortest path distance d(P,Q) on the hypercube H(X) is the Hamming distance between sets P and Q: d(P,Q) = |P∆Q| for P,Q ∈ Pf . (2.1) The set Pf (X) is a metric space with the metric d. Definition 2.2. A graph G is a partial cube if it can be isometrically embedded into a hypercube H(X) for some set X . We often identify G with its isometric image in the hypercube H(X), and say that G is a partial cube on the set X . Figure 2.1: A graph and its isometric embedding into Q3. An example of a partial cube and its isometric embedding into the cube Q3 is shown in Figure 2.1. Clearly, a family F of finite subsets of X induces a partial cube on X if and only if for any two distinct subsets P,Q ∈ F there is a sequence R0 = P,R1, . . . , Rn = Q of sets in F such that d(Ri, Ri+1) = 1 for all 0 ≤ i < n, and d(P,Q) = n. (2.2) The families of sets satisfying condition (2.2) are known as well graded fam- ilies of sets [10]. Note that a sequence (Ri) satisfying (2.2) is a shortest path from P to Q in H(X) (and in the subgraph induced by F). Definition 2.3. A family F of arbitrary subsets ofX is a wg-family (well graded family of sets) if, for any two distinct subsets P,Q ∈ F, the set P∆Q is finite and there is a sequence R0 = P,R1, . . . , Rn = Q of sets in F such that |Ri∆Ri+1| = 1 for all 0 ≤ i < n and |P∆Q| = n. Example 2.1. The induced graph can be a partial cube on a different set if the family F is not well graded. Consider, for instance, the family F = {∅, {a}, {a, b}, {a, b, c}, {b, c}} of subsets of X = {a, b, c}. The graph induced by this family is a path of length 4 in the cube Q3 (cf. Figure 2.2). Clearly, F is not well graded. On the other hand, as it can be easily seen, any path is a partial cube. Figure 2.2: A nonisometric path in the cube Q3. Any family F of subsets of X defines a graph GF = (F, EF), where EF = {{P,Q} ⊆ F : |P∆Q| = 1}. Theorem 2.1. The graph GF defined by a family F of subsets of a set X is isomorphic to a partial cube on X if and only if the family F is well graded. Proof. We need to prove sufficiency only. Let S be a fixed set in F. We define a mapping f : F → Pf (X) by f(R) = R∆S for R ∈ F. Then d(f(R), f(T )) = |(R∆S)∆(T∆S)| = |R∆T |. Thus f is an isometric embedding of F into Pf (X). Let (Ri) be a sequence of sets in F such that R0 = P , Rn = Q, |P∆Q| = n, and |Ri∆Ri+1| = 1 for all 0 ≤ i < n. Then the sequence (f(Ri)) satisfies conditions (2.2). The result follows. A set R ∈ Pf (X) is said to be lattice between sets P,Q ∈ Pf (X) if P ∩Q ⊆ R ⊆ P ∪Q. It is metrically between P and Q if d(P,R) + d(R,Q) = d(P,Q). The following theorem is a well-known result about these two betweenness re- lations on Pf (X) (see, for instance, [3]). Theorem 2.2. Lattice and metric betweenness relations coincide on Pf (X). Let F be a family of finite subsets of X . The set of all R ∈ F that are between P,Q ∈ F is the interval I(P,Q) between P and Q in F. Thus, I(P,Q) = F ∩ [P ∩Q,P ∪Q], where [P ∩Q,P ∪Q] is the usual interval in the lattice Pf . Two distinct sets P,Q ∈ F are adjacent in F if J(P,Q) = {P,Q}. If sets P and Q form an edge in the graph induced by F, then P and Q are adjacent in F, but, generally speaking, not vice versa. For instance, in Example 2.1, the vertices ∅ and {b, c} are adjacent in F but do not define an edge in the induced graph (cf. Figure 2.2). The following theorem is a ‘local’ characterization of wg-families of sets. Theorem 2.3. A family F ⊆ Pf (X) is well graded if and only if d(P,Q) = 1 for any two sets P and Q that are adjacent in F. Proof. (Necessity.) Let F be a wg-family of sets. Suppose that P and Q are adjacent in F. There is a sequence R0 = P,R1, . . . , Rn = Q that satisfies conditions (2.2). Since the sequence (Ri) is a shortest path in F, we have d(P, Pi) + d(Pi, Q) = d(P,Q) for all 0 ≤ i ≤ n. Thus, Pi ∈ I(P,Q) = {P,Q}. It follows that d(P,Q) = n = 1. (Sufficiency.) Let P and Q be two distinct sets in F. We prove by induction on n = d(P,Q) that there is a sequence (Ri) ∈ F satisfying conditions (2.2). The statement is trivial for n = 1. Suppose that n > 1 and that the statement is true for all k < n. Let P and Q be two sets in F such that d(P,Q) = n. Since d(P,Q) > 1, the sets P and Q are not adjacent in F. Therefore there exists R ∈ F that lies between P and Q and is distinct from these two sets. Then d(P,R) + d(R,Q) = d(P,Q) and both distances d(P,R) and d(R,Q) are less than n. By the induction hypothesis, there is a sequence (Ri) ∈ F such that P = R0, R = Rj , Q = Rn for some 0 < j < n, satisfying conditions (2.2) for 0 ≤ i < j and j ≤ i < n. It follows that F is a wg-family of sets. We conclude this section with two examples of infinite partial cubes (more examples are found in [17]). Example 2.2. Let Z be the graph on the set Z of integers with edges defined by pairs of consecutive integers. This graph is a partial cube since its vertex set is isometric to the wg-family of intervals {(−∞,m) : m ∈ Z} in Z. Example 2.3. Let us consider Zn as a metric space with respect to the ℓ1- metric. The graph Zn has Zn as the vertex set; two vertices in Zn are connected if they are on the unit distance from each other. We will show in Section 6 (Corollary 6.1) that Zn is a partial cube. 3 Characterizations Only connected graphs are considered in this section. Definition 3.1. Let G = (V,E) be a graph and d be its distance function. For any two adjacent vertices a, b ∈ V let Wab be the set of vertices that are closer to a than to b: Wab = {w ∈ V : d(w, a) < d(w, b)}. Following [11], we call the sets Wab and induced subgraphs 〈Wab〉 semicubes of the graph G. The semicubes Wab and Wba are called opposite semicubes. Remark 3.1. The subscript ab in Wab stands for an ordered pair of vertices, not for an edge of G. In his original paper [9], Djoković uses notation G(a, b) (cf. [8]). We use the notation from [15]. Clearly, two opposite semicubes are disjoint. They can be used to charac- terize bipartite graphs as follows. Theorem 3.1. A graph G = (V,E) is bipartite if and only if the semicubes Wab and Wba form a partition of V for any edge ab ∈ E. Proof. Let us recall that a connected graph G is bipartite if and only if for every vertex x there is no edge ab with d(x, a) = d(x, b) (see, for instance, [1]). For any edge ab ∈ E and vertex x ∈ V we clearly have d(x, a) = d(x, b) ⇔ x /∈ Wab ∪Wba. The result follows. The following lemma is instrumental and will be used frequently in the rest of the paper. Lemma 3.1. Let G = (V,E) be a graph and w ∈ Wab for some edge ab ∈ E. d(w, b) = d(w, a) + 1. Accordingly, Wab = {w ∈ V : d(w, b) = d(w, a) + 1}. Proof. By the triangle inequality, we have d(w, a) < d(w, b) ≤ d(w, a) + d(a, b) = d(w, a) + 1. The result follows, since d takes values in N. There are two binary relations on the set of edges of a graph that play a central role in characterizing partial cubes. Definition 3.2. Let G = (V,E) be a graph and e = xy and f = uv be two edges of G. (i) (Djoković [9]) The relation θ on E is defined by e θf ⇔ f joins a vertex in Wxy with a vertex in Wyx. The notation can be chosen such that u ∈Wxy and v ∈ Wyx. (ii) (Winkler [20]) The relation Θ on E is defined by eΘf ⇔ d(x, u) + d(y, v) 6= d(x, v) + d(y, u). It is clear that both relations θ and Θ are reflexive and Θ is symmetric. Lemma 3.2. The relation θ is a symmetric relation on E. Proof. Suppose that xy θ uv with u ∈ Wxy and v ∈ Wyx. By Lemma 3.1 and the triangle inequality, we have d(u, x) = d(u, y)− 1 ≤ d(u, v) + d(v, y)− 1 = d(v, y) = = d(v, x)− 1 ≤ d(v, u) + d(u, x) − 1 = d(u, x). Hence, d(u, x) = d(v, x) − 1 and d(v, y) = d(u, y)− 1. Therefore, x ∈ Wuv and y ∈ Wvu. It follows that uv θ xy. Lemma 3.3. θ ⊆ Θ. Proof. Suppose that xy θ uv with u ∈Wxy, v ∈ Wyx. By Lemma 3.1, d(x, u) + d(y, v) = d(x, v) − 1 + d(y, u)− 1 6= d(x, v) + d(y, u). Hence, xyΘ uv. Example 3.1. It is easy to verify that θ is the identity relation on the set of edges of the cycle C3. On the other hand, any two edges of C3 stand in the relation Θ. Thus, θ 6= Θ in this case. Bipartite graphs can be characterized in terms of relations θ and Θ as follows. Theorem 3.2. A graph G = (V,E) is bipartite if and only if θ = Θ. Proof. (Necessity.) Suppose that G is a bipartite graph, two edges xy and uv stand in the relation Θ, that is, d(x, u) + d(y, v) 6= d(x, v) + d(y, u), and that edges xy and uv do not stand in the relation θ. By Theorem 3.1, we may assume that u, v ∈ Wxy. By Lemma 3.1, we have d(x, u) + d(y, v) = d(y, u)− 1 + d(x, v) + 1 = d(x, v) + d(y, u), a contradiction. It follows that Θ ⊆ θ. By Lemma 3.3, θ = Θ. (Sufficiency.) Suppose that G is not bipartite. By Theorem 3.1, there is an edge xy such that Wxy ∪Wyx is a proper subset of V . Since G is connected, there is an edge uv with u /∈ Wxy ∪Wyx and v ∈ Wxy ∪Wyx. Clearly, uv does not stand in the relation θ to xy. On the other hand, d(x, u) + d(y, v) 6= d(x, v) + d(y, u), since u /∈ Wxy ∪Wyx and v ∈ Wxy ∪Wyx. Thus, xyΘ uv, a contradiction, since we assumed that θ = Θ. By Theorem 3.2, the relations θ and Θ coincide on bipartite graphs. For this reason we use the relation θ in the rest of the paper. Lemma 3.4. Let G = (V,E) be a bipartite graph such that all its semicubes are convex sets. Then two edges xy and uv stand in the relation θ if and only if the corresponding pairs of mutually opposite semicubes form equal partitions of V : xy θ uv ⇔ {Wxy,Wyx} = {Wuv,Wvu}. Proof. (Necessity) We assume that the notation is chosen such that u ∈ Wxy and v ∈ Wyx. Let z ∈ Wxy ∩Wvu. By Lemma 3.1, d(z, u) = d(z, v) + d(v, u). Since z, u ∈ Wxy and Wxy is convex, we have v ∈ Wxy, a contradiction to the assumption that v ∈Wyx. Thus Wxy ∩Wvu = ∅. Since two opposite semicubes in a bipartite graph form a partition of V , we haveWuv =Wxy andWvu =Wyx. A similar argument shows that Wuv = Wyx and Wvu = Wxy, if u ∈ Wyx and v ∈ Wxy. (Sufficiency.) Follows from the definition of the relation θ. We need another general property of the relation θ (cf. Lemma 2.2 in [15]). Lemma 3.5. Let P be a shortest path in a graph G. Then no two distinct edges of P stand in the relation θ. Proof. Let i < j and xixi+1 and xjxj+1 be two edges in a shortest path P from x0 to xn. Then d(xi, xj) < d(xi, xj+1) and d(xi+1, xj) < d(xi+1, xj+1), so xi, xi+1 ∈ Wxjxj+1 . It follows that edges xixi+1 and xjxj+1 do not stand in the relation θ. The converse statement is true for bipartite graphs (we omit the proof); a counterexample is the cycle C5 which is not bipartite. Lemma 3.6. Let G = (V,E) be a bipartite graph. The following statements are equivalent (i) All semicubes of G are convex. (ii) The relation θ is an equivalence relation on E. Proof. (i) ⇒ (ii). Follows from Lemma 3.4. (ii) ⇒ (i). Suppose that θ is transitive and there is a nonconvex semicube Wab. Then there are two vertices u, v ∈ Wab and a shortest path P from u to v that intersects Wba. This path contains two distinct edges e and f joining vertices of semicubes Wab and Wba. The edges e and f stand in the relation θ to the edge ab. By transitivity of θ, we have e θf . This contradicts the result of Lemma 3.5. Thus all semicubes of G are convex. We now establish some basic properties of partial cubes. Theorem 3.3. Let G = (V,E) be a partial cube. Then (i) G is a bipartite graph. (ii) Each pair of opposite semicubes form a partition of V . (iii) All semicubes are convex subsets of V . (iv) θ is an equivalence relation on E. Proof. We may assume that G is an isometric subgraph of some hypercube H(X), that is, G = (F, EF) for a wg-family F of finite subsets of X . (i) It suffices to note that if two sets in H(X) are connected by an edge then they have different parity. Thus, H(X) is a bipartite graph and so is G. (ii) Follows from (i) and Theorem 3.1. (iii) LetWAB be a semicube of G. By Lemma 3.1 and Theorem 2.2, we have WAB = {S ∈ F : S ∩B ⊆ A ⊆ S ∪B}. Let Q,R ∈WAB and P be a vertex of G such that d(Q,P ) + d(P,R) = d(Q,R). By Theorem 2.2, Q ∩R ⊆ P ⊆ Q ∪R. Since Q,R ∈WAB , we have Q ∩B ⊆ A ⊆ Q ∪B and R ∩B ⊆ A ⊆ R ∪B, which implies P ∩B ⊆ (Q ∪R) ∩B ⊆ A ⊆ (Q ∩R) ∪B ⊆ S ∪B. Hence, P ∈ WAB, and the result follows. (iv) Follows from (iii) and Lemma 3.6. Remark 3.2. Since semicubes of a partial cube G = (V,E) are convex subsets of the metric space V , they are half-spaces in V [19]. This terminology is used in [6, 7]. The following theorem presents four characterizations of partial cubes. The first two are due to Djoković [9] and Winkler [20] (cf. Theorem 2.10 in [15]). Theorem 3.4. Let G = (V,E) be a connected graph. The following statements are equivalent: (i) G is a partial cube. (ii) G is bipartite and all semicubes of G are convex. (iii) G is bipartite and θ is an equivalence relation. (iv) G is bipartite and, for all xy, uv ∈ E, xy θ uv ⇒ {Wxy,Wyx} = {Wuv,Wvu}. (3.1) (v) G is bipartite and, for any pair of adjacent vertices of G, there is a unique pair of opposite semicubes separating these two vertices. Proof. By Lemma 3.6, the statements (ii) and (iii) are equivalent and, by The- orem 3.3, (i) implies both (ii) and (iii). (iii) ⇒ (i). By Theorem 3.1, each pair {Wab,Wba} of opposite semicubes of G form a partition of V . We orient these partitions by calling, in an arbitrary way, one of the two opposite semicubes in each partition a positive semicube. Let us assign to each x ∈ V the set W+(x) of all positive semicubes containing x. In the next paragraph we prove that the family F = {W+(x)}x∈V is well graded and that the assignment x 7→ W+(x) is an isometry between V and F. Let x and y be two distinct vertices of G. We say that a positive semicube Wab separates x and y if either x ∈ Wab, y ∈ Wba or x ∈ Wba, y ∈ Wab. It is clear that Wab separates x and Y if and only if Wab ∈ W +(x)∆W+(y). Let P be a shortest path x0 = x, x1, . . . , xn = y from x to y. By Lemma 3.5, no two distinct edges of P stand in the relation θ. By Lemma 3.4, distinct edges of P define distinct positive semicubes; clearly, these semicubes separate x and y. Let Wab be a positive semicube separating x and y, and, say, x ∈Wab and y ∈Wba. There is an edge f ∈ P that joins vertices in Wab and Wba. Hence, f stands in the relation θ to ab and, by Lemma 3.4, Wab is defined by f . It follows that any semicube inW+(x)∆W+(y) is defined by a unique edge in P and any edge in P defines a semicube in W+(x)∆W+(y). Therefore, d(W+(x),W+(y)) = d(x, y), that is x 7→W+(x) is an isometry. Clearly, F is a wg-family of sets. By Theorem 2.1, the family F is isometric to a wg-family of finite sets. Hence, G is a partial cube. (iv) ⇒ (ii). Suppose that there exist an edge ab such that semicube Wba is not convex. Let p and q be two vertices in Wba such that there is a shortest path P from p to q that intersects Wab. There are two distinct edges xy and uv in P such that x, u ∈ Wab and y, v ∈ Wba. Since ab θ xy and ab θ uv, we have, by (3.1), Wab =Wxy =Wuv. Hence, u ∈ Wxy and v ∈Wyx. By Lemma 3.1, d(x, u) = d(x, v) − 1 = 1 + d(v, y)− 1 = d(v, y), a contradiction, since P is a shortest path from p to q. (ii) ⇒ (iv). Follows from Lemma 3.4. It is clear that (iv) and (v) are equivalent. 4 Fundamental sets in partial cubes Semicubes played an important role in the previous section. In this section we introduce three more classes of useful subsets of graphs. We also establish one more characterization of partial cubes. Let G = (V,E) be a connected graph. For a given edge e = ab ∈ E, we define the following sets (cf. [15, 16]): Fab = {f ∈ E : e θf} = {uv ∈ E : u ∈Wab, v ∈Wba}, Uab = {w ∈Wab : w is adjacent to a vertex in Wba}, Uba = {w ∈Wba : w is adjacent to a vertex in Wab}. The five sets are schematically shown in Figure 4.1. Figure 4.1: Fundamental sets in a partial cube. Remark 4.1. In the case of a partial cube G = (V,E), the semicubes Wab and Wba are complementary half-spaces in the metric space V (cf. Remark 3.2). Then the set Fab can be regarded as a ‘hyperplane’ separating these half-spaces (see [17] where this analogy is formalized in the context of hyperplane arrange- ments). The following theorem generalizes the result obtained in [16] for median graphs (see also [15]). Theorem 4.1. Let ab be an edge of a connected bipartite graph G. If the semicubes Wab and Wba are convex, then the set Fab is a matching and induces an isomorphism between the graphs 〈Uab〉 and 〈Uba〉. Proof. Suppose that Fab is not a matching. Then there are distinct edges xu and xv with, say, x ∈ Uab and u, v ∈ Uba. By the triangle inequality, d(u, v) ≤ 2. Since G does not have triangles, d(u, v) 6= 1. Hence, d(u, v) = 2, which implies that x lies between u and v. This contradicts convexity of Wba, since x ∈ Wab. Therefore Fab is a matching. To show that Fab induces an isomorphism, let xy, uv ∈ Fab and xu ∈ E, where x, u ∈ Uab and y, v ∈ Uba. Since G does not have odd cycles, d(v, y) 6= 2. By the triangle inequality, d(v, y) ≤ d(v, u) + d(u, x) + d(x, y) = 3. Since Wba is convex, d(v, y) 6= 3. Thus d(v, y) = 1, that is, vy is an edge. The result follows by symmetry. By Theorem 3.4(ii), we have the following corollary. Corollary 4.1. Let G = (V,E) be a partial cube. For any edge ab the set Fab is a matching and induces an isomorphism between induced graphs 〈Uab〉 and 〈Uba〉. Figure 4.2: Graph G. Example 4.1. Let G be the graph depicted in Figure 4.2. The set Fab = {ab, xu, yv} is a matching and defines an isomorphism between the graphs induced by subsets Uab = {a, x, y} and Uba = {b, u, v}. The set Wba is not convex, so G is not a partial cube. Thus the converse of Corollary 4.1 does not hold. We now establish another characterization of partial cubes that utilizes a geometric property of families Fab. Theorem 4.2. For a connected graph G the following statements are equivalent: (i) G is a partial cube. (ii) G is bipartite and d(x, u) = d(y, v) and d(x, v) = d(y, u), (4.1) for any ab ∈ E and xy, uv ∈ Fab. Proof. (i)⇒(ii). We may assume that x, u ∈ Wab and y, v ∈ Wba. Since θ is an equivalence relation, we have xy θ uv θab. By Lemma 3.4, Wuv = Wxy = Wab. By Lemma 3.1, d(x, u) = d(x, v) − 1 = d(v, y) + 1− 1 = d(y, v). We also have d(x, v) = d(y, v) + 1 = d(y, u), by the same lemma. (ii)⇒(i). Suppose that G is not a partial cube. Then, by Theorem 3.4, there exist an edge ab such that, say, semicube Wba is not convex. Let p and q be two vertices in Wba such that there is a shortest path P from p to q that intersects Wab. Let uv be the first edge in P which belongs to Fab and xy be the last edge in P with the same property (see Figure 4.3). Figure 4.3: An illustration to the proof of theorem 4.2. Since P is a shortest path, we have d(v, y) = d(v, u) + d(u, x) + d(x, y) 6= d(x, u), which contradicts condition (4.1). Thus all semicubes of G are convex. By Theorem 3.4, G is a partial cube. Remark 4.2. One can say that four vertices satisfying conditions (4.1) define a rectangle in G. Then Theorem 4.2 states that a connected graph is a partial cube if and only if it is bipartite and for any edge ab pairs of edges in Fab define rectangles in G. 5 Dimensions of partial cubes There are many different ways in which a given partial cube can be isometrically embedded into a hypercube. For instance, the graph K2 can be isometrically embedded in different ways into any hypercube H(X) with |X | > 2. Following Djoković [9] (see also [8]), we define the isometric dimension, dimI(G), of a partial cube G as the minimum possible dimension of a hypercube H(X) in which G is isometrically embeddable. Recall (see Section 2) that the dimension of H(X) is the cardinality of the set X . Theorem 5.1. (Theorem 2 in [9].) Let G = (V,E) be a partial cube. Then dimI(G) = |E/θ|, (5.1) where θ is Djoković’s equivalence relation on E and E/θ is the set of its equiv- alence classes (the quotient-set). The quotient-set E/θ can be identified with the family of all distinct sets Fab (see Section 4). If G is a finite partial cube, we may consider it as an isometric subgraph of some hypercube Qn. Then the edges in each family Fab are parallel edges in Qn (cf. Theorem 4.2). This observation essentially proves (5.1) in the finite case. Let G be a partial cube on a set X . The vertex set of G is a wg-family F of finite subsets of X (see Section 2). We define the retraction of F as a family F′ of subsets of X ′ = ∪F \ ∩F consisting of the intersections of sets in F with X ′. It is clear that F′ satisfies conditions ∩ F′ = ∅ and ∪ F′ = X ′. (5.2) Proposition 5.1. The partial cubes induced by a wg-family F and its retraction F′ are isomorphic. Proof. It suffices to prove that metric spaces F and F′ are isometric. Clearly, α : P 7→ P ∩X ′ is a mapping from F onto F′. For P,Q ∈ F, we have (P ∩X ′)∆(Q ∩X ′) = (P∆Q) ∩X ′ = (P∆Q) ∩ (∪F \ ∩F) = P∆Q. Thus, d(α(P ), α(Q)) = d(P,Q). Consequently, α is an isometry. Let G be a partial cube on some set X induced by a wg-family F satisfying conditions (5.2), and let PQ be an edge of G. By definition, there is x ∈ X such that P∆Q = {x}. The following two lemmas are instrumental. Lemma 5.1. Let PQ be an edge of a partial cube G on X and let P∆Q = {x}. The two sets {R ∈ F : x ∈ R} and {R ∈ F : x /∈ R} form the same bipartition of the family F as semicubes WPQ and WQP . Proof. We may assume that Q = P + {x}. Then, for any R ∈ F, R∆Q = R∆(P + {x}) = (R∆P ) + {x}, if x ∈ R, R∆P, if x /∈ R. Hence, |R∆P | < |R∆Q| if and only if x ∈ R. It follows that WPQ = {R ∈ F : x ∈ R}. A similar argument shows that WQP = {R ∈ F : x /∈ R}. Lemma 5.2. If F is a wg-family of sets satisfying conditions (5.2), then for any x ∈ X there are sets P,Q ∈ F such that P∆Q = {x}. Proof. By conditions 5.2, for a given x ∈ X there are sets S and T in F such that x ∈ S and x /∈ T . Let R0 = S,R1, . . . , Rn = T be a sequence of sets in F satisfying conditions (2.2). It is clear that there is i such that x ∈ Ri and x /∈ Ri+1. Hence, Ri∆Ri+1 = {x}, so we can choose P = Ri and Q = Ri+1. By Lemmas 5.1 and 5.2, there is one-to-one correspondence between the set X and the quotient-set E/θ. From Theorem 5.1 we obtain the following result. Theorem 5.2. Let F be a wg-family of finite subsets of a set X such that ∩F = ∅ and ∪F = X, and let G be a partial cube on X induced by F. Then dimI(G) = |X |. Clearly, a graph which is isometrically embeddable into a partial cube is a partial cube itself. We will show in Section 6 (Corollary 6.1) that the integer lattice Zn is a partial cube. Thus a graph which is isometrically embeddable into an integer lattice is a partial cube. It follows that a finite graph is a partial cube if and only if it is embeddable in some integer lattice. Examples of infinite partial cubes isometrically embeddable into a finite dimensional integer lattice are found in [17]. We call the minimum possible dimension n of an integer lattice Zn, in which a given graph G is isometrically embeddable, its lattice dimension and denote it dimZ(G). The lattice dimension of a partial cube can be expressed in terms of maximum matchings in so-called semicube graphs [11]. Definition 5.1. The semicube graph Sc(G) has all semicubes in G as the set of its vertices. Two vertices Wab and Wcd are connected in Sc(G) if Wab ∪Wcd = V and Wab ∩Wcd 6= ∅. (5.3) If G is a partial cube, then condition (5.3) is equivalent to each of the two equivalent conditions: Wba ⊂Wcd ⇔ Wdc ⊂Wab, (5.4) where ⊂ stands for the proper inclusion. Theorem 5.3. (Theorem 1 in [11].) Let G be a finite partial cube. Then dimZ(G) = dimI(G) − |M |, where M is a maximum matching in the semicube graph Sc(G). Example 5.1. Let G be the graph shown in Figure 2.1. It is easy to see that dimI(G) = 3 and dimZ(G) = 2. Example 5.2. Let T be a tree with n edges and m leaves. Then dimI(T ) = n and dimZ(T ) = ⌈m/2⌉ (cf. [8] and [14], respectively). Example 5.3. For the cycle C6 we have (see Figure 8.2) dimI(C6) = dimZ(C6) = 3. 6 Subcubes and Cartesian products Let G be a partial cube. We say that G′ is a subcube of G if it is an isometric subgraph of G. Clearly, a subcube is itself a partial cube. The converse does not hold; a subgraph of a graph G can be a partial cube but not an isometric subgraph of G (cf. Example 2.1). If G′ is a subcube of a partial cube G, then dimI(G ′) ≤ dimI(G) and dimZ(G ′) ≤ dimZ(G). In general, the two inequalities are not strict. For instance, the cycle C6 is an isometric subgraph of the cube Q3 (see Figure 8.2) dimI(C6) = dimZ(C6) = dimI(Q3) = dimZ(Q3) = 3. Semicubes of a partial cube are examples of subcubes. Indeed, by Theo- rem 3.4, semicubes are convex subgraphs and therefore isometric. In general, the converse is not true; a path connecting two opposite vertices in C6 is an isometric subgraph but not a convex one. Another common way of constructing new partial cubes from old ones is by forming their Cartesian products (see [15] for details and proofs). Definition 6.1. Given two graphs G1 = (V1, E1) and G2 = (V2, E2), their Cartesian product G = G1�G2 has vertex set V = V1 × V2; a vertex u = (u1, u2) is adjacent to a vertex v = (v1, v2) if and only if u1v1 ∈ E1 and u2 = v2, or u1 = v1 and u2v2 ∈ E2. The operation � is associative, so we can write G = G1� · · ·�Gn = for the Cartesian product of graphs G1, . . . , Gn. A Cartesian product i=1Gi is connected if and only if the factors are connected. Then we have dG(u, v) = dGi(ui, vi). (6.1) Example 6.1. Let {Xi} i=1 be a family of sets and Y = i=1 be their sum. Then the Cartesian product of the hypercubes H(Xi) is isomorphic to the hy- percube H(Y ). The isomorphism is established by the mapping f : (P1, . . . , Pn) 7→ Formula (6.1) yields immediately the following results. Proposition 6.1. Let Hi be isometric subgraphs of graphs Gi for all 1 ≤ i ≤ n. Then the Cartesian product i=1Hi is an isometric subgraph of the Cartesian product i=1Gi. Corollary 6.1. The Cartesian product of a finite family of partial cubes is a partial cube. In particular, the integer lattice Zn (cf. Examples 2.2 and 2.3) is a partial cube. The results of the next two theorems can be easily extended to arbitrary finite products of finite partial cubes. Theorem 6.1. Let G = G1�G2 be the Cartesian product of two finite partial cubes. Then dimI(G) = dimI(G1) + dimI(G2). Proof. We may assume that G1 (resp. G2) is induced by a wg-family F1 (resp. F2) of subsets of a finite set X1 (resp. X2) such that ∩F1 = ∅ and ∪F1 = X1 (resp. ∩F2 = ∅ and ∪F2 = X1) (see Section 5). By Theorem 5.2, dimI(G1) = |X1| and dimI(G2) = |X2|. It is clear that the graph G is induced by the wg-family F = F1 +F2 of subsets of the set X = X1 + X2 (cf. Example 6.1) with ∩F = ∅, ∪F = X . By Theorem 5.2, dimI(G) = |X | = |X1|+ |X2| = dimI(G1) + dimI(G2). Theorem 6.2. Let G = (V,E) be the Cartesian product of two finite partial cubes G1 = (V1, E1) and G2 = (V2, E2). Then dimZ(G) = dimZ(G1) + dimZ(G2). Proof. Let W(a,b)(c,d) be a semicube of the graph G. There are two possible cases: (i) c = a, bd ∈ E2. Let (x, y) be a vertex of G. Then, by (6.1), dG((x, y), (a, b)) = dG1(x, a) + dG2(y, b) dG((x, y), (c, d)) = dG1(x, c) + dG2(y, d). Hence, dG((x, y), (a, b)) < dG((x, y), (c, d)) ⇔ dG2(y, b) < dG2(y, d). It follows that W(a,b)(c,d) = V1 ×Wbd. (6.2) (ii) d = b, ac ∈ E1. Like in (i), we have W(a,b)(c,d) =Wac × V2. (6.3) Clearly, two semicubes given by (6.2) form an edge in the semicube graph Sc(G) if and only if their second factors form an edge in the semicube graph Sc(G2). The same is true for semicubes in the form (6.3) with respect to their first factors. It is also clear that semicubes in the form (6.2) and in the form (6.3) are not connected by an edge in Sc(G). Therefore the semicube graph Sc(G) is isomorphic to the disjoint union of semicube graphs Sc(G1) and Sc(G2). If M1 is a maximum matching in Sc(G1) and M2 is a maximum matching in Sc(G2), then M =M1 ∪M2 is a maximum matching in Sc(G). The result follows from theorems 5.3 and 6.1. Remark 6.1. The result of Corollary 6.1 does not hold for infinite Cartesian products of partial cubes, as these products are disconnected. On the other hand, it can be shown that arbitrary weak Cartesian products (connected com- ponents of Cartesian products [15]) of partial cubes are partial cubes. 7 Pasting partial cubes In this section we use the set pasting technique [5, ch.I, §2.5] to build new partial cubes from old ones. Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs, H1 = (U1, F1) and H2 = (U2, F2) be two isomorphic subgraphs of G1 and G2, respectively, and ψ : U1 → U2 be a bijection defining an isomorphism between H1 and H2. The bijection ψ defines an equivalence relation R on the sum V1+V2 as follows: any element in (V1 \U1)∪ (V2 \U2) is equivalent to itself only and elements u1 ∈ U1 and u2 ∈ U2 are equivalent if and only if u2 = ψ(u1). We say that the quotient set V = (V1 + V2)/R is obtained by pasting together the sets V1 and V2 along the subsets U1 and U2. Since the graphs H1 and H2 are isomorphic, the pasting of the sets V1 and V2 can be naturally extended to a pasting of sets of edges E1 and E2 resulting in the set E of edges joining vertices in V . We say that the graph G = (E, V ) is obtained by pasting together the graphs G1 and G2 along the isomorphic subgraphs H1 and H2. The pasting construction allows for identifying in a natural way the graphs G1 and G2 with subgraphs of G, and the isomorphic graphs H1 and H2 with a common subgraph H of both graphs G1 and G2. We often follow this convention below. Remark 7.1. Note that in the above construction the resulting graph G de- pends not only on graphs G1 and G2 and their isomorphic subgraphs H1 and H2 but also on the bijection ψ defining an isomorphism from H1 onto H2 (see the drawings in Figures 7.1 and 7.2). Figure 7.1: Pasting of two trees. Figure 7.2: Another pasting of the same trees. In general, pasting of two partial cubes G1 and G2 along two isomorphic subgraphs H1 and H2 does not produce a partial cube even under strong as- sumptions about these subgraphs as the next example illustrates. Figure 7.3: Pasting partial cubes G1 and G2. Example 7.1. Pasting of two partial cubes G1 = C6 and G2 = C6 along subgraphs H1 and H2 is shown in Figure 7.3. The resulting graph G is not a partial cube. Indeed, the semicubeWab is not a convex set. Note that subgraphs H1 and H2 are convex subgraphs of the respective partial cubes. In this section we study two simple pastings of connected graphs together, the vertex-pasting and the edge-pasting, and show that these pastings produce partial cubes from partial cubes. We also compute the isometric and lattice dimensions of the resulting graphs. Let G1 = (V1, E1) and G2 = (V2, E2) be two connected graphs, a1 ∈ V1, a2 ∈ V2, and H1 = ({a1},∅), H2 = ({a2},∅). Let G be the graph obtained by pasting G1 and G2 along subgraphs H1 and H2. In this case we say that the graph G is obtained from graphs G1 and G2 by vertex-pasting. We also say that G is obtained from G1 and G2 by identifying vertices a1 and a2. Figure 7.4 illustrates this construction. Note that the vertex a = {a1, a2} is a cut vertex of G, since G1 ∪ G2 = G and G1 ∩ G2 = {a}. (We follow our convention and identify graphs G1 and G2 with subgraphs of G.) Figure 7.4: An example of vertex-pasting. In what follows we use superscripts to distinguish subgraphs of the graphs G1 and G2. For instance, W stands for the semicube of G2 defined by two adjacent vertices a, b ∈ V2. Theorem 7.1. A graph G = (V,E) obtained by vertex-pasting from partial cubes G1 = (V1, E1) and G2 = (V2, E2) is a partial cube. Proof. We denote a = {a1, a2} the vertex of G obtained by identifying vertices a1 ∈ V1 and a2 ∈ V2. Clearly, G is a bipartite graph. Let xy be an edge of G. Without loss of generality we may assume that xy ∈ E1 and a ∈ Wxy. Note that any path between vertices in V1 and V2 must go through a. Since a ∈Wxy, we have, for any v ∈ V2, d(v, x) = d(v, a) + d(a, x) < d(v, a) + d(a, y) = d(v, y), which implies V2 ⊆ Wxy and Wyx ⊆ V1. It follows that Wxy = W xy ∪ V2 and Wyx = W yx . The sets W xy , W yx and V2 are convex subsets of V . Since xy ∩ V2 = {a}, the set Wxy = W xy ∪ V2 is also convex. By Theorem 3.4(ii), the graph G is a partial cube. The vertex-pasting construction introduced above can be generalized as follows. Let G = {Gi = (Vi, Ei)}i∈J be a family of connected graphs and A = {ai ∈ Gi}i∈J be a family of distinguished vertices of these graphs. Let G be the graph obtained from the graphs Gi by identifying vertices in the set A. We say that G is obtained by vertex-pasting together the graphs Gi (along the set A). Example 7.2. Let J = {1, . . . , n} with n ≥ 2, G = {Gi = ({ai, bi}, {aibi})}i∈J , and A = {ai}i∈J . Clearly, each Gi is K2. By vertex-pasting these graphs along A, we obtain the n-star graph K1,n. Since the star K1,n is a tree it can be also obtained from K1 by successive vertex-pasting as in Example 7.3. Example 7.3. Let G1 be a tree and G2 = K2. By vertex-pasting these graphs we obtain a new tree. Conversely, let G be a tree and v be its leaf. Let G1 be a tree obtained from G by deleting the leaf v. Clearly, G can be obtained by vertex-pasting G1 and K2. It follows that any tree can obtained from the graph K1 by successive vertex-pasting of copies of K2 (cf. Theorem 2.3(e) in [12]). Any connected graph G can be constructed by successive vertex-pasting of its blocks using its block cut-vertex tree [4] structure. Let G1 be an endblock of G with a cut vertex v and G2 be the union of the remaining blocks of G. Then G can be obtained from G1 and G2 by vertex-pasting along the vertex v. It follows that any connected graph can be obtained from its blocks by successive vertex-pastings. Let G = (V,E) be a partial cube. We recall that the isometric dimension dimI(G) of G is the cardinality of the quotient set E/θ, where θ is Djoković’s equivalence relation on the set E (cf. formula (5.1)). Theorem 7.2. Let G = (V,E) be a partial cube obtained by vertex-pasting together partial cubes G1 = (V1, E1) and G2 = (V2, E2). Then dimI(G) = dimI(G1) + dimI(G2). Proof. It suffices to prove that there are no edges xy ∈ E1 and uv ∈ E2 which are in Djoković’s relation θ with each other. Suppose that G1 and G2 are vertex-pasted along vertices a1 ∈ E1 and a2 ∈ E2 and let a = {a1, a2} ∈ E. Let xy ∈ E1 and uv ∈ E2 be two edges in E. We may assume that u ∈ Wxy. Since a is a cut-vertex of G and u ∈Wxy, we have d(u, a) + d(a, x) = d(u, x) < d(u, y) = d(u, a) + d(a, y). Hence, d(a, x) < d(a, y), which implies d(v, x) = d(v, a) + d(a, x) < d(v, a) + d(a, y) = d(v, y). It follows that v ∈ Wxy. Therefore the edge xy does not stand in the relation θ to the vertex uv. The next result follows immediately from the previous theorem. Note that blocks of a partial cube are partial cubes themselves. Corollary 7.1. Let G be a partial cube and {G1, . . . , Gn} be the family of its blocks. Then dimI(G) = dimI(Gi). In the case of the lattice dimension of a partial cube we can claim only much weaker result than one stated in Theorem 7.2 for the isometric dimension. We omit the proof. Theorem 7.3. Let G be a partial cube obtained by vertex-pasting together partial cubes G1 and G2. Then max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) ≤ dimZ(G1) + dimZ(G2). The following example illustrate possible cases for inequalities in Theo- rem 7.3. Let us recall that the lattice dimension of a tree with m leaves is ⌈m/2⌉ (cf. [14]). Example 7.4. The star K1,6 can be obtained from the stars K1,2 and K1,4 by vertex-pasting these two stars along their centers. Clearly, max{dimZ(K1,2), dimZ(K1,4)} < dimZ(K1,6) = dimZ(K1,2) + dimZ(K1,4). The same star K1,6 is obtained from two copies of the star K1,3 by vertex- pasting along their centers. We have dimZ(K1,3) = 2, dimZ(K1,6) = 3, so max{dimZ(K1,3), dimZ(K1,3)} < dimZ(K1,6) < dimZ(K1,3) + dimZ(K1,3). Let us vertex-paste two stars K1,3 along their two leaves. The resulting graph T is a tree with four vertices. Therefore, max{dimZ(K1,3), dimZ(K1,3)} = dimZ(T ) < dimZ(K1,3) + dimZ(K1,3). We now consider another simple way of pasting two graphs together. Let G1 = (V1, E1) and G2 = (V2, E2) be two connected graphs, a1b1 ∈ E1, a2b2 ∈ E2, and H1 = ({a1, b1}, {a1b1}), H2 = ({a2, b2}, {a2b2}). Let G be the graph obtained by pasting G1 and G2 along subgraphs H1 and H2. In this case we say that the graph G is obtained from graphs G1 and G2 by edge-pasting. Figures 7.1, 7.2, and 7.5 illustrate this construction. Figure 7.5: An example of edge-pasting. As before, we identify the graphs G1 and G2 with subgraphs of the graph G and denote a = {a1, a2}, b = {b1, b2} the two vertices obtained by pasting together vertices a1 and a2 and, respectively, b1 and b2. The edge ab ∈ E is obtained by pasting together edges a1b1 ∈ E1 and a2b2 ∈ E2 (cf. Figure 7.5). Then G = G1∪G2, V1∩V2 = {a, b} and E1∩E2 = {ab}. We use these notations in the rest of this section. Proposition 7.1. A graph G obtained by edge-pasting together bipartite graphs G1 and G2 is bipartite. Proof. Let C be a cycle in G. If C ⊆ G1 or C ⊆ G2, then the length of C is even, since the graphs G1 and G2 are bipartite. Otherwise, the vertices a and b separate C into two paths each of odd length. Therefore C is a cycle of even length. The result follows. The following lemma is instrumental; it describes the semicubes of the graph G in terms of semicubes of graphs G1 and G2. Lemma 7.1. Let uv be an edge of G. Then (i) For uv ∈ E1, a, b ∈ Wuv ⇒ Wuv =W uv ∪ V2, Wvu =W (ii) For uv ∈ E2, a, b ∈ Wuv ⇒ Wuv =W uv ∪ V1, Wvu =W (iii) a ∈ Wuv, b ∈Wvu ⇒ Wuv =Wab. Figure 7.6: Edge-pasting of graphs G1 and G2. Proof. We prove parts (i) and (iii) (see Figure 7.6). (i) Since any path from w ∈ V2 to u or v contains a or b and a, b ∈Wuv, we have w ∈Wuv. Hence, Wuv =W uv ∪ V2 and Wvu =W (iii) Since ab θ uv in G1, we have W uv = W , by Theorem 3.4(iv). Let w be a vertex in W uv . Then, by the triangle inequality, d(w, u) < d(w, v) ≤ d(w, b) + d(b, v) < d(w, b) + d(b, u). Since any shortest path from w to u contains a or b, we have d(w, a) + d(a, u) = d(w, u). Therefore, d(w, a) + d(a, u) < d(w, b) + d(b, u). Since ab θ uv in G1, we have d(a, u) = d(b, v), by Theorem 4.2. It follows that d(w, a) < d(w, b), that is, w ∈ W . We proved that W uv ⊆ W symmetry, W vu ⊆ W . Since two opposite semicubes form a partition of V2, we have W uv =W . The result follows. Theorem 7.4. A graph G obtained by edge-pasting together partial cubes G1 and G2 is a partial cube. Proof. By Theorem 3.4(ii) and Proposition 7.1, we need to show that for any edge uv of G the semicube Wuv is a convex subset of V . There are two possible cases. (i) uv = ab. The semicube Wab is the union of semicubes W and W which are convex subsets of V1 and V2, respectively. It is clear that any shortest path connecting a vertex in W with a vertex in W contains vertex a and therefore is contained in Wab. Hence, Wab is a convex set. A similar argument proves that the set Wba is convex. (ii) uv 6= ab. We may assume that uv ∈ E1. To prove that the semicube Wuv is a convex set, we consider two cases. (a) a, b ∈ Wuv. (The case when a, b ∈ Wvu is treated similarly.) By Lemma 7.1(i), the semicube Wuv is the union of the semicube W uv and the set V2 which are both convex sets. Any shortest path P from a vertex in V2 to a vertex in W uv contains either a or b. It follows that P ⊆ W uv ∪ V2 = Wuv. Therefore the semicube Wuv is convex. (b) a ∈ Wuv, b ∈ Wvu. (The case when b ∈ Wuv , a ∈ Wvu is treated similarly.) By Lemma 7.1(ii), Wuv = Wab. The result follows from part (i) of the proof. Theorem 7.5. Let G be a graph obtained by edge-pasting together finite partial cubes G1 and G2. Then dimI(G) = dimI(G1) + dimI(G2)− 1. Proof. Let θ, θ1, and θ2 be Djoković’s relations on E, E1, and E2, respectively. By Lemma 7.1, for uv, xy ∈ E1 (resp. uv, xy ∈ E2) we have uv θ xy ⇔ uv θ1xy (resp. uv θ xy ⇔ uv θ2xy). Let uv ∈ E1, xy ∈ E2, and uv θ xy. Suppose that (uv, ab) /∈ θ. We may assume that a, b ∈ Wuv . By Lemma 7.1(i), V2 ⊂ Wuv, a contradiction, since xy ∈ E2. Hence, uv θ xy θ ab. It follows that each equivalence class of the relation θ is either an equivalence class of θ1, an equivalence class of θ2 or the class containing the edge ab. Therefore |E/θ| = |E1/θ1|+ |E2/θ2| − 1. The result follows, since the isometric dimension of a partial cube is equal to the cardinality of the set of equivalence classes of Djoković’s relation (formula (5.1)). We need some results about semicube graphs in order to prove an analog of Theorem 7.3 for a partial cube obtained by edge-pasting of two partial cubes. Lemma 7.2. Let G be a partial cube and WpqWuv , WqpWxy be two edges in the graph Sc(G). Then WxyWuv is an edge in Sc(G). Proof. By condition (5.4), Wqp ⊂ Wuv and Wyx ⊂ Wqp. Hence, Wyx ⊂ Wuv. By the same condition, WxyWuv ∈ Sc(G). As before, we identify partial cubes G1 and G2 with subgraphs of the partial cube G. Then G1 ∪G2 = G and G1 ∩G2 = ({a, b}, {ab}) = K2 (cf. Figure 7.6). Lemma 7.3. Let G be a partial cube obtained by edge-pasting together partial cubes G1 and G2. Let W xy (resp. W xy ) be an edge in the semicube Sc(G1) (resp. Sc(G2)). Then WuvWxy is an edge in Sc(G). Figure 7.7: Semicubes forming an edge in Sc(G1). Proof. It suffices to consider the case of Sc(G1) (see Figure 7.7). By condi- tion (5.4),W vu ⊂W xy andW yx ⊂W uv . Suppose that a ∈ W vu and b ∈W (the case when b ∈ W vu and a ∈ W yx is treated similarly). Then ab θ1xy and ab θ1uv. By transitivity of θ1, we have uv θ1xy, a contradiction, since semicubes uv and W xy are distinct. Therefore we may assume that, say, a, b ∈ W Then, by Lemma 7.1, Wvu = W vu ⊂ V1. Since W vu ⊂ W xy ⊆ Wxy, we have Wvu ⊂Wxy. By condition (5.4), WuvWxy is an edge in Sc(G). Lemma 7.4. LetM1 andM2 be matchings in graphs Sc(G1) and Sc(G2). There is a matching M in Sc(G) such that |M | ≥ |M1|+ |M2| − 1. Proof. By Lemma 7.3, M1 and M2 induce matchings in Sc(G) which we denote by the same symbols. The intersection M1 ∩M2 is either empty or a subgraph of the empty graph with vertices Wab and Wba. If M1 ∩M2 is empty, then M = M1 ∪M2 is a matching in Sc(G) and the result follows. If M1 ∩M2 is an empty graph with a single vertex, say, in M1, we remove fromM1 the edge that has this vertex as its end vertex, resulting in the matching M ′1. Clearly, M =M 1 ∪M2 is a matching in Sc(G) and |M | = |M1|+ |M2| − 1. Suppose now that M1 ∩M2 is the empty graph with vertices Wab and Wba. Let WabWuv, WbaWpq (resp. WabWxy, WbaWrs) be edges in M1 (resp. M2). By Lemma 7.2, WxyWrs is an edge in Sc(G2). Let us replace edgesWabWxy and WbaWrs in M2 by a single edge WxyWrs, resulting in the matching M 2. Then M =M1 ∪M 2 is a matching in Sc(G) and |M | = |M1|+ |M2| − 1. Corollary 7.2. Let M1 and M2 be maximum matchings in Sc(G1) and Sc(G2), respectively, and M be a maximum matching in Sc(G). Then |M | ≥ |M1|+ |M2| − 1. (7.1) By Theorem 5.3, we have dimI(G1) = dimZ(G1) + |M1|, dimI(G2) = dimZ(G2) + |M2|, dimI(G) = dimZ(G) + |M |, where M1 and M2 are maximum matchings in Sc(G1) and Sc(G2), respectively, and M is a maximum matching in Sc(G). Therefore, by Theorem 7.5 and (7.1), we have the following result (cf. Theorem 7.3). Theorem 7.6. Let G be a partial cube obtained by edge-pasting from partial cubes G1 and G2. Then max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) ≤ dimZ(G1) + dimZ(G2). Example 7.5. Let us consider two edge-pastings of the stars G1 = K1,3 and G2 = K1,3 of lattice dimension 2 shown in figures 7.1 and 7.2. In the first case the resulting graph is the star G = K1,5 of lattice dimension 3. Then we have max{dimZ(G1), dimZ(G2)} < dimZ(G) < dimZ(G1) + dimZ(G2). In the second case the resulting graph is a tree with 4 leaves. Therefore, max{dimZ(G1), dimZ(G2)} = dimZ(G) < dimZ(G1) + dimZ(G2). Let c1a1 and c2a2 be edges of stars G1 = K1,4 and G2 = K1,4 (each of which has lattice dimension 2), where c1 and c2 are centers of the respective stars. Let us edge-paste these two graphs by identifying c1 with c2 and a1 with a2, respectively. The resulting graph G is the star K1,7 of lattice dimension 4. Thus, max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) = dimZ(G1) + dimZ(G2). 8 Expansions and contractions of partial cubes The graph expansion procedure was introduced by Mulder in [16], where it is shown that a graph is a median graph if and only if it can be obtained from K1 by a sequence of convex expansions (see also [15]). A similar result for partial cubes was established in [6] (see also [7]) as a corollary to a more general result concerning isometric embeddability into Hamming graphs; it was also established in [13] in the framework of oriented matroids theory. In this section we investigate properties of (isometric) expansion and con- traction operations and, in particular, prove in two different ways that a graph is a partial cube if and only if it can be obtained from the graph K1 by a sequence of expansions. A remark about notations is in order. In the product {1, 2} × (V1 ∪ V2), we denote V ′i = {i} × Vi and x i = (i, x) for x ∈ Vi, where i, j = 1, 2. Definition 8.1. Let G = (V,E) be a connected graph, and let G1 = (V1, E1) and G2 = (V2, E2) be two isometric subgraphs of G such that G = G1 ∪ G2. The expansion of G with respect to G1 and G2 is the graph G ′ = (V ′, E′) constructed as follows from G (see Figure 8.1): (i) V ′ = V1 + V2 = V 1 ∪ V (ii) E′ = E1 + E2 +M , where M is the matching x∈V1∩V2 {x1x2}. In this case, we also say that G is a contraction of G′. Figure 8.1: Expansion/contraction processes. It is clear that the graphs G1 and 〈V 1〉 are isomorphic, as well as the graphs G2 and 〈V We define a projection p : V ′ → V by p(xi) = x for x ∈ V . Clearly, the restriction of p to V ′1 is a bijection p1 : V 1 → V1 and its restriction to V 2 is a bijection p2 : V 2 → V2. These bijections define isomorphisms 〈V 1〉 → G1 and 〈V ′2〉 → G2. Let P ′ be a path in G′. The vertices of G obtained from the vertices in P ′ under the projection p define a walk P in G; we call this walk P the projection of the path P ′. It is clear that ℓ(P ) = ℓ(P ′), if P ′ ⊆ 〈V ′1〉 or P ′ ⊆ 〈V ′2〉. (8.1) In this case, P is a path in G and either P = p1(P ′) or P = p2(P ′). On the other hand, ℓ(P ) < ℓ(P ′), if P ′ ∩ 〈V ′1〉 6= ∅ and P ′ ∩ 〈V ′2 〉 6= ∅, (8.2) and P is not necessarily a path. We will frequently use the results of the following lemma in this section. Lemma 8.1. (i) For u1, v1 ∈ V ′1 , any shortest path Pu1v1 in G ′ belongs to 〈V ′1 〉 and its projection Puv = p1(Pu1v1) is a shortest path in G. Accordingly, dG′(u 1, v1) = dG(u, v) and 〈V ′1〉 is a convex subgraph of G ′. A similar statement holds for u2, v2 ∈ V ′2 . (ii) For u1 ∈ V ′1 and v 2 ∈ V ′2 , dG′(u 1, v2) = dG(u, v) + 1. Let Pu1v2 be a shortest path in G ′. There is a unique edge x1x2 ∈M such that x1, x2 ∈ Pu1v2 and the sections Pu1x1 and Px2v2 of the path Pu1v2 are shortest paths in 〈V ′1 〉 and 〈V 2 〉, respectively. The projection Puv of Pu1v2 in G ′ is a shortest path in G. Proof. (i) Let Pu1v1 be a path in G ′ that intersects V ′2 . Since 〈V1〉 is an isometric subgraph of G, there is a path Puv in G that belongs to 〈V1〉. Then p 1 (Puv) is a path in 〈V ′1 〉 of the same length as Puv. By (8.1) and (8.2), ℓ(p−11 (Puv)) < ℓ(Pu1v1). Therefore any shortest path Pu1v1 in G ′ belongs to 〈V ′1 〉. The result follows. (ii) Let Pu1v2 be a shortest path in G ′ and Puv be its projection to V . By (8.2), dG′(u 1, v2) = ℓ(Pu1v2) > ℓ(Puv) ≥ dG(u, v). Since there is no edge of G joining vertices in V1 \ V2 and V2 \ V1, a shortest path in G from u to v must contain a vertex x ∈ V1 ∩ V2. Since G1 and G2 are isometric subgraphs, there are shortest paths Pux in G1 and Pxv in G2 such that their union is a shortest path from u to v. Then, by the triangle inequality and part (i) of the proof, we have (cf. Figure 8.1) dG′(u 1, v2) ≤ dG′(u 1, x1) + dG′(x 1, x2) + dG′(x 2, v2) = dG(u, v) + 1. The last two displayed formulas imply dG′(u 1, v2) = dG(u, v) + 1. Since u1 ∈ V ′1 and v 2 ∈ V ′2 the path Pu1v2 must contain an edge, say x 1x2, in M . Since this path is a shortest path in G′, this edge is unique. Then the sec- tions Pu1x1 and Px2v2 of Pu1v2 are shortest paths in 〈V 1 〉 and 〈V 2〉, respectively. Clearly, Puv is a shortest path in G. Let a1a2 be an edge in the matchingM = ∪x∈V1∩V2{x 1x2}. This edge defines five fundamental sets (cf. Section 4): the semicubes Wa1a2 and Wa2a1 , the sets of vertices Ua1a2 and Ua2a1 , and the set of edges Fa1a2 . The next theorem follows immediately from Lemma 8.1. It gives a hint to a connection between the expansion process and partial cubes. Theorem 8.1. Let G′ be an expansion of a connected graph G and notations are chosen as above. Then (i) Wa1a2 = V 1 and Wa2a1 = V 2 are convex semicubes of G (ii) Fa1a2 =M defines an isomorphism between induced subgraphs 〈Ua1a2〉 and 〈Ua2a1〉, which are isomorphic to the subgraph G1 ∩G2. The result of Theorem 8.1 justifies the following constructive definition of the contraction process. Definition 8.2. Let ab be an edge of a connected graph G′ = (V ′, E′) such (i) semicubes Wab and Wba are convex and form a partition of V (ii) the set Fab is a matching and defines an isomorphism between subgraphs 〈Uab〉 and 〈Uba〉. A graph G obtained from the graphs 〈Wab〉 and 〈Wba〉 by pasting them along subgraphs 〈Uab〉 and 〈Uba〉 is said to be a contraction of the graph G Remark 8.1. If G′ is bipartite, then semicubesWab andWba form a partition of its vertex set. Then, by Theorem 4.1, condition (i) implies condition (ii). Thus any pair of opposite convex semicubes in a connected bipartite graph defines a contraction of this graph. By Theorem 8.1, a graph is a contraction of its expansion. It is not difficult to see that any connected graph is also an expansion of its contraction. The following three examples give geometric illustrations for the expansion and contraction procedures. Example 8.1. Let a and b be two opposite vertices in the graph G = C4. Clearly, the two distinct paths P1 and P2 from a to b are isometric subgraphs of G defining an expansion G′ = C6 of G (see Figure 8.2). Note that P1 and P2 are not convex subsets of V . Example 8.2. Another isometric expansion of the graph G = C4 is shown in Figure 8.3. Here, the path P1 is the same as in the previous example and G2 = G. Example 8.3. Lemma 8.1 claims, in particular, that the projection of a shortest path in an extension G′ of a graphG is a shortest path in G. Generally speaking, Figure 8.2: An expansion of the cycle C4. Figure 8.3: Another isometric expansion of the cycle C4. the converse is not true. Consider the graph G shown in Figure 8.4 and two paths in G: V1 = abcef and V2 = bde. The graph G′ in Figure 8.4 is the convex expansion of G with respect to V1 and V2. The path abdef is a shortest path in G; it is not a projection of a shortest path in G′. Figure 8.4: A shortest path which is not a projection of a shortest path. One can say that, in the case of finite partial cubes, the contraction procedure is defined by an orthogonal projection of a hypercube onto one of its facets. By Theorem 8.1, the sets V ′1 and V 2 are opposite semicubes of the graph G defined by edges in M . Their projections are the sets V1 and V2 which are not necessarily semicubes of G. For other semicubes in G′ we have the following result. Lemma 8.2. For any two adjacent vertices u, v ∈ V , Wuivi = p −1(Wuv) for u, v ∈ Vi and i = 1, 2. Proof. By Lemma 8.1, dG′(x j , ui) < dG′(x j , vi) ⇔ dG(x, u) < dG(x, v) for x ∈ V and i, j = 1, 2. The result follows. Corollary 8.1. If uv is an edge of G1 ∩G2, then Wu1v1 =Wu2v2 . The following lemma is an immediate consequence of Lemma 8.1. We shall use it implicitly in our arguments later. Lemma 8.3. Let u, v ∈ V1 and x ∈ V1 ∩ V2. Then x1 ∈Wu1v1 ⇔ x 2 ∈Wu1v1 . The same result holds for semicubes in the form Wu2v2 . Generally speaking, the projection of a convex subgraph of G′ is not a con- vex subgraph of G. For instance, the projection of the convex path b2d2e2 in Figure 8.4 is the path bde which is not a convex subgraph of G. On the other hand, we have the following result. Theorem 8.2. Let G′ = (V ′, E′) be an expansion of a graph G = (V,E) with respect to subgraphs G1 = (V1, E1) and G2 = (V2, E2). The projection of a convex semicube of G′ different from 〈V ′1〉 and 〈V 2 〉 is a convex semicube of G. Proof. It suffices to consider the case when Wuv = p(Wu1v1) for u, v ∈ V1 (cf. Theorem 8.2). Let x, y ∈Wuv and z ∈ V be a vertex such that dG(x, z) + dG(z, y) = dG(x, y). We need to show that z ∈Wuv. Figure 8.5: A shortest path from x to y. (i) x, y ∈ V1 (the case when x, y ∈ V2 is treated similarly). Suppose that z ∈ V1. Then x 1, y1, z1 ∈ V ′1 and, by Lemma 8.1, dG′(x 1, z1) + dG′(z 1, y1) = dG′(z 1, y1). Since x1, y1 ∈ Wu1v1 and Wu1v1 is convex, z 1 ∈ Wu1v1 . Hence, z ∈Wuv. Suppose now that z ∈ V2 \ V1. Consider a shortest path Pxy in G from x to y containing z. This path contains vertices x′, y′ ∈ V1 ∩ V2 such that (see Figure 8.5) dG(x, x ′) + dG(x ′, z) = dG(x, z) and dG(y, y ′) + dG(y ′, z) = dG(y, z). Since Pxy is a shortest path in G, we have dG(x, x ′) + dG(x ′, y) = dG(x, y), dG(x, y ′) + dG(y ′, y) = dG(x, y), ′, z) + dG(z, y ′) = dG(x ′, y′). Since x, x′, y ∈ V1, we have x 1, x′1, y1 ∈ V ′1 . Because x 1, y1 ∈ Wu1v1 and Wu1v1 is convex, x′1 ∈ Wu1v1 . Hence, x ′ ∈ Wuv and, similarly, y ′ ∈ Wuv. Since x′2, y′2, z2 ∈ V ′2 and Wu1v1 is convex, z 2 ∈Wu1v1 . Hence, z ∈Wuv. (ii) x ∈ V1 \V2 and y ∈ V2 \V1. We may assume that z ∈ V1. By Lemma 8.1, dG′(x 1, y2) = dG(x, y) + 1 = dG(x, z) + dG(z, y) + 1 = dG′(x 1, z1) + dG′(z 1, y2). Since x1, y2 ∈ Wu1v1 and Wu1v1 is convex, z 1 ∈ Wu1v1 . Hence, z ∈Wuv. By using the results of Lemma 8.1, it is not difficult to show that the class of connected bipartite graphs is closed under the expansion and contraction operations. The next theorem establishes this result for the class of partial cubes. Theorem 8.3. (i) An expansion G′ of a partial cube G is a partial cube. (ii) A contraction G of a partial cube G′ is a partial cube. Proof. (i) Let G = (V,E) be a partial cube and G′ = (V ′, E′) be its expansion with respect to isometric subgraphs G1 = (V1, E1) and G2 = (V2, E2). By Theorem 3.4(ii), it suffices to show that the semicubes of G′ are convex. By Lemma 8.1, the semicubes 〈V ′1〉 and 〈V 2〉 are convex, so we consider a semicube in the formWu1v1 where uv ∈ E1 (the other case is treated similarly). Let Px′y′ be a shortest path connecting two vertices in Wu1v1 and Pxy be its projection to G. By Theorem 8.2, x, y ∈ Wuv and, by Lemma 8.1, Pxy is a shortest path in G. Since Wuv is convex, Pxy belongs to Wuv. Let z ′ be a vertex in Px′y′ and z = p(z ′) ∈ Pxy. By Lemma 8.1, dG(z, u) < dG(z, v) ⇒ dG′(z ′, u1) ≤ dG′(z ′, v1). Since G′ is a bipartite graph, dG′(z ′, u1) < dG′(z ′, v1). Hence, Px′y′ ⊆ Wu1v1 , so Wu1v1 is convex. (ii) Let G = (V,E) be a contraction of a partial cube G′ = (V ′, E′). By Theorem 3.4, we need to show that the semicubes of G are convex. By The- orem 8.2, all semicubes of G are projections of semicubes of G′ distinct from 〈V ′1〉 and 〈V 2〉. By Theorem 8.2, the semicubes of G are convex. Corollary 8.2. (i) A finite connected graph is a partial cube if and only if it can be obtained from K1 by a sequence of expansions. (ii) The number of expansions needed to produce a partial cube G from K1 is dimI(G). Proof. (i) Follows immediately from Theorem 8.3. (ii) Follows from theorems 8.2 and 5.1 (see the discussion in Section 5 just before Theorem 5.2 ). The processes of expansion and contraction admit useful descriptions in the case of partial cubes on a set. Let G = (V,E) be a partial cube on a set X , that is an isometric subgraph of the hypercube H(X). Then it is induced by some wg-family F of finite subsets of X (cf. Theorem 2.1). We may assume (see Section 5) that ∩F = ∅ and ∪F = X . In what follows we present proofs of the results of Theorem 8.3 and Corol- lary 8.2 given in terms of wg-families of sets. The expansion process for a partial cube G on X can be described as follows: Let F1 and F2 be wg-families of finite subsets of X such that F1 ∩ F2 6= ∅, F1∪F2 = F, and the distance between any two sets P ∈ F1 \F2 and Q ∈ F2 \F1 is greater than one. Note that 〈F1〉 and 〈F2〉 are partial cubes, 〈F1〉∩ 〈F2〉 6= ∅, and 〈F1〉 ∪ 〈F2〉 = 〈F〉 = G. Let X ′ = X + {p}, where p /∈ X , and 2 = {Q+ {p} : Q ∈ F2}, F ′ = F1 ∪ F It is quite clear that the graphs 〈F′2〉 and 〈F2〉 are isomorphic and the graph G′ = 〈F′〉 is an isometric expansion of the graph G. Theorem 8.4. An expansion of a partial cube is a partial cube. Proof. We need to verify that F′ is a wg-family of finite subsets of X ′. By Theorem 2.3, it suffices to show that the distance between any two adjacent sets in F′ is 1. It is obvious if each of these two sets belong to one of the families F1 or F 2. Suppose that P ∈ F1 and Q+ {p} ∈ F 2 are adjacent, that is, for any S ∈ F′ we have P ∩ (Q+ {p}) ⊆ S ⊆ P ∪ (Q+ {p}) ⇒ S = P or S = Q+ {p}. (8.3) If Q ∈ F1, then P ∩ (Q + {p}) ⊆ Q ⊆ P ∪ (Q+ {p}), since p /∈ P . By (8.3), Q = P implying d(P,Q + {p}) = 1. If Q ∈ F2 \ F1, there is R ∈ F1 ∩ F2 such that d(P,R) + d(R,Q) = d(P,Q), since F is well graded. By Theorem 2.2, P ∩Q ⊆ R ⊆ P ∪Q, which implies P ∩ (Q + {p}) ⊆ R+ {p} ⊆ P ∪ (Q+ {p}). By (8.3), R + {p} = Q+ {p}, a contradiction. It is easy to recognize the fundamental sets (cf. Section 4) in an isometric expansion G′ of a partial cube G = 〈F〉. Let P ∈ F1∩F2 and Q = P +{p} ∈ F be two vertices defining an edge in G′ according to Definition 8.1(ii). Clearly, the families F1 and F 2 are the semicubes WPQ and WQP of the graph G ′ (cf. Lemma 5.1) and therefore are convex subsets of F′. The set FPQ is the set of edges defined by p as in Lemma 5.1. In addition, UPQ = F1 ∩ F2 and UQP = {R+ {p} : R ∈ F1 ∩ F2}. Let G be a partial cube induced by a wg-family F of finite subsets of a set X . As before, we assume that ∩F = ∅ and ∪F = X . Let PQ be an edge of G. We may assume that Q = P + {p} for some p /∈ P . Then (see Lemma 5.1) WPQ = {R ∈ F : p /∈ R} and WQP = {R ∈ F : p ∈ R}. Let X ′ = X \ {p} and F′ = {R \ {p} : R ∈ F}. It is clear that the graph G′ induced by the family F′ is isomorphic to the contraction of G defined by the edge PQ. Geometrically, the graph G′ is the orthogonal projection of the graph G along the edge PQ (cf. figures 8.2 and 8.3). Theorem 8.5. (i) A contraction G′ of a partial cube G is a partial cube. (ii) If G is finite, then dimI(G ′) = dimI(G)− 1. Proof. (i) For p ∈ X we define F1 = {R ∈ F : p /∈ R}, F2 = {R ∈ F : p ∈ R}, and F′2 = {R \ {p} ∈ F : p ∈ R}. Note that F1 and F2 are semicubes of G and F′2 is isometric to F2. Hence, F1 and F 2 are wg-families of finite subsets of X ′. We need to prove that F′ = F1 ∪ F 2 is a wg-family. By Theorem 2.3, it suffices to show that d(P,Q) = 1 for any two adjacent sets P,Q ∈ F′. This is true if P,Q ∈ F1 or P,Q ∈ F 2, since these two families are well graded. For P ∈ F1 \ F 2 and Q ∈ F 2 \ F1, the sets P and Q + {p} are not adjacent in F, since F is well graded and Q /∈ F. Hence there is R ∈ F1 such that P ∩ (Q+ {p}) ⊆ R ⊆ P ∪ (Q + {p}) and R 6= P . Since p /∈ R, we have P ∩Q ⊆ R ⊆ P ∪Q. Since R 6= P and R 6= Q, the sets P and Q are not adjacent in F′. The result follows. (ii) If G is a finite partial cube, then, by Theorem 5.2, dimI(G ′) = |X ′| = |X | − 1 = dimI(G)− 1. 9 Conclusion The paper focuses on two themes of a rather general mathematical nature. 1. The characterization problem. It is a common practice in mathematics to characterize a particular class of object in different terms. We present new characterizations of the classes of bipartite graphs and partial cubes, and give new proofs for known characterization results. 2. Constructions. The problem of constructing new objects from old ones is a standard topic in many branches of mathematics. For the class of partial cubes, we discuss operations of forming the Cartesian product, expansion and contraction, and pasting. It is shown that the class of partial cubes is closed under these operations. Because partial cubes are defined as graphs isometrically embeddable into hypercubes, the theory of partial cubes has a distinctive geometric flavor. The three main structures on a graph—semicubes and Djoković’s and Winkler’s relations—are defined in terms of the metric structure on a graph. One can say that this theory is a branch of discrete metric geometry. Not surprisingly, geo- metric structures play an important role in our treatment of the characterization and construction problems. References [1] A.S. Asratian, T.M.J. Denley, and R. Häggkvist, Bipartite Graphs and their Applications, Cambridge University Press, 1998. [2] D. Avis, Hypermetric spaces and the Hamming cone, Canadian Journal of Mathematics 33 (1981) 795–802. [3] L. 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Combinatorics 26 (2005) 585–592, doi: 10.1016/j.ejc.2004.05.001. [12] A. Frank, Connectivity and network flows, in: R.L. Graham, M. Grötshel, and L. Lovász (Eds.), Handbook of Combinatorics, The MIT Press, Cam- bridge, Massachusetts, 1995, pp. 111–177. [13] K. Fukuda and K. Handa, Antipodal graphs and oriented matroids, Dis- crete Mathematics 111 (1993) 245–256. [14] F. Hadlock and F. Hoffman, Manhattan trees, Util. Math. 13 (1978) 55–67. [15] W. Imrich and S. Klavžar, Product Graphs, John Wiley & Sons, 2000. [16] H.M. Mulder, The Interval Function of a Graph, Mathematical Centre Tracts 132, Mathematisch Centrum, Amsterdam, 1980. [17] S. Ovchinnikov, Media theory: representations and examples, Discrete Ap- plied Mathematics, (in review, e-print available at http://arxiv.org/abs/math.CO/0512282). [18] R.I. Roth and P.M. Winkler, Collapse of the metric hierarchy for bipartite graphs, European Journal of Combinatorics 7 (1986) 371–375. [19] M.L.J. van de Vel, Theory of Convex Structures, Elsevier, The Netherlands, 1993. [20] P.M. Winkler, Isometric embedding in products of complete graphs, Dis- crete Appl. Math. 8 (1984) 209–212. http://arxiv.org/abs/math.CO/0512282 Introduction Hypercubes and partial cubes Characterizations Fundamental sets in partial cubes Dimensions of partial cubes Subcubes and Cartesian products Pasting partial cubes Expansions and contractions of partial cubes Conclusion
Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djokovi\'{c}'s and Winkler's relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated.
Introduction A hypercube H(X) on a set X is a graph which vertices are the finite subsets of X ; two vertices are joined by an edge if they differ by a singleton. A partial cube is a graph that can be isometrically embedded into a hypercube. There are three general graph-theoretical structures that play a prominent role in the theory of partial cubes; namely, semicubes, Djoković’s relation θ, and Winkler’s relation Θ. We use these structures, in particular, to characterize bi- partite graphs and partial cubes. The characterization problem for partial cubes was considered as an important one and many characterizations are known. We list contributions in the chronological order: Djoković [9] (1973), Avis [2] (1981), Winkler [20] (1984), Roth and Winkler [18] (1986), Chepoi [6, 7] (1988 and 1994). In the paper, we present new proofs for the results of Djoković [9], Winkler [20], and Chepoi [6], and obtain two more characterizations of partial cubes. http://arxiv.org/abs/0704.0010v1 The paper is also concerned with some ways of constructing new partial cubes from old ones. Properties of subcubes, the Cartesian product of partial cubes, and expansion and contraction of a partial cube are investigated. We introduce a construction based on pasting two graphs together and show how new partial cubes can be obtained from old ones by pasting them together. The paper is organized as follows. Hypercubes and partial cubes are introduced in Section 2 together with two basic examples of infinite partial cubes. Vertex sets of partial cubes are described in terms of well graded families of finite sets. In Section 3 we introduce the concepts of a semicube, Djoković’s θ and Win- kler’s Θ relations, and establish some of their properties. Bipartite graphs and partial cubes are characterized by means of these structures. One more charac- terization of partial cubes is obtained in Section 4, where so-called fundamental sets in a graph are introduced. The rest of the paper is devoted to constructions: subcubes and the Carte- sian product (Section 6), pasting (Section 7), and expansions and contractions (Section 8). We show that these constructions produce new partial cubes from old ones. Isometric and lattice dimensions of new partial cubes are calculated. These dimensions are introduced in Section 5. Few words about conventions used in the paper are in order. The sum (disjoint union) A+B of two sets A and B is the union ({1} ×A) ∪ ({2} ×B). All graphs in the paper are simple undirected graphs. In the notation G = (V,E), the symbol V stands for the set of vertices of the graph G and E stands for its set of edges. By abuse of language, we often write ab for an edge in a graph; if this is the case, ab is an unordered pair of distinct vertices. We denote 〈U〉 the graph induced by the set of vertices U ⊆ V . If G is a connected graph, then dG(a, b) stands for the distance between two vertices a and b of the graph G. Wherever it is clear from the context which graph is under consideration, we drop the subscript G in dG(a, b). A subgraph H ⊆ G is an isometric subgraph if dH(a, b) = dG(a, b) for all vertices a and b of H ; it is convex if any shortest path in G between vertices of H belongs to H . 2 Hypercubes and partial cubes Let X be a set. We denote Pf (X) the set of all finite subsets of X . Definition 2.1. A graph H(X) has the set Pf (X) as the set of its vertices; a pair of vertices PQ is an edge of H(X) if the symmetric difference P∆Q is a singleton. The graph H(X) is called the hypercube on X [9]. If X is a finite set of cardinality n, then the graph H(X) is the n-cube Qn. The dimension of the hypercube H(X) is the cardinality of the set X . The shortest path distance d(P,Q) on the hypercube H(X) is the Hamming distance between sets P and Q: d(P,Q) = |P∆Q| for P,Q ∈ Pf . (2.1) The set Pf (X) is a metric space with the metric d. Definition 2.2. A graph G is a partial cube if it can be isometrically embedded into a hypercube H(X) for some set X . We often identify G with its isometric image in the hypercube H(X), and say that G is a partial cube on the set X . Figure 2.1: A graph and its isometric embedding into Q3. An example of a partial cube and its isometric embedding into the cube Q3 is shown in Figure 2.1. Clearly, a family F of finite subsets of X induces a partial cube on X if and only if for any two distinct subsets P,Q ∈ F there is a sequence R0 = P,R1, . . . , Rn = Q of sets in F such that d(Ri, Ri+1) = 1 for all 0 ≤ i < n, and d(P,Q) = n. (2.2) The families of sets satisfying condition (2.2) are known as well graded fam- ilies of sets [10]. Note that a sequence (Ri) satisfying (2.2) is a shortest path from P to Q in H(X) (and in the subgraph induced by F). Definition 2.3. A family F of arbitrary subsets ofX is a wg-family (well graded family of sets) if, for any two distinct subsets P,Q ∈ F, the set P∆Q is finite and there is a sequence R0 = P,R1, . . . , Rn = Q of sets in F such that |Ri∆Ri+1| = 1 for all 0 ≤ i < n and |P∆Q| = n. Example 2.1. The induced graph can be a partial cube on a different set if the family F is not well graded. Consider, for instance, the family F = {∅, {a}, {a, b}, {a, b, c}, {b, c}} of subsets of X = {a, b, c}. The graph induced by this family is a path of length 4 in the cube Q3 (cf. Figure 2.2). Clearly, F is not well graded. On the other hand, as it can be easily seen, any path is a partial cube. Figure 2.2: A nonisometric path in the cube Q3. Any family F of subsets of X defines a graph GF = (F, EF), where EF = {{P,Q} ⊆ F : |P∆Q| = 1}. Theorem 2.1. The graph GF defined by a family F of subsets of a set X is isomorphic to a partial cube on X if and only if the family F is well graded. Proof. We need to prove sufficiency only. Let S be a fixed set in F. We define a mapping f : F → Pf (X) by f(R) = R∆S for R ∈ F. Then d(f(R), f(T )) = |(R∆S)∆(T∆S)| = |R∆T |. Thus f is an isometric embedding of F into Pf (X). Let (Ri) be a sequence of sets in F such that R0 = P , Rn = Q, |P∆Q| = n, and |Ri∆Ri+1| = 1 for all 0 ≤ i < n. Then the sequence (f(Ri)) satisfies conditions (2.2). The result follows. A set R ∈ Pf (X) is said to be lattice between sets P,Q ∈ Pf (X) if P ∩Q ⊆ R ⊆ P ∪Q. It is metrically between P and Q if d(P,R) + d(R,Q) = d(P,Q). The following theorem is a well-known result about these two betweenness re- lations on Pf (X) (see, for instance, [3]). Theorem 2.2. Lattice and metric betweenness relations coincide on Pf (X). Let F be a family of finite subsets of X . The set of all R ∈ F that are between P,Q ∈ F is the interval I(P,Q) between P and Q in F. Thus, I(P,Q) = F ∩ [P ∩Q,P ∪Q], where [P ∩Q,P ∪Q] is the usual interval in the lattice Pf . Two distinct sets P,Q ∈ F are adjacent in F if J(P,Q) = {P,Q}. If sets P and Q form an edge in the graph induced by F, then P and Q are adjacent in F, but, generally speaking, not vice versa. For instance, in Example 2.1, the vertices ∅ and {b, c} are adjacent in F but do not define an edge in the induced graph (cf. Figure 2.2). The following theorem is a ‘local’ characterization of wg-families of sets. Theorem 2.3. A family F ⊆ Pf (X) is well graded if and only if d(P,Q) = 1 for any two sets P and Q that are adjacent in F. Proof. (Necessity.) Let F be a wg-family of sets. Suppose that P and Q are adjacent in F. There is a sequence R0 = P,R1, . . . , Rn = Q that satisfies conditions (2.2). Since the sequence (Ri) is a shortest path in F, we have d(P, Pi) + d(Pi, Q) = d(P,Q) for all 0 ≤ i ≤ n. Thus, Pi ∈ I(P,Q) = {P,Q}. It follows that d(P,Q) = n = 1. (Sufficiency.) Let P and Q be two distinct sets in F. We prove by induction on n = d(P,Q) that there is a sequence (Ri) ∈ F satisfying conditions (2.2). The statement is trivial for n = 1. Suppose that n > 1 and that the statement is true for all k < n. Let P and Q be two sets in F such that d(P,Q) = n. Since d(P,Q) > 1, the sets P and Q are not adjacent in F. Therefore there exists R ∈ F that lies between P and Q and is distinct from these two sets. Then d(P,R) + d(R,Q) = d(P,Q) and both distances d(P,R) and d(R,Q) are less than n. By the induction hypothesis, there is a sequence (Ri) ∈ F such that P = R0, R = Rj , Q = Rn for some 0 < j < n, satisfying conditions (2.2) for 0 ≤ i < j and j ≤ i < n. It follows that F is a wg-family of sets. We conclude this section with two examples of infinite partial cubes (more examples are found in [17]). Example 2.2. Let Z be the graph on the set Z of integers with edges defined by pairs of consecutive integers. This graph is a partial cube since its vertex set is isometric to the wg-family of intervals {(−∞,m) : m ∈ Z} in Z. Example 2.3. Let us consider Zn as a metric space with respect to the ℓ1- metric. The graph Zn has Zn as the vertex set; two vertices in Zn are connected if they are on the unit distance from each other. We will show in Section 6 (Corollary 6.1) that Zn is a partial cube. 3 Characterizations Only connected graphs are considered in this section. Definition 3.1. Let G = (V,E) be a graph and d be its distance function. For any two adjacent vertices a, b ∈ V let Wab be the set of vertices that are closer to a than to b: Wab = {w ∈ V : d(w, a) < d(w, b)}. Following [11], we call the sets Wab and induced subgraphs 〈Wab〉 semicubes of the graph G. The semicubes Wab and Wba are called opposite semicubes. Remark 3.1. The subscript ab in Wab stands for an ordered pair of vertices, not for an edge of G. In his original paper [9], Djoković uses notation G(a, b) (cf. [8]). We use the notation from [15]. Clearly, two opposite semicubes are disjoint. They can be used to charac- terize bipartite graphs as follows. Theorem 3.1. A graph G = (V,E) is bipartite if and only if the semicubes Wab and Wba form a partition of V for any edge ab ∈ E. Proof. Let us recall that a connected graph G is bipartite if and only if for every vertex x there is no edge ab with d(x, a) = d(x, b) (see, for instance, [1]). For any edge ab ∈ E and vertex x ∈ V we clearly have d(x, a) = d(x, b) ⇔ x /∈ Wab ∪Wba. The result follows. The following lemma is instrumental and will be used frequently in the rest of the paper. Lemma 3.1. Let G = (V,E) be a graph and w ∈ Wab for some edge ab ∈ E. d(w, b) = d(w, a) + 1. Accordingly, Wab = {w ∈ V : d(w, b) = d(w, a) + 1}. Proof. By the triangle inequality, we have d(w, a) < d(w, b) ≤ d(w, a) + d(a, b) = d(w, a) + 1. The result follows, since d takes values in N. There are two binary relations on the set of edges of a graph that play a central role in characterizing partial cubes. Definition 3.2. Let G = (V,E) be a graph and e = xy and f = uv be two edges of G. (i) (Djoković [9]) The relation θ on E is defined by e θf ⇔ f joins a vertex in Wxy with a vertex in Wyx. The notation can be chosen such that u ∈Wxy and v ∈ Wyx. (ii) (Winkler [20]) The relation Θ on E is defined by eΘf ⇔ d(x, u) + d(y, v) 6= d(x, v) + d(y, u). It is clear that both relations θ and Θ are reflexive and Θ is symmetric. Lemma 3.2. The relation θ is a symmetric relation on E. Proof. Suppose that xy θ uv with u ∈ Wxy and v ∈ Wyx. By Lemma 3.1 and the triangle inequality, we have d(u, x) = d(u, y)− 1 ≤ d(u, v) + d(v, y)− 1 = d(v, y) = = d(v, x)− 1 ≤ d(v, u) + d(u, x) − 1 = d(u, x). Hence, d(u, x) = d(v, x) − 1 and d(v, y) = d(u, y)− 1. Therefore, x ∈ Wuv and y ∈ Wvu. It follows that uv θ xy. Lemma 3.3. θ ⊆ Θ. Proof. Suppose that xy θ uv with u ∈Wxy, v ∈ Wyx. By Lemma 3.1, d(x, u) + d(y, v) = d(x, v) − 1 + d(y, u)− 1 6= d(x, v) + d(y, u). Hence, xyΘ uv. Example 3.1. It is easy to verify that θ is the identity relation on the set of edges of the cycle C3. On the other hand, any two edges of C3 stand in the relation Θ. Thus, θ 6= Θ in this case. Bipartite graphs can be characterized in terms of relations θ and Θ as follows. Theorem 3.2. A graph G = (V,E) is bipartite if and only if θ = Θ. Proof. (Necessity.) Suppose that G is a bipartite graph, two edges xy and uv stand in the relation Θ, that is, d(x, u) + d(y, v) 6= d(x, v) + d(y, u), and that edges xy and uv do not stand in the relation θ. By Theorem 3.1, we may assume that u, v ∈ Wxy. By Lemma 3.1, we have d(x, u) + d(y, v) = d(y, u)− 1 + d(x, v) + 1 = d(x, v) + d(y, u), a contradiction. It follows that Θ ⊆ θ. By Lemma 3.3, θ = Θ. (Sufficiency.) Suppose that G is not bipartite. By Theorem 3.1, there is an edge xy such that Wxy ∪Wyx is a proper subset of V . Since G is connected, there is an edge uv with u /∈ Wxy ∪Wyx and v ∈ Wxy ∪Wyx. Clearly, uv does not stand in the relation θ to xy. On the other hand, d(x, u) + d(y, v) 6= d(x, v) + d(y, u), since u /∈ Wxy ∪Wyx and v ∈ Wxy ∪Wyx. Thus, xyΘ uv, a contradiction, since we assumed that θ = Θ. By Theorem 3.2, the relations θ and Θ coincide on bipartite graphs. For this reason we use the relation θ in the rest of the paper. Lemma 3.4. Let G = (V,E) be a bipartite graph such that all its semicubes are convex sets. Then two edges xy and uv stand in the relation θ if and only if the corresponding pairs of mutually opposite semicubes form equal partitions of V : xy θ uv ⇔ {Wxy,Wyx} = {Wuv,Wvu}. Proof. (Necessity) We assume that the notation is chosen such that u ∈ Wxy and v ∈ Wyx. Let z ∈ Wxy ∩Wvu. By Lemma 3.1, d(z, u) = d(z, v) + d(v, u). Since z, u ∈ Wxy and Wxy is convex, we have v ∈ Wxy, a contradiction to the assumption that v ∈Wyx. Thus Wxy ∩Wvu = ∅. Since two opposite semicubes in a bipartite graph form a partition of V , we haveWuv =Wxy andWvu =Wyx. A similar argument shows that Wuv = Wyx and Wvu = Wxy, if u ∈ Wyx and v ∈ Wxy. (Sufficiency.) Follows from the definition of the relation θ. We need another general property of the relation θ (cf. Lemma 2.2 in [15]). Lemma 3.5. Let P be a shortest path in a graph G. Then no two distinct edges of P stand in the relation θ. Proof. Let i < j and xixi+1 and xjxj+1 be two edges in a shortest path P from x0 to xn. Then d(xi, xj) < d(xi, xj+1) and d(xi+1, xj) < d(xi+1, xj+1), so xi, xi+1 ∈ Wxjxj+1 . It follows that edges xixi+1 and xjxj+1 do not stand in the relation θ. The converse statement is true for bipartite graphs (we omit the proof); a counterexample is the cycle C5 which is not bipartite. Lemma 3.6. Let G = (V,E) be a bipartite graph. The following statements are equivalent (i) All semicubes of G are convex. (ii) The relation θ is an equivalence relation on E. Proof. (i) ⇒ (ii). Follows from Lemma 3.4. (ii) ⇒ (i). Suppose that θ is transitive and there is a nonconvex semicube Wab. Then there are two vertices u, v ∈ Wab and a shortest path P from u to v that intersects Wba. This path contains two distinct edges e and f joining vertices of semicubes Wab and Wba. The edges e and f stand in the relation θ to the edge ab. By transitivity of θ, we have e θf . This contradicts the result of Lemma 3.5. Thus all semicubes of G are convex. We now establish some basic properties of partial cubes. Theorem 3.3. Let G = (V,E) be a partial cube. Then (i) G is a bipartite graph. (ii) Each pair of opposite semicubes form a partition of V . (iii) All semicubes are convex subsets of V . (iv) θ is an equivalence relation on E. Proof. We may assume that G is an isometric subgraph of some hypercube H(X), that is, G = (F, EF) for a wg-family F of finite subsets of X . (i) It suffices to note that if two sets in H(X) are connected by an edge then they have different parity. Thus, H(X) is a bipartite graph and so is G. (ii) Follows from (i) and Theorem 3.1. (iii) LetWAB be a semicube of G. By Lemma 3.1 and Theorem 2.2, we have WAB = {S ∈ F : S ∩B ⊆ A ⊆ S ∪B}. Let Q,R ∈WAB and P be a vertex of G such that d(Q,P ) + d(P,R) = d(Q,R). By Theorem 2.2, Q ∩R ⊆ P ⊆ Q ∪R. Since Q,R ∈WAB , we have Q ∩B ⊆ A ⊆ Q ∪B and R ∩B ⊆ A ⊆ R ∪B, which implies P ∩B ⊆ (Q ∪R) ∩B ⊆ A ⊆ (Q ∩R) ∪B ⊆ S ∪B. Hence, P ∈ WAB, and the result follows. (iv) Follows from (iii) and Lemma 3.6. Remark 3.2. Since semicubes of a partial cube G = (V,E) are convex subsets of the metric space V , they are half-spaces in V [19]. This terminology is used in [6, 7]. The following theorem presents four characterizations of partial cubes. The first two are due to Djoković [9] and Winkler [20] (cf. Theorem 2.10 in [15]). Theorem 3.4. Let G = (V,E) be a connected graph. The following statements are equivalent: (i) G is a partial cube. (ii) G is bipartite and all semicubes of G are convex. (iii) G is bipartite and θ is an equivalence relation. (iv) G is bipartite and, for all xy, uv ∈ E, xy θ uv ⇒ {Wxy,Wyx} = {Wuv,Wvu}. (3.1) (v) G is bipartite and, for any pair of adjacent vertices of G, there is a unique pair of opposite semicubes separating these two vertices. Proof. By Lemma 3.6, the statements (ii) and (iii) are equivalent and, by The- orem 3.3, (i) implies both (ii) and (iii). (iii) ⇒ (i). By Theorem 3.1, each pair {Wab,Wba} of opposite semicubes of G form a partition of V . We orient these partitions by calling, in an arbitrary way, one of the two opposite semicubes in each partition a positive semicube. Let us assign to each x ∈ V the set W+(x) of all positive semicubes containing x. In the next paragraph we prove that the family F = {W+(x)}x∈V is well graded and that the assignment x 7→ W+(x) is an isometry between V and F. Let x and y be two distinct vertices of G. We say that a positive semicube Wab separates x and y if either x ∈ Wab, y ∈ Wba or x ∈ Wba, y ∈ Wab. It is clear that Wab separates x and Y if and only if Wab ∈ W +(x)∆W+(y). Let P be a shortest path x0 = x, x1, . . . , xn = y from x to y. By Lemma 3.5, no two distinct edges of P stand in the relation θ. By Lemma 3.4, distinct edges of P define distinct positive semicubes; clearly, these semicubes separate x and y. Let Wab be a positive semicube separating x and y, and, say, x ∈Wab and y ∈Wba. There is an edge f ∈ P that joins vertices in Wab and Wba. Hence, f stands in the relation θ to ab and, by Lemma 3.4, Wab is defined by f . It follows that any semicube inW+(x)∆W+(y) is defined by a unique edge in P and any edge in P defines a semicube in W+(x)∆W+(y). Therefore, d(W+(x),W+(y)) = d(x, y), that is x 7→W+(x) is an isometry. Clearly, F is a wg-family of sets. By Theorem 2.1, the family F is isometric to a wg-family of finite sets. Hence, G is a partial cube. (iv) ⇒ (ii). Suppose that there exist an edge ab such that semicube Wba is not convex. Let p and q be two vertices in Wba such that there is a shortest path P from p to q that intersects Wab. There are two distinct edges xy and uv in P such that x, u ∈ Wab and y, v ∈ Wba. Since ab θ xy and ab θ uv, we have, by (3.1), Wab =Wxy =Wuv. Hence, u ∈ Wxy and v ∈Wyx. By Lemma 3.1, d(x, u) = d(x, v) − 1 = 1 + d(v, y)− 1 = d(v, y), a contradiction, since P is a shortest path from p to q. (ii) ⇒ (iv). Follows from Lemma 3.4. It is clear that (iv) and (v) are equivalent. 4 Fundamental sets in partial cubes Semicubes played an important role in the previous section. In this section we introduce three more classes of useful subsets of graphs. We also establish one more characterization of partial cubes. Let G = (V,E) be a connected graph. For a given edge e = ab ∈ E, we define the following sets (cf. [15, 16]): Fab = {f ∈ E : e θf} = {uv ∈ E : u ∈Wab, v ∈Wba}, Uab = {w ∈Wab : w is adjacent to a vertex in Wba}, Uba = {w ∈Wba : w is adjacent to a vertex in Wab}. The five sets are schematically shown in Figure 4.1. Figure 4.1: Fundamental sets in a partial cube. Remark 4.1. In the case of a partial cube G = (V,E), the semicubes Wab and Wba are complementary half-spaces in the metric space V (cf. Remark 3.2). Then the set Fab can be regarded as a ‘hyperplane’ separating these half-spaces (see [17] where this analogy is formalized in the context of hyperplane arrange- ments). The following theorem generalizes the result obtained in [16] for median graphs (see also [15]). Theorem 4.1. Let ab be an edge of a connected bipartite graph G. If the semicubes Wab and Wba are convex, then the set Fab is a matching and induces an isomorphism between the graphs 〈Uab〉 and 〈Uba〉. Proof. Suppose that Fab is not a matching. Then there are distinct edges xu and xv with, say, x ∈ Uab and u, v ∈ Uba. By the triangle inequality, d(u, v) ≤ 2. Since G does not have triangles, d(u, v) 6= 1. Hence, d(u, v) = 2, which implies that x lies between u and v. This contradicts convexity of Wba, since x ∈ Wab. Therefore Fab is a matching. To show that Fab induces an isomorphism, let xy, uv ∈ Fab and xu ∈ E, where x, u ∈ Uab and y, v ∈ Uba. Since G does not have odd cycles, d(v, y) 6= 2. By the triangle inequality, d(v, y) ≤ d(v, u) + d(u, x) + d(x, y) = 3. Since Wba is convex, d(v, y) 6= 3. Thus d(v, y) = 1, that is, vy is an edge. The result follows by symmetry. By Theorem 3.4(ii), we have the following corollary. Corollary 4.1. Let G = (V,E) be a partial cube. For any edge ab the set Fab is a matching and induces an isomorphism between induced graphs 〈Uab〉 and 〈Uba〉. Figure 4.2: Graph G. Example 4.1. Let G be the graph depicted in Figure 4.2. The set Fab = {ab, xu, yv} is a matching and defines an isomorphism between the graphs induced by subsets Uab = {a, x, y} and Uba = {b, u, v}. The set Wba is not convex, so G is not a partial cube. Thus the converse of Corollary 4.1 does not hold. We now establish another characterization of partial cubes that utilizes a geometric property of families Fab. Theorem 4.2. For a connected graph G the following statements are equivalent: (i) G is a partial cube. (ii) G is bipartite and d(x, u) = d(y, v) and d(x, v) = d(y, u), (4.1) for any ab ∈ E and xy, uv ∈ Fab. Proof. (i)⇒(ii). We may assume that x, u ∈ Wab and y, v ∈ Wba. Since θ is an equivalence relation, we have xy θ uv θab. By Lemma 3.4, Wuv = Wxy = Wab. By Lemma 3.1, d(x, u) = d(x, v) − 1 = d(v, y) + 1− 1 = d(y, v). We also have d(x, v) = d(y, v) + 1 = d(y, u), by the same lemma. (ii)⇒(i). Suppose that G is not a partial cube. Then, by Theorem 3.4, there exist an edge ab such that, say, semicube Wba is not convex. Let p and q be two vertices in Wba such that there is a shortest path P from p to q that intersects Wab. Let uv be the first edge in P which belongs to Fab and xy be the last edge in P with the same property (see Figure 4.3). Figure 4.3: An illustration to the proof of theorem 4.2. Since P is a shortest path, we have d(v, y) = d(v, u) + d(u, x) + d(x, y) 6= d(x, u), which contradicts condition (4.1). Thus all semicubes of G are convex. By Theorem 3.4, G is a partial cube. Remark 4.2. One can say that four vertices satisfying conditions (4.1) define a rectangle in G. Then Theorem 4.2 states that a connected graph is a partial cube if and only if it is bipartite and for any edge ab pairs of edges in Fab define rectangles in G. 5 Dimensions of partial cubes There are many different ways in which a given partial cube can be isometrically embedded into a hypercube. For instance, the graph K2 can be isometrically embedded in different ways into any hypercube H(X) with |X | > 2. Following Djoković [9] (see also [8]), we define the isometric dimension, dimI(G), of a partial cube G as the minimum possible dimension of a hypercube H(X) in which G is isometrically embeddable. Recall (see Section 2) that the dimension of H(X) is the cardinality of the set X . Theorem 5.1. (Theorem 2 in [9].) Let G = (V,E) be a partial cube. Then dimI(G) = |E/θ|, (5.1) where θ is Djoković’s equivalence relation on E and E/θ is the set of its equiv- alence classes (the quotient-set). The quotient-set E/θ can be identified with the family of all distinct sets Fab (see Section 4). If G is a finite partial cube, we may consider it as an isometric subgraph of some hypercube Qn. Then the edges in each family Fab are parallel edges in Qn (cf. Theorem 4.2). This observation essentially proves (5.1) in the finite case. Let G be a partial cube on a set X . The vertex set of G is a wg-family F of finite subsets of X (see Section 2). We define the retraction of F as a family F′ of subsets of X ′ = ∪F \ ∩F consisting of the intersections of sets in F with X ′. It is clear that F′ satisfies conditions ∩ F′ = ∅ and ∪ F′ = X ′. (5.2) Proposition 5.1. The partial cubes induced by a wg-family F and its retraction F′ are isomorphic. Proof. It suffices to prove that metric spaces F and F′ are isometric. Clearly, α : P 7→ P ∩X ′ is a mapping from F onto F′. For P,Q ∈ F, we have (P ∩X ′)∆(Q ∩X ′) = (P∆Q) ∩X ′ = (P∆Q) ∩ (∪F \ ∩F) = P∆Q. Thus, d(α(P ), α(Q)) = d(P,Q). Consequently, α is an isometry. Let G be a partial cube on some set X induced by a wg-family F satisfying conditions (5.2), and let PQ be an edge of G. By definition, there is x ∈ X such that P∆Q = {x}. The following two lemmas are instrumental. Lemma 5.1. Let PQ be an edge of a partial cube G on X and let P∆Q = {x}. The two sets {R ∈ F : x ∈ R} and {R ∈ F : x /∈ R} form the same bipartition of the family F as semicubes WPQ and WQP . Proof. We may assume that Q = P + {x}. Then, for any R ∈ F, R∆Q = R∆(P + {x}) = (R∆P ) + {x}, if x ∈ R, R∆P, if x /∈ R. Hence, |R∆P | < |R∆Q| if and only if x ∈ R. It follows that WPQ = {R ∈ F : x ∈ R}. A similar argument shows that WQP = {R ∈ F : x /∈ R}. Lemma 5.2. If F is a wg-family of sets satisfying conditions (5.2), then for any x ∈ X there are sets P,Q ∈ F such that P∆Q = {x}. Proof. By conditions 5.2, for a given x ∈ X there are sets S and T in F such that x ∈ S and x /∈ T . Let R0 = S,R1, . . . , Rn = T be a sequence of sets in F satisfying conditions (2.2). It is clear that there is i such that x ∈ Ri and x /∈ Ri+1. Hence, Ri∆Ri+1 = {x}, so we can choose P = Ri and Q = Ri+1. By Lemmas 5.1 and 5.2, there is one-to-one correspondence between the set X and the quotient-set E/θ. From Theorem 5.1 we obtain the following result. Theorem 5.2. Let F be a wg-family of finite subsets of a set X such that ∩F = ∅ and ∪F = X, and let G be a partial cube on X induced by F. Then dimI(G) = |X |. Clearly, a graph which is isometrically embeddable into a partial cube is a partial cube itself. We will show in Section 6 (Corollary 6.1) that the integer lattice Zn is a partial cube. Thus a graph which is isometrically embeddable into an integer lattice is a partial cube. It follows that a finite graph is a partial cube if and only if it is embeddable in some integer lattice. Examples of infinite partial cubes isometrically embeddable into a finite dimensional integer lattice are found in [17]. We call the minimum possible dimension n of an integer lattice Zn, in which a given graph G is isometrically embeddable, its lattice dimension and denote it dimZ(G). The lattice dimension of a partial cube can be expressed in terms of maximum matchings in so-called semicube graphs [11]. Definition 5.1. The semicube graph Sc(G) has all semicubes in G as the set of its vertices. Two vertices Wab and Wcd are connected in Sc(G) if Wab ∪Wcd = V and Wab ∩Wcd 6= ∅. (5.3) If G is a partial cube, then condition (5.3) is equivalent to each of the two equivalent conditions: Wba ⊂Wcd ⇔ Wdc ⊂Wab, (5.4) where ⊂ stands for the proper inclusion. Theorem 5.3. (Theorem 1 in [11].) Let G be a finite partial cube. Then dimZ(G) = dimI(G) − |M |, where M is a maximum matching in the semicube graph Sc(G). Example 5.1. Let G be the graph shown in Figure 2.1. It is easy to see that dimI(G) = 3 and dimZ(G) = 2. Example 5.2. Let T be a tree with n edges and m leaves. Then dimI(T ) = n and dimZ(T ) = ⌈m/2⌉ (cf. [8] and [14], respectively). Example 5.3. For the cycle C6 we have (see Figure 8.2) dimI(C6) = dimZ(C6) = 3. 6 Subcubes and Cartesian products Let G be a partial cube. We say that G′ is a subcube of G if it is an isometric subgraph of G. Clearly, a subcube is itself a partial cube. The converse does not hold; a subgraph of a graph G can be a partial cube but not an isometric subgraph of G (cf. Example 2.1). If G′ is a subcube of a partial cube G, then dimI(G ′) ≤ dimI(G) and dimZ(G ′) ≤ dimZ(G). In general, the two inequalities are not strict. For instance, the cycle C6 is an isometric subgraph of the cube Q3 (see Figure 8.2) dimI(C6) = dimZ(C6) = dimI(Q3) = dimZ(Q3) = 3. Semicubes of a partial cube are examples of subcubes. Indeed, by Theo- rem 3.4, semicubes are convex subgraphs and therefore isometric. In general, the converse is not true; a path connecting two opposite vertices in C6 is an isometric subgraph but not a convex one. Another common way of constructing new partial cubes from old ones is by forming their Cartesian products (see [15] for details and proofs). Definition 6.1. Given two graphs G1 = (V1, E1) and G2 = (V2, E2), their Cartesian product G = G1�G2 has vertex set V = V1 × V2; a vertex u = (u1, u2) is adjacent to a vertex v = (v1, v2) if and only if u1v1 ∈ E1 and u2 = v2, or u1 = v1 and u2v2 ∈ E2. The operation � is associative, so we can write G = G1� · · ·�Gn = for the Cartesian product of graphs G1, . . . , Gn. A Cartesian product i=1Gi is connected if and only if the factors are connected. Then we have dG(u, v) = dGi(ui, vi). (6.1) Example 6.1. Let {Xi} i=1 be a family of sets and Y = i=1 be their sum. Then the Cartesian product of the hypercubes H(Xi) is isomorphic to the hy- percube H(Y ). The isomorphism is established by the mapping f : (P1, . . . , Pn) 7→ Formula (6.1) yields immediately the following results. Proposition 6.1. Let Hi be isometric subgraphs of graphs Gi for all 1 ≤ i ≤ n. Then the Cartesian product i=1Hi is an isometric subgraph of the Cartesian product i=1Gi. Corollary 6.1. The Cartesian product of a finite family of partial cubes is a partial cube. In particular, the integer lattice Zn (cf. Examples 2.2 and 2.3) is a partial cube. The results of the next two theorems can be easily extended to arbitrary finite products of finite partial cubes. Theorem 6.1. Let G = G1�G2 be the Cartesian product of two finite partial cubes. Then dimI(G) = dimI(G1) + dimI(G2). Proof. We may assume that G1 (resp. G2) is induced by a wg-family F1 (resp. F2) of subsets of a finite set X1 (resp. X2) such that ∩F1 = ∅ and ∪F1 = X1 (resp. ∩F2 = ∅ and ∪F2 = X1) (see Section 5). By Theorem 5.2, dimI(G1) = |X1| and dimI(G2) = |X2|. It is clear that the graph G is induced by the wg-family F = F1 +F2 of subsets of the set X = X1 + X2 (cf. Example 6.1) with ∩F = ∅, ∪F = X . By Theorem 5.2, dimI(G) = |X | = |X1|+ |X2| = dimI(G1) + dimI(G2). Theorem 6.2. Let G = (V,E) be the Cartesian product of two finite partial cubes G1 = (V1, E1) and G2 = (V2, E2). Then dimZ(G) = dimZ(G1) + dimZ(G2). Proof. Let W(a,b)(c,d) be a semicube of the graph G. There are two possible cases: (i) c = a, bd ∈ E2. Let (x, y) be a vertex of G. Then, by (6.1), dG((x, y), (a, b)) = dG1(x, a) + dG2(y, b) dG((x, y), (c, d)) = dG1(x, c) + dG2(y, d). Hence, dG((x, y), (a, b)) < dG((x, y), (c, d)) ⇔ dG2(y, b) < dG2(y, d). It follows that W(a,b)(c,d) = V1 ×Wbd. (6.2) (ii) d = b, ac ∈ E1. Like in (i), we have W(a,b)(c,d) =Wac × V2. (6.3) Clearly, two semicubes given by (6.2) form an edge in the semicube graph Sc(G) if and only if their second factors form an edge in the semicube graph Sc(G2). The same is true for semicubes in the form (6.3) with respect to their first factors. It is also clear that semicubes in the form (6.2) and in the form (6.3) are not connected by an edge in Sc(G). Therefore the semicube graph Sc(G) is isomorphic to the disjoint union of semicube graphs Sc(G1) and Sc(G2). If M1 is a maximum matching in Sc(G1) and M2 is a maximum matching in Sc(G2), then M =M1 ∪M2 is a maximum matching in Sc(G). The result follows from theorems 5.3 and 6.1. Remark 6.1. The result of Corollary 6.1 does not hold for infinite Cartesian products of partial cubes, as these products are disconnected. On the other hand, it can be shown that arbitrary weak Cartesian products (connected com- ponents of Cartesian products [15]) of partial cubes are partial cubes. 7 Pasting partial cubes In this section we use the set pasting technique [5, ch.I, §2.5] to build new partial cubes from old ones. Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs, H1 = (U1, F1) and H2 = (U2, F2) be two isomorphic subgraphs of G1 and G2, respectively, and ψ : U1 → U2 be a bijection defining an isomorphism between H1 and H2. The bijection ψ defines an equivalence relation R on the sum V1+V2 as follows: any element in (V1 \U1)∪ (V2 \U2) is equivalent to itself only and elements u1 ∈ U1 and u2 ∈ U2 are equivalent if and only if u2 = ψ(u1). We say that the quotient set V = (V1 + V2)/R is obtained by pasting together the sets V1 and V2 along the subsets U1 and U2. Since the graphs H1 and H2 are isomorphic, the pasting of the sets V1 and V2 can be naturally extended to a pasting of sets of edges E1 and E2 resulting in the set E of edges joining vertices in V . We say that the graph G = (E, V ) is obtained by pasting together the graphs G1 and G2 along the isomorphic subgraphs H1 and H2. The pasting construction allows for identifying in a natural way the graphs G1 and G2 with subgraphs of G, and the isomorphic graphs H1 and H2 with a common subgraph H of both graphs G1 and G2. We often follow this convention below. Remark 7.1. Note that in the above construction the resulting graph G de- pends not only on graphs G1 and G2 and their isomorphic subgraphs H1 and H2 but also on the bijection ψ defining an isomorphism from H1 onto H2 (see the drawings in Figures 7.1 and 7.2). Figure 7.1: Pasting of two trees. Figure 7.2: Another pasting of the same trees. In general, pasting of two partial cubes G1 and G2 along two isomorphic subgraphs H1 and H2 does not produce a partial cube even under strong as- sumptions about these subgraphs as the next example illustrates. Figure 7.3: Pasting partial cubes G1 and G2. Example 7.1. Pasting of two partial cubes G1 = C6 and G2 = C6 along subgraphs H1 and H2 is shown in Figure 7.3. The resulting graph G is not a partial cube. Indeed, the semicubeWab is not a convex set. Note that subgraphs H1 and H2 are convex subgraphs of the respective partial cubes. In this section we study two simple pastings of connected graphs together, the vertex-pasting and the edge-pasting, and show that these pastings produce partial cubes from partial cubes. We also compute the isometric and lattice dimensions of the resulting graphs. Let G1 = (V1, E1) and G2 = (V2, E2) be two connected graphs, a1 ∈ V1, a2 ∈ V2, and H1 = ({a1},∅), H2 = ({a2},∅). Let G be the graph obtained by pasting G1 and G2 along subgraphs H1 and H2. In this case we say that the graph G is obtained from graphs G1 and G2 by vertex-pasting. We also say that G is obtained from G1 and G2 by identifying vertices a1 and a2. Figure 7.4 illustrates this construction. Note that the vertex a = {a1, a2} is a cut vertex of G, since G1 ∪ G2 = G and G1 ∩ G2 = {a}. (We follow our convention and identify graphs G1 and G2 with subgraphs of G.) Figure 7.4: An example of vertex-pasting. In what follows we use superscripts to distinguish subgraphs of the graphs G1 and G2. For instance, W stands for the semicube of G2 defined by two adjacent vertices a, b ∈ V2. Theorem 7.1. A graph G = (V,E) obtained by vertex-pasting from partial cubes G1 = (V1, E1) and G2 = (V2, E2) is a partial cube. Proof. We denote a = {a1, a2} the vertex of G obtained by identifying vertices a1 ∈ V1 and a2 ∈ V2. Clearly, G is a bipartite graph. Let xy be an edge of G. Without loss of generality we may assume that xy ∈ E1 and a ∈ Wxy. Note that any path between vertices in V1 and V2 must go through a. Since a ∈Wxy, we have, for any v ∈ V2, d(v, x) = d(v, a) + d(a, x) < d(v, a) + d(a, y) = d(v, y), which implies V2 ⊆ Wxy and Wyx ⊆ V1. It follows that Wxy = W xy ∪ V2 and Wyx = W yx . The sets W xy , W yx and V2 are convex subsets of V . Since xy ∩ V2 = {a}, the set Wxy = W xy ∪ V2 is also convex. By Theorem 3.4(ii), the graph G is a partial cube. The vertex-pasting construction introduced above can be generalized as follows. Let G = {Gi = (Vi, Ei)}i∈J be a family of connected graphs and A = {ai ∈ Gi}i∈J be a family of distinguished vertices of these graphs. Let G be the graph obtained from the graphs Gi by identifying vertices in the set A. We say that G is obtained by vertex-pasting together the graphs Gi (along the set A). Example 7.2. Let J = {1, . . . , n} with n ≥ 2, G = {Gi = ({ai, bi}, {aibi})}i∈J , and A = {ai}i∈J . Clearly, each Gi is K2. By vertex-pasting these graphs along A, we obtain the n-star graph K1,n. Since the star K1,n is a tree it can be also obtained from K1 by successive vertex-pasting as in Example 7.3. Example 7.3. Let G1 be a tree and G2 = K2. By vertex-pasting these graphs we obtain a new tree. Conversely, let G be a tree and v be its leaf. Let G1 be a tree obtained from G by deleting the leaf v. Clearly, G can be obtained by vertex-pasting G1 and K2. It follows that any tree can obtained from the graph K1 by successive vertex-pasting of copies of K2 (cf. Theorem 2.3(e) in [12]). Any connected graph G can be constructed by successive vertex-pasting of its blocks using its block cut-vertex tree [4] structure. Let G1 be an endblock of G with a cut vertex v and G2 be the union of the remaining blocks of G. Then G can be obtained from G1 and G2 by vertex-pasting along the vertex v. It follows that any connected graph can be obtained from its blocks by successive vertex-pastings. Let G = (V,E) be a partial cube. We recall that the isometric dimension dimI(G) of G is the cardinality of the quotient set E/θ, where θ is Djoković’s equivalence relation on the set E (cf. formula (5.1)). Theorem 7.2. Let G = (V,E) be a partial cube obtained by vertex-pasting together partial cubes G1 = (V1, E1) and G2 = (V2, E2). Then dimI(G) = dimI(G1) + dimI(G2). Proof. It suffices to prove that there are no edges xy ∈ E1 and uv ∈ E2 which are in Djoković’s relation θ with each other. Suppose that G1 and G2 are vertex-pasted along vertices a1 ∈ E1 and a2 ∈ E2 and let a = {a1, a2} ∈ E. Let xy ∈ E1 and uv ∈ E2 be two edges in E. We may assume that u ∈ Wxy. Since a is a cut-vertex of G and u ∈Wxy, we have d(u, a) + d(a, x) = d(u, x) < d(u, y) = d(u, a) + d(a, y). Hence, d(a, x) < d(a, y), which implies d(v, x) = d(v, a) + d(a, x) < d(v, a) + d(a, y) = d(v, y). It follows that v ∈ Wxy. Therefore the edge xy does not stand in the relation θ to the vertex uv. The next result follows immediately from the previous theorem. Note that blocks of a partial cube are partial cubes themselves. Corollary 7.1. Let G be a partial cube and {G1, . . . , Gn} be the family of its blocks. Then dimI(G) = dimI(Gi). In the case of the lattice dimension of a partial cube we can claim only much weaker result than one stated in Theorem 7.2 for the isometric dimension. We omit the proof. Theorem 7.3. Let G be a partial cube obtained by vertex-pasting together partial cubes G1 and G2. Then max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) ≤ dimZ(G1) + dimZ(G2). The following example illustrate possible cases for inequalities in Theo- rem 7.3. Let us recall that the lattice dimension of a tree with m leaves is ⌈m/2⌉ (cf. [14]). Example 7.4. The star K1,6 can be obtained from the stars K1,2 and K1,4 by vertex-pasting these two stars along their centers. Clearly, max{dimZ(K1,2), dimZ(K1,4)} < dimZ(K1,6) = dimZ(K1,2) + dimZ(K1,4). The same star K1,6 is obtained from two copies of the star K1,3 by vertex- pasting along their centers. We have dimZ(K1,3) = 2, dimZ(K1,6) = 3, so max{dimZ(K1,3), dimZ(K1,3)} < dimZ(K1,6) < dimZ(K1,3) + dimZ(K1,3). Let us vertex-paste two stars K1,3 along their two leaves. The resulting graph T is a tree with four vertices. Therefore, max{dimZ(K1,3), dimZ(K1,3)} = dimZ(T ) < dimZ(K1,3) + dimZ(K1,3). We now consider another simple way of pasting two graphs together. Let G1 = (V1, E1) and G2 = (V2, E2) be two connected graphs, a1b1 ∈ E1, a2b2 ∈ E2, and H1 = ({a1, b1}, {a1b1}), H2 = ({a2, b2}, {a2b2}). Let G be the graph obtained by pasting G1 and G2 along subgraphs H1 and H2. In this case we say that the graph G is obtained from graphs G1 and G2 by edge-pasting. Figures 7.1, 7.2, and 7.5 illustrate this construction. Figure 7.5: An example of edge-pasting. As before, we identify the graphs G1 and G2 with subgraphs of the graph G and denote a = {a1, a2}, b = {b1, b2} the two vertices obtained by pasting together vertices a1 and a2 and, respectively, b1 and b2. The edge ab ∈ E is obtained by pasting together edges a1b1 ∈ E1 and a2b2 ∈ E2 (cf. Figure 7.5). Then G = G1∪G2, V1∩V2 = {a, b} and E1∩E2 = {ab}. We use these notations in the rest of this section. Proposition 7.1. A graph G obtained by edge-pasting together bipartite graphs G1 and G2 is bipartite. Proof. Let C be a cycle in G. If C ⊆ G1 or C ⊆ G2, then the length of C is even, since the graphs G1 and G2 are bipartite. Otherwise, the vertices a and b separate C into two paths each of odd length. Therefore C is a cycle of even length. The result follows. The following lemma is instrumental; it describes the semicubes of the graph G in terms of semicubes of graphs G1 and G2. Lemma 7.1. Let uv be an edge of G. Then (i) For uv ∈ E1, a, b ∈ Wuv ⇒ Wuv =W uv ∪ V2, Wvu =W (ii) For uv ∈ E2, a, b ∈ Wuv ⇒ Wuv =W uv ∪ V1, Wvu =W (iii) a ∈ Wuv, b ∈Wvu ⇒ Wuv =Wab. Figure 7.6: Edge-pasting of graphs G1 and G2. Proof. We prove parts (i) and (iii) (see Figure 7.6). (i) Since any path from w ∈ V2 to u or v contains a or b and a, b ∈Wuv, we have w ∈Wuv. Hence, Wuv =W uv ∪ V2 and Wvu =W (iii) Since ab θ uv in G1, we have W uv = W , by Theorem 3.4(iv). Let w be a vertex in W uv . Then, by the triangle inequality, d(w, u) < d(w, v) ≤ d(w, b) + d(b, v) < d(w, b) + d(b, u). Since any shortest path from w to u contains a or b, we have d(w, a) + d(a, u) = d(w, u). Therefore, d(w, a) + d(a, u) < d(w, b) + d(b, u). Since ab θ uv in G1, we have d(a, u) = d(b, v), by Theorem 4.2. It follows that d(w, a) < d(w, b), that is, w ∈ W . We proved that W uv ⊆ W symmetry, W vu ⊆ W . Since two opposite semicubes form a partition of V2, we have W uv =W . The result follows. Theorem 7.4. A graph G obtained by edge-pasting together partial cubes G1 and G2 is a partial cube. Proof. By Theorem 3.4(ii) and Proposition 7.1, we need to show that for any edge uv of G the semicube Wuv is a convex subset of V . There are two possible cases. (i) uv = ab. The semicube Wab is the union of semicubes W and W which are convex subsets of V1 and V2, respectively. It is clear that any shortest path connecting a vertex in W with a vertex in W contains vertex a and therefore is contained in Wab. Hence, Wab is a convex set. A similar argument proves that the set Wba is convex. (ii) uv 6= ab. We may assume that uv ∈ E1. To prove that the semicube Wuv is a convex set, we consider two cases. (a) a, b ∈ Wuv. (The case when a, b ∈ Wvu is treated similarly.) By Lemma 7.1(i), the semicube Wuv is the union of the semicube W uv and the set V2 which are both convex sets. Any shortest path P from a vertex in V2 to a vertex in W uv contains either a or b. It follows that P ⊆ W uv ∪ V2 = Wuv. Therefore the semicube Wuv is convex. (b) a ∈ Wuv, b ∈ Wvu. (The case when b ∈ Wuv , a ∈ Wvu is treated similarly.) By Lemma 7.1(ii), Wuv = Wab. The result follows from part (i) of the proof. Theorem 7.5. Let G be a graph obtained by edge-pasting together finite partial cubes G1 and G2. Then dimI(G) = dimI(G1) + dimI(G2)− 1. Proof. Let θ, θ1, and θ2 be Djoković’s relations on E, E1, and E2, respectively. By Lemma 7.1, for uv, xy ∈ E1 (resp. uv, xy ∈ E2) we have uv θ xy ⇔ uv θ1xy (resp. uv θ xy ⇔ uv θ2xy). Let uv ∈ E1, xy ∈ E2, and uv θ xy. Suppose that (uv, ab) /∈ θ. We may assume that a, b ∈ Wuv . By Lemma 7.1(i), V2 ⊂ Wuv, a contradiction, since xy ∈ E2. Hence, uv θ xy θ ab. It follows that each equivalence class of the relation θ is either an equivalence class of θ1, an equivalence class of θ2 or the class containing the edge ab. Therefore |E/θ| = |E1/θ1|+ |E2/θ2| − 1. The result follows, since the isometric dimension of a partial cube is equal to the cardinality of the set of equivalence classes of Djoković’s relation (formula (5.1)). We need some results about semicube graphs in order to prove an analog of Theorem 7.3 for a partial cube obtained by edge-pasting of two partial cubes. Lemma 7.2. Let G be a partial cube and WpqWuv , WqpWxy be two edges in the graph Sc(G). Then WxyWuv is an edge in Sc(G). Proof. By condition (5.4), Wqp ⊂ Wuv and Wyx ⊂ Wqp. Hence, Wyx ⊂ Wuv. By the same condition, WxyWuv ∈ Sc(G). As before, we identify partial cubes G1 and G2 with subgraphs of the partial cube G. Then G1 ∪G2 = G and G1 ∩G2 = ({a, b}, {ab}) = K2 (cf. Figure 7.6). Lemma 7.3. Let G be a partial cube obtained by edge-pasting together partial cubes G1 and G2. Let W xy (resp. W xy ) be an edge in the semicube Sc(G1) (resp. Sc(G2)). Then WuvWxy is an edge in Sc(G). Figure 7.7: Semicubes forming an edge in Sc(G1). Proof. It suffices to consider the case of Sc(G1) (see Figure 7.7). By condi- tion (5.4),W vu ⊂W xy andW yx ⊂W uv . Suppose that a ∈ W vu and b ∈W (the case when b ∈ W vu and a ∈ W yx is treated similarly). Then ab θ1xy and ab θ1uv. By transitivity of θ1, we have uv θ1xy, a contradiction, since semicubes uv and W xy are distinct. Therefore we may assume that, say, a, b ∈ W Then, by Lemma 7.1, Wvu = W vu ⊂ V1. Since W vu ⊂ W xy ⊆ Wxy, we have Wvu ⊂Wxy. By condition (5.4), WuvWxy is an edge in Sc(G). Lemma 7.4. LetM1 andM2 be matchings in graphs Sc(G1) and Sc(G2). There is a matching M in Sc(G) such that |M | ≥ |M1|+ |M2| − 1. Proof. By Lemma 7.3, M1 and M2 induce matchings in Sc(G) which we denote by the same symbols. The intersection M1 ∩M2 is either empty or a subgraph of the empty graph with vertices Wab and Wba. If M1 ∩M2 is empty, then M = M1 ∪M2 is a matching in Sc(G) and the result follows. If M1 ∩M2 is an empty graph with a single vertex, say, in M1, we remove fromM1 the edge that has this vertex as its end vertex, resulting in the matching M ′1. Clearly, M =M 1 ∪M2 is a matching in Sc(G) and |M | = |M1|+ |M2| − 1. Suppose now that M1 ∩M2 is the empty graph with vertices Wab and Wba. Let WabWuv, WbaWpq (resp. WabWxy, WbaWrs) be edges in M1 (resp. M2). By Lemma 7.2, WxyWrs is an edge in Sc(G2). Let us replace edgesWabWxy and WbaWrs in M2 by a single edge WxyWrs, resulting in the matching M 2. Then M =M1 ∪M 2 is a matching in Sc(G) and |M | = |M1|+ |M2| − 1. Corollary 7.2. Let M1 and M2 be maximum matchings in Sc(G1) and Sc(G2), respectively, and M be a maximum matching in Sc(G). Then |M | ≥ |M1|+ |M2| − 1. (7.1) By Theorem 5.3, we have dimI(G1) = dimZ(G1) + |M1|, dimI(G2) = dimZ(G2) + |M2|, dimI(G) = dimZ(G) + |M |, where M1 and M2 are maximum matchings in Sc(G1) and Sc(G2), respectively, and M is a maximum matching in Sc(G). Therefore, by Theorem 7.5 and (7.1), we have the following result (cf. Theorem 7.3). Theorem 7.6. Let G be a partial cube obtained by edge-pasting from partial cubes G1 and G2. Then max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) ≤ dimZ(G1) + dimZ(G2). Example 7.5. Let us consider two edge-pastings of the stars G1 = K1,3 and G2 = K1,3 of lattice dimension 2 shown in figures 7.1 and 7.2. In the first case the resulting graph is the star G = K1,5 of lattice dimension 3. Then we have max{dimZ(G1), dimZ(G2)} < dimZ(G) < dimZ(G1) + dimZ(G2). In the second case the resulting graph is a tree with 4 leaves. Therefore, max{dimZ(G1), dimZ(G2)} = dimZ(G) < dimZ(G1) + dimZ(G2). Let c1a1 and c2a2 be edges of stars G1 = K1,4 and G2 = K1,4 (each of which has lattice dimension 2), where c1 and c2 are centers of the respective stars. Let us edge-paste these two graphs by identifying c1 with c2 and a1 with a2, respectively. The resulting graph G is the star K1,7 of lattice dimension 4. Thus, max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) = dimZ(G1) + dimZ(G2). 8 Expansions and contractions of partial cubes The graph expansion procedure was introduced by Mulder in [16], where it is shown that a graph is a median graph if and only if it can be obtained from K1 by a sequence of convex expansions (see also [15]). A similar result for partial cubes was established in [6] (see also [7]) as a corollary to a more general result concerning isometric embeddability into Hamming graphs; it was also established in [13] in the framework of oriented matroids theory. In this section we investigate properties of (isometric) expansion and con- traction operations and, in particular, prove in two different ways that a graph is a partial cube if and only if it can be obtained from the graph K1 by a sequence of expansions. A remark about notations is in order. In the product {1, 2} × (V1 ∪ V2), we denote V ′i = {i} × Vi and x i = (i, x) for x ∈ Vi, where i, j = 1, 2. Definition 8.1. Let G = (V,E) be a connected graph, and let G1 = (V1, E1) and G2 = (V2, E2) be two isometric subgraphs of G such that G = G1 ∪ G2. The expansion of G with respect to G1 and G2 is the graph G ′ = (V ′, E′) constructed as follows from G (see Figure 8.1): (i) V ′ = V1 + V2 = V 1 ∪ V (ii) E′ = E1 + E2 +M , where M is the matching x∈V1∩V2 {x1x2}. In this case, we also say that G is a contraction of G′. Figure 8.1: Expansion/contraction processes. It is clear that the graphs G1 and 〈V 1〉 are isomorphic, as well as the graphs G2 and 〈V We define a projection p : V ′ → V by p(xi) = x for x ∈ V . Clearly, the restriction of p to V ′1 is a bijection p1 : V 1 → V1 and its restriction to V 2 is a bijection p2 : V 2 → V2. These bijections define isomorphisms 〈V 1〉 → G1 and 〈V ′2〉 → G2. Let P ′ be a path in G′. The vertices of G obtained from the vertices in P ′ under the projection p define a walk P in G; we call this walk P the projection of the path P ′. It is clear that ℓ(P ) = ℓ(P ′), if P ′ ⊆ 〈V ′1〉 or P ′ ⊆ 〈V ′2〉. (8.1) In this case, P is a path in G and either P = p1(P ′) or P = p2(P ′). On the other hand, ℓ(P ) < ℓ(P ′), if P ′ ∩ 〈V ′1〉 6= ∅ and P ′ ∩ 〈V ′2 〉 6= ∅, (8.2) and P is not necessarily a path. We will frequently use the results of the following lemma in this section. Lemma 8.1. (i) For u1, v1 ∈ V ′1 , any shortest path Pu1v1 in G ′ belongs to 〈V ′1 〉 and its projection Puv = p1(Pu1v1) is a shortest path in G. Accordingly, dG′(u 1, v1) = dG(u, v) and 〈V ′1〉 is a convex subgraph of G ′. A similar statement holds for u2, v2 ∈ V ′2 . (ii) For u1 ∈ V ′1 and v 2 ∈ V ′2 , dG′(u 1, v2) = dG(u, v) + 1. Let Pu1v2 be a shortest path in G ′. There is a unique edge x1x2 ∈M such that x1, x2 ∈ Pu1v2 and the sections Pu1x1 and Px2v2 of the path Pu1v2 are shortest paths in 〈V ′1 〉 and 〈V 2 〉, respectively. The projection Puv of Pu1v2 in G ′ is a shortest path in G. Proof. (i) Let Pu1v1 be a path in G ′ that intersects V ′2 . Since 〈V1〉 is an isometric subgraph of G, there is a path Puv in G that belongs to 〈V1〉. Then p 1 (Puv) is a path in 〈V ′1 〉 of the same length as Puv. By (8.1) and (8.2), ℓ(p−11 (Puv)) < ℓ(Pu1v1). Therefore any shortest path Pu1v1 in G ′ belongs to 〈V ′1 〉. The result follows. (ii) Let Pu1v2 be a shortest path in G ′ and Puv be its projection to V . By (8.2), dG′(u 1, v2) = ℓ(Pu1v2) > ℓ(Puv) ≥ dG(u, v). Since there is no edge of G joining vertices in V1 \ V2 and V2 \ V1, a shortest path in G from u to v must contain a vertex x ∈ V1 ∩ V2. Since G1 and G2 are isometric subgraphs, there are shortest paths Pux in G1 and Pxv in G2 such that their union is a shortest path from u to v. Then, by the triangle inequality and part (i) of the proof, we have (cf. Figure 8.1) dG′(u 1, v2) ≤ dG′(u 1, x1) + dG′(x 1, x2) + dG′(x 2, v2) = dG(u, v) + 1. The last two displayed formulas imply dG′(u 1, v2) = dG(u, v) + 1. Since u1 ∈ V ′1 and v 2 ∈ V ′2 the path Pu1v2 must contain an edge, say x 1x2, in M . Since this path is a shortest path in G′, this edge is unique. Then the sec- tions Pu1x1 and Px2v2 of Pu1v2 are shortest paths in 〈V 1 〉 and 〈V 2〉, respectively. Clearly, Puv is a shortest path in G. Let a1a2 be an edge in the matchingM = ∪x∈V1∩V2{x 1x2}. This edge defines five fundamental sets (cf. Section 4): the semicubes Wa1a2 and Wa2a1 , the sets of vertices Ua1a2 and Ua2a1 , and the set of edges Fa1a2 . The next theorem follows immediately from Lemma 8.1. It gives a hint to a connection between the expansion process and partial cubes. Theorem 8.1. Let G′ be an expansion of a connected graph G and notations are chosen as above. Then (i) Wa1a2 = V 1 and Wa2a1 = V 2 are convex semicubes of G (ii) Fa1a2 =M defines an isomorphism between induced subgraphs 〈Ua1a2〉 and 〈Ua2a1〉, which are isomorphic to the subgraph G1 ∩G2. The result of Theorem 8.1 justifies the following constructive definition of the contraction process. Definition 8.2. Let ab be an edge of a connected graph G′ = (V ′, E′) such (i) semicubes Wab and Wba are convex and form a partition of V (ii) the set Fab is a matching and defines an isomorphism between subgraphs 〈Uab〉 and 〈Uba〉. A graph G obtained from the graphs 〈Wab〉 and 〈Wba〉 by pasting them along subgraphs 〈Uab〉 and 〈Uba〉 is said to be a contraction of the graph G Remark 8.1. If G′ is bipartite, then semicubesWab andWba form a partition of its vertex set. Then, by Theorem 4.1, condition (i) implies condition (ii). Thus any pair of opposite convex semicubes in a connected bipartite graph defines a contraction of this graph. By Theorem 8.1, a graph is a contraction of its expansion. It is not difficult to see that any connected graph is also an expansion of its contraction. The following three examples give geometric illustrations for the expansion and contraction procedures. Example 8.1. Let a and b be two opposite vertices in the graph G = C4. Clearly, the two distinct paths P1 and P2 from a to b are isometric subgraphs of G defining an expansion G′ = C6 of G (see Figure 8.2). Note that P1 and P2 are not convex subsets of V . Example 8.2. Another isometric expansion of the graph G = C4 is shown in Figure 8.3. Here, the path P1 is the same as in the previous example and G2 = G. Example 8.3. Lemma 8.1 claims, in particular, that the projection of a shortest path in an extension G′ of a graphG is a shortest path in G. Generally speaking, Figure 8.2: An expansion of the cycle C4. Figure 8.3: Another isometric expansion of the cycle C4. the converse is not true. Consider the graph G shown in Figure 8.4 and two paths in G: V1 = abcef and V2 = bde. The graph G′ in Figure 8.4 is the convex expansion of G with respect to V1 and V2. The path abdef is a shortest path in G; it is not a projection of a shortest path in G′. Figure 8.4: A shortest path which is not a projection of a shortest path. One can say that, in the case of finite partial cubes, the contraction procedure is defined by an orthogonal projection of a hypercube onto one of its facets. By Theorem 8.1, the sets V ′1 and V 2 are opposite semicubes of the graph G defined by edges in M . Their projections are the sets V1 and V2 which are not necessarily semicubes of G. For other semicubes in G′ we have the following result. Lemma 8.2. For any two adjacent vertices u, v ∈ V , Wuivi = p −1(Wuv) for u, v ∈ Vi and i = 1, 2. Proof. By Lemma 8.1, dG′(x j , ui) < dG′(x j , vi) ⇔ dG(x, u) < dG(x, v) for x ∈ V and i, j = 1, 2. The result follows. Corollary 8.1. If uv is an edge of G1 ∩G2, then Wu1v1 =Wu2v2 . The following lemma is an immediate consequence of Lemma 8.1. We shall use it implicitly in our arguments later. Lemma 8.3. Let u, v ∈ V1 and x ∈ V1 ∩ V2. Then x1 ∈Wu1v1 ⇔ x 2 ∈Wu1v1 . The same result holds for semicubes in the form Wu2v2 . Generally speaking, the projection of a convex subgraph of G′ is not a con- vex subgraph of G. For instance, the projection of the convex path b2d2e2 in Figure 8.4 is the path bde which is not a convex subgraph of G. On the other hand, we have the following result. Theorem 8.2. Let G′ = (V ′, E′) be an expansion of a graph G = (V,E) with respect to subgraphs G1 = (V1, E1) and G2 = (V2, E2). The projection of a convex semicube of G′ different from 〈V ′1〉 and 〈V 2 〉 is a convex semicube of G. Proof. It suffices to consider the case when Wuv = p(Wu1v1) for u, v ∈ V1 (cf. Theorem 8.2). Let x, y ∈Wuv and z ∈ V be a vertex such that dG(x, z) + dG(z, y) = dG(x, y). We need to show that z ∈Wuv. Figure 8.5: A shortest path from x to y. (i) x, y ∈ V1 (the case when x, y ∈ V2 is treated similarly). Suppose that z ∈ V1. Then x 1, y1, z1 ∈ V ′1 and, by Lemma 8.1, dG′(x 1, z1) + dG′(z 1, y1) = dG′(z 1, y1). Since x1, y1 ∈ Wu1v1 and Wu1v1 is convex, z 1 ∈ Wu1v1 . Hence, z ∈Wuv. Suppose now that z ∈ V2 \ V1. Consider a shortest path Pxy in G from x to y containing z. This path contains vertices x′, y′ ∈ V1 ∩ V2 such that (see Figure 8.5) dG(x, x ′) + dG(x ′, z) = dG(x, z) and dG(y, y ′) + dG(y ′, z) = dG(y, z). Since Pxy is a shortest path in G, we have dG(x, x ′) + dG(x ′, y) = dG(x, y), dG(x, y ′) + dG(y ′, y) = dG(x, y), ′, z) + dG(z, y ′) = dG(x ′, y′). Since x, x′, y ∈ V1, we have x 1, x′1, y1 ∈ V ′1 . Because x 1, y1 ∈ Wu1v1 and Wu1v1 is convex, x′1 ∈ Wu1v1 . Hence, x ′ ∈ Wuv and, similarly, y ′ ∈ Wuv. Since x′2, y′2, z2 ∈ V ′2 and Wu1v1 is convex, z 2 ∈Wu1v1 . Hence, z ∈Wuv. (ii) x ∈ V1 \V2 and y ∈ V2 \V1. We may assume that z ∈ V1. By Lemma 8.1, dG′(x 1, y2) = dG(x, y) + 1 = dG(x, z) + dG(z, y) + 1 = dG′(x 1, z1) + dG′(z 1, y2). Since x1, y2 ∈ Wu1v1 and Wu1v1 is convex, z 1 ∈ Wu1v1 . Hence, z ∈Wuv. By using the results of Lemma 8.1, it is not difficult to show that the class of connected bipartite graphs is closed under the expansion and contraction operations. The next theorem establishes this result for the class of partial cubes. Theorem 8.3. (i) An expansion G′ of a partial cube G is a partial cube. (ii) A contraction G of a partial cube G′ is a partial cube. Proof. (i) Let G = (V,E) be a partial cube and G′ = (V ′, E′) be its expansion with respect to isometric subgraphs G1 = (V1, E1) and G2 = (V2, E2). By Theorem 3.4(ii), it suffices to show that the semicubes of G′ are convex. By Lemma 8.1, the semicubes 〈V ′1〉 and 〈V 2〉 are convex, so we consider a semicube in the formWu1v1 where uv ∈ E1 (the other case is treated similarly). Let Px′y′ be a shortest path connecting two vertices in Wu1v1 and Pxy be its projection to G. By Theorem 8.2, x, y ∈ Wuv and, by Lemma 8.1, Pxy is a shortest path in G. Since Wuv is convex, Pxy belongs to Wuv. Let z ′ be a vertex in Px′y′ and z = p(z ′) ∈ Pxy. By Lemma 8.1, dG(z, u) < dG(z, v) ⇒ dG′(z ′, u1) ≤ dG′(z ′, v1). Since G′ is a bipartite graph, dG′(z ′, u1) < dG′(z ′, v1). Hence, Px′y′ ⊆ Wu1v1 , so Wu1v1 is convex. (ii) Let G = (V,E) be a contraction of a partial cube G′ = (V ′, E′). By Theorem 3.4, we need to show that the semicubes of G are convex. By The- orem 8.2, all semicubes of G are projections of semicubes of G′ distinct from 〈V ′1〉 and 〈V 2〉. By Theorem 8.2, the semicubes of G are convex. Corollary 8.2. (i) A finite connected graph is a partial cube if and only if it can be obtained from K1 by a sequence of expansions. (ii) The number of expansions needed to produce a partial cube G from K1 is dimI(G). Proof. (i) Follows immediately from Theorem 8.3. (ii) Follows from theorems 8.2 and 5.1 (see the discussion in Section 5 just before Theorem 5.2 ). The processes of expansion and contraction admit useful descriptions in the case of partial cubes on a set. Let G = (V,E) be a partial cube on a set X , that is an isometric subgraph of the hypercube H(X). Then it is induced by some wg-family F of finite subsets of X (cf. Theorem 2.1). We may assume (see Section 5) that ∩F = ∅ and ∪F = X . In what follows we present proofs of the results of Theorem 8.3 and Corol- lary 8.2 given in terms of wg-families of sets. The expansion process for a partial cube G on X can be described as follows: Let F1 and F2 be wg-families of finite subsets of X such that F1 ∩ F2 6= ∅, F1∪F2 = F, and the distance between any two sets P ∈ F1 \F2 and Q ∈ F2 \F1 is greater than one. Note that 〈F1〉 and 〈F2〉 are partial cubes, 〈F1〉∩ 〈F2〉 6= ∅, and 〈F1〉 ∪ 〈F2〉 = 〈F〉 = G. Let X ′ = X + {p}, where p /∈ X , and 2 = {Q+ {p} : Q ∈ F2}, F ′ = F1 ∪ F It is quite clear that the graphs 〈F′2〉 and 〈F2〉 are isomorphic and the graph G′ = 〈F′〉 is an isometric expansion of the graph G. Theorem 8.4. An expansion of a partial cube is a partial cube. Proof. We need to verify that F′ is a wg-family of finite subsets of X ′. By Theorem 2.3, it suffices to show that the distance between any two adjacent sets in F′ is 1. It is obvious if each of these two sets belong to one of the families F1 or F 2. Suppose that P ∈ F1 and Q+ {p} ∈ F 2 are adjacent, that is, for any S ∈ F′ we have P ∩ (Q+ {p}) ⊆ S ⊆ P ∪ (Q+ {p}) ⇒ S = P or S = Q+ {p}. (8.3) If Q ∈ F1, then P ∩ (Q + {p}) ⊆ Q ⊆ P ∪ (Q+ {p}), since p /∈ P . By (8.3), Q = P implying d(P,Q + {p}) = 1. If Q ∈ F2 \ F1, there is R ∈ F1 ∩ F2 such that d(P,R) + d(R,Q) = d(P,Q), since F is well graded. By Theorem 2.2, P ∩Q ⊆ R ⊆ P ∪Q, which implies P ∩ (Q + {p}) ⊆ R+ {p} ⊆ P ∪ (Q+ {p}). By (8.3), R + {p} = Q+ {p}, a contradiction. It is easy to recognize the fundamental sets (cf. Section 4) in an isometric expansion G′ of a partial cube G = 〈F〉. Let P ∈ F1∩F2 and Q = P +{p} ∈ F be two vertices defining an edge in G′ according to Definition 8.1(ii). Clearly, the families F1 and F 2 are the semicubes WPQ and WQP of the graph G ′ (cf. Lemma 5.1) and therefore are convex subsets of F′. The set FPQ is the set of edges defined by p as in Lemma 5.1. In addition, UPQ = F1 ∩ F2 and UQP = {R+ {p} : R ∈ F1 ∩ F2}. Let G be a partial cube induced by a wg-family F of finite subsets of a set X . As before, we assume that ∩F = ∅ and ∪F = X . Let PQ be an edge of G. We may assume that Q = P + {p} for some p /∈ P . Then (see Lemma 5.1) WPQ = {R ∈ F : p /∈ R} and WQP = {R ∈ F : p ∈ R}. Let X ′ = X \ {p} and F′ = {R \ {p} : R ∈ F}. It is clear that the graph G′ induced by the family F′ is isomorphic to the contraction of G defined by the edge PQ. Geometrically, the graph G′ is the orthogonal projection of the graph G along the edge PQ (cf. figures 8.2 and 8.3). Theorem 8.5. (i) A contraction G′ of a partial cube G is a partial cube. (ii) If G is finite, then dimI(G ′) = dimI(G)− 1. Proof. (i) For p ∈ X we define F1 = {R ∈ F : p /∈ R}, F2 = {R ∈ F : p ∈ R}, and F′2 = {R \ {p} ∈ F : p ∈ R}. Note that F1 and F2 are semicubes of G and F′2 is isometric to F2. Hence, F1 and F 2 are wg-families of finite subsets of X ′. We need to prove that F′ = F1 ∪ F 2 is a wg-family. By Theorem 2.3, it suffices to show that d(P,Q) = 1 for any two adjacent sets P,Q ∈ F′. This is true if P,Q ∈ F1 or P,Q ∈ F 2, since these two families are well graded. For P ∈ F1 \ F 2 and Q ∈ F 2 \ F1, the sets P and Q + {p} are not adjacent in F, since F is well graded and Q /∈ F. Hence there is R ∈ F1 such that P ∩ (Q+ {p}) ⊆ R ⊆ P ∪ (Q + {p}) and R 6= P . Since p /∈ R, we have P ∩Q ⊆ R ⊆ P ∪Q. Since R 6= P and R 6= Q, the sets P and Q are not adjacent in F′. The result follows. (ii) If G is a finite partial cube, then, by Theorem 5.2, dimI(G ′) = |X ′| = |X | − 1 = dimI(G)− 1. 9 Conclusion The paper focuses on two themes of a rather general mathematical nature. 1. The characterization problem. It is a common practice in mathematics to characterize a particular class of object in different terms. We present new characterizations of the classes of bipartite graphs and partial cubes, and give new proofs for known characterization results. 2. Constructions. The problem of constructing new objects from old ones is a standard topic in many branches of mathematics. For the class of partial cubes, we discuss operations of forming the Cartesian product, expansion and contraction, and pasting. It is shown that the class of partial cubes is closed under these operations. Because partial cubes are defined as graphs isometrically embeddable into hypercubes, the theory of partial cubes has a distinctive geometric flavor. The three main structures on a graph—semicubes and Djoković’s and Winkler’s relations—are defined in terms of the metric structure on a graph. One can say that this theory is a branch of discrete metric geometry. Not surprisingly, geo- metric structures play an important role in our treatment of the characterization and construction problems. References [1] A.S. Asratian, T.M.J. Denley, and R. Häggkvist, Bipartite Graphs and their Applications, Cambridge University Press, 1998. [2] D. Avis, Hypermetric spaces and the Hamming cone, Canadian Journal of Mathematics 33 (1981) 795–802. [3] L. Blumenthal, Theory and Applications of Distance Geometry, Oxford University Press, London, Great Britain, 1953. [4] J.A. Bondy, Basic graph theory: Paths and circuits, in: R.L. Graham, M. Grötshel, and L. Lovász (Eds.), Handbook of Combinatorics, The MIT Press, Cambridge, Massachusetts, 1995, pp. 3–110. [5] N. Bourbaki, General Topology, Addison-Wesley Publ. Co., 1966. [6] V. Chepoi, Isometric subgraphs of Hamming graphs and d-convexity, Con- trol and Cybernetics 24 (1988) 6–11. [7] V. Chepoi, Separation of two convex sets in convexity structures, Journal of Geometry 50 (1994) 30–51. [8] M.M. Deza and M. Laurent, Geometry of Cuts and Metrics, Springer, 1997. [9] D.Ž. Djoković, Distance preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (1973) 263–267. [10] J.-P. Doignon and J.-Cl. Falmagne, Well-graded families of relations, Dis- crete Math. 173 (1997) 35–44. [11] D. Eppstein, The lattice dimension of a graph, European J. Combinatorics 26 (2005) 585–592, doi: 10.1016/j.ejc.2004.05.001. [12] A. Frank, Connectivity and network flows, in: R.L. Graham, M. Grötshel, and L. Lovász (Eds.), Handbook of Combinatorics, The MIT Press, Cam- bridge, Massachusetts, 1995, pp. 111–177. [13] K. Fukuda and K. Handa, Antipodal graphs and oriented matroids, Dis- crete Mathematics 111 (1993) 245–256. [14] F. Hadlock and F. Hoffman, Manhattan trees, Util. Math. 13 (1978) 55–67. [15] W. Imrich and S. Klavžar, Product Graphs, John Wiley & Sons, 2000. [16] H.M. Mulder, The Interval Function of a Graph, Mathematical Centre Tracts 132, Mathematisch Centrum, Amsterdam, 1980. [17] S. Ovchinnikov, Media theory: representations and examples, Discrete Ap- plied Mathematics, (in review, e-print available at http://arxiv.org/abs/math.CO/0512282). [18] R.I. Roth and P.M. Winkler, Collapse of the metric hierarchy for bipartite graphs, European Journal of Combinatorics 7 (1986) 371–375. [19] M.L.J. van de Vel, Theory of Convex Structures, Elsevier, The Netherlands, 1993. [20] P.M. Winkler, Isometric embedding in products of complete graphs, Dis- crete Appl. Math. 8 (1984) 209–212. http://arxiv.org/abs/math.CO/0512282 Introduction Hypercubes and partial cubes Characterizations Fundamental sets in partial cubes Dimensions of partial cubes Subcubes and Cartesian products Pasting partial cubes Expansions and contractions of partial cubes Conclusion
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COMPUTING GENUS 2 HILBERT-SIEGEL MODULAR FORMS OVER Q( 5) VIA THE JACQUET-LANGLANDS CORRESPONDENCE CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ Abstract. In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field Q( 5). In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms. Introduction Let F be a real quadratic field of narrow class number one and let B be the unique (up to isomorphism) quaternion algebra over F which is ramified at both archimedean places of F and unramified everywhere else. Let GU2(B) be the unitary similitude group of B⊕2. This is the set of Q-rational points of an algebraic group GB defined over Q. The group GB is an inner form of G := ResF/Q(GSp4) such that G B(R) is compact modulo its centre. (These notions are reviewed at the beginning of Section 1.) In this paper we develop an algorithm which computes automorphic forms on GB in the following sense: given an idealN inOF and an integer k greater than 2, the algorithm returns the Hecke eigensystems of all automorphic forms f of level N and parallel weight k. More precisely, given a prime p in OF , the algorithm returns the Hecke eigenvalues of f at p, and hence the Euler factor Lp(f, s), for each eigenform f of level N and parallel weight k. The algorithm is a generalization of the one developed in [D1 2005] to the genus 2 case. Although we have only described the algorithm in the case of a real quadratic field in this paper, it should be clear from our presentation that it can be adapted to any totally real number field of narrow class number one. The Jacquet-Langlands Correspondence of the title refers to the conjec- tural map JL : Π(GB) → Π(G) from automorphic representations of GB to automorphic representations of G, which is injective, matches L-functions and enjoys other properties compatible with the principle of functoriality; Date: October 29, 2018. 1991 Mathematics Subject Classification. Primary: 11F41 (Hilbert and Hilbert-Siegel modular forms). Key words and phrases. Hilbert-Siegel modular forms, Jacquet-Langlands Correspon- dence, Brandt matrices, Satake parameters. http://arxiv.org/abs/0704.0011v3 2 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ in particular, the image of the Jacquet-Langlands Correspondence is to be contained in the space of holomorphic automorphic representations. If we admit this conjecture, then the algorithm above provides a way to produce examples of cuspidal Hilbert-Siegel modular forms of genus 2 over F and allows us to compute the L-factors of the corresponding automorphic repre- sentations for arbitrary finite primes p of F . In fact, we are also able to use these calculations to provide evidence for the Jacquet-Langlands Correspondence itself by comparing the Euler factors we find with those of known Hilbert-Siegel modular forms obtained by lifting. This we do in the final section of the paper where we observe that some of the Euler factors we compute match those of lifts of Hilbert modular forms, for the primes we computed. Although this does not definitively establish that these Hilbert-Siegel modular forms are indeed lifts, in principle one can establish equality in this way, using an analogue of the Sturm bound. The first systematic approach to Siegel modular forms from a computa- tional viewpoint is due to Skoruppa [Sk 1992] who used Jacobi symbols to generate spaces of such forms. His algorithm, which has been extensively exploited by Ryan [R 2006], applies only to the case of full level structure. More recently, Faber and van der Geer [FvdG1 2004] and [FvdG2 2004] also produced examples of Siegel modular forms by counting points on hy- perelliptic curves of genus 2; again their results are available only in the full level structure case. The most substantial progress toward the com- putation of Siegel modular forms for proper level structure is by Gunnells [Gu 2000] who extended the theory of modular symbols to the symplectic group Sp4/Q. However, this work does not see the cuspidal cohomology, which is the only part of the cohomology which is relevant to arithmetic geometric applications. To the best of our knowledge, there are no numer- ical examples of Hilbert-Siegel modular forms for proper level structure in the literature, with the exception of those produced from liftings of Hilbert modular forms. The outline of the paper is as follows. In Section 1 we recall the basic properties of Hilbert-Siegel modular forms and algebraic automorphic forms together with the Jacquet-Langlands Correspondence. In Section 2 we give a detailed description of our algorithm. Finally, in Section 3 we present numerical results for the quadratic field Q( Acknowledgements. During the course of the preparation of this paper, the second author had helpful email exchanges with several people includ- ing Alexandru Ghitza, David Helm, Marc-Hubert Nicole, David Pollack, Jacques Tilouine and Eric Urban. The authors wish to thank them all. Also, we would like to thank William Stein for allowing us to use the SAGE computer cluster at the University of Washington. And finally, the sec- ond author would like to thank the PIMS institute for their postdoctoral fellowship support, and the University of Calgary for its hospitality. COMPUTING HILBERT-SIEGEL MODULAR FORMS 3 1. Hilbert-Siegel modular forms and the Jacquet-Langlands correspondence Throughout this paper, F denotes a real quadratic field of narrow class number one. The two archimedean places of F and the real embeddings of F will both be denoted v0 and v1. For every a ∈ F , we write a0 (resp. a1) for the image of a under v0 (resp. v1). The ring of integers of F is denoted by OF . For every prime ideal p in OF , the completion of F and OF at p will be denoted by Fp and OFp , respectively. Let B be the unique (up to isomorphism) totally definite quaternion al- gebra over F which is unramified at all finite primes of F . We fix a maximal order OB of B. Also, we choose a splitting field K/F of B that is Ga- lois over Q and such that there exists an isomorphism j : OB ⊗Z OK ∼= M2(OK)⊕M2(OK), where M2(A) denotes the ring of 2× 2-matrices with entries from a ring A. For every finite prime p in F , we fix an isomorphism Bp ∼= M2(Fp) which restricts to an isomorphism from OB, p onto M2(OFp ). The algebraic group G = ResF/Q(GSp4) is defined as follows. For any Q-algebra A, the set of A-rational points of G is given by G(A) = γ ∈ GL4(A⊗Q F ) t = νG(γ)J2 νG(γ) ∈ (A⊗Q F )× where −12 0 This group admits an integral model with A-rational points for every Z- algebra A given by GZ(A) = γ ∈ GL4(A⊗Z OF ) t = νG(γ)J2 νG(γ) ∈ (A⊗Z OF )× For any Q-algebra A, the conjugation on B extends in a natural way to the matrix algebra M2(B ⊗Q A). The algebraic group GB/Q is defined as follows. For any Q-algebra A, the set of A-rational points of GB is given by GB(A) = γ ∈ M2(B ⊗Q A) γγ̄t = νGB(γ)12 νGB (γ) ∈ (A⊗Q F )× This group also admits an integral model with A-rational points for every Z-algebra given by GBZ (A) = γ ∈ M2(OB ⊗Z A) γγ̄t = νGB(γ)12 νGB (γ) ∈ (A⊗Z OF )× The group GB/Q is an inner form of G/Q such that GB(R) is compact modulo its center. Combining the isomorphism j (see above) with con- jugation by a permutation matrix, we obtain an isomorphism GBZ (OK) ∼= 4 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ GZ(OK), which we fix from now on. For every prime ideal p in F , the split- ting of GB at p amounts to the splitting of the quaternion algebra B at p; we refer to [D1 2005] for further details. By the choice of the quaternion algebra B, we have GB(Q̂) ∼= G(Q̂). (We denote the finite adèles of Q (resp. Z) by Q̂ (resp. Ẑ)). 1.1. Hilbert-Siegel modular forms. We fix an integer k ≥ 3 and, for simplicity, we restrict ourselves to Hilbert-Siegel modular forms of parallel weight k. The real embeddings v0 and v1 of F extend to G(Q) = GSp4(F ) in a natural way. We denote by GSp+4 (F ) the subgroup of elements γ with totally positive similitude factor νG(γ). We recall that the Siegel upper-half plane of genus 2 is defined by H2 = {γ ∈ GL2(C) ∣ γt = γ and Im(γ) is positive definite }. We also recall that GSp+4 (F ) acts on H (τ0, τ1) := (a0τ0 + b0)(c0τ0 + d0) −1, (a1τ1 + b1)(c1τ1 + d1) This induces an action on the space of functions f : H22 → C by , f |kγ(τ) = νG(γi) det(ciτi + di)k f(τ). Let N be an ideal in OF and set Γ0(N) = ∈ GSp+4 (OF ) ∣ c ≡ 0(N) A Hilbert-Siegel modular form of level N and parallel weight k is a holomorphic function f : H22 → C such that ∀γ ∈ Γ0(N), f |kγ = f. The space of Hilbert-Siegel modular forms of parallel weight k and level N is denoted Mk(N). Each f ∈Mk(N) admits a Fourier expansion, which by the Koecher principle takes the form ∀τ ∈ H22, f(τ) = {Q}∪{0} 2πiTr(Qτ), where Q ∈ M2(F ) runs over all symmetric totally positive and semi-definite matrices. A Hilbert-Siegel modular forms f is a cusp form if, for all γ ∈ 4 (F ), the constant term in the Fourier expansion of f |kγ is zero. The space of Hilbert-Siegel cusp forms is denoted Sk(N). COMPUTING HILBERT-SIEGEL MODULAR FORMS 5 1.2. The Hecke algebra. The space Sk(N) comes equipped with a Hecke action, which we now recall. Take u ∈ GSp+4 (F ) ∩M4(OF ), and write the finite disjoint union Γ0(N)uΓ0(N) = Γ0(N)ui. Then the Hecke operator [Γ0(N)uΓ0(N)] on Sk(N) is given by [Γ0(N)uΓ0(N)]f = f |kui. Let p be a prime ideal in OF and let πp be a totally positive generator of p; let T1(p) and T2(p) be the Hecke operators corresponding to the double Γ0(N)-cosets of the symplectic similitude matrices 1 0 0 0 0 1 0 0 0 0 πp 0 0 0 0 πp 1 0 0 0 0 πp 0 0 0 0 π2p 0 0 0 0 πp respectively. (We remind the reader of the symplectic form J2 fixed at the beginning of Section 1.) The Hecke algebra Tk(N) is the Z-algebra generated by the operators T1(p) and T2(p), where p runs over all primes not dividing N . 1.3. Algebraic Hilbert-Siegel autormorphic forms. We only consider level structure of Siegel type. Namely, we define the compact open subgroup U0(N) of G(Q̂) by U0(N) = GSp4(OFp )× ep ), where N = p|N p ep and ep ) := ∈ GSp4(OFp ) ∣ c ≡ 0 mod pep The weight representation is defined as follows. Let Lk be the repre- sentation of GSp4(C) of highest weight (k− 3, k− 3). We let Vk = Lk ⊗Lk and define the complex representation (ρk, Vk) by ρk : G B(R) −→ GL(Vk), where the action on the first factor is via v0, and the action on the second one is via v1. The space of algebraic Hilbert-Siegel modular forms of weight k and level N is given by MBk (N) := f : GB(Q̂)/U0(N) → Vk ∣ ∀γ ∈ GB(Q), f |kγ = f 6 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ where f |kγ(x) = f(γx)γ, for all x ∈ GB(Q̂)/U0(N). When k = 3, we let IBk (N) := f : GB(Q)\GB(Q̂)/U0(N) → C ∣ f is constant Then, the space of algebraic Hilbert-Siegel cusp forms of weight k and level N is defined by SBk (N) := MBk (N) if k > 3, MBk (N)/I k (N) if k = 3. The action of the Hecke algebra on SBk (N) is given as follows. For any u ∈ G(Q̂), write the finite disjoint union U0(N)uU0(N) = uiU0(N), and define [U0(N)uU0(N)] : S k (N) → SBk (N) f 7→ f |k[U0(N)uU0(N)], f |k[U0(N)uU0(N)](x) = f(xui), x ∈ G(Q̂). For any prime p ∤ N , let ̟p be a local uniformizer at p. The local Hecke alge- bra at p is generated by the Hecke operators T1(p) and T2(p) corresponding to the double U0(N)-cosets ∆1(p) and ∆2(p) of the matrices 1 0 0 0 0 1 0 0 0 0 ̟p 0 0 0 0 ̟p 1 0 0 0 0 ̟p 0 0 0 0 ̟2 0 0 0 ̟p respectively. We let TBk (N) be the Hecke algebra generated by T1(p) and T2(p) for all primes p ∤ N . 1.4. The Jacquet-Langlands Correspondence. The Hecke modules Sk(N) and SBk (N) are related by the following conjecture known as the Jacquet- Langlands Correspondence for symplectic similitude groups. Conjecture 1. The Hecke algebras Tk(N) and T k (N) are isomorphic and there is a compatible isomorphism of Hecke modules Sk(N) ∼−→ SBk (N). It is common, but perhaps not entirely accurate, to attribute this con- jecture to Jacquet-Langlands. To the best of our knowledge, the correspon- dence in this form was first discussed by Ihara [Ih 1964] in the case F = Q. In [Ib 1984], Ibukiyama provided some numerical evidence. On the other hand, it is appropriate to refer to Conjecture 1 as the Jacquet-Langlands Corre- spondence (for GSp(4)) since it is an analogue of the Jacquet-Langlands COMPUTING HILBERT-SIEGEL MODULAR FORMS 7 Correspondence (for GL(2)) which relates automorphic representations of the multiplicative group of a quaternion algebra with certain automorphic representations of GL(2) (see [JL 1970]). Both correspondences are, in turn, special consequences of the principle of functoriality, as expounded by Lang- lands. Finally, it appears that Conjecture 1 may soon be a theorem due to the work of [So 2008] and the forthcoming book by James Arthur on auto- morphic representations of classical groups. 2. The Algorithm In this section, we present the algorithm we used in order to compute the Hecke module of (algebraic) Hilbert-Siegel modular forms. The main assumption in this section is that the class number of the principal genus of GB is 1. (We refer to [D3 2007] to see how one can relax this condition on the class number.) We recall that since B is totally definite, GB satis- fies Proposition 1.4 in Gross [Gr 1999]. Thus the group GB(R) is compact modulo its centre, and Γ = GB(Z)/O×F is finite. For any prime p in F , let Fp = OF /p be the residue field at p and define the reduction map M2(OB, p) → M4(Fp) g 7→ g̃, where we use the splitting of OB,p that was fixed at the beginning of Sec- tion 1. Now, choose a totally positive generator πp of p and put Θ1(p) := Γ\ u ∈ M2(OB) ∣ uūt = πp12and rank(g̃) = 2 Θ2(p) := Γ\ u ∈ M2(OB) ∣ uūt = π2 12 and rank(g̃) = 1 We let H20(N) = G(Ẑ)/U0(N). Then the group Γ acts on H20(N), thus on the space of functions f : H20(N) → Vk by ∀x ∈ H20(N),∀γ ∈ Γ, f |kγ(x) := f(γx)γ. Theorem 2. There is an isomorphism of Hecke modules MBk (N) f : H20(N) → Vk ∣ f |kγ = f, γ ∈ Γ where the Hecke action on the right hand side is given by f |kT1(p) = u∈Θ1(p) f |ku, f |kT2(p) = u∈Θ2(p) f |ku. Proof. The canonical map φ : GB(Z)\GB(Ẑ)/U0(N) → GB(Q)\GB(Q̂)/U0(N) is an injection. Making use of the fact that the class number in the principal genus of GB is one (GB(Q̂) = GB(Q)GBZ (Ẑ)), we see that φ is in fact a 8 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ bijection. Since each element f ∈ MBk (N) is determined by its values on a set of coset representatives of GB(Q)\GB(Q̂)/U0(N), the map φ induces an isomorphism of complex vector spaces MBk (N) f : H20(N) → Vk ∣ f |kγ = f, γ ∈ Γ f 7−→ f ◦ φ. We make this into a Hecke module isomorphism by defining the Hecke action on the right hand side as indicated in the statement of the theorem. � In the rest of this section, we explain the main steps of the algorithm provided by Theorem 2. 2.1. The quotient H20(N). Keeping the notations of the previous section, we recall that N = p|N p ep . Let p be a prime dividing N and consider the rank 4 free OFp/pep -module L = OFp/pep endowed with the symplectic pairing 〈 , 〉 given by the matrix −12 0 where 12 is the identity matrix in M2(OFp/pep ). Let M be a rank 2 OFp/pep -submodule which is a direct factor in L. We say that M is isotropic if 〈u, v〉 = 0 for all u, v ∈ M . We recall that GSp4(OFp ) acts transitively on the set of rank 2, isotropic OFp/pep -submodules of L and that the stabilizer of the submodule generated by e1 = (1, 0, 0, 0) T and e2 = (0, 1, 0, 0) T is U0(p ep ). The quotient H20(pep ) = GSp4(OFp )/U0(pep ) is the set of rank 2, isotropic OFp/pep -submodules of L. Via the reduction map ÔF → OF /N , the quotient GZ(Ẑ)/U0(N) can be identified with the product H20(N) = H20(pep ). The cardinality of H20(N) is extremely useful and is determined using the following lemma. Lemma 1. Let p be a prime in F and ep ≥ 1 an integer. Then, the cardi- nality of the set H20(pep ) is given by #H20(pep ) = N(p)3(ep−1)(N(p) + 1)(N(p)2 + 1). Proof. For ep = 1, the cardinality of the Lagrange variety over the finite field Fp = OF /p is given by (N(p) + 1)(N(p)2 + 1). Proceed by induction on ep. � We have more to say about elements of H20(pep ) in Subsection 2.5. COMPUTING HILBERT-SIEGEL MODULAR FORMS 9 2.2. Brandt matrices. Let F = {x1, . . . , xh} be a fundamental domain for the action of Γ on H20(N) and, for each i, let Γi be the stabilizer of xi. Then, every element in MBk (N) is completely determined by its values on F . Thus, there is an isomorphism of complex spaces MBk (N) → f 7→ (f(xi)), where V is the subspace of Γi-invariants in Vk. For any x, y ∈ H20(N), we let Θ1(x, y, p) := u ∈ Θ1(p) ∣ ∃γ ∈ Γ, ux = γy Θ2(x, y, p) := u ∈ Θ2(p) ∣ ∃γ ∈ Γ, ux = γy Proposition 3. The actions of the Hecke operators Ts(p), s = 1, 2, are given by the Brandt matrices Bs(p) = (bsij(p)), where bsji(p) : V k → V v 7→ v · u∈Θs(xi, xj ,p) γ−1u u Proof. The proof of Proposition 3 follows the lines of [D1 2005, §3]. � 2.3. Computing the group GB(Z). It is enough to compute the subgroup Γ consisting of the elements in GB(Z) with similitude factor 1. But it is easy to see that u, v ∈ O1B u, v ∈ O1B where O1B is the group of norm 1 elements. 2.4. Computing the sets Θ1(p) and Θ2(p). Let us consider the quadratic form on the vector space V = B2 given by V → F (a, b) 7→ ||(a, b)|| := nr(a) + nr(b), where nr is the reduced norm on B. This determines an inner form V × V → F (u, v) 7→ 〈u, v〉. An element of Θ1(p) (resp. Θ2(p)) is a unitary matrix γ ∈ M2(OB) with respect to this inner form such that the norm of each row is πp (resp. π and the rank of the reduced matrix is 1). So we first start by computing all the vectors u = (a, b) ∈ O2B such that ||u|| = πp (resp. ||u|| = π2p). And for each such vector u, we compute the vectors v = (c, d) ∈ O2B of the same 10 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ norm such that 〈u, v〉 = 0. The corresponding matrix γ = belongs to Θ1(p) (resp. Θ2(p)) when its reduction mod p has the appropriate rank. We list all these matrices up to equivalence and stop when we reach the right cardinality. 2.5. The implementation of the algorithm. The implementation of the algorithm is similar to that of [D1 2005]. However, it is important to note how we represent elements in H20(N) so that we can retrieve them easily once stored. As in [D1 2005] we choose to work with the product H20(N) = H20(pep ). Using Plucker’s coordinates, we can view H20(pep ) as a closed subspace of P5(OFp/pep ). We then represent each element in H20(pep ) by choosing a point x = (a0 : · · · : a5) = [u ∧ v] ∈ P5(OFp/pep ) such that the submodule M generated by u and v is a Lagrange submodule, and the first invertible coordinate is scaled to 1. Remark 1. In [LP 2002], Lansky and Pollack describe an algorithm which computes algebraic modular forms on the same inner form of GSp4/Q that we use. We would like to note that there are some differences between the two algorithms. Although [LP 2002] also uses the flag variety H20(N) in order to determine the double coset space GB(Q)\GB(Q̂)/U0(N), it later returns to the adelic setting in order to compute the Brandt matrices. In contrast, Theorem 2 and Proposition 3 allow us to avoid that unnecessary step by describing the Hecke action on the flag variety H20(N) directly. As a result, we get an algorithm that is more efficient. 3. Numerical examples: F = Q( 5) and B = −1,−1 In this section, we provide some numerical examples using the quadratic field F = Q( 5). It is proven in K. Hashimoto and T. Ibukiyama [HI 1980] that, for the Hamilton quaternion algebra B over F , the class number of the principal genus of GB is one. We use our algorithm to compute all the systems of Hecke eigenvalues of Hilbert-Siegel cusp forms of weight 3 and level N that are defined over real quadratic fields, where N runs over all prime ideals of norm less than 50. We then determine which of the forms we obtained are possible lifts of Hilbert cusp forms by comparing the Hecke eigenvalues for those primes. 3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3. In Table 1 we list all the systems of eigenvalues of Hilbert-Siegel cusp forms of weight 3 and level N that are defined over real quadratic fields, where N runs over all prime ideals in F of norm less than 50. Here are the conventions we use in the tables. COMPUTING HILBERT-SIEGEL MODULAR FORMS 11 (1) For a quadratic field K of discriminant D, we let ωD be a generator of the ring of integers OK of K. (2) The first row contains the level N , given in the format (Norm(N), α) for some generator α ∈ F of N , and the dimensions of the relevant spaces. (3) The second row lists the Hecke operators that have been computed. (4) For each eigenform f , the Hecke eigenvalues are given in a row, and the last entry of that row indicates if the form f is a probable lift. (5) The levels and the eigenforms are both listed up to Galois conjuga- tion. For an eigenform f and a given prime p ∤ N , let a1(p, f) and a2(p, f) be the eigenvalues of the Hecke operators T1(p) and T2(p), respectively. Then the Euler factor Lp(f, s) is given (for example, in [AS 2001, §3.4]) by Lp(f, s) = Qp(q −s)−1, where Qp(x) = 1− a1(p, f)x+ b1(p, f)x2 − a1(p, f)q2k−3x3 + q4k−6x4, b1(p, f) = a1(p, f) 2 − a2(p, f)− q2k−4, q = N(p). 3.2. Tables of Hilbert cusp forms of parellel weight 4. In Table 2, we list all the Hilbert cusp forms of parallel weight 4 and level N that are defined over real quadratic fields, with N running over all prime ideals of norm less than 50. (They are computed by using the algorithm in [D1 2005]). We use this data in order to determine the forms in Table 1 that are possible lifts from GL2. 3.3. Lifts. There are two types of lifts from GL2 to GSp4. The first one corresponds to the homomorphism of L-groups determined by the long root embedding into GSp4, and the second one by the short root embedding. (See [LP 2002] for more details). Let f be a Hilbert cusp form of parallel weight k and level N with Hecke eigenvalues a(p, f), where p is a prime not dividing N . Let φ be the lift of f to GSp4 via the long root, and ψ the one via the short root. Then the Hecke eigenvalues of φ are given by a1(p, φ) = a(p, f) N(p) 2 +N(p)2 +N(p) a2(p, φ) = a(p, f) N(p) 2 (N(p) + 1) +N(p)2 − 1, and the Hecke eigenvalues of ψ are given by a1(p, ψ) = a(p, f) 2 − 2 a(p, f) N(p) a2(p, ψ) = a(p, f) N(p)4−2k − 3 a(p, f)2 N(p)3−k +N(p)2 − 1. The second lift ψ is the so-called symmetric cube lifting. 12 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ N = (4, 2) : dimMB (N) = 2, dimSB (N) = 1 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 −4 0 20 −36 140 580 yes N = (5, 2 + ω5) : dimM (N) = 2, dimSB (N) = 1 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 20 15 −5 0 40 −420 yes N = (9, 3) : dimMB (N) = 3, dimSB (N) = 2 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 25− 3ω41 40− 15ω41 30 + 6ω41 24 + 36ω41 −9 0 yes N = (11, 3 + ω5) : dimM (N) = 3, dimSB (N) = 2 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 24 35 34 48 88 60 yes f2 −20 35 −10 4 0 60 no N = (19, 4 + ω5) : dimM (N) = 5, dimSB (N) = 4 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 4 11 −20 28 6 76 no f2 7 −50 15 −66 73 −90 yes f3 24 + ω161 35 + 5ω161 36− ω161 60− 6ω161 98− 3ω161 160− 30ω161 yes N = (29, 5 + ω5) : dimM (N) = 9, dimSB (N) = 8 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 −4 11 10 20 30 60 no f2 8 −45 30 24 50 −320 yes f3 17 0 9 −102 86 40 yes N = (31, 5 + 2ω5) : dimM (N) = 12, dimSB (N) = 11 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 13 −20 20 −36 76 −60 yes N = (41, 6 + ω5) : dimM (N) = 19, dimSB (N) = 18 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 10 20 −10 29 30 −20 no f2 −1 1 5 14 −2 −56 no f3 27 50 40 84 124 420 yes f4 −12 19 30 65 0 0 no f5 16− 2ω21 −5− 10ω21 21 + 4ω21 −30 + 24ω21 72− 2ω21 −100− 20ω21 yes f6 2− 6ω5 11− 2ω5 8 + 4ω5 11− 4ω5 −12 + 54ω5 160 + 40ω5 no N = (49, 7) : dimMB (N) = 26, dimSB (N) = 25 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 5 −60 46 120 40 −420 yes f2 4 + 4ω65 32 + 3ω65 12− 4ω65 44− 4ω65 −6− 12ω65 145 + 8ω65 no Table 1. Hilbert-Siegel eigenforms of weight 3 COMPUTING HILBERT-SIEGEL MODULAR FORMS 13 N (4, 2) (5, 2 + ω5) (9, 3) (11, 3 + ω5) N(p) p a(p, f1) a(p, f1) a(p, f1) a(p, f1) 4 2 −4 0 5− 3ω41 4 5 2 + ω5 −10 −5 6ω41 4 9 3 50 −50 −9 −2 11 3 + 2ω5 −28 32 −18− 6ω41 −10 11 3 + ω5 −28 32 −18− 6ω41 −11 19 4 + 3ω5 60 100 −40 + 24ω41 −94 19 4 + ω5 60 100 −40 + 24ω41 28 N (19, 4 + ω5) (29, 5 + ω5) N(p) p a(p, f1) a(p, f2) a(p, f1) a(p, f2) 4 2 −13 5− ω161 −12 −3 5 2 + ω5 −15 5 + ω161 0 −21 9 3 −17 5 + 3ω161 −40 −4 11 3 + 2ω5 −6 2 + 8ω161 −68 37 11 3 + ω5 33 7− 7ω161 30 −66 19 4 + 3ω5 −139 −15− 9ω161 −28 −40 19 4 + ω5 19 −19 84 −9 N (31, 5 + 2ω5) (41, 6 + ω5) N(p) p a(p, f1) a(p, f1) a(p, f2) 4 2 −7 7 −4− 2ω21 5 2 + ω5 −10 10 −9 + 4ω21 9 3 −14 34 −18− 2ω21 11 3 + 2ω5 −20 −60 −19 11 3 + ω5 −28 −2 −24− 4ω21 19 4 + 3ω5 −12 74 4− 50ω21 19 4 + ω5 28 16 −29 + 44ω21 N (49, 7) N(p) p a(p, f1) a(p, f2) 4 2 −15 −2 5 2 + ω5 16 −10 9 3 −50 −11 11 3 + 2ω5 −8 −7− 28ω13 11 3 + ω5 −8 −35 + 28ω13 19 4 + 3ω5 −110 −26 + 14ω13 19 4 + ω5 −110 −12− 14ω13 Table 2. Hilbert eigenforms of weight 4 Remark 2. So far, our algorithm has been implemented only for congruence subgroups of Siegel type. We intend to improve the implementation in the near future so as to include more additional level structures such as the Klingen type. Indeed, Ramakrishnan and Shahidi [RS 2007] recently showed the existence of symmetric cube lifts for non-CM elliptic curves E/Q to GSp4/Q. And their result should hold for other totally real number fields, with the level structures of the lifts being of Klingen type. Unfortunately, 14 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ those lifts cannot be seen in our current tables. For example, there are modular elliptic curves over Q( 5) whose conductors have norm 31, 41 and 49, but the corresponding symmetric cubic lifts do not appear in Table 1. We would like to remedy that in our next implementation. References [D1 2005] L. Dembélé, Explicit computations of Hilbert modular forms on Q( 5). Exper- iment. Math. 14 (2005), no. 4, 457–466. [D2 2007] L. Dembélé, Quaternionic M -symbols, Brandt matrices and Hilbert modular forms. Math. Comp. 76, no 258, (2007), 1039-1057. Also available electronically. [D3 2007] L. Dembélé, On the computation of algebraic modular forms (submitted). [AS 2001] Mahdi Asgari and Ralf Schmidt, Siegel modular forms and representations, Manuscripta Math. 104 (2001), 173–200. [FvdG1 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes. I, C. R. Math. Acad. Sci. Paris 338 (2004), no. 5, 381–384. [FvdG2 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes. II, C. R. Math. Acad. Sci. Paris 338 (2004), no. 6, 467–470. [JL 1970] Hervé Jacquet and Robert Langlands, Automorphic forms on GL(2), Lecture notes in mathematics 114 and 278, 1970. [Gr 1999] Benedict H. Gross, Algebraic modular forms. Israel J. Math. 113 (1999), 61–93. [Gu 2000] P. Gunnells, Symplectic modular symbols, Duke Math. J. 102 (2000), no. 2, 329-350. [HI 1980] K. Hashimoto and T. Ibukiyama, On the class numbers of positive definite binary quaternion hermitian forms. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 549-601. [Ib 1984] T. Ibukiyama, On symplectic Euler factors of genus 2. J. Fac. Sci. Univ. Tokyo 30 (1984), 587614. [Ih 1964] Y. Ihara, On certain Dirichlet series, J. Math. Soc. Japan 16 (1964), 214-225. [LP 2002] J. Lansky and D. Pollack, Hecke algebras and automorphic forms. Compositio Math. 130 (2002), no. 1, 21–48. [RS 2007] Dinakar Ramakrishnan and Freydoon Shahidi, Siegel modular forms of genus 2 attached to elliptic curves (preprint). Available at www.math.arxiv. [R 2006] N. C. Ryan, Computing the Satake p-parameters of Siegel modular forms. (sub- mitted). [Sk 1992] Nils-Peter Skoruppa, Computations of Siegel modular forms of genus two. Math. Comp. 58 (1992), no. 197, 381–398. [So 2008] Claus M. Sorensen, Potential level-lowering for GSp(4), arXive:0804.0588v1. Department of Mathematics, University of Calgary E-mail address: cunning@math.ucalgary.ca Institut für Experimentelle Mathematik, Universität Duisburg-Essen E-mail address: lassina.dembele@uni-duisburg-essen.de Introduction 1. Hilbert-Siegel modular forms and the Jacquet-Langlands correspondence 1.1. Hilbert-Siegel modular forms 1.2. The Hecke algebra 1.3. Algebraic Hilbert-Siegel autormorphic forms 1.4. The Jacquet-Langlands Correspondence 2. The Algorithm 2.1. The quotient H02(N) 2.2. Brandt matrices 2.3. Computing the group GB(Z) 2.4. Computing the sets 1(p) and 2(p) 2.5. The implementation of the algorithm 3. Numerical examples: F=Q(5) and B=(-1,-1F) 3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3 3.2. Tables of Hilbert cusp forms of parellel weight 4 3.3. Lifts References
In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\Q(\sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms.
Introduction Let F be a real quadratic field of narrow class number one and let B be the unique (up to isomorphism) quaternion algebra over F which is ramified at both archimedean places of F and unramified everywhere else. Let GU2(B) be the unitary similitude group of B⊕2. This is the set of Q-rational points of an algebraic group GB defined over Q. The group GB is an inner form of G := ResF/Q(GSp4) such that G B(R) is compact modulo its centre. (These notions are reviewed at the beginning of Section 1.) In this paper we develop an algorithm which computes automorphic forms on GB in the following sense: given an idealN inOF and an integer k greater than 2, the algorithm returns the Hecke eigensystems of all automorphic forms f of level N and parallel weight k. More precisely, given a prime p in OF , the algorithm returns the Hecke eigenvalues of f at p, and hence the Euler factor Lp(f, s), for each eigenform f of level N and parallel weight k. The algorithm is a generalization of the one developed in [D1 2005] to the genus 2 case. Although we have only described the algorithm in the case of a real quadratic field in this paper, it should be clear from our presentation that it can be adapted to any totally real number field of narrow class number one. The Jacquet-Langlands Correspondence of the title refers to the conjec- tural map JL : Π(GB) → Π(G) from automorphic representations of GB to automorphic representations of G, which is injective, matches L-functions and enjoys other properties compatible with the principle of functoriality; Date: October 29, 2018. 1991 Mathematics Subject Classification. Primary: 11F41 (Hilbert and Hilbert-Siegel modular forms). Key words and phrases. Hilbert-Siegel modular forms, Jacquet-Langlands Correspon- dence, Brandt matrices, Satake parameters. http://arxiv.org/abs/0704.0011v3 2 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ in particular, the image of the Jacquet-Langlands Correspondence is to be contained in the space of holomorphic automorphic representations. If we admit this conjecture, then the algorithm above provides a way to produce examples of cuspidal Hilbert-Siegel modular forms of genus 2 over F and allows us to compute the L-factors of the corresponding automorphic repre- sentations for arbitrary finite primes p of F . In fact, we are also able to use these calculations to provide evidence for the Jacquet-Langlands Correspondence itself by comparing the Euler factors we find with those of known Hilbert-Siegel modular forms obtained by lifting. This we do in the final section of the paper where we observe that some of the Euler factors we compute match those of lifts of Hilbert modular forms, for the primes we computed. Although this does not definitively establish that these Hilbert-Siegel modular forms are indeed lifts, in principle one can establish equality in this way, using an analogue of the Sturm bound. The first systematic approach to Siegel modular forms from a computa- tional viewpoint is due to Skoruppa [Sk 1992] who used Jacobi symbols to generate spaces of such forms. His algorithm, which has been extensively exploited by Ryan [R 2006], applies only to the case of full level structure. More recently, Faber and van der Geer [FvdG1 2004] and [FvdG2 2004] also produced examples of Siegel modular forms by counting points on hy- perelliptic curves of genus 2; again their results are available only in the full level structure case. The most substantial progress toward the com- putation of Siegel modular forms for proper level structure is by Gunnells [Gu 2000] who extended the theory of modular symbols to the symplectic group Sp4/Q. However, this work does not see the cuspidal cohomology, which is the only part of the cohomology which is relevant to arithmetic geometric applications. To the best of our knowledge, there are no numer- ical examples of Hilbert-Siegel modular forms for proper level structure in the literature, with the exception of those produced from liftings of Hilbert modular forms. The outline of the paper is as follows. In Section 1 we recall the basic properties of Hilbert-Siegel modular forms and algebraic automorphic forms together with the Jacquet-Langlands Correspondence. In Section 2 we give a detailed description of our algorithm. Finally, in Section 3 we present numerical results for the quadratic field Q( Acknowledgements. During the course of the preparation of this paper, the second author had helpful email exchanges with several people includ- ing Alexandru Ghitza, David Helm, Marc-Hubert Nicole, David Pollack, Jacques Tilouine and Eric Urban. The authors wish to thank them all. Also, we would like to thank William Stein for allowing us to use the SAGE computer cluster at the University of Washington. And finally, the sec- ond author would like to thank the PIMS institute for their postdoctoral fellowship support, and the University of Calgary for its hospitality. COMPUTING HILBERT-SIEGEL MODULAR FORMS 3 1. Hilbert-Siegel modular forms and the Jacquet-Langlands correspondence Throughout this paper, F denotes a real quadratic field of narrow class number one. The two archimedean places of F and the real embeddings of F will both be denoted v0 and v1. For every a ∈ F , we write a0 (resp. a1) for the image of a under v0 (resp. v1). The ring of integers of F is denoted by OF . For every prime ideal p in OF , the completion of F and OF at p will be denoted by Fp and OFp , respectively. Let B be the unique (up to isomorphism) totally definite quaternion al- gebra over F which is unramified at all finite primes of F . We fix a maximal order OB of B. Also, we choose a splitting field K/F of B that is Ga- lois over Q and such that there exists an isomorphism j : OB ⊗Z OK ∼= M2(OK)⊕M2(OK), where M2(A) denotes the ring of 2× 2-matrices with entries from a ring A. For every finite prime p in F , we fix an isomorphism Bp ∼= M2(Fp) which restricts to an isomorphism from OB, p onto M2(OFp ). The algebraic group G = ResF/Q(GSp4) is defined as follows. For any Q-algebra A, the set of A-rational points of G is given by G(A) = γ ∈ GL4(A⊗Q F ) t = νG(γ)J2 νG(γ) ∈ (A⊗Q F )× where −12 0 This group admits an integral model with A-rational points for every Z- algebra A given by GZ(A) = γ ∈ GL4(A⊗Z OF ) t = νG(γ)J2 νG(γ) ∈ (A⊗Z OF )× For any Q-algebra A, the conjugation on B extends in a natural way to the matrix algebra M2(B ⊗Q A). The algebraic group GB/Q is defined as follows. For any Q-algebra A, the set of A-rational points of GB is given by GB(A) = γ ∈ M2(B ⊗Q A) γγ̄t = νGB(γ)12 νGB (γ) ∈ (A⊗Q F )× This group also admits an integral model with A-rational points for every Z-algebra given by GBZ (A) = γ ∈ M2(OB ⊗Z A) γγ̄t = νGB(γ)12 νGB (γ) ∈ (A⊗Z OF )× The group GB/Q is an inner form of G/Q such that GB(R) is compact modulo its center. Combining the isomorphism j (see above) with con- jugation by a permutation matrix, we obtain an isomorphism GBZ (OK) ∼= 4 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ GZ(OK), which we fix from now on. For every prime ideal p in F , the split- ting of GB at p amounts to the splitting of the quaternion algebra B at p; we refer to [D1 2005] for further details. By the choice of the quaternion algebra B, we have GB(Q̂) ∼= G(Q̂). (We denote the finite adèles of Q (resp. Z) by Q̂ (resp. Ẑ)). 1.1. Hilbert-Siegel modular forms. We fix an integer k ≥ 3 and, for simplicity, we restrict ourselves to Hilbert-Siegel modular forms of parallel weight k. The real embeddings v0 and v1 of F extend to G(Q) = GSp4(F ) in a natural way. We denote by GSp+4 (F ) the subgroup of elements γ with totally positive similitude factor νG(γ). We recall that the Siegel upper-half plane of genus 2 is defined by H2 = {γ ∈ GL2(C) ∣ γt = γ and Im(γ) is positive definite }. We also recall that GSp+4 (F ) acts on H (τ0, τ1) := (a0τ0 + b0)(c0τ0 + d0) −1, (a1τ1 + b1)(c1τ1 + d1) This induces an action on the space of functions f : H22 → C by , f |kγ(τ) = νG(γi) det(ciτi + di)k f(τ). Let N be an ideal in OF and set Γ0(N) = ∈ GSp+4 (OF ) ∣ c ≡ 0(N) A Hilbert-Siegel modular form of level N and parallel weight k is a holomorphic function f : H22 → C such that ∀γ ∈ Γ0(N), f |kγ = f. The space of Hilbert-Siegel modular forms of parallel weight k and level N is denoted Mk(N). Each f ∈Mk(N) admits a Fourier expansion, which by the Koecher principle takes the form ∀τ ∈ H22, f(τ) = {Q}∪{0} 2πiTr(Qτ), where Q ∈ M2(F ) runs over all symmetric totally positive and semi-definite matrices. A Hilbert-Siegel modular forms f is a cusp form if, for all γ ∈ 4 (F ), the constant term in the Fourier expansion of f |kγ is zero. The space of Hilbert-Siegel cusp forms is denoted Sk(N). COMPUTING HILBERT-SIEGEL MODULAR FORMS 5 1.2. The Hecke algebra. The space Sk(N) comes equipped with a Hecke action, which we now recall. Take u ∈ GSp+4 (F ) ∩M4(OF ), and write the finite disjoint union Γ0(N)uΓ0(N) = Γ0(N)ui. Then the Hecke operator [Γ0(N)uΓ0(N)] on Sk(N) is given by [Γ0(N)uΓ0(N)]f = f |kui. Let p be a prime ideal in OF and let πp be a totally positive generator of p; let T1(p) and T2(p) be the Hecke operators corresponding to the double Γ0(N)-cosets of the symplectic similitude matrices 1 0 0 0 0 1 0 0 0 0 πp 0 0 0 0 πp 1 0 0 0 0 πp 0 0 0 0 π2p 0 0 0 0 πp respectively. (We remind the reader of the symplectic form J2 fixed at the beginning of Section 1.) The Hecke algebra Tk(N) is the Z-algebra generated by the operators T1(p) and T2(p), where p runs over all primes not dividing N . 1.3. Algebraic Hilbert-Siegel autormorphic forms. We only consider level structure of Siegel type. Namely, we define the compact open subgroup U0(N) of G(Q̂) by U0(N) = GSp4(OFp )× ep ), where N = p|N p ep and ep ) := ∈ GSp4(OFp ) ∣ c ≡ 0 mod pep The weight representation is defined as follows. Let Lk be the repre- sentation of GSp4(C) of highest weight (k− 3, k− 3). We let Vk = Lk ⊗Lk and define the complex representation (ρk, Vk) by ρk : G B(R) −→ GL(Vk), where the action on the first factor is via v0, and the action on the second one is via v1. The space of algebraic Hilbert-Siegel modular forms of weight k and level N is given by MBk (N) := f : GB(Q̂)/U0(N) → Vk ∣ ∀γ ∈ GB(Q), f |kγ = f 6 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ where f |kγ(x) = f(γx)γ, for all x ∈ GB(Q̂)/U0(N). When k = 3, we let IBk (N) := f : GB(Q)\GB(Q̂)/U0(N) → C ∣ f is constant Then, the space of algebraic Hilbert-Siegel cusp forms of weight k and level N is defined by SBk (N) := MBk (N) if k > 3, MBk (N)/I k (N) if k = 3. The action of the Hecke algebra on SBk (N) is given as follows. For any u ∈ G(Q̂), write the finite disjoint union U0(N)uU0(N) = uiU0(N), and define [U0(N)uU0(N)] : S k (N) → SBk (N) f 7→ f |k[U0(N)uU0(N)], f |k[U0(N)uU0(N)](x) = f(xui), x ∈ G(Q̂). For any prime p ∤ N , let ̟p be a local uniformizer at p. The local Hecke alge- bra at p is generated by the Hecke operators T1(p) and T2(p) corresponding to the double U0(N)-cosets ∆1(p) and ∆2(p) of the matrices 1 0 0 0 0 1 0 0 0 0 ̟p 0 0 0 0 ̟p 1 0 0 0 0 ̟p 0 0 0 0 ̟2 0 0 0 ̟p respectively. We let TBk (N) be the Hecke algebra generated by T1(p) and T2(p) for all primes p ∤ N . 1.4. The Jacquet-Langlands Correspondence. The Hecke modules Sk(N) and SBk (N) are related by the following conjecture known as the Jacquet- Langlands Correspondence for symplectic similitude groups. Conjecture 1. The Hecke algebras Tk(N) and T k (N) are isomorphic and there is a compatible isomorphism of Hecke modules Sk(N) ∼−→ SBk (N). It is common, but perhaps not entirely accurate, to attribute this con- jecture to Jacquet-Langlands. To the best of our knowledge, the correspon- dence in this form was first discussed by Ihara [Ih 1964] in the case F = Q. In [Ib 1984], Ibukiyama provided some numerical evidence. On the other hand, it is appropriate to refer to Conjecture 1 as the Jacquet-Langlands Corre- spondence (for GSp(4)) since it is an analogue of the Jacquet-Langlands COMPUTING HILBERT-SIEGEL MODULAR FORMS 7 Correspondence (for GL(2)) which relates automorphic representations of the multiplicative group of a quaternion algebra with certain automorphic representations of GL(2) (see [JL 1970]). Both correspondences are, in turn, special consequences of the principle of functoriality, as expounded by Lang- lands. Finally, it appears that Conjecture 1 may soon be a theorem due to the work of [So 2008] and the forthcoming book by James Arthur on auto- morphic representations of classical groups. 2. The Algorithm In this section, we present the algorithm we used in order to compute the Hecke module of (algebraic) Hilbert-Siegel modular forms. The main assumption in this section is that the class number of the principal genus of GB is 1. (We refer to [D3 2007] to see how one can relax this condition on the class number.) We recall that since B is totally definite, GB satis- fies Proposition 1.4 in Gross [Gr 1999]. Thus the group GB(R) is compact modulo its centre, and Γ = GB(Z)/O×F is finite. For any prime p in F , let Fp = OF /p be the residue field at p and define the reduction map M2(OB, p) → M4(Fp) g 7→ g̃, where we use the splitting of OB,p that was fixed at the beginning of Sec- tion 1. Now, choose a totally positive generator πp of p and put Θ1(p) := Γ\ u ∈ M2(OB) ∣ uūt = πp12and rank(g̃) = 2 Θ2(p) := Γ\ u ∈ M2(OB) ∣ uūt = π2 12 and rank(g̃) = 1 We let H20(N) = G(Ẑ)/U0(N). Then the group Γ acts on H20(N), thus on the space of functions f : H20(N) → Vk by ∀x ∈ H20(N),∀γ ∈ Γ, f |kγ(x) := f(γx)γ. Theorem 2. There is an isomorphism of Hecke modules MBk (N) f : H20(N) → Vk ∣ f |kγ = f, γ ∈ Γ where the Hecke action on the right hand side is given by f |kT1(p) = u∈Θ1(p) f |ku, f |kT2(p) = u∈Θ2(p) f |ku. Proof. The canonical map φ : GB(Z)\GB(Ẑ)/U0(N) → GB(Q)\GB(Q̂)/U0(N) is an injection. Making use of the fact that the class number in the principal genus of GB is one (GB(Q̂) = GB(Q)GBZ (Ẑ)), we see that φ is in fact a 8 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ bijection. Since each element f ∈ MBk (N) is determined by its values on a set of coset representatives of GB(Q)\GB(Q̂)/U0(N), the map φ induces an isomorphism of complex vector spaces MBk (N) f : H20(N) → Vk ∣ f |kγ = f, γ ∈ Γ f 7−→ f ◦ φ. We make this into a Hecke module isomorphism by defining the Hecke action on the right hand side as indicated in the statement of the theorem. � In the rest of this section, we explain the main steps of the algorithm provided by Theorem 2. 2.1. The quotient H20(N). Keeping the notations of the previous section, we recall that N = p|N p ep . Let p be a prime dividing N and consider the rank 4 free OFp/pep -module L = OFp/pep endowed with the symplectic pairing 〈 , 〉 given by the matrix −12 0 where 12 is the identity matrix in M2(OFp/pep ). Let M be a rank 2 OFp/pep -submodule which is a direct factor in L. We say that M is isotropic if 〈u, v〉 = 0 for all u, v ∈ M . We recall that GSp4(OFp ) acts transitively on the set of rank 2, isotropic OFp/pep -submodules of L and that the stabilizer of the submodule generated by e1 = (1, 0, 0, 0) T and e2 = (0, 1, 0, 0) T is U0(p ep ). The quotient H20(pep ) = GSp4(OFp )/U0(pep ) is the set of rank 2, isotropic OFp/pep -submodules of L. Via the reduction map ÔF → OF /N , the quotient GZ(Ẑ)/U0(N) can be identified with the product H20(N) = H20(pep ). The cardinality of H20(N) is extremely useful and is determined using the following lemma. Lemma 1. Let p be a prime in F and ep ≥ 1 an integer. Then, the cardi- nality of the set H20(pep ) is given by #H20(pep ) = N(p)3(ep−1)(N(p) + 1)(N(p)2 + 1). Proof. For ep = 1, the cardinality of the Lagrange variety over the finite field Fp = OF /p is given by (N(p) + 1)(N(p)2 + 1). Proceed by induction on ep. � We have more to say about elements of H20(pep ) in Subsection 2.5. COMPUTING HILBERT-SIEGEL MODULAR FORMS 9 2.2. Brandt matrices. Let F = {x1, . . . , xh} be a fundamental domain for the action of Γ on H20(N) and, for each i, let Γi be the stabilizer of xi. Then, every element in MBk (N) is completely determined by its values on F . Thus, there is an isomorphism of complex spaces MBk (N) → f 7→ (f(xi)), where V is the subspace of Γi-invariants in Vk. For any x, y ∈ H20(N), we let Θ1(x, y, p) := u ∈ Θ1(p) ∣ ∃γ ∈ Γ, ux = γy Θ2(x, y, p) := u ∈ Θ2(p) ∣ ∃γ ∈ Γ, ux = γy Proposition 3. The actions of the Hecke operators Ts(p), s = 1, 2, are given by the Brandt matrices Bs(p) = (bsij(p)), where bsji(p) : V k → V v 7→ v · u∈Θs(xi, xj ,p) γ−1u u Proof. The proof of Proposition 3 follows the lines of [D1 2005, §3]. � 2.3. Computing the group GB(Z). It is enough to compute the subgroup Γ consisting of the elements in GB(Z) with similitude factor 1. But it is easy to see that u, v ∈ O1B u, v ∈ O1B where O1B is the group of norm 1 elements. 2.4. Computing the sets Θ1(p) and Θ2(p). Let us consider the quadratic form on the vector space V = B2 given by V → F (a, b) 7→ ||(a, b)|| := nr(a) + nr(b), where nr is the reduced norm on B. This determines an inner form V × V → F (u, v) 7→ 〈u, v〉. An element of Θ1(p) (resp. Θ2(p)) is a unitary matrix γ ∈ M2(OB) with respect to this inner form such that the norm of each row is πp (resp. π and the rank of the reduced matrix is 1). So we first start by computing all the vectors u = (a, b) ∈ O2B such that ||u|| = πp (resp. ||u|| = π2p). And for each such vector u, we compute the vectors v = (c, d) ∈ O2B of the same 10 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ norm such that 〈u, v〉 = 0. The corresponding matrix γ = belongs to Θ1(p) (resp. Θ2(p)) when its reduction mod p has the appropriate rank. We list all these matrices up to equivalence and stop when we reach the right cardinality. 2.5. The implementation of the algorithm. The implementation of the algorithm is similar to that of [D1 2005]. However, it is important to note how we represent elements in H20(N) so that we can retrieve them easily once stored. As in [D1 2005] we choose to work with the product H20(N) = H20(pep ). Using Plucker’s coordinates, we can view H20(pep ) as a closed subspace of P5(OFp/pep ). We then represent each element in H20(pep ) by choosing a point x = (a0 : · · · : a5) = [u ∧ v] ∈ P5(OFp/pep ) such that the submodule M generated by u and v is a Lagrange submodule, and the first invertible coordinate is scaled to 1. Remark 1. In [LP 2002], Lansky and Pollack describe an algorithm which computes algebraic modular forms on the same inner form of GSp4/Q that we use. We would like to note that there are some differences between the two algorithms. Although [LP 2002] also uses the flag variety H20(N) in order to determine the double coset space GB(Q)\GB(Q̂)/U0(N), it later returns to the adelic setting in order to compute the Brandt matrices. In contrast, Theorem 2 and Proposition 3 allow us to avoid that unnecessary step by describing the Hecke action on the flag variety H20(N) directly. As a result, we get an algorithm that is more efficient. 3. Numerical examples: F = Q( 5) and B = −1,−1 In this section, we provide some numerical examples using the quadratic field F = Q( 5). It is proven in K. Hashimoto and T. Ibukiyama [HI 1980] that, for the Hamilton quaternion algebra B over F , the class number of the principal genus of GB is one. We use our algorithm to compute all the systems of Hecke eigenvalues of Hilbert-Siegel cusp forms of weight 3 and level N that are defined over real quadratic fields, where N runs over all prime ideals of norm less than 50. We then determine which of the forms we obtained are possible lifts of Hilbert cusp forms by comparing the Hecke eigenvalues for those primes. 3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3. In Table 1 we list all the systems of eigenvalues of Hilbert-Siegel cusp forms of weight 3 and level N that are defined over real quadratic fields, where N runs over all prime ideals in F of norm less than 50. Here are the conventions we use in the tables. COMPUTING HILBERT-SIEGEL MODULAR FORMS 11 (1) For a quadratic field K of discriminant D, we let ωD be a generator of the ring of integers OK of K. (2) The first row contains the level N , given in the format (Norm(N), α) for some generator α ∈ F of N , and the dimensions of the relevant spaces. (3) The second row lists the Hecke operators that have been computed. (4) For each eigenform f , the Hecke eigenvalues are given in a row, and the last entry of that row indicates if the form f is a probable lift. (5) The levels and the eigenforms are both listed up to Galois conjuga- tion. For an eigenform f and a given prime p ∤ N , let a1(p, f) and a2(p, f) be the eigenvalues of the Hecke operators T1(p) and T2(p), respectively. Then the Euler factor Lp(f, s) is given (for example, in [AS 2001, §3.4]) by Lp(f, s) = Qp(q −s)−1, where Qp(x) = 1− a1(p, f)x+ b1(p, f)x2 − a1(p, f)q2k−3x3 + q4k−6x4, b1(p, f) = a1(p, f) 2 − a2(p, f)− q2k−4, q = N(p). 3.2. Tables of Hilbert cusp forms of parellel weight 4. In Table 2, we list all the Hilbert cusp forms of parallel weight 4 and level N that are defined over real quadratic fields, with N running over all prime ideals of norm less than 50. (They are computed by using the algorithm in [D1 2005]). We use this data in order to determine the forms in Table 1 that are possible lifts from GL2. 3.3. Lifts. There are two types of lifts from GL2 to GSp4. The first one corresponds to the homomorphism of L-groups determined by the long root embedding into GSp4, and the second one by the short root embedding. (See [LP 2002] for more details). Let f be a Hilbert cusp form of parallel weight k and level N with Hecke eigenvalues a(p, f), where p is a prime not dividing N . Let φ be the lift of f to GSp4 via the long root, and ψ the one via the short root. Then the Hecke eigenvalues of φ are given by a1(p, φ) = a(p, f) N(p) 2 +N(p)2 +N(p) a2(p, φ) = a(p, f) N(p) 2 (N(p) + 1) +N(p)2 − 1, and the Hecke eigenvalues of ψ are given by a1(p, ψ) = a(p, f) 2 − 2 a(p, f) N(p) a2(p, ψ) = a(p, f) N(p)4−2k − 3 a(p, f)2 N(p)3−k +N(p)2 − 1. The second lift ψ is the so-called symmetric cube lifting. 12 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ N = (4, 2) : dimMB (N) = 2, dimSB (N) = 1 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 −4 0 20 −36 140 580 yes N = (5, 2 + ω5) : dimM (N) = 2, dimSB (N) = 1 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 20 15 −5 0 40 −420 yes N = (9, 3) : dimMB (N) = 3, dimSB (N) = 2 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 25− 3ω41 40− 15ω41 30 + 6ω41 24 + 36ω41 −9 0 yes N = (11, 3 + ω5) : dimM (N) = 3, dimSB (N) = 2 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 24 35 34 48 88 60 yes f2 −20 35 −10 4 0 60 no N = (19, 4 + ω5) : dimM (N) = 5, dimSB (N) = 4 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 4 11 −20 28 6 76 no f2 7 −50 15 −66 73 −90 yes f3 24 + ω161 35 + 5ω161 36− ω161 60− 6ω161 98− 3ω161 160− 30ω161 yes N = (29, 5 + ω5) : dimM (N) = 9, dimSB (N) = 8 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 −4 11 10 20 30 60 no f2 8 −45 30 24 50 −320 yes f3 17 0 9 −102 86 40 yes N = (31, 5 + 2ω5) : dimM (N) = 12, dimSB (N) = 11 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 13 −20 20 −36 76 −60 yes N = (41, 6 + ω5) : dimM (N) = 19, dimSB (N) = 18 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 10 20 −10 29 30 −20 no f2 −1 1 5 14 −2 −56 no f3 27 50 40 84 124 420 yes f4 −12 19 30 65 0 0 no f5 16− 2ω21 −5− 10ω21 21 + 4ω21 −30 + 24ω21 72− 2ω21 −100− 20ω21 yes f6 2− 6ω5 11− 2ω5 8 + 4ω5 11− 4ω5 −12 + 54ω5 160 + 40ω5 no N = (49, 7) : dimMB (N) = 26, dimSB (N) = 25 T1(2) T2(2) T1( 5) T2( 5) T1(3) T2(3) Lift? f1 5 −60 46 120 40 −420 yes f2 4 + 4ω65 32 + 3ω65 12− 4ω65 44− 4ω65 −6− 12ω65 145 + 8ω65 no Table 1. Hilbert-Siegel eigenforms of weight 3 COMPUTING HILBERT-SIEGEL MODULAR FORMS 13 N (4, 2) (5, 2 + ω5) (9, 3) (11, 3 + ω5) N(p) p a(p, f1) a(p, f1) a(p, f1) a(p, f1) 4 2 −4 0 5− 3ω41 4 5 2 + ω5 −10 −5 6ω41 4 9 3 50 −50 −9 −2 11 3 + 2ω5 −28 32 −18− 6ω41 −10 11 3 + ω5 −28 32 −18− 6ω41 −11 19 4 + 3ω5 60 100 −40 + 24ω41 −94 19 4 + ω5 60 100 −40 + 24ω41 28 N (19, 4 + ω5) (29, 5 + ω5) N(p) p a(p, f1) a(p, f2) a(p, f1) a(p, f2) 4 2 −13 5− ω161 −12 −3 5 2 + ω5 −15 5 + ω161 0 −21 9 3 −17 5 + 3ω161 −40 −4 11 3 + 2ω5 −6 2 + 8ω161 −68 37 11 3 + ω5 33 7− 7ω161 30 −66 19 4 + 3ω5 −139 −15− 9ω161 −28 −40 19 4 + ω5 19 −19 84 −9 N (31, 5 + 2ω5) (41, 6 + ω5) N(p) p a(p, f1) a(p, f1) a(p, f2) 4 2 −7 7 −4− 2ω21 5 2 + ω5 −10 10 −9 + 4ω21 9 3 −14 34 −18− 2ω21 11 3 + 2ω5 −20 −60 −19 11 3 + ω5 −28 −2 −24− 4ω21 19 4 + 3ω5 −12 74 4− 50ω21 19 4 + ω5 28 16 −29 + 44ω21 N (49, 7) N(p) p a(p, f1) a(p, f2) 4 2 −15 −2 5 2 + ω5 16 −10 9 3 −50 −11 11 3 + 2ω5 −8 −7− 28ω13 11 3 + ω5 −8 −35 + 28ω13 19 4 + 3ω5 −110 −26 + 14ω13 19 4 + ω5 −110 −12− 14ω13 Table 2. Hilbert eigenforms of weight 4 Remark 2. So far, our algorithm has been implemented only for congruence subgroups of Siegel type. We intend to improve the implementation in the near future so as to include more additional level structures such as the Klingen type. Indeed, Ramakrishnan and Shahidi [RS 2007] recently showed the existence of symmetric cube lifts for non-CM elliptic curves E/Q to GSp4/Q. And their result should hold for other totally real number fields, with the level structures of the lifts being of Klingen type. Unfortunately, 14 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ those lifts cannot be seen in our current tables. For example, there are modular elliptic curves over Q( 5) whose conductors have norm 31, 41 and 49, but the corresponding symmetric cubic lifts do not appear in Table 1. We would like to remedy that in our next implementation. References [D1 2005] L. Dembélé, Explicit computations of Hilbert modular forms on Q( 5). Exper- iment. Math. 14 (2005), no. 4, 457–466. [D2 2007] L. Dembélé, Quaternionic M -symbols, Brandt matrices and Hilbert modular forms. Math. Comp. 76, no 258, (2007), 1039-1057. Also available electronically. [D3 2007] L. Dembélé, On the computation of algebraic modular forms (submitted). [AS 2001] Mahdi Asgari and Ralf Schmidt, Siegel modular forms and representations, Manuscripta Math. 104 (2001), 173–200. [FvdG1 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes. I, C. R. Math. Acad. Sci. Paris 338 (2004), no. 5, 381–384. [FvdG2 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes. II, C. R. Math. Acad. Sci. Paris 338 (2004), no. 6, 467–470. [JL 1970] Hervé Jacquet and Robert Langlands, Automorphic forms on GL(2), Lecture notes in mathematics 114 and 278, 1970. [Gr 1999] Benedict H. Gross, Algebraic modular forms. Israel J. Math. 113 (1999), 61–93. [Gu 2000] P. Gunnells, Symplectic modular symbols, Duke Math. J. 102 (2000), no. 2, 329-350. [HI 1980] K. Hashimoto and T. Ibukiyama, On the class numbers of positive definite binary quaternion hermitian forms. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 549-601. [Ib 1984] T. Ibukiyama, On symplectic Euler factors of genus 2. J. Fac. Sci. Univ. Tokyo 30 (1984), 587614. [Ih 1964] Y. Ihara, On certain Dirichlet series, J. Math. Soc. Japan 16 (1964), 214-225. [LP 2002] J. Lansky and D. Pollack, Hecke algebras and automorphic forms. Compositio Math. 130 (2002), no. 1, 21–48. [RS 2007] Dinakar Ramakrishnan and Freydoon Shahidi, Siegel modular forms of genus 2 attached to elliptic curves (preprint). Available at www.math.arxiv. [R 2006] N. C. Ryan, Computing the Satake p-parameters of Siegel modular forms. (sub- mitted). [Sk 1992] Nils-Peter Skoruppa, Computations of Siegel modular forms of genus two. Math. Comp. 58 (1992), no. 197, 381–398. [So 2008] Claus M. Sorensen, Potential level-lowering for GSp(4), arXive:0804.0588v1. Department of Mathematics, University of Calgary E-mail address: cunning@math.ucalgary.ca Institut für Experimentelle Mathematik, Universität Duisburg-Essen E-mail address: lassina.dembele@uni-duisburg-essen.de Introduction 1. Hilbert-Siegel modular forms and the Jacquet-Langlands correspondence 1.1. Hilbert-Siegel modular forms 1.2. The Hecke algebra 1.3. Algebraic Hilbert-Siegel autormorphic forms 1.4. The Jacquet-Langlands Correspondence 2. The Algorithm 2.1. The quotient H02(N) 2.2. Brandt matrices 2.3. Computing the group GB(Z) 2.4. Computing the sets 1(p) and 2(p) 2.5. The implementation of the algorithm 3. Numerical examples: F=Q(5) and B=(-1,-1F) 3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3 3.2. Tables of Hilbert cusp forms of parellel weight 4 3.3. Lifts References
704.001
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS OF A MODULAR FORM OF HALF INTEGRAL WEIGHT MODULO PRIMES D. CHOI Abstract. Recently, Bruinier and Ono classified cusp forms f(z) := af (n)q Sλ+ 1 (Γ0(N), χ) ∩ Z[[q]] that does not satisfy a certain distribution property for modulo odd primes p. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes p ≥ 5. As applications of our main theorem we derive distribution properties, for modulo primes p ≥ 5, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman’s conjecture for overpartitions. 1. Introduction and Results Let Mλ+ 1 (Γ0(N), χ) and Sλ+ 1 (Γ0(N), χ) be the spaces, respectively, of modular forms and cusp forms of weight λ + 1 on Γ0(N) with a Dirichlet character χ whose conductor divides N . If f(z) ∈Mλ+ 1 (Γ0(N), χ), then f(z) has the form f(z) = a(n)qn, where q := e2πiz. It is well-known that the coefficients of f are related to interesting objects in number theory such as the special values of L-function, class number, traces of singular moduli and so on. In this paper, we study congruence properties of the Fourier coefficients of f(z) ∈Mλ+ 1 (Γ0(N), χ) ∩ Z[[q]] and their applications. Recently, Bruinier and Ono proved in [3] that g(z) ∈ Sλ+ 1 (Γ0(N), χ) ∩ Z[[q]] has a special form (see (2.1)) by modulo p when p is an odd prime and the coefficients of f(z) do not satisfy the following property for p: Property A. IfM is a positive integer, we say that a sequence α(n) ∈ Z satisfies Property A for M if for every integer r ♯{ 1 ≤ n ≤ X | α(n) ≡ r (mod M) and gcd(M,n) = 1} if r 6≡ 0 (mod M), X if r ≡ 0 (mod M). 2000 Mathematics Subject Classification. 11F11,11F33. Key words and phrases. Modular forms, Congruences. http://arxiv.org/abs/0704.0012v1 2 D. CHOI θ(f(z)) := f(z) = n · a(n)qn. Using Rankin-Cohen Bracket (see (2.3)), we prove that there exists f̃(z) ∈ Sλ+p+1+ 1 (Γ0(4N), χ) ∩ Z[[q]] such that θ(f(z)) ≡ f̃(z) (mod p). We extend the results in [3] to modular forms of half integral weight. Theorem 1. Let λ be a non-negative integer. We assume that f(z) = n=0 a(n)q Mλ+ 1 (Γ0(4N), χ) ∩ Z[[q]], where χ is a real Dirichlet character. If p ≥ 5 is a prime and there exists a positive integer n for which gcd(a(n), p) = 1 and gcd(n, p) = 1, then at least one of the following is true: (1) The coefficients of θp−1(f(z)) satisfies Property A for p. (2) There are finitely many square-free integers n1, n2, · · · , nt for which (1.1) θp−1(f(z)) ≡ a(nim 2)qnim (mod p). Moreover, if gcd(4N, p) = 1 and an odd prime ℓ divides some ni, then p|(ℓ− 1)ℓ(ℓ+ 1)N or ℓ | N. Remark 1.1. Note that for every odd prime p ≥ 5, θp−1(f(z)) ≡ a(n)qn (mod p). As an applications of Theorem 1, we study the distribution of traces of singular moduli modulo primes p ≥ 5. Let j(z) be the usual j-invariant function. We denote by Fd the set of positive definite binary quadratic forms F (x, y) = ax2 + bxy + cy2 = [a, b, c] with discriminant −d = b2−4ac. For each F (x, y), let αF be the unique complex number in the complex upper half plane, which is a root of F (x, 1). We define ωF ∈ {1, 2, 3} as ωF := 2 if F ∼Γ [a, 0, a], 3 if F ∼Γ [a, a, a], 1 otherwise, where Γ := SL2(Z). Here, F ∼Γ [a, b, c] denotes that F (x, y) is equivalent to [a, b, c]. From these notations, we define the Hecke trace of singular moduli. DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 3 Definition 1.2. If m ≥ 1, then we define the mth Hecke trace of the singular moduli of discriminant −d as tm(d) := F∈Fd/Γ jm(αF ) where Fd/Γ denotes a set of Γ−equivalence classes of Fd and jm(z) := j(z)|T0(m) = az + b Here, T0(m) denotes the normalized mth weight zero Hecke operator. Note that t1(d) = t(d), where t(d) := F∈Fd/Γ j(αF )− 744 is the usual trace of singular moduli. Let h(z) := η(z)2 η(2z) · E4(4z) η(4z)6 and Bm(1, d) denote the coefficient of q d in h(z)|T (m2, 1, χ0), where E4(z) := 1 + 240 d3qn, η(z) := q (1− qn) , and χ0 is a trivial character. Here, T (m 2, λ, χ) denotes the mth Hecke operator of weight λ + 1 with a Dirichlet chracter χ (see VI. §3. in [5] or (2.5)). Zagier proved in [11] that for all m and d (1.2) tm(d) = −Bm(1, d). Using these generating functions, Ahlgren and Ono studied the divisibility properties of traces and Hecke traces of singular moduli in terms of the factorization of primes in imaginary quadratic fields (see [2]). For example, they proved that a positive proportion of the primes ℓ has the property that tm(ℓ 3n) ≡ 0 (mod ps) for every positive integer n coprime to ℓ such that p is inert or ramified in Q . Here, p is an odd prime, and s and m are integers with p ∤ m. In the following theorem, we give the distribution of traces and Hecke traces of singular moduli modulo primes p. 4 D. CHOI Theorem 2. Suppose that p ≥ 5 is a prime such that p ≡ 2 (mod 3). (1) Then, for every integer r, p ∤ r, ♯{ 1 ≤ n ≤ X | t1(n) ≡ r (mod p)} ≫r,p if r 6≡ 0 (mod p) X if r ≡ 0 (mod p). (2) Then, a positive proportion of the primes ℓ has the property that ♯{ 1 ≤ n ≤ X | tℓ(n) ≡ r (mod p)} ≫r,p if r 6≡ 0 (mod p) X if r ≡ 0 (mod p). for every integer r, p ∤ r. As another application we study the distribution of Hurwitz class number modulo primes p ≥ 5. The Hurwitz class number H(−N) is defined as follows: the class number of quadratic forms of the discriminant −N where each class C is counted with multiplicity Aut(C) . The following theorem gives the distribution of Hurwitz class number modulo primes p ≥ 5. Theorem 3. Suppose that p ≥ 5 is a prime. Then, for every integer r ♯{ 1 ≤ n ≤ X | H(n) ≡ r (mod p)} ≫r,p if r 6≡ 0 (mod p), X if r ≡ 0 (mod p). We also use the main theorem to study an analogue of Newman’s conjecture for overpar- titions. Newman’s conjecture concerns the distribution of the ordinary partition function modulo primes p. Newman’s Conjecture. Let P (n) be an ordinary partition function. If M is a positive integer, then for every integer r there are infinitely many nonnegative integer n for which P (n) ≡ r (mod M). This conjecture was already studied by many mathematicians (see Chapter 5. in [8]). The overpartition of a natural number n is a partition of n in which the first occurrence of a number may be overlined. Let P̄ (n) be the number of the overpartition of an integer n. As an analogue of Newman’s conjecture, the following theorem gives a distribution property of P̄ (n) modulo odd primes p. Theorem 4. Suppose that p ≥ 5 is a prime such that p ≡ 2 (mod 3). Then, for every integer r, ♯{ 1 ≤ n ≤ X | P̄ (n) ≡ r (mod p)} ≫r,p if r 6≡ 0 (mod p), X if r ≡ 0 (mod p). Remark 1.3. When r ≡ 0 (mod p), Theorem 2, 3 and 4 were proved in [2] and [10]. DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 5 Next sections are detailed proofs of theorems: Section 2 gives a proof of Theorem 1. In Section 3, we give the proofs of Theorem 2, 3, and 4. 2. Proof of Theorem 1 We begin by stating the following theorem proved in [3]. Theorem 2.1 ([3]). Let λ be a non-negative integer. Suppose that g(z) = n=0 ag(n)q Sλ+ 1 (Γ0(4N), χ) ∩ Z[[q]], where χ is a real Dirichlet character. If p is an odd prime and a positive integer n exists for which gcd(ag(n), p) = 1, then at least one of the following is true: (1) If 0 ≤ r < p, then ♯{ 1 ≤ n ≤ X | ag(n) ≡ r (mod p)} ≫r,M if r 6≡ 0 (mod p), X if r ≡ 0 (mod p). (2) There are finitely many square-free integers n1, n2, · · · , nt for which (2.1) g(z) ≡ ag(nim 2)qnim (mod p). Moreover, if gcd(p, 4N) = 1, ǫ ∈ {±1}, and ℓ ∤ 4Np is a prime with ∈ {0, ǫ} for 1 ≤ i ≤ t, then (ℓ−1)g(z) is an eigenform modulo p of the half-integral weight Hecke operator T (ℓ2, λ, χ). In particular, we have (2.2) (ℓ− 1)g(z)|T (ℓ2, λ, χ) ≡ ǫχ(p) (−1)λ ℓλ + ℓλ−1 (ℓ− 1)g(z) (mod p). Recall that f(z) = a(n)qn ∈ Mλ+ 1 (Γ0(4N), χ) ∩ Z[[q]]. Thus, to apply Theorem 2.1, we show that there exists a cusp form f̃(z) such that f̃(z) ≡ θp−1(f(z)) (mod p) for a prime p ≥ 5. Lemma 2.2. Suppose that p ≥ 5 is a prime and f(z) = a(n)qn ∈Mλ+ 1 (Γ0(N), χ) ∩ Z[[q]]. Then, there exists a cusp form f̃(z) ∈ Sλ+(p+1)(p−1)+ 1 (Γ0(N), χ) ∩ Z[[q]] such that f̃(z) ≡ θp−1(f(z)) (mod p). Proof of Lemma 2.2. For F (z) ∈Mk1 (Γ0(N), χ1) and G(z) ∈Mk2 (Γ0(N), χ2), let (2.3) [F (z), G(z)]1 := θ(F (z)) ·G(z)− F (z) · θ(G(z)). This operator is referred to as a Rankin-Cohen 1-bracket, and it was proved in [4] that [F (z), G(z)]1 ∈ S k1+k2 (Γ0(N), χ1χ2χ 6 D. CHOI where χ′ = 1 if k1 and k2 ∈ Z, χ′(d) = 2 if ki ∈ Z and k3−i + Z, and χ′(d) = ) k1+k2 2 if k1 and k2 For even k ≥ 4, let Ek(z) := 1− dk−1qn be the usual normalized Eisenstein series of weight k. Here, the number Bk denotes the kth Bernoulli number. The function Ek(z) is a modular form of weight k on SL2(Z), and (2.4) Ep−1(z) ≡ 1 (mod p) (see [6]). From (2.3) and (2.4), we have [Ep−1(z), f(z)]1 ≡ θ(f(z)) (mod p) and [Ep−1(z), f(z)]1 ∈ Sλ+p+1+ 1 (Γ0(N), χ). Repeating this method p− 1 times, we com- plete the proof. � Using the following lemma, we can deal with the divisibility of ag(n) for positive integers n, p ∤ n, where g(z) = n=1 ag(n)q n ∈ Sλ+ 1 (Γ0(N), χ) ∩ Z[[q]]. Lemma 2.3 (see Chapter 3 in [8]). Suppose that g(z) = n=1 ag(n)q n ∈ Sλ+ 1 (Γ0(N), χ) has coefficients in OK , the algebraic integers of some number field K. Furthermore, suppose that λ ≥ 1 and that m ⊂ OK is an ideal norm M . (1) Then, a positive proportion of the primes Q ≡ −1 (mod 4MN) has the property g(z)|T (Q2, λ, χ) ≡ 0 (mod m). (2) Then a positive proportion of the primes Q ≡ 1 (mod 4MN) has the property that g(z)|T (Q2, λ, χ) ≡ 2g(z) (mod m). We can now prove Theorem 1. Proof of Theorem 1. From Lemma 2.2, there exists a cusp form f̃(z) ∈ Sλ+(p+1)(p−1)+ 1 (Γ0(N), χ) ∩ Z[[q]] such that f̃(z) ≡ θp−1(f(z)) (mod p). Note that, for F (z) = n=0 aF (n)q n ∈ Mk+ 1 (Γ0(N), χ) and each prime Q ∤ N , the half-integral weight Hecke operator T (Q2, λ, χ) is defined as (2.5) F (z)|T (Q2, k, χ) aF (Q 2n) + χ∗(Q) Qk−1aF (n) + χ ∗(Q2)Q2k−1aF (n/Q DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 7 where χ∗(n) := χ∗(n) (−1)k and aF (n/Q 2) = 0 if Q2 ∤ n. If F (z)|T (Q2, k, χ) ≡ 0 (mod p) for a prime Q ∤ N , then we have aF (Q 2 ·Qn) + χ∗(Q) Qk−1aF (Qn) + χ ∗(Q2)Q2k−1aF Qn/Q2 ≡ aF (Q3n) ≡ 0 (mod p) for every positive integer n such that gcd(Q, n) = 1. Thus, we have the following by Lemma 2.3-(1): ♯{ 1 ≤ n ≤ X | a(n) ≡ 0 (mod p) and gcd(p, n) = 1} ≫ X. We apply Theorem 2.1 with f̃(z). Then the purpose of the remaining part of the proof is to show the following: if gcd(p, 4N) = 1, an odd prime ℓ divides some ni, and (2.6) θp−1(f(z)) ≡ a(nim 2)qnim (mod p), then p|(ℓ− 1)ℓ(ℓ+ 1)N or ℓ | N . We assume that there exists a prime ℓ1 such that ℓ1|n1, p ∤ (ℓ1 − 1)ℓ1(ℓ1 + 1)N and ℓ | N . We also assume that nt = 1 and that ni ∤ n1 for every i, 2 ≤ i ≤ t − 1. Then, we can take a prime ℓi for each i, 2 ≤ i ≤ t − 1, such that ℓi|ni and ℓi ∤ n1. For convention, we define (−1)(n−1)2/8 if n is odd, 0 otherwise, and χQ(d) := for a prime Q. Let ψ(d) := i=2 χℓi(d). We take a prime β such that ψ(n1)χβ(n1) = −1. If we denote the ψ-twist of f̃(z) by f̃ψ(z) and the ψχβ-twist of f̃(z) by f̃ψχβ(z), then f̃ψχ2 (z)− f̃ψχβ(z) ≡ 2 gcd(m,β ℓj)=1 a(n1m 2)qn1m (mod p) and f̃ψχβ(z) ∈ Sλ+(p+1)(p−1)+ 1 (Γ0(Nα 2β2), χ) ∩ Z[[q]] (see Chapter 3 in [8]). Note that gcd(Nα2β2, p) = gcd(Nα2β2, ℓ1) = 1. Thus, (f̃ψ(z)− f̃ψχβ(z))|T (ℓ21, λ+ (p+ 1)(p− 1), χ) satisfies the formula (2.2) of Theorem 2.1 for both of ǫ = 1 and ǫ = −1. This results in a contradiction since (f̃ψ(z)− f̃ψχβ(z))|T (ℓ 1, λ+ (p+ 1)(p− 1), χ) 6≡ 0 (mod p) and p ≥ 5. Thus, we complete the proof. � 8 D. CHOI 3. Proofs of Theorem 2, 3, and 4 3.1. Proof of Theorem 2. Note that h(z) = η(z)2 η(2z) ·E4(4z) η(4z)6 is a meromorphic modular form. In [2] it was obtained a holomorphic modular form on Γ0(4p 2) whose Fourier coefficients generate traces of singular moduli modulo p (see the formula (3.1) and (3.2)). Since the level of this modular form is not relatively prime to p, we need the following proposition. Proposition 3.1 ([1]). Suppose that p ≥ 5 is a prime. Also, suppose that p ∤ N , j ≥ 1 is an integer, and g(z) = a(n)qn ∈ Sλ+ 1 (Γ0(Np j)) ∩ Z[[q]]. Then, there exists a cusp form G(z) ∈ Sλ′+ 1 (Γ0(N)) ∩ Z[[q]] such that G(z) ≡ g(z) (mod p), where λ′ + 1 = (λ+ 1 )pj + pe(p− 1) for a sufficiently large e ∈ N. Using Theorem 1 and Proposition 3.1, we give the proof of Theorem 2. Proof of Theorem 2. Let (3.1) h1,p(z) := h(z)− hχp(z), where hχp(z) is the χp-twist of h(z). From (1.2), we have h1,p(z) := −2 − 0<d≡0,3 (mod 4) t1(d)q d − 2 0<d≡0,3 (mod 4) (−dp )=−1 t1(d)q hm,p(z) := h1,p(z)|T (m2, 1, χ0) = −2 − 0<d≡0,3 (mod 4) tm(d)q d − 2 0<d≡0,3 (mod 4) (−dp )=−1 tm(d)q for every positive integer m. Let Fp(z) := η(4z)p η(4pz) It was proved in [2] that if α is a sufficiently large positive integer, then h1,p(z)Fp(z) (Γ0(4p 2)) and (3.2) h1,p(z)Fp(z) α ≡ h1,p(z) (mod p), DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 9 where k0 = α · p . Lemma 2.2 and Proposition 3.1 imply that there exists f1,p(z) ∈ Sλ′+ 1 (Γ0(4)) ∩ Z[[q]] such that f1,p(z) ≡ −2 0<d≡0,3 (mod 4) (−dp )=−1 tm(d)q d (mod p), where λ′ = (k0 + 1 + (p− 1)(p+ 1) + 12)p 2 + pe(p− 1) for a sufficiently large e ∈ N. We assume that the coefficients of f1,p(z) do not satisfy Property A for an odd prime p ≡ 2 (mod 3). Note that = −1 and that p ∤ (3−1)3(3+1). So, Theorem 1 implies 2t1(3) ≡ 0 (mod p). This results in a contradiction since 2t1(3) = 2 4 ·31. Thus, we obtain a proof when m = 1. For every odd prime ℓ, we have f1,p(z)|T (ℓ2, λ′, χ0) ≡ θp−1(h1,p(z))|T (ℓ2, λ′, χ0) ≡ θp−1(h1,p(z)|T (ℓ2, 1, χ0)) ≡ θp−1(hℓ,p(z)) (mod p). Moreover, Lemma 2.3 implies that a positive proportion of the primes ℓ satisfies the property f1,p(z)|T (ℓ2, λ′, χ0) ≡ 2f1,p (mod p). This completes the proof. � 3.2. Proofs of Theorem 3. The following theorem gives the formula for the Hurwitz class number in terms of the Fourier coefficients of a modular form of half integral weight. Theorem 3.2. Let T (z) := 1 + 2 n=1 q n2. If integers r3(n) are defined as r3(n)q n := T (z)3, r(n) =   12H(−4n) if n ≡ 1, 2 (mod 4), 24H(−n) if n ≡ 3 (mod 8), r(n/4) if n ≡ 0 (mod 4), 0 if n ≡ 7 (mod 8). Note that T (z) is a half integral weight modular form of weight 1 on Γ0(4). Combining Theorem 1 and Theorem 3.2, we derive the proof of Theorem 3. Proof of Theorem 3. Let G(z) be the -twist of T (z)3. Then, from Theorem 3.2, we G(z) = 1 + n≡1 (mod 4) 12H(−4n)qn + n≡3 (mod 8) 24H(−n)qn 10 D. CHOI and G(z) ∈ M 3 (Γ0(16)). Note that 24H(−3) = 8. This gives the complete proof by Theorem 1. � 3.3. Proofs of Theorem 4. In the following, we prove Theorem 4. Proof of Theorem 4. Let W (z) := η(2z) η(z)2 It is known that W (z) = P̄ (n)qn and that W (z) is a weakly holomorphic modular form on Γ0(16). Let G(z) := W (z)− Wχp(z) Fp(z) where Fp(z) = η(4z)p η(4p2z) and β are positive integers. Then we have G(z) ≡ 2 (−np )=−1 P̄ (n)qn + P̄ (n)qn (mod p). We claim that there exists a positive integer β such that G(z) is a holomorphic modular form of half integral weight on Γ0(16p 2). To prove our claim, we follow the arguments of Ahlgren and Ono ([1], Lemma 4.2). Note that, by a well-known criterion, Fp(z) is a holomorphic modular form on Γ0(4p 2) that vanishes at each cusp a ∈ Q for which p2 ∤ c (see [7]). This implies that G(z) is a weakly holomorphic modular form on Γ0(16p 2). If β is sufficiently large, then G(z) is holomorphic except at each cusp a for which p2|c′. Thus, we prove that G(z) is holomorphic at 1 for 0 ≤ m ≤ 3. Let, for odd d, ǫd := 1 if d ≡ 1 (mod 4), i if d ≡ 3 (mod 4). If f(z) is a function on the complex upper half plane, λ ∈ Z, and γ = ( a bc d ) ∈ Γ0(4), then we define the usual slash operator by f(z) |λ+ 1 )2λ+1 ǫ−1−2λd (cz + d) −λ− 1 az + b cz + d Let g := e2πiv/p be the usual Gauss sum. Note that Wχp(z) = W (z)|− 1 1 −v/p Choose an integer kv satisfying 16kv ≡ 15v (mod p). DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 11 Then, we have (3.3) 2mp2 1 = γv,m 2mp2 1 1 −16v + 16kv where γv,m = 1− 2m+4p(v + kv + 2mv2p− 2mvkvp) 1p(15v − 16kv − 2 m+4(v2p+ vkvp)) 22mp2(−16vp+ 16kvp) 2m+4vp− 2m+4kvp+ 1 Note that W (z) has its only pole at z ∼ 0 up to Γ0(16). Since γv,m ∈ Γ0(16), the formula (3.3) implies that Wχp(z) is holomorphic at 2 mp2 for 1 ≤ m ≤ 3. Thus, G(z) is holomorphic at 2mp2 for 1 ≤ m ≤ 3. If m = 0, then we have W (z)|− 1 γv,0 = −16vp3 + 16kvp3 16vp− 16kvp+ 1 W (z) = p2(−vp+ kvp) 16vp− 16kvp + 1 W (z) = W (z). Note that (3.4) W (z)|− 1 = α · q− 16 +O(1), where α is a nonzero complex number. The q-expansion of Wχp(z) at is given by (3.5) Wχp(z)|− 1 Using (3.3) and (3.4), the only term in (3.5) with a negative exponent on q is the term (v−kv). If N is defined by 16N ≡ 1 (mod p), then we have (v−kv) = Thus, we have that (W (z)−Wχp(z))|− 1 = O(1). This implies that G(z) is a holomorphic modular form of half integral weight on Γ0(16p Noting that P̄ (3) = 8, the remaining part of the proof is similar to that in Theorem 3. Thus, it is omitted. � 12 D. CHOI References [1] S. Ahlgren and M. Boylan, Central Critical Values of Modular L-functions and Coeffients of Half Integral Weight Modular Forms Modulo ℓ, to appear in Amer. J. Math. [2] S. Ahlgren and K. Ono, Arithmetic of singular moduli and class polynomials, Compos. Math. 141 (2005), no. 2, 293–312. [3] J. H. Bruinier and K. Ono, Coefficients of half-integral weight modular forms, J. Number Theory 99 (2003), no. 1, 164–179. [4] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. [5] N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag New York, GTM 97, 1993. [6] S. Lang, Introduction to Modular Forms, Grundl. d. Math. Wiss. no. 222, Springer: Berlin Heidelberg New York, 1976 Berlin, 1995. [7] B. Gordon and K. Hughes, Multiplicative properties of eta-product, Cont. Math. 143 (1993), 415-430. [8] K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, Amer. Math. Soc., CBMS Regional Conf. Series in Math., vol. 102, 2004. [9] J.-P. Serre, Divisibilite de certaines fonctions arithmetiques, Enseignement Math. (2) 22 (1976), no. 3-4, 227–260. [10] S. Treneer, Congruences for the Coefficients of Weakly Holomorphic Modular Forms, to appear in the Proceedings of the London Mathematical Society. [11] D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002, pp.211-244. School of Mathematics, KIAS, 207-43 Cheongnyangni 2-dong 130-722, Korea E-mail address : choija@postech.ac.kr 1. Introduction and Results 2. Proof of Theorem ?? 3. Proofs of Theorem ??, ??, and ?? 3.1. Proof of Theorem ?? 3.2. Proofs of Theorem ?? 3.3. Proofs of Theorem ?? References
Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{\lambda+1/2}(\Gamma_0(N),\chi)\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p \geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $p\geq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman's conjecture for overpartitions.
Introduction and Results Let Mλ+ 1 (Γ0(N), χ) and Sλ+ 1 (Γ0(N), χ) be the spaces, respectively, of modular forms and cusp forms of weight λ + 1 on Γ0(N) with a Dirichlet character χ whose conductor divides N . If f(z) ∈Mλ+ 1 (Γ0(N), χ), then f(z) has the form f(z) = a(n)qn, where q := e2πiz. It is well-known that the coefficients of f are related to interesting objects in number theory such as the special values of L-function, class number, traces of singular moduli and so on. In this paper, we study congruence properties of the Fourier coefficients of f(z) ∈Mλ+ 1 (Γ0(N), χ) ∩ Z[[q]] and their applications. Recently, Bruinier and Ono proved in [3] that g(z) ∈ Sλ+ 1 (Γ0(N), χ) ∩ Z[[q]] has a special form (see (2.1)) by modulo p when p is an odd prime and the coefficients of f(z) do not satisfy the following property for p: Property A. IfM is a positive integer, we say that a sequence α(n) ∈ Z satisfies Property A for M if for every integer r ♯{ 1 ≤ n ≤ X | α(n) ≡ r (mod M) and gcd(M,n) = 1} if r 6≡ 0 (mod M), X if r ≡ 0 (mod M). 2000 Mathematics Subject Classification. 11F11,11F33. Key words and phrases. Modular forms, Congruences. http://arxiv.org/abs/0704.0012v1 2 D. CHOI θ(f(z)) := f(z) = n · a(n)qn. Using Rankin-Cohen Bracket (see (2.3)), we prove that there exists f̃(z) ∈ Sλ+p+1+ 1 (Γ0(4N), χ) ∩ Z[[q]] such that θ(f(z)) ≡ f̃(z) (mod p). We extend the results in [3] to modular forms of half integral weight. Theorem 1. Let λ be a non-negative integer. We assume that f(z) = n=0 a(n)q Mλ+ 1 (Γ0(4N), χ) ∩ Z[[q]], where χ is a real Dirichlet character. If p ≥ 5 is a prime and there exists a positive integer n for which gcd(a(n), p) = 1 and gcd(n, p) = 1, then at least one of the following is true: (1) The coefficients of θp−1(f(z)) satisfies Property A for p. (2) There are finitely many square-free integers n1, n2, · · · , nt for which (1.1) θp−1(f(z)) ≡ a(nim 2)qnim (mod p). Moreover, if gcd(4N, p) = 1 and an odd prime ℓ divides some ni, then p|(ℓ− 1)ℓ(ℓ+ 1)N or ℓ | N. Remark 1.1. Note that for every odd prime p ≥ 5, θp−1(f(z)) ≡ a(n)qn (mod p). As an applications of Theorem 1, we study the distribution of traces of singular moduli modulo primes p ≥ 5. Let j(z) be the usual j-invariant function. We denote by Fd the set of positive definite binary quadratic forms F (x, y) = ax2 + bxy + cy2 = [a, b, c] with discriminant −d = b2−4ac. For each F (x, y), let αF be the unique complex number in the complex upper half plane, which is a root of F (x, 1). We define ωF ∈ {1, 2, 3} as ωF := 2 if F ∼Γ [a, 0, a], 3 if F ∼Γ [a, a, a], 1 otherwise, where Γ := SL2(Z). Here, F ∼Γ [a, b, c] denotes that F (x, y) is equivalent to [a, b, c]. From these notations, we define the Hecke trace of singular moduli. DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 3 Definition 1.2. If m ≥ 1, then we define the mth Hecke trace of the singular moduli of discriminant −d as tm(d) := F∈Fd/Γ jm(αF ) where Fd/Γ denotes a set of Γ−equivalence classes of Fd and jm(z) := j(z)|T0(m) = az + b Here, T0(m) denotes the normalized mth weight zero Hecke operator. Note that t1(d) = t(d), where t(d) := F∈Fd/Γ j(αF )− 744 is the usual trace of singular moduli. Let h(z) := η(z)2 η(2z) · E4(4z) η(4z)6 and Bm(1, d) denote the coefficient of q d in h(z)|T (m2, 1, χ0), where E4(z) := 1 + 240 d3qn, η(z) := q (1− qn) , and χ0 is a trivial character. Here, T (m 2, λ, χ) denotes the mth Hecke operator of weight λ + 1 with a Dirichlet chracter χ (see VI. §3. in [5] or (2.5)). Zagier proved in [11] that for all m and d (1.2) tm(d) = −Bm(1, d). Using these generating functions, Ahlgren and Ono studied the divisibility properties of traces and Hecke traces of singular moduli in terms of the factorization of primes in imaginary quadratic fields (see [2]). For example, they proved that a positive proportion of the primes ℓ has the property that tm(ℓ 3n) ≡ 0 (mod ps) for every positive integer n coprime to ℓ such that p is inert or ramified in Q . Here, p is an odd prime, and s and m are integers with p ∤ m. In the following theorem, we give the distribution of traces and Hecke traces of singular moduli modulo primes p. 4 D. CHOI Theorem 2. Suppose that p ≥ 5 is a prime such that p ≡ 2 (mod 3). (1) Then, for every integer r, p ∤ r, ♯{ 1 ≤ n ≤ X | t1(n) ≡ r (mod p)} ≫r,p if r 6≡ 0 (mod p) X if r ≡ 0 (mod p). (2) Then, a positive proportion of the primes ℓ has the property that ♯{ 1 ≤ n ≤ X | tℓ(n) ≡ r (mod p)} ≫r,p if r 6≡ 0 (mod p) X if r ≡ 0 (mod p). for every integer r, p ∤ r. As another application we study the distribution of Hurwitz class number modulo primes p ≥ 5. The Hurwitz class number H(−N) is defined as follows: the class number of quadratic forms of the discriminant −N where each class C is counted with multiplicity Aut(C) . The following theorem gives the distribution of Hurwitz class number modulo primes p ≥ 5. Theorem 3. Suppose that p ≥ 5 is a prime. Then, for every integer r ♯{ 1 ≤ n ≤ X | H(n) ≡ r (mod p)} ≫r,p if r 6≡ 0 (mod p), X if r ≡ 0 (mod p). We also use the main theorem to study an analogue of Newman’s conjecture for overpar- titions. Newman’s conjecture concerns the distribution of the ordinary partition function modulo primes p. Newman’s Conjecture. Let P (n) be an ordinary partition function. If M is a positive integer, then for every integer r there are infinitely many nonnegative integer n for which P (n) ≡ r (mod M). This conjecture was already studied by many mathematicians (see Chapter 5. in [8]). The overpartition of a natural number n is a partition of n in which the first occurrence of a number may be overlined. Let P̄ (n) be the number of the overpartition of an integer n. As an analogue of Newman’s conjecture, the following theorem gives a distribution property of P̄ (n) modulo odd primes p. Theorem 4. Suppose that p ≥ 5 is a prime such that p ≡ 2 (mod 3). Then, for every integer r, ♯{ 1 ≤ n ≤ X | P̄ (n) ≡ r (mod p)} ≫r,p if r 6≡ 0 (mod p), X if r ≡ 0 (mod p). Remark 1.3. When r ≡ 0 (mod p), Theorem 2, 3 and 4 were proved in [2] and [10]. DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 5 Next sections are detailed proofs of theorems: Section 2 gives a proof of Theorem 1. In Section 3, we give the proofs of Theorem 2, 3, and 4. 2. Proof of Theorem 1 We begin by stating the following theorem proved in [3]. Theorem 2.1 ([3]). Let λ be a non-negative integer. Suppose that g(z) = n=0 ag(n)q Sλ+ 1 (Γ0(4N), χ) ∩ Z[[q]], where χ is a real Dirichlet character. If p is an odd prime and a positive integer n exists for which gcd(ag(n), p) = 1, then at least one of the following is true: (1) If 0 ≤ r < p, then ♯{ 1 ≤ n ≤ X | ag(n) ≡ r (mod p)} ≫r,M if r 6≡ 0 (mod p), X if r ≡ 0 (mod p). (2) There are finitely many square-free integers n1, n2, · · · , nt for which (2.1) g(z) ≡ ag(nim 2)qnim (mod p). Moreover, if gcd(p, 4N) = 1, ǫ ∈ {±1}, and ℓ ∤ 4Np is a prime with ∈ {0, ǫ} for 1 ≤ i ≤ t, then (ℓ−1)g(z) is an eigenform modulo p of the half-integral weight Hecke operator T (ℓ2, λ, χ). In particular, we have (2.2) (ℓ− 1)g(z)|T (ℓ2, λ, χ) ≡ ǫχ(p) (−1)λ ℓλ + ℓλ−1 (ℓ− 1)g(z) (mod p). Recall that f(z) = a(n)qn ∈ Mλ+ 1 (Γ0(4N), χ) ∩ Z[[q]]. Thus, to apply Theorem 2.1, we show that there exists a cusp form f̃(z) such that f̃(z) ≡ θp−1(f(z)) (mod p) for a prime p ≥ 5. Lemma 2.2. Suppose that p ≥ 5 is a prime and f(z) = a(n)qn ∈Mλ+ 1 (Γ0(N), χ) ∩ Z[[q]]. Then, there exists a cusp form f̃(z) ∈ Sλ+(p+1)(p−1)+ 1 (Γ0(N), χ) ∩ Z[[q]] such that f̃(z) ≡ θp−1(f(z)) (mod p). Proof of Lemma 2.2. For F (z) ∈Mk1 (Γ0(N), χ1) and G(z) ∈Mk2 (Γ0(N), χ2), let (2.3) [F (z), G(z)]1 := θ(F (z)) ·G(z)− F (z) · θ(G(z)). This operator is referred to as a Rankin-Cohen 1-bracket, and it was proved in [4] that [F (z), G(z)]1 ∈ S k1+k2 (Γ0(N), χ1χ2χ 6 D. CHOI where χ′ = 1 if k1 and k2 ∈ Z, χ′(d) = 2 if ki ∈ Z and k3−i + Z, and χ′(d) = ) k1+k2 2 if k1 and k2 For even k ≥ 4, let Ek(z) := 1− dk−1qn be the usual normalized Eisenstein series of weight k. Here, the number Bk denotes the kth Bernoulli number. The function Ek(z) is a modular form of weight k on SL2(Z), and (2.4) Ep−1(z) ≡ 1 (mod p) (see [6]). From (2.3) and (2.4), we have [Ep−1(z), f(z)]1 ≡ θ(f(z)) (mod p) and [Ep−1(z), f(z)]1 ∈ Sλ+p+1+ 1 (Γ0(N), χ). Repeating this method p− 1 times, we com- plete the proof. � Using the following lemma, we can deal with the divisibility of ag(n) for positive integers n, p ∤ n, where g(z) = n=1 ag(n)q n ∈ Sλ+ 1 (Γ0(N), χ) ∩ Z[[q]]. Lemma 2.3 (see Chapter 3 in [8]). Suppose that g(z) = n=1 ag(n)q n ∈ Sλ+ 1 (Γ0(N), χ) has coefficients in OK , the algebraic integers of some number field K. Furthermore, suppose that λ ≥ 1 and that m ⊂ OK is an ideal norm M . (1) Then, a positive proportion of the primes Q ≡ −1 (mod 4MN) has the property g(z)|T (Q2, λ, χ) ≡ 0 (mod m). (2) Then a positive proportion of the primes Q ≡ 1 (mod 4MN) has the property that g(z)|T (Q2, λ, χ) ≡ 2g(z) (mod m). We can now prove Theorem 1. Proof of Theorem 1. From Lemma 2.2, there exists a cusp form f̃(z) ∈ Sλ+(p+1)(p−1)+ 1 (Γ0(N), χ) ∩ Z[[q]] such that f̃(z) ≡ θp−1(f(z)) (mod p). Note that, for F (z) = n=0 aF (n)q n ∈ Mk+ 1 (Γ0(N), χ) and each prime Q ∤ N , the half-integral weight Hecke operator T (Q2, λ, χ) is defined as (2.5) F (z)|T (Q2, k, χ) aF (Q 2n) + χ∗(Q) Qk−1aF (n) + χ ∗(Q2)Q2k−1aF (n/Q DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 7 where χ∗(n) := χ∗(n) (−1)k and aF (n/Q 2) = 0 if Q2 ∤ n. If F (z)|T (Q2, k, χ) ≡ 0 (mod p) for a prime Q ∤ N , then we have aF (Q 2 ·Qn) + χ∗(Q) Qk−1aF (Qn) + χ ∗(Q2)Q2k−1aF Qn/Q2 ≡ aF (Q3n) ≡ 0 (mod p) for every positive integer n such that gcd(Q, n) = 1. Thus, we have the following by Lemma 2.3-(1): ♯{ 1 ≤ n ≤ X | a(n) ≡ 0 (mod p) and gcd(p, n) = 1} ≫ X. We apply Theorem 2.1 with f̃(z). Then the purpose of the remaining part of the proof is to show the following: if gcd(p, 4N) = 1, an odd prime ℓ divides some ni, and (2.6) θp−1(f(z)) ≡ a(nim 2)qnim (mod p), then p|(ℓ− 1)ℓ(ℓ+ 1)N or ℓ | N . We assume that there exists a prime ℓ1 such that ℓ1|n1, p ∤ (ℓ1 − 1)ℓ1(ℓ1 + 1)N and ℓ | N . We also assume that nt = 1 and that ni ∤ n1 for every i, 2 ≤ i ≤ t − 1. Then, we can take a prime ℓi for each i, 2 ≤ i ≤ t − 1, such that ℓi|ni and ℓi ∤ n1. For convention, we define (−1)(n−1)2/8 if n is odd, 0 otherwise, and χQ(d) := for a prime Q. Let ψ(d) := i=2 χℓi(d). We take a prime β such that ψ(n1)χβ(n1) = −1. If we denote the ψ-twist of f̃(z) by f̃ψ(z) and the ψχβ-twist of f̃(z) by f̃ψχβ(z), then f̃ψχ2 (z)− f̃ψχβ(z) ≡ 2 gcd(m,β ℓj)=1 a(n1m 2)qn1m (mod p) and f̃ψχβ(z) ∈ Sλ+(p+1)(p−1)+ 1 (Γ0(Nα 2β2), χ) ∩ Z[[q]] (see Chapter 3 in [8]). Note that gcd(Nα2β2, p) = gcd(Nα2β2, ℓ1) = 1. Thus, (f̃ψ(z)− f̃ψχβ(z))|T (ℓ21, λ+ (p+ 1)(p− 1), χ) satisfies the formula (2.2) of Theorem 2.1 for both of ǫ = 1 and ǫ = −1. This results in a contradiction since (f̃ψ(z)− f̃ψχβ(z))|T (ℓ 1, λ+ (p+ 1)(p− 1), χ) 6≡ 0 (mod p) and p ≥ 5. Thus, we complete the proof. � 8 D. CHOI 3. Proofs of Theorem 2, 3, and 4 3.1. Proof of Theorem 2. Note that h(z) = η(z)2 η(2z) ·E4(4z) η(4z)6 is a meromorphic modular form. In [2] it was obtained a holomorphic modular form on Γ0(4p 2) whose Fourier coefficients generate traces of singular moduli modulo p (see the formula (3.1) and (3.2)). Since the level of this modular form is not relatively prime to p, we need the following proposition. Proposition 3.1 ([1]). Suppose that p ≥ 5 is a prime. Also, suppose that p ∤ N , j ≥ 1 is an integer, and g(z) = a(n)qn ∈ Sλ+ 1 (Γ0(Np j)) ∩ Z[[q]]. Then, there exists a cusp form G(z) ∈ Sλ′+ 1 (Γ0(N)) ∩ Z[[q]] such that G(z) ≡ g(z) (mod p), where λ′ + 1 = (λ+ 1 )pj + pe(p− 1) for a sufficiently large e ∈ N. Using Theorem 1 and Proposition 3.1, we give the proof of Theorem 2. Proof of Theorem 2. Let (3.1) h1,p(z) := h(z)− hχp(z), where hχp(z) is the χp-twist of h(z). From (1.2), we have h1,p(z) := −2 − 0<d≡0,3 (mod 4) t1(d)q d − 2 0<d≡0,3 (mod 4) (−dp )=−1 t1(d)q hm,p(z) := h1,p(z)|T (m2, 1, χ0) = −2 − 0<d≡0,3 (mod 4) tm(d)q d − 2 0<d≡0,3 (mod 4) (−dp )=−1 tm(d)q for every positive integer m. Let Fp(z) := η(4z)p η(4pz) It was proved in [2] that if α is a sufficiently large positive integer, then h1,p(z)Fp(z) (Γ0(4p 2)) and (3.2) h1,p(z)Fp(z) α ≡ h1,p(z) (mod p), DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 9 where k0 = α · p . Lemma 2.2 and Proposition 3.1 imply that there exists f1,p(z) ∈ Sλ′+ 1 (Γ0(4)) ∩ Z[[q]] such that f1,p(z) ≡ −2 0<d≡0,3 (mod 4) (−dp )=−1 tm(d)q d (mod p), where λ′ = (k0 + 1 + (p− 1)(p+ 1) + 12)p 2 + pe(p− 1) for a sufficiently large e ∈ N. We assume that the coefficients of f1,p(z) do not satisfy Property A for an odd prime p ≡ 2 (mod 3). Note that = −1 and that p ∤ (3−1)3(3+1). So, Theorem 1 implies 2t1(3) ≡ 0 (mod p). This results in a contradiction since 2t1(3) = 2 4 ·31. Thus, we obtain a proof when m = 1. For every odd prime ℓ, we have f1,p(z)|T (ℓ2, λ′, χ0) ≡ θp−1(h1,p(z))|T (ℓ2, λ′, χ0) ≡ θp−1(h1,p(z)|T (ℓ2, 1, χ0)) ≡ θp−1(hℓ,p(z)) (mod p). Moreover, Lemma 2.3 implies that a positive proportion of the primes ℓ satisfies the property f1,p(z)|T (ℓ2, λ′, χ0) ≡ 2f1,p (mod p). This completes the proof. � 3.2. Proofs of Theorem 3. The following theorem gives the formula for the Hurwitz class number in terms of the Fourier coefficients of a modular form of half integral weight. Theorem 3.2. Let T (z) := 1 + 2 n=1 q n2. If integers r3(n) are defined as r3(n)q n := T (z)3, r(n) =   12H(−4n) if n ≡ 1, 2 (mod 4), 24H(−n) if n ≡ 3 (mod 8), r(n/4) if n ≡ 0 (mod 4), 0 if n ≡ 7 (mod 8). Note that T (z) is a half integral weight modular form of weight 1 on Γ0(4). Combining Theorem 1 and Theorem 3.2, we derive the proof of Theorem 3. Proof of Theorem 3. Let G(z) be the -twist of T (z)3. Then, from Theorem 3.2, we G(z) = 1 + n≡1 (mod 4) 12H(−4n)qn + n≡3 (mod 8) 24H(−n)qn 10 D. CHOI and G(z) ∈ M 3 (Γ0(16)). Note that 24H(−3) = 8. This gives the complete proof by Theorem 1. � 3.3. Proofs of Theorem 4. In the following, we prove Theorem 4. Proof of Theorem 4. Let W (z) := η(2z) η(z)2 It is known that W (z) = P̄ (n)qn and that W (z) is a weakly holomorphic modular form on Γ0(16). Let G(z) := W (z)− Wχp(z) Fp(z) where Fp(z) = η(4z)p η(4p2z) and β are positive integers. Then we have G(z) ≡ 2 (−np )=−1 P̄ (n)qn + P̄ (n)qn (mod p). We claim that there exists a positive integer β such that G(z) is a holomorphic modular form of half integral weight on Γ0(16p 2). To prove our claim, we follow the arguments of Ahlgren and Ono ([1], Lemma 4.2). Note that, by a well-known criterion, Fp(z) is a holomorphic modular form on Γ0(4p 2) that vanishes at each cusp a ∈ Q for which p2 ∤ c (see [7]). This implies that G(z) is a weakly holomorphic modular form on Γ0(16p 2). If β is sufficiently large, then G(z) is holomorphic except at each cusp a for which p2|c′. Thus, we prove that G(z) is holomorphic at 1 for 0 ≤ m ≤ 3. Let, for odd d, ǫd := 1 if d ≡ 1 (mod 4), i if d ≡ 3 (mod 4). If f(z) is a function on the complex upper half plane, λ ∈ Z, and γ = ( a bc d ) ∈ Γ0(4), then we define the usual slash operator by f(z) |λ+ 1 )2λ+1 ǫ−1−2λd (cz + d) −λ− 1 az + b cz + d Let g := e2πiv/p be the usual Gauss sum. Note that Wχp(z) = W (z)|− 1 1 −v/p Choose an integer kv satisfying 16kv ≡ 15v (mod p). DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 11 Then, we have (3.3) 2mp2 1 = γv,m 2mp2 1 1 −16v + 16kv where γv,m = 1− 2m+4p(v + kv + 2mv2p− 2mvkvp) 1p(15v − 16kv − 2 m+4(v2p+ vkvp)) 22mp2(−16vp+ 16kvp) 2m+4vp− 2m+4kvp+ 1 Note that W (z) has its only pole at z ∼ 0 up to Γ0(16). Since γv,m ∈ Γ0(16), the formula (3.3) implies that Wχp(z) is holomorphic at 2 mp2 for 1 ≤ m ≤ 3. Thus, G(z) is holomorphic at 2mp2 for 1 ≤ m ≤ 3. If m = 0, then we have W (z)|− 1 γv,0 = −16vp3 + 16kvp3 16vp− 16kvp+ 1 W (z) = p2(−vp+ kvp) 16vp− 16kvp + 1 W (z) = W (z). Note that (3.4) W (z)|− 1 = α · q− 16 +O(1), where α is a nonzero complex number. The q-expansion of Wχp(z) at is given by (3.5) Wχp(z)|− 1 Using (3.3) and (3.4), the only term in (3.5) with a negative exponent on q is the term (v−kv). If N is defined by 16N ≡ 1 (mod p), then we have (v−kv) = Thus, we have that (W (z)−Wχp(z))|− 1 = O(1). This implies that G(z) is a holomorphic modular form of half integral weight on Γ0(16p Noting that P̄ (3) = 8, the remaining part of the proof is similar to that in Theorem 3. Thus, it is omitted. � 12 D. CHOI References [1] S. Ahlgren and M. Boylan, Central Critical Values of Modular L-functions and Coeffients of Half Integral Weight Modular Forms Modulo ℓ, to appear in Amer. J. Math. [2] S. Ahlgren and K. Ono, Arithmetic of singular moduli and class polynomials, Compos. Math. 141 (2005), no. 2, 293–312. [3] J. H. Bruinier and K. Ono, Coefficients of half-integral weight modular forms, J. Number Theory 99 (2003), no. 1, 164–179. [4] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. [5] N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag New York, GTM 97, 1993. [6] S. Lang, Introduction to Modular Forms, Grundl. d. Math. Wiss. no. 222, Springer: Berlin Heidelberg New York, 1976 Berlin, 1995. [7] B. Gordon and K. Hughes, Multiplicative properties of eta-product, Cont. Math. 143 (1993), 415-430. [8] K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, Amer. Math. Soc., CBMS Regional Conf. Series in Math., vol. 102, 2004. [9] J.-P. Serre, Divisibilite de certaines fonctions arithmetiques, Enseignement Math. (2) 22 (1976), no. 3-4, 227–260. [10] S. Treneer, Congruences for the Coefficients of Weakly Holomorphic Modular Forms, to appear in the Proceedings of the London Mathematical Society. [11] D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002, pp.211-244. School of Mathematics, KIAS, 207-43 Cheongnyangni 2-dong 130-722, Korea E-mail address : choija@postech.ac.kr 1. Introduction and Results 2. Proof of Theorem ?? 3. Proofs of Theorem ??, ??, and ?? 3.1. Proof of Theorem ?? 3.2. Proofs of Theorem ?? 3.3. Proofs of Theorem ?? References
704.001
p-ADIC LIMIT OF THE FOURIER COEFFICIENTS OF WEAKLY HOLOMORPHIC MODULAR FORMS OF HALF INTEGRAL WEIGHT D. CHOI AND Y. CHOIE Abstract. Serre obtained the p-adic limit of the integral Fourier coefficients of modular forms on SL2(Z) for p = 2, 3, 5, 7. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ0(4N) for N = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications of our main result, we obtain congruences on various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space. November 4, 2018 1. Introduction and Statement of Main Results Serre obtained the p-adic limits of the integral Fourier coefficients of modular forms on SL2(Z) for p = 2, 3, 5, 7 (see Théorème 7 and Lemma 8 in [20]). In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ0(4N) forN = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications of our main result, we obtain congruences for various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space. For odd d, let := γtΓ0(4N)tγ where γt = ( c d ) ∈ Γ(1) and γt(t) = ∞. We denote the q-expansion of a modular form f ∈Mλ+ 1 (Γ0(4N)) at each cusp t of Γ0(4N) by (1.1) (f |λ+ 1 γt)(z) = (cz + d) −λ− 1 az + b cz + d atf (n)q t , qt := q where (1.2) r(t) ∈ 2000 Mathematics Subject Classification. 11F11,11F33. Key words and phrases. modular forms, p-adic limit, Borcherds exponents, Maass space . This work was partially supported by KOSEF R01-2003-00011596-0 , ITRC and BRSI-POSTECH. http://arxiv.org/abs/0704.0013v2 2 D. CHOI AND Y. CHOIE When t ∼ ∞, we denote atf (n) by af (n). Note that the number r(t) is independent of the choice of f ∈Mλ+ 1 (Γ0(4N)) and λ. We call t a regular cusp if r(t) = 0 (see Chapter IV. §1. of [15] for a more general definition of a λ-regular cusp ). Remark 1.1. Our definition of a regular cusp is different from the usual one. Let U4N := {t1, · · · , tν(4N)} be the set of all inequivalent regular cusps of Γ0(4N). Note that the genus of Γ0(4N) is zero if and only if 1 ≤ N ≤ 4. LetMλ+ 1 (Γ0(4N)) be the space of weakly holomorphic modular forms of weight λ + 1 on Γ0(4N) and let M0λ+ 1 (Γ0(N)) denote the set of f(z) ∈ Mλ+ 1 (Γ0(N)) such that the constant term of its q-expansion at each cusp is zero. Let Up be the operator defined by (f |Up)(z) := af(pn)q Let OL be the ring of integers of a number field L with a prime ideal p ⊂ OL. For f(z) := af(n)q n and g(z) := ag(n)q n ∈ L[[q−1, q]] we write f(z) ≡ g(z) (mod p) if and only if af (n)− ag(n) ∈ p for every integer n. With these notations we state the following theorem. Theorem 1. For N = 1, 2, 4 consider f(z) := af (n)q n ∈ M0 (Γ0(4N)) ∩ L[[q−1, q]]. Suppose that p ⊂ OL is any prime ideal such that p|p, p prime, and that af(n) is p-integral for every integer n ≥ n0. (1) If p = 2 and af (0) = 0, then there exists a positive integer b such that (f |(Up)b)(z) ≡ 0 (mod pj) for each j ∈ N. (2) If p ≥ 3 and f(z) ∈ M0 (Γ0(4N)) with λ ≡ 2 or 2+ (mod p−1 ), then there exists a positive integer b such that (f |(Up)b)(z) ≡ 0 (mod pj) for each j ∈ N. Remark 1.2. The p-adic limit of a sum of Fourier coefficients of f ∈ M 3 (Γ0(4N)) was studied in [13]. Our method only allows to prove a weaker result if f(z) 6∈ M0 (Γ0(4N)). THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 3 Theorem 2. For N = 1, 2 or 4, let f(z) := af (n)q n ∈ Mλ+ 1 (Γ0(4N)) ∩ L[[q−1, q]]. Suppose that p ⊂ OL is any prime ideal with p|p, p prime, p ≥ 5, and that af (n) is p-integral for every integer n ≥ n0. If λ ≡ 2 or 2 + (mod p−1 ), then there exists a positive integer b0 such that p2b−m(p:λ) t∈U4N ∆4N,3−α(p:λ)(z) R4N (z) e·ω(4N) (0)atf (0) (mod p) for every positive integer b > b0 (see Section 3 for detailed notation ). Example 1.3. Recall that the generating function of the overpartition P̄ (n) of n(see [11]) P̄ (n)qn = η(2z) η(z)2 is in M− 1 (Γ0(16)), where η(z) := q n=1(1− qn). Therefore, theorem 2 implies that P̄ (52b) ≡ 1 (mod 5), ∀b ∈ N. 2. Applications: More Congruences In this section, we study congruences for various modular objects such as those for Borcherds exponents and for quotients of Eisenstein series. 2.1. p-adic Limits of Borcherds Exponents. Let MH denote the set of meromorphic modular forms of integral weight on SL2(Z) with Heegner divisor, integer coefficients and leading coefficient 1. Let (Γ0(4)) := {f(z) = af(n)q n ∈ M 1 (Γ0(4)) | a(n) = 0 for n ≡ 2, 3 (mod 4)}. If f(z) = af(n)q n ∈ M+1 (Γ0(4)), then define Ψ(f(z)) by Ψ(f(z)) := q−h (1− qn)af (n2), where h = − 1 af(0) + 1<n≡0,1 (mod 4) af (−n)H(−n). Here H(−n) denotes the usual Hurwitz class number of discriminant −n. The following was proved by Borcherds. Theorem 2.1 ([4]). The map Ψ is an isomorphism from M+1 (Γ0(4)) to MH , and the weight of Ψ(f(z)) is af (0). 4 D. CHOI AND Y. CHOIE Let j(z) be the usual j-invariant function with the product expansion j(z) = q−1 (1− qn)A(n). Let F (z) := q−h n=1(1 − qn)c(n) be a meromorphic modular form of weight k in MH . The p-adic limit of d|n d · c(d) was studied in [5] for p = 2, 3, 5, 7. Here we obtain the p-adic limit of c(d) for p = 2, 3, 5, 7. Theorem 3. Let F (z) := q−h n=1(1− qn)c(n) be a meromorphic modular form of weight k in MH . (1) If p = 2, then for each j ∈ N there exists a positive integer b such that c(mpb) ≡ 2k (mod pj) for every positive integer m. (2) If p ∈ {3, 5, 7}, then, for each j ∈ N there exists a positive integer b such that 5c(mpb)−̟(F )A(mpb) ≡ 10k (mod pj) for every positive integer m. Here, ̟(F ) is a constant determined by the constant term of the q-expansion of Ψ−1(F ) at 0. 2.2. Sums of n-Squares. For u ∈ Z>0, let rn(u) := ♯{(s1, · · · , sn) ∈ Zn : s21 + · · ·+ s2n = u}. Theorem 4. Suppose that p ≥ 5 is a prime. If λ ≡ 2 or 3 (mod p−1 ), then there exists a positive integer C0 such that r2λ+1 p2b−m(p:λ) ≡ − (14− 4α (p : λ)) + 16 )[ λp−1 ]+α(p:λ)m(p:λ) (mod p), for every b > C0. Remark 2.2. As for an example, if λ ≡ 2 (mod p− 1) and p is an odd prime, then there exists a positive integer C0 such that r2λ+1 ≡ 10 (mod p), ∀b > C0 2.3. Quotients of Eisenstein Series. Congruences for the coefficients of quotients of elliptic Eisenstein series have been studied in [3]. Let us consider the Cohen Eisenstein series Hr+ 1 (z) := N=0H(r,N)q n of weight r+ 1 , r ≥ 2 (see [7]). We derive congruences for the coefficients of quotients of Hr+ 1 (z) and Eisenstein series. THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 5 Theorem 5. Let F (z) := E4(z) aF (n)q G(z) := E6(z) aG(n)q W (z) := E6(z) aW (n)q Then there exists a positive integer C0 such that aF (11 2b+1) ≡ 1 (mod 11), aG(11 2b+1) ≡ 6 (mod 11), aW (11 2b+1) ≡ 2 (mod 11) for every integer b > C0. 2.4. The Maass Space. Next we deal with congruences for the Fourier coefficients of a Siegel modular form in the Maass space. To define the Maass space, let us introduce notations given in [17]: let T ∈ M2g(Q) be a rational, half-integral, symmetric, non- degenerate matrix of size 2g with discriminant DT := (−1)g det(2T ). Let DT = DT,0f T , where DT,0 is the corresponding fundamental discriminant. Further- more, let G8 :=  2 0 −1 0 0 0 0 0 0 2 0 −1 0 0 0 0 −1 0 2 −1 0 0 0 0 0 −1 −1 2 −1 0 0 0 0 0 0 −1 2 −1 0 0 0 0 0 0 −1 2 −1 0 0 0 0 0 0 −1 2 −1 0 0 0 0 0 0 −1 2  and G7 be the upper (7, 7)-submatrix of G8. Define Sg := (g−1)/8 2, if g ≡ 1 (mod 8), (g−7)/8 G7, if g ≡ −1 (mod 8). 6 D. CHOI AND Y. CHOIE For each m ∈ N such that (−1)gm ≡ 0, 1 (mod 4), define a rational, half-integral, sym- metric, positive definite matrix Tm of size 2g by Tm :=   0 m/4 , if m ≡ 0 (mod 4), e2g−1 e′2g−1 [m+ 2 + (−1)n]/4 , if m ≡ (−1)g (mod 4) Here e2g−1 ∈ Z(2n−1,1) is the standard column vector and e′2g−1 is its transpose. Definition 2.3. (The Maass Space) Take g, k ∈ N such that g ≡ 0, 1 (mod 4) and g ≡ k (mod 2). Let SMaassk+g (Γ2g) F (Z) = A(T )qtr(TZ) ∈ Sk+g(Γ2g) ∣∣∣∣∣∣ A(T ) = ak−1φ(a;T )A(T|DT |/a2) (see (6.2) for details). This space is called the Maass space of genus 2g and weight g + k. In [17] it was proved that the Maass space is the same as the image of the Ikeda lifting when g ≡ 0, 1 (mod 4). Using this fact together with Theorem 1, we derive the following congruences for the Fourier coefficients of F (Z) in SMaassk+g (Γ2g). Theorem 6. For g ≡ 0, 1 (mod 4), let F (Z) := A(T )qtr(TZ) ∈ SMaassk+g (Γ2g) with integral coefficients A(T ), T > 0. If k ≡ 2 or 3 (mod p−1 ) for some prime p, then, for each j ∈ N, there exists a positive integer b for which A(T ) ≡ 0 (mod pj) for every T > 0, det(2T ) ≡ 0 (mod pb). This paper is organized as follows. Section 3 gives a linear relation among Fourier coefficients of modular forms of half integral weight. The remaining sections contain detailed proofs of the main theorems. 3. Linear Relation among Fourier Coefficients of modular forms of Half Integral Weight Let V (N ; k, n) be the subspace of Cn generated by the first n coefficients of the q- expansion of f at ∞ for f ∈ Sk(Γ0(N)), where Sk(Γ0(N)) denotes the space of cusp forms of weight k ∈ Z on Γ0(N). Let L(N ; k, n) be the orthogonal complement of V (N ; k, n) THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 7 in Cn with the usual inner product of Cn. The vector space L(1; k, d(k) + 1), d(k) = dim(Sk(Γ(1))), was studied by Siegel to evaluate the value of the Dedekind zeta function at a certain point. The vector space L(1; k, n) is explicitly described in terms of the principal part of negative weight modular forms in [9]. These results were extended in [8] to the groups Γ0(N) of genus zero. For 1 ≤ N ≤ 4, let 4N, λ+ at1f (0), · · · , a tν(4N) f (0), af(1), · · · , af(n) ∈ Cn+ν(4n) ∣∣∣ f ∈Mλ+ 1 (Γ0(4N)) where U4N := {t1, · · · , tν(4N)} is the set of all inequivalent regular cusps of Γ0(4N). We define EL(4N, λ+ 1 ;n) to be the orthogonal complement of EV (4N, λ+ 1 ;n) in Cn+ν(4N). Let ∆4N,λ := q δλ(4N)+O(qδλ(4N)+1) be inMλ+ 1 (Γ0(4N) with the maximum order at ∞, that is, its order at ∞ is bigger than that of any other modular form of the same level and weight. Furthermore, let R4(z) := η(4z)8 η(2z)4 , R8(z) := η(8z)8 η(4z)4 R12(z) := η(12z)12η(2z)2 η(6z)6η(4z)4 and R16(z) := η(16z)8 η(8z)4 For ℓ, n ∈ N, define m(ℓ : n) := ≡ 0 (mod 2) ≡ 1 (mod 2) α(ℓ : n) := n− ℓ− 1 Let ω(4N) be the order of zero of R4N (z) at ∞. Note that R4N (z) ∈ M2(Γ0(4N)) has its only zero at ∞. So, using the definition of η(z) = q 124 n=1(1− qn), we find that (3.1) ω(4) = 1, ω(8) = 2, ω(12) = 4, ω(16) = 4. For each g ∈Mr+ 1 (Γ0(4N)) and e ∈ N, let (3.2) R4N (z)e e·ω(4N)∑ b(4N, e, g; ν)q−ν +O(1) at ∞. With these notations we state the following theorem: Theorem 3.1. Suppose that λ ≥ 0 is an integer and 1 ≤ N ≤ 4. For each e ∈ N such that e ≥ λ − 1, take r = 2e − λ + 1. The linear map Φr,e(4N) : Mr+ 1 (Γ0(4N)) → 8 D. CHOI AND Y. CHOIE EL(4N, λ+ 1 ; e · ω(4N)), defined by Φr,e(4N)(g) R4N (z) (0), · · · , htν(4N)a tν(4N) R4N (z) (0), b(4N, e, g; 1), · · · , b(4N, e, g; e · ω(4N)) is an isomorphism. Proof of Theorem 3.1. Suppose that G(z) is a meromorphic modular form of weight 2 on Γ0(4N). For τ ∈ H∪C4N , let Dτ be the image of τ under the canonical map from H∪C4N to a compact Riemann surface X0(4N). Here H is the usual complex upper half plane, and C4N denotes the set of all inequivalent cusps of Γ0(4N). The residue ResDτGdz of G(z) at Dτ ∈ X0(4N) is well-defined since we have a canonical correspondence between a meromorphic modular form of weight 2 on Γ0(4N) and a meromorphic 1-form of X0(4N). If ResτG denotes the residue of G at τ on H, then ResDτGdz = ResτG. Here lτ is the order of the isotropy group at τ . The residue of G at each cusp t ∈ C4N is (3.3) ResDtGdz = ht · atG(0) Now we give a proof of Theorem 3.1. To prove Theorem 3.1, take G(z) = R4N (z)e f(z), where g ∈Mr+ 1 (Γ0(4N)) and f(z) = n=1 af(n)q n ∈Mλ+ 1 (Γ0(4N)). Note that G(z) is holomorphic on H. Since g(z), R4N (z) and f(z) are holomorphic and R4N (z) has no zero on H, it is enough to compute the residues of G(z) only at all inequivalent cusps to apply the Residue Theorem. The q-expansion of R4N (z) ef(z) at ∞ is R4N(z)e f(z) = e·ω(4N)∑ b(4N, e, g; ν)q−ν + a g(z) R4N (z) (0) +O(q) af(n)q Since R4N (z) has no zero at t ≁ ∞, we have R4N (z)e γt = a R4N (z) (0)af(0) +O(qt). Further note that, for an irregular cusp t, at g(z) R4N (z) (0)af(0) = 0. THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 9 So the Residue Theorem and (3.3) imply that (3.4) t∈U4N e·ω(4N) (0)atf(0) + e·ω(4N)∑ b(4N, e, g; ν)af(ν) = 0. This shows that Φr,e(4N) is well-defined. The linearity of the map Φr,e(4N) is clear. It remains to check that Φr,e(4N) is an isomorphism. Since there exists no holomorphic modular form of negative weight except the zero function, we obtain the injectivity of Φr,e(4N). Note that for e ≥ λ−12 , 4N ;λ+ , e · ω(4N) = e · ω(4N) + ν(4N)− dimC Mλ+ 1 (Γ0(4N)) However, the set C4N , 1 ≤ N ≤ 4, of all inequivalent cusps of Γ0(4N) are ∞, 0, 1 ∞, 0, 1 C12 = ∞, 0, 1 C16 = ∞, 0, 1 and it can be checked that (3.5) ν(4) = 2, ν(8) = 3, ν(12) = 4, ν(16) = 6 (see §1 of Chapter 4. in [15] for details). The dimension formula of Mλ+ 1 (Γ0(4N)) (see Table 1) together with the results in (3.1) and (3.5), implies that 4N, λ+ ; e · ω(N) = dimC(Mr+ 1 (Γ0(4N))) since r = 2e− λ+ 1. Table 1. Dimension Formula for Mk(Γ0(4N)) N k = 2n + 1 k = 2n+ 3 k = 2n N = 1 n + 1 n + 1 n + 1 N = 2 2n+ 1 2n+ 2 2n+ 1 N = 3 4n+ 1 4n+ 3 4n+ 1 N = 4 4n+ 2 4n+ 4 4n+ 1 So Φr,e(4N) is surjective since the map Φr,e(4N) is injective. This completes our claim. 10 D. CHOI AND Y. CHOIE 4. Proofs of Theorem 1 and 2 4.1. Proof of Theorem 1. First, we obtain linear relations among Fourier coefficients of modular forms of half integral weight modulo p. Let Op := {α ∈ L | α is p-integral}. M̃λ+ 1 , p(Γ0(4N)) := {H(z) = aH(n)q n ∈ Op/pOp[[q−1, q]] | H ≡ h (mod p) for some h ∈ Op[[q−1, q]] ∩Mλ+ 1 (Γ0(4N))}. S̃λ+ 1 , p(Γ0(4N)) := {H(z) = aH(n)q n ∈ Op/pOp[[q−1, q]] | H ≡ h (mod p) for some h ∈ Op[[q−1, q]] ∩ Sλ+ 1 (Γ0(4N))}. The following lemma gives the dimension of M̃λ+ 1 , p(Γ0(4N)). Lemma 4.1. Take λ ∈ N, 1 ≤ N ≤ 4 and a prime p such that p ≥ 3 if N = 1, 2, 4, p ≥ 5 if N = 3. Now take any prime ideal p ⊂ OL, p|p. Then dim M̃λ+ 1 , p(Γ0(4N)) = dimMλ+ 1 (Γ0(4N)) dim S̃λ+ 1 , p(Γ0(4N)) = dimSλ+ 1 (Γ0(4N)). Proof. Let j4N (z) = q −1 +O(q) be a meromorphic modular function with a pole only at ∞. Explicitly, these functions j4(z) = η(z)8 η(4z)8 + 8, j8(z) = η(4z)12 η(2z)4η(8z)8 j12(z) = η(4z)4η(6z)2 η(2z)2η(12z)4 , j16(z) = η2(z)η(8z) η(2z)η2(16z) Since the Fourier coefficients of η(z) and 1 are integral, the q-expansion of j4N (z) has integral coefficients. Recall that ∆4N,λ = q δλ(4N) + O(qδλ(4N)+1) is the modular form of weight λ + 1 Γ0(4N) such that the order of its zero at ∞ is higher than that of any other modular form THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 11 of the same level and weight. Denote the order of zero of ∆4N,λ at ∞ by δλ(4N). Then the basis of Mλ+ 1 (Γ0(4N)) can be chosen as (4.1) {∆4N,λ(z)j4N (z)e | 0 ≤ e ≤ δλ(4N)} . If ∆4N,λ(z) is p-integral, then {∆4N,λ(z)j4N (z)e | 0 ≤ e ≤ δλ(4N)} also forms a basis of M̃λ+ 1 ,p(Γ0(4N)). Note that δλ(4N) = dimMλ+ 1 (Γ0(4N))− 1. So from Table 1 we have (4.2) ∆4N,λ(z) = ∆4N,j(z)R4N (z) where λ ≡ j (mod 2), j ∈ {0, 1}. More precisely, one can choose ∆4N,j(z) as followings: ∆4,0(z) = θ(z), ∆4,1(z) = θ(z) ∆8,0(z) = θ(z), ∆8,1(z) = (θ(z)3 − θ(z)θ(2z)2) , ∆12,0(z) = θ(z), ∆12,1(z) = x,y,z∈Z q 3x2+2(y2+z2+yz) − x,y,z∈Z q 3x2+4y2+4z2+4yz ∆16,0(z) = (θ(z)− θ(4z)) , ∆16,1(z) = 18 (θ(z) 3 − 3θ(z)2θ(4z) + 3θ(z)θ(4z)2 − θ(4z)3) . Since θ(z) = 1+ 2 n=1 q n, the coefficients of the q-expansion of ∆4N,j(z), j ∈ {0, 1}, are p-integral. This completes the proof. � Remark 4.2. The proof of Lemma 4.1 implies that the spaces of Mλ+ 1 (Γ0(4N)) for N = 1, 2, 4 are generated by eta-quotients since θ(z) = η(2z)5 η(z)2η(4z)2 For 1 ≤ N ≤ 4 set 4N, λ+ (af(1), · · · , af(n)) ∈ Fnp | f ∈ S̃λ+ 1 (Γ0(4N)) ,Fp := Op/pOp. We define L̃S(4N, λ + ;n) to be the orthogonal complement of ṼS(4N, λ + ;n) in Fn Using Lemma 4.1, we obtain the following proposition. Proposition 4.3. Suppose that λ is a positive integer and 1 ≤ N ≤ 4. For each e ∈ N, e ≥ λ − 1, take r = 2e−λ+1. The linear map ψ̃r,e(4N) : M̃r+ 1 ,p(Γ0(4N)) → L̃S(4N, λ+ ; e · ω(4N)), defined by ψ̃r,e(4N)(g) = (b(4N, e, g; 1), · · · , b(N, e, g; e · ω(4N))) , is an isomorphism. Here b(4N, e, g; ν) is defined in (3.2). Proof. Note that dimS 3 (4N) = 0 and that dimSλ+ 1 (4N) +N + 1 + = dimMλ+ 1 (see [10]). So, from Lemma 4.1 and Table 1, it is enough to show that ψr,e(4N) is injective. If g is in the kernel of ψr,e(4N), then R4N (z) e · R4N (z)e ≡ 0 (mod p) by Sturm’s formula (see [21]). So we have g(z) ≡ 0 (mod p) since R4N(z)e 6≡ 0 (mod p). This completes the proof. � 12 D. CHOI AND Y. CHOIE Theorem 4.4. Take a prime p,N = 1, 2, 4 and f(z) := af(n)q n ∈ Sλ+ 1 (Γ0(4N)) ∩ L[[q]]. Suppose that p ⊂ OL is any prime ideal with p|p and that af (n) is p-integral for every integer n ≥ n0. If λ ≡ 2 or 2 + (mod p−1 ) or p = 2, then there exists a positive integer b such that ≡ 0 (mod p), ∀n ∈ N. Proof of Theorem 4.4. i) First, suppose that p ≥ 3: Take positive integers ℓ and b such (4.3) 3− 2α(p : λ) p2b + pm(p:λ) + ℓ(p− 1) = 2. Note that if b is large enough, that is, b > logp 3−2α(p:λ) pm(p:λ) − 2 , then there exists a positive integer ℓ satisfying (4.3). Also note that atf(0) = 0 for every cusp t of Γ0(4N) since f(z) is a cusp form. So, if r = 2e− α(p : λ) + 1, then Theorem 3.1 implies that, for g(z) ∈ M̃r+ 1 (Γ0(4N)), e·ω(4N)∑ b(4N, e, g; ν)af(νp 2b−m(p:λ)) ≡ 0 (mod p), since R4N (z)e f(z)p m(p:λ) Eℓp−1(z) e·ω(4N)∑ b(4N, e, g; ν)q−νp + a g(z) R4N (z) (0) + a g(z) R4N (z) (n)qnp af(n)q npm(p:λ) (mod p). So Proposition 4.3 implies that p2b−m(p:λ) 2p2b−m(p:λ) , · · · , a e · ω(4N)p2b−m(p:λ) ∈ ṼS 4N,α(p : λ) + 1 If α(p : λ) = 2 or 2 + , then dimSα(p:λ)+ 1 (Γ0(4N)) = dim ṼS 4N,α(p : λ) + THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 13 ii) p = 2: Note that ∆4N,1(z) R4N (z) = q−1+O(1) for N = 1, 2, 4. So, there exists a polynomial F (X) ∈ Z[X ] such that F (j4N(z)) ∆4N,1(z) R4N (z) = q−n +O(1). For an integer b, 22 > λ+ 2, let G(z) := F (j4N(z)) ∆4N,1(z) R4N(z) f(z)θ(z)2 1+2b−2λ+3. Since θ(z) ≡ 1 (mod 2), Theorem 3.1 implies that af(2b · n) ≡ 0 (mod p). � To apply Theorem 4.4, we need the following two propositions. Proposition 4.5 (Proposition 3.2 in [22]). Suppose that p is an odd prime, k and N are integers with (N, p) = 1. Let f(z) = a(n)qn ∈ Mλ+ 1 (Γ0(4N)). Suppose that ξ := cp2 d , with ac > 0. Then there exist n0, h0 ∈ N with h0|N, a sequence {a0(n)}n≥n0 and r0 ∈ {0, 1, 2, 3} such that (f |Upm|λ+ 1 ξ)(z) = 4n+r0≡0 (mod p a0(n)q 4n+r0 m , ∀m ≥ 1. Proposition 4.6 (Proposition 5.1 in [1]). Suppose that p is an odd prime such that p ∤ N and consider g(z) = a(n)qn ∈ Sλ+ 1 (Γ0(4Np j)) ∩ L[[q]], for each j ∈ N. Suppose further that p ⊂ OL is any prime ideal with p|p and that a(n) is p-integral for every integer n ≥ 1. Then there exists G(z) ∈ Sλ′+ 1 (Γ0(4N)) ∩OL[[q]] such that G(z) ≡ g(z) (mod p), where λ′ + 1 = (λ+ 1 )pj + pe(p− 1) with eN large. Remark 4.7. Proposition 4.6 was proved for p ≥ 5 in [1]. One can check that this holds also for p = 3. Now we prove Theorem 1. Proof of Theorem 1. Take Gp(z) := η(8z)48 η(16z)24 ∈M12(Γ0(16)) if p = 2, η(z)27 η(9z)3 ∈M12(Γ0(9)) if p = 3, η(4z)p η(4p2z) ∈M p2−1 (Γ0(p 2)) if p ≥ 5. 14 D. CHOI AND Y. CHOIE Using properties of eta-quotients (see [12]), note that Gp(z) vanishes at every cusp of Γ0(16) except ∞ if p = 2, and vanishes at every cusp ac of Γ0(4Np 2) with p2 ∤ N if p ≥ 3. Thus, Proposition 4.5 implies that there exist positive integers ℓ,m, k such that (f |Upm)(z)Gp(z)ℓ ∈ Sk+ 1 (Γ0(16)) if p = 2, (f |Upm)(z)Gp(z)ℓ ∈ Sk+ 1 (Γ0(4p 2N)) if p ≥ 3. Note that k ≡ λ (mod p− 1). Using Proposition 4.6, we can find F (z) ∈ Sk′+ 1 (Γ0(4N)) such that F (z) ≡ (f(z)|Upm)Gp(z)ℓ ≡ (f |Upm)(z) (mod p) and k′ ≡ k (mod p − 1). Theorem 4.4 implies that there exists a positive integer b such that (F |Up2b)(z) ≡ 0 (mod p). Thus, we have shown so far that if ρ ∈ p \ p2, all the Fourier coefficients of · F (z)|Upm+2b are p-integral. Repeat this argument to complete our claim. � 4.2. Proof of Theorem 2. Theorem 2 can be derived from Theorem 3.1 by taking a special modular form. Proof of Theorem 2. Take a positive integer ℓ and a positive even integer u such that 3− 2α(p : λ) pm(p:λ) + ℓ(p− 1) = 2. Let F (z) := ∆4N,3−α(p:λ)(z) R4N (z) and G(z) := Ep−1(z) ℓf(z)p m(p:λ) . Since Ep−1(z) ≡ 1 (mod p), we have F (z)G(z) ≡ a∆4N,3−α(p:λ)(z) R4N (z) (n)qnp af(n)q nm(p:λ) (mod p). If Fourier coefficients of f(z) at each cusp are p-integral, then ((F ·G)|2γt) (z) ≡ atF (n)q atG(n)q atf (n)q at∆4N,3−α(p:λ)(z) R4N (z) (mod p) for t ≁ ∞. Since aF (z)G(z)(0) ≡ a∆4N,3−α(p:λ)(z) R4N (z) (0)af (0) + af (p u−m(p:λ)) (mod p) , F (z)G(z) (0) ≡ at∆4N,3−α(p:λ)(z) R4N (z) (0)atf (0) (mod p) for t ≁ ∞, for large u, the Residue Theorem implies Theorem 2 by letting u = 2b. Therefore it is enough to check a p-integral property of Fourier coefficients of f(z) at each cusp: take a positive integer e such that ∆(z)ef(z) is a holomorphic modular form, where THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 15 ∆(z) := q n=1(1− qn)24. Note that the q-expansions of j4N (z) and ∆4N,12e+λ(z) at each cusp are p-integral. Thus (4.1) implies that ∆(z)ef(z) = δ12e+λ(4N)∑ cnj4N (z) n∆4N,12e+λ(z). Moreover, cn is p-integral since j4N (z) n∆4N,12e+λ(z) = q δ12e+λ(4N)−n +O qδ12e+λ(4N)−n+1 and f(z) ∈ OL[[q, q−1]]. Note that p ∤ 4N since 1 ≤ N ≤ 4 and p ≥ 5 is a prime. So Fourier coefficients of j4N (z), ∆N,12e+λ(z) and at each cusp are p-integral. This completes our claim. � 5. Proof of Theorem 3 Theorem 3 follows from Theorem 1 and Theorem 2.1. Proof of Theorem 3. Note that j(z) ∈ MH . Let g(z) := Ψ−1(j(z)) and f(z) := Ψ−1(F (z)) = af (n)q It is known (see §14 in [4]) that g(z) = (θ(z))E10(4z) 4πi∆(4z) θ(z) d (E10(4z)) 80πi∆(4z) θ(z). Since the constant terms of the q-expansions at ∞ of f(z), θ(z) and g(z) are 0, a0 (0) = and a0g(0) = · 456 , respectively, we have f(z)− kθ(z)− a0f(0) + k(1− i)/2 a0g(0) g(z) ∈ M01 (Γ0(4)). Applying Theorem 1, one obtains the result. � 6. Proofs of Theorem 4 and 5 We begin with the following proposition. Proposition 6.1. Let p be an odd prime and f(z) := af (n)q n ∈Mλ+ 1 (Γ0(4)) ∩ Zp[[q]]. If λ ≡ 2 or 3 (mod p−1 ), then p2b−m(p:λ) ≡ −(14 − 4α(p : λ))af(0) + 28 2−1 − 2−1i )pb(7−2α(p:λ)) a0f (0) (mod p) 16 D. CHOI AND Y. CHOIE for every integer b > logp 2α(p:λ)−3 pm(p:λ) + 2 Proof of Proposition 6.1. For ν ∈ Z≥0, pm(p:λ) := ν · (p− 1) + α(p : λ) + 1 For an integer b with 3− 2α(p : λ) pm(p:λ) − 2 there exists an ℓ ∈ N such that 3− 2α(p : λ) p2b + pm(p:λ) + ℓ(p− 1) = 2, since 3− 2α(p : λ) p2b + pm(p:λ) − 2 = 3− 2α(p : λ) (p2b − 1) + ν(p− 1). We have F (z) ≡ n=0 af (n)q npm(p:λ) (mod p), G(z) ≡ q−pb + 14− 4α(p : λ) + aG(1)q + · · · (mod p). Note that aG(n) is p-integral for every integer n. Moreover, we obtain F (z)G(z)|2 ( 0 −11 0 ) ≡ a0f (0) + · · · −26pb )pb(7−2α(p:λ)) + · · · (mod p), where a0f (0) is given in (1.1). Note that ∞, 0, 1 is the set of cusps of Γ0(4), so Theorem 2 implies that af (p 2b−m(p:n)) + (14− 4α(p : λ))af(0)− 28a0f (0) )pb(7−2α(p:λ)) ≡ 0 (mod p). This proves Proposition 6.1. � 6.1. Proof of Theorem 4. Now we prove Theorem 4. Proof of Theorem 4. Take f(z) := θ2λ+1(z) = 1 + r2λ+1(ℓ)q af(n)q Note that f(z) ∈Mλ+ 1 (Γ0(4)). Since (θ| 1 ( 0 −11 0 ))(z) = , we obtain af(0) = 1 and a f (0) = )2λ+1 THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 17 Since λ ≡ 2, 3 (mod p−1 ) and , we have )p2u(7−2α(p:λ)) a0f (0) pm(p:λ) )p2u(7−2α(p:λ)) ( )pm(p:λ)(2α(p:λ)+(p−1)(2[ λp−1 ]+m(p:λ))+1) )(7−2α(p:λ))(p2u−1)( )8+2(p−1)[ λp−1 ]+m(p:λ)pm(p:λ)(p−1)+(pm(p:λ)−1)(1+2α(p:λ)) )8+2[ λp−1 ](p−1)+2α(p:λ)(pm(p:λ)−1) )[ λp−1 ]+α(p:λ)m(p:λ) (mod p), for some u ∈ N. Applying Proposition 6.1, we obtain the result. � 6.2. Proof of Theorem 5. Consider the Cohen Eisenstein seriesHr+ 1 (z) := N=0H(r,N)q of weight r + 1 , where r ≥ 2 is an integer. If (−1)rN ≡ 0, 1 (mod 4), then H(r,N) = 0. If N = 0, then H(r, 0) = −B2r . If N is a positive integer and Df 2 = (−1)rN , where D is a fundamental discriminant, then (6.1) H(r,N) = L(1− r, χD) µ(d)χD(d)d r−1σ2r−1(f/d). Here µ(d) is the Möbius function. The following theorem implies that the Fourier coeffi- cients of Hr+ 1 (z) are p-integral if p−1 Theorem 6.2 ([6]). Let D be a fundamental discriminant. If D is divisible by at least two different primes, then L(1−n, χD) is an integer for every positive integer n. If D = p, p > 2, then L(1−n, χD) is an integer for every positive integer n unless gcd(p, 1−χD(g)gn) 6= 1, where g is a primitive root (mod p). Proof of Theorem 5. Note that E10(z) = E4(z)E6(z). So, E10(z)F (z), E10(z)G(z) and E10(z)W (z) are modular forms of weights, 8 · 12 , 7 · and 8 · 1 respectively. Moreover, the Fourier coefficients of those modular forms are 11-integral, since the Fourier coefficients of H 5 (z), H 7 (z) and H 9 (z) are 11-integral by Theorem 6.2. We have E10(z)F (z) = +O(q), E10(z)F (z)| 17 ( 0 −11 0 ) = (1 + i)(2i)−5 +O E10(z)G(z) = +O(q), E10(z)G(z)| 15 ( 0 −11 0 ) = (1− i)(2i)−7 +O E10(z)W (z) = +O(q), E10(z)W (z)| 17 ( 0 −11 0 ) = (1 + i)(2i)−9 +O 18 D. CHOI AND Y. CHOIE where B2r is the 2rth Bernoulli number. The conclusion now follows from Proposition 6.1. � 6.3. Proof of Theorem 6. We begin by introducing some notations (see [17]). Let V := (F2np , Q) be the quadratic space over Fp, where Q is the quadratic form obtained from a quadratic form x 7→ T [x](x ∈ Z2np ) by reducing modulo p. We denote by < x, y >:= Q(x, y)−Q(x)−Q(y), x, y ∈ F2np , the associated bilinear form and let R(V ) := {x ∈ F2np : < x, y >= 0, ∀y ∈ F2np , Q(x) = 0} be the radical of R(V ). Following [14], define a polynomial Hn,p(T ;X) := 1 if sp = 0,∏[(sp−1)/2] j=1 (1− p2j−1X2) if sp > 0, sp odd, (1 + λp(T )p (sp−1)/2X) ∏[(sp−1)/2] j=1 (1− p2j−1X2) if sp > 0, sp even, where for even sp we denote λp(T ) := 1 if W is a hyperbolic space or sp = 2n, −1 otherwise. Following [16], for a nonnegative integer µ, define ρT (p µ) by ρT (p µ)Xµ := (1−X2)Hn,p(T ;X), if p|fT , 1 otherwise. We extend the functions ρT multiplicatively to natural numbers N by defining ρT (p µ)X−µ := ((1−X2)Hn,p(T ;X)). D(T ) := GL2n(Z) \ {G ∈M2n(Z) ∩GL2n(Q) : T [G−1] half-integral}, where GL2n(Z) operates by left-multiplication and T [G −1] = T ′G−1T . Then D(T ) is finite. For a ∈ N with a|fT , let (6.2) φ(a;T ) := G∈D(T ),|det(G)|=d ρT [G−1](a/d Note that φ(a;T ) ∈ Z for all a. With these notations we state the following theorem: Theorem 6.3 ([17]). Suppose that g ≡ 0, 1 (mod 4) and let k ∈ N with g ≡ k (mod 2). A Siegel modular form F is in SMaassk+n (Γ2g) if and only if there exists a modular form f(z) = c(n)qn ∈ Sk+ 1 (Γ0(4)) THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 19 such that A(T ) = ak−1φ(a;T )c |DT | for all T . Here, DT := (−1)g · det(2T ) and DT = DT,0f T with DT,0 the corresponding fundamental discriminant and fT ∈ N. Remark 6.4. A proof of Theorem 6.3 given in [17] implies that if A(T ) ∈ Z for all T , then c(m) ∈ Z for all m ∈ N. Proof of Theorem 6. From Theorem 6.3 we can take f(z) = c(n)qn ∈ Sk+ 1 (Γ0(4)) ∩ Zp[[q]] such that F (Z) = A(T )qtr(TZ) = ak−1φ(a;T )c |DT | qtr(TZ). By Theorem 1, there exists a positive integer b such that, for every positive integer m, c(pbm) ≡ 0 (mod pj), since k ≡ 2 or 3 (mod p−1 ). Suppose that pb+2j ||DT |. If pj|a and a|fT , then ak−1φ(a;T )c |DT | ≡ 0 (mod pj). If pj ∤ a and a|fT , then pb ∣∣∣ |DT |a2 and a k−1φ(a;T )c |DT | ≡ 0 (mod pj). � Acknowledgement We thank the referee for many helpful comments which have improved our exposition. References [1] S. Ahlgren and M. Boylan Central Critical Values of Modular L-functions and Coeffients of Half Integral Weight Modular Forms Modulo ℓ, Amer. J. Math. 129 (2007), no. 2, 429–454. [2] A. Balog, H. Darmon, K. Ono, Congruences for Fourier coefficients of half-integer weight modu- lar forms and special values of L-functions, Analytic Number Theory, 105–128. Progr. Math. 138 Birkhauser, 1996. [3] B. Berndt and A. Yee, Congruences for the coefficients of quotients of Eisenstein series, Acta Arith. 104 (2002), no. 3, 297–308. [4] R. E. Borcherds, Automorphic forms on Os+2,2(R) and infinite products, Invent. Math. 120 (1995) 161–213. [5] J. H. Bruinier, K. Ono, The arithmetic of Borcherds’ exponents, Math. Ann. 327 (2003), no. 2, 293–303. [6] L. Carlitz, Arithmetic properties of generalized Bernoulli numbers, J. Reine Angew. Math. 202 1959 174–182. 20 D. CHOI AND Y. CHOIE [7] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. [8] D. Choi and Y. Choie, Linear Relations among the Fourier Coefficients of Modular Forms on Groups Γ0(N) of Genus Zero and Their Applications, to appear in J. Math. Anal. Appl. 326 (2007), no. 1, 655–666. [9] Y. Choie, W. Kohnen, K. Ono, Linear relations between modular form coefficients and non-ordinary primes, Bull. London Math. Soc. 37 (2005), no. 3, 335–341. [10] H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Lecture Notes in Mathe- matics, 627 (1977), 69–78. [11] S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004) 1623–1635. [12] B. Gordon and K. Hughes, Multiplicative properties of eta-product, Cont. Math. 143 (1993), 415-430. [13] P. Guerzhoy, The Borcherds-Zagier isomorphism and a p-adic version of the Kohnen-Shimura map, Int. Math. Res. Not. 2005, no. 13, 799–814. [14] Y. Kitaoka, Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J. 95 (1984), 73–84. [15] N. Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, 97. Springer-Verlag, New York, 1993 [16] W. Kohnen, Lifting modular forms of half-integral weight to Siegel modular forms of even genus, Math. Ann. 322 (2002), 787–809. [17] W. Kohnen and H. Kojima, A Maass space in higher genus, Compos. Math. 141 (2005), no. 2, 313–322. [18] P. Jenkins and K. Ono, Divisibility criteria for class numbers of imaginary quadratic fields, Acta Arith. 125 (2006), no. 3, 285–289. [19] T. Miyake, Modular forms, Translated from the Japanese by Yoshitaka Maeda, Springer-Verlag, Berlin, 1989 [20] J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, Lecture Notes in Math. 350, Modular Functions of One Variable III. Springer, Berlin Heidelberg, 1973, pp. 191–268. [21] J. Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985), 275–280, Lecture Notes in Math., 1240, Springer, Berlin, 1987. [22] S. Treneer, Congruences for the Coefficients of Weakly Holomorphic Modular Forms, to appear in the Proceedings of the London Mathematical Society. [23] D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002, pp.211–244. School of Liberal Arts and Sciences, Korea Aerospace University, 200-1, Hwajeon- dong, Goyang, Gyeonggi, 412-791, Korea E-mail address : choija@postech.ac.kr Department of Mathematics and Pohang Mathematical Institute, POSTECH, Pohang, 790–784, Korea E-mail address : yjc@postech.ac.kr 1. Introduction and Statement of Main Results 2. Applications: More Congruences 2.1. p-adic Limits of Borcherds Exponents 2.2. Sums of n-Squares 2.3. Quotients of Eisenstein Series 2.4. The Maass Space 3. Linear Relation among Fourier Coefficients of modular forms of Half Integral Weight 4. Proofs of Theorem ?? and ?? 4.1. Proof of Theorem ?? 4.2. Proof of Theorem ?? 5. Proof of Theorem ?? 6. Proofs of Theorem ?? and ?? 6.1. Proof of Theorem ?? 6.2. Proof of Theorem ?? 6.3. Proof of Theorem ?? Acknowledgement References
Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(\mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $\Gamma_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coefficients of Siegel modular forms on Maass Space are obtained using Ikeda lifting.
Introduction and Statement of Main Results Serre obtained the p-adic limits of the integral Fourier coefficients of modular forms on SL2(Z) for p = 2, 3, 5, 7 (see Théorème 7 and Lemma 8 in [20]). In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ0(4N) forN = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications of our main result, we obtain congruences for various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space. For odd d, let := γtΓ0(4N)tγ where γt = ( c d ) ∈ Γ(1) and γt(t) = ∞. We denote the q-expansion of a modular form f ∈Mλ+ 1 (Γ0(4N)) at each cusp t of Γ0(4N) by (1.1) (f |λ+ 1 γt)(z) = (cz + d) −λ− 1 az + b cz + d atf (n)q t , qt := q where (1.2) r(t) ∈ 2000 Mathematics Subject Classification. 11F11,11F33. Key words and phrases. modular forms, p-adic limit, Borcherds exponents, Maass space . This work was partially supported by KOSEF R01-2003-00011596-0 , ITRC and BRSI-POSTECH. http://arxiv.org/abs/0704.0013v2 2 D. CHOI AND Y. CHOIE When t ∼ ∞, we denote atf (n) by af (n). Note that the number r(t) is independent of the choice of f ∈Mλ+ 1 (Γ0(4N)) and λ. We call t a regular cusp if r(t) = 0 (see Chapter IV. §1. of [15] for a more general definition of a λ-regular cusp ). Remark 1.1. Our definition of a regular cusp is different from the usual one. Let U4N := {t1, · · · , tν(4N)} be the set of all inequivalent regular cusps of Γ0(4N). Note that the genus of Γ0(4N) is zero if and only if 1 ≤ N ≤ 4. LetMλ+ 1 (Γ0(4N)) be the space of weakly holomorphic modular forms of weight λ + 1 on Γ0(4N) and let M0λ+ 1 (Γ0(N)) denote the set of f(z) ∈ Mλ+ 1 (Γ0(N)) such that the constant term of its q-expansion at each cusp is zero. Let Up be the operator defined by (f |Up)(z) := af(pn)q Let OL be the ring of integers of a number field L with a prime ideal p ⊂ OL. For f(z) := af(n)q n and g(z) := ag(n)q n ∈ L[[q−1, q]] we write f(z) ≡ g(z) (mod p) if and only if af (n)− ag(n) ∈ p for every integer n. With these notations we state the following theorem. Theorem 1. For N = 1, 2, 4 consider f(z) := af (n)q n ∈ M0 (Γ0(4N)) ∩ L[[q−1, q]]. Suppose that p ⊂ OL is any prime ideal such that p|p, p prime, and that af(n) is p-integral for every integer n ≥ n0. (1) If p = 2 and af (0) = 0, then there exists a positive integer b such that (f |(Up)b)(z) ≡ 0 (mod pj) for each j ∈ N. (2) If p ≥ 3 and f(z) ∈ M0 (Γ0(4N)) with λ ≡ 2 or 2+ (mod p−1 ), then there exists a positive integer b such that (f |(Up)b)(z) ≡ 0 (mod pj) for each j ∈ N. Remark 1.2. The p-adic limit of a sum of Fourier coefficients of f ∈ M 3 (Γ0(4N)) was studied in [13]. Our method only allows to prove a weaker result if f(z) 6∈ M0 (Γ0(4N)). THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 3 Theorem 2. For N = 1, 2 or 4, let f(z) := af (n)q n ∈ Mλ+ 1 (Γ0(4N)) ∩ L[[q−1, q]]. Suppose that p ⊂ OL is any prime ideal with p|p, p prime, p ≥ 5, and that af (n) is p-integral for every integer n ≥ n0. If λ ≡ 2 or 2 + (mod p−1 ), then there exists a positive integer b0 such that p2b−m(p:λ) t∈U4N ∆4N,3−α(p:λ)(z) R4N (z) e·ω(4N) (0)atf (0) (mod p) for every positive integer b > b0 (see Section 3 for detailed notation ). Example 1.3. Recall that the generating function of the overpartition P̄ (n) of n(see [11]) P̄ (n)qn = η(2z) η(z)2 is in M− 1 (Γ0(16)), where η(z) := q n=1(1− qn). Therefore, theorem 2 implies that P̄ (52b) ≡ 1 (mod 5), ∀b ∈ N. 2. Applications: More Congruences In this section, we study congruences for various modular objects such as those for Borcherds exponents and for quotients of Eisenstein series. 2.1. p-adic Limits of Borcherds Exponents. Let MH denote the set of meromorphic modular forms of integral weight on SL2(Z) with Heegner divisor, integer coefficients and leading coefficient 1. Let (Γ0(4)) := {f(z) = af(n)q n ∈ M 1 (Γ0(4)) | a(n) = 0 for n ≡ 2, 3 (mod 4)}. If f(z) = af(n)q n ∈ M+1 (Γ0(4)), then define Ψ(f(z)) by Ψ(f(z)) := q−h (1− qn)af (n2), where h = − 1 af(0) + 1<n≡0,1 (mod 4) af (−n)H(−n). Here H(−n) denotes the usual Hurwitz class number of discriminant −n. The following was proved by Borcherds. Theorem 2.1 ([4]). The map Ψ is an isomorphism from M+1 (Γ0(4)) to MH , and the weight of Ψ(f(z)) is af (0). 4 D. CHOI AND Y. CHOIE Let j(z) be the usual j-invariant function with the product expansion j(z) = q−1 (1− qn)A(n). Let F (z) := q−h n=1(1 − qn)c(n) be a meromorphic modular form of weight k in MH . The p-adic limit of d|n d · c(d) was studied in [5] for p = 2, 3, 5, 7. Here we obtain the p-adic limit of c(d) for p = 2, 3, 5, 7. Theorem 3. Let F (z) := q−h n=1(1− qn)c(n) be a meromorphic modular form of weight k in MH . (1) If p = 2, then for each j ∈ N there exists a positive integer b such that c(mpb) ≡ 2k (mod pj) for every positive integer m. (2) If p ∈ {3, 5, 7}, then, for each j ∈ N there exists a positive integer b such that 5c(mpb)−̟(F )A(mpb) ≡ 10k (mod pj) for every positive integer m. Here, ̟(F ) is a constant determined by the constant term of the q-expansion of Ψ−1(F ) at 0. 2.2. Sums of n-Squares. For u ∈ Z>0, let rn(u) := ♯{(s1, · · · , sn) ∈ Zn : s21 + · · ·+ s2n = u}. Theorem 4. Suppose that p ≥ 5 is a prime. If λ ≡ 2 or 3 (mod p−1 ), then there exists a positive integer C0 such that r2λ+1 p2b−m(p:λ) ≡ − (14− 4α (p : λ)) + 16 )[ λp−1 ]+α(p:λ)m(p:λ) (mod p), for every b > C0. Remark 2.2. As for an example, if λ ≡ 2 (mod p− 1) and p is an odd prime, then there exists a positive integer C0 such that r2λ+1 ≡ 10 (mod p), ∀b > C0 2.3. Quotients of Eisenstein Series. Congruences for the coefficients of quotients of elliptic Eisenstein series have been studied in [3]. Let us consider the Cohen Eisenstein series Hr+ 1 (z) := N=0H(r,N)q n of weight r+ 1 , r ≥ 2 (see [7]). We derive congruences for the coefficients of quotients of Hr+ 1 (z) and Eisenstein series. THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 5 Theorem 5. Let F (z) := E4(z) aF (n)q G(z) := E6(z) aG(n)q W (z) := E6(z) aW (n)q Then there exists a positive integer C0 such that aF (11 2b+1) ≡ 1 (mod 11), aG(11 2b+1) ≡ 6 (mod 11), aW (11 2b+1) ≡ 2 (mod 11) for every integer b > C0. 2.4. The Maass Space. Next we deal with congruences for the Fourier coefficients of a Siegel modular form in the Maass space. To define the Maass space, let us introduce notations given in [17]: let T ∈ M2g(Q) be a rational, half-integral, symmetric, non- degenerate matrix of size 2g with discriminant DT := (−1)g det(2T ). Let DT = DT,0f T , where DT,0 is the corresponding fundamental discriminant. Further- more, let G8 :=  2 0 −1 0 0 0 0 0 0 2 0 −1 0 0 0 0 −1 0 2 −1 0 0 0 0 0 −1 −1 2 −1 0 0 0 0 0 0 −1 2 −1 0 0 0 0 0 0 −1 2 −1 0 0 0 0 0 0 −1 2 −1 0 0 0 0 0 0 −1 2  and G7 be the upper (7, 7)-submatrix of G8. Define Sg := (g−1)/8 2, if g ≡ 1 (mod 8), (g−7)/8 G7, if g ≡ −1 (mod 8). 6 D. CHOI AND Y. CHOIE For each m ∈ N such that (−1)gm ≡ 0, 1 (mod 4), define a rational, half-integral, sym- metric, positive definite matrix Tm of size 2g by Tm :=   0 m/4 , if m ≡ 0 (mod 4), e2g−1 e′2g−1 [m+ 2 + (−1)n]/4 , if m ≡ (−1)g (mod 4) Here e2g−1 ∈ Z(2n−1,1) is the standard column vector and e′2g−1 is its transpose. Definition 2.3. (The Maass Space) Take g, k ∈ N such that g ≡ 0, 1 (mod 4) and g ≡ k (mod 2). Let SMaassk+g (Γ2g) F (Z) = A(T )qtr(TZ) ∈ Sk+g(Γ2g) ∣∣∣∣∣∣ A(T ) = ak−1φ(a;T )A(T|DT |/a2) (see (6.2) for details). This space is called the Maass space of genus 2g and weight g + k. In [17] it was proved that the Maass space is the same as the image of the Ikeda lifting when g ≡ 0, 1 (mod 4). Using this fact together with Theorem 1, we derive the following congruences for the Fourier coefficients of F (Z) in SMaassk+g (Γ2g). Theorem 6. For g ≡ 0, 1 (mod 4), let F (Z) := A(T )qtr(TZ) ∈ SMaassk+g (Γ2g) with integral coefficients A(T ), T > 0. If k ≡ 2 or 3 (mod p−1 ) for some prime p, then, for each j ∈ N, there exists a positive integer b for which A(T ) ≡ 0 (mod pj) for every T > 0, det(2T ) ≡ 0 (mod pb). This paper is organized as follows. Section 3 gives a linear relation among Fourier coefficients of modular forms of half integral weight. The remaining sections contain detailed proofs of the main theorems. 3. Linear Relation among Fourier Coefficients of modular forms of Half Integral Weight Let V (N ; k, n) be the subspace of Cn generated by the first n coefficients of the q- expansion of f at ∞ for f ∈ Sk(Γ0(N)), where Sk(Γ0(N)) denotes the space of cusp forms of weight k ∈ Z on Γ0(N). Let L(N ; k, n) be the orthogonal complement of V (N ; k, n) THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 7 in Cn with the usual inner product of Cn. The vector space L(1; k, d(k) + 1), d(k) = dim(Sk(Γ(1))), was studied by Siegel to evaluate the value of the Dedekind zeta function at a certain point. The vector space L(1; k, n) is explicitly described in terms of the principal part of negative weight modular forms in [9]. These results were extended in [8] to the groups Γ0(N) of genus zero. For 1 ≤ N ≤ 4, let 4N, λ+ at1f (0), · · · , a tν(4N) f (0), af(1), · · · , af(n) ∈ Cn+ν(4n) ∣∣∣ f ∈Mλ+ 1 (Γ0(4N)) where U4N := {t1, · · · , tν(4N)} is the set of all inequivalent regular cusps of Γ0(4N). We define EL(4N, λ+ 1 ;n) to be the orthogonal complement of EV (4N, λ+ 1 ;n) in Cn+ν(4N). Let ∆4N,λ := q δλ(4N)+O(qδλ(4N)+1) be inMλ+ 1 (Γ0(4N) with the maximum order at ∞, that is, its order at ∞ is bigger than that of any other modular form of the same level and weight. Furthermore, let R4(z) := η(4z)8 η(2z)4 , R8(z) := η(8z)8 η(4z)4 R12(z) := η(12z)12η(2z)2 η(6z)6η(4z)4 and R16(z) := η(16z)8 η(8z)4 For ℓ, n ∈ N, define m(ℓ : n) := ≡ 0 (mod 2) ≡ 1 (mod 2) α(ℓ : n) := n− ℓ− 1 Let ω(4N) be the order of zero of R4N (z) at ∞. Note that R4N (z) ∈ M2(Γ0(4N)) has its only zero at ∞. So, using the definition of η(z) = q 124 n=1(1− qn), we find that (3.1) ω(4) = 1, ω(8) = 2, ω(12) = 4, ω(16) = 4. For each g ∈Mr+ 1 (Γ0(4N)) and e ∈ N, let (3.2) R4N (z)e e·ω(4N)∑ b(4N, e, g; ν)q−ν +O(1) at ∞. With these notations we state the following theorem: Theorem 3.1. Suppose that λ ≥ 0 is an integer and 1 ≤ N ≤ 4. For each e ∈ N such that e ≥ λ − 1, take r = 2e − λ + 1. The linear map Φr,e(4N) : Mr+ 1 (Γ0(4N)) → 8 D. CHOI AND Y. CHOIE EL(4N, λ+ 1 ; e · ω(4N)), defined by Φr,e(4N)(g) R4N (z) (0), · · · , htν(4N)a tν(4N) R4N (z) (0), b(4N, e, g; 1), · · · , b(4N, e, g; e · ω(4N)) is an isomorphism. Proof of Theorem 3.1. Suppose that G(z) is a meromorphic modular form of weight 2 on Γ0(4N). For τ ∈ H∪C4N , let Dτ be the image of τ under the canonical map from H∪C4N to a compact Riemann surface X0(4N). Here H is the usual complex upper half plane, and C4N denotes the set of all inequivalent cusps of Γ0(4N). The residue ResDτGdz of G(z) at Dτ ∈ X0(4N) is well-defined since we have a canonical correspondence between a meromorphic modular form of weight 2 on Γ0(4N) and a meromorphic 1-form of X0(4N). If ResτG denotes the residue of G at τ on H, then ResDτGdz = ResτG. Here lτ is the order of the isotropy group at τ . The residue of G at each cusp t ∈ C4N is (3.3) ResDtGdz = ht · atG(0) Now we give a proof of Theorem 3.1. To prove Theorem 3.1, take G(z) = R4N (z)e f(z), where g ∈Mr+ 1 (Γ0(4N)) and f(z) = n=1 af(n)q n ∈Mλ+ 1 (Γ0(4N)). Note that G(z) is holomorphic on H. Since g(z), R4N (z) and f(z) are holomorphic and R4N (z) has no zero on H, it is enough to compute the residues of G(z) only at all inequivalent cusps to apply the Residue Theorem. The q-expansion of R4N (z) ef(z) at ∞ is R4N(z)e f(z) = e·ω(4N)∑ b(4N, e, g; ν)q−ν + a g(z) R4N (z) (0) +O(q) af(n)q Since R4N (z) has no zero at t ≁ ∞, we have R4N (z)e γt = a R4N (z) (0)af(0) +O(qt). Further note that, for an irregular cusp t, at g(z) R4N (z) (0)af(0) = 0. THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 9 So the Residue Theorem and (3.3) imply that (3.4) t∈U4N e·ω(4N) (0)atf(0) + e·ω(4N)∑ b(4N, e, g; ν)af(ν) = 0. This shows that Φr,e(4N) is well-defined. The linearity of the map Φr,e(4N) is clear. It remains to check that Φr,e(4N) is an isomorphism. Since there exists no holomorphic modular form of negative weight except the zero function, we obtain the injectivity of Φr,e(4N). Note that for e ≥ λ−12 , 4N ;λ+ , e · ω(4N) = e · ω(4N) + ν(4N)− dimC Mλ+ 1 (Γ0(4N)) However, the set C4N , 1 ≤ N ≤ 4, of all inequivalent cusps of Γ0(4N) are ∞, 0, 1 ∞, 0, 1 C12 = ∞, 0, 1 C16 = ∞, 0, 1 and it can be checked that (3.5) ν(4) = 2, ν(8) = 3, ν(12) = 4, ν(16) = 6 (see §1 of Chapter 4. in [15] for details). The dimension formula of Mλ+ 1 (Γ0(4N)) (see Table 1) together with the results in (3.1) and (3.5), implies that 4N, λ+ ; e · ω(N) = dimC(Mr+ 1 (Γ0(4N))) since r = 2e− λ+ 1. Table 1. Dimension Formula for Mk(Γ0(4N)) N k = 2n + 1 k = 2n+ 3 k = 2n N = 1 n + 1 n + 1 n + 1 N = 2 2n+ 1 2n+ 2 2n+ 1 N = 3 4n+ 1 4n+ 3 4n+ 1 N = 4 4n+ 2 4n+ 4 4n+ 1 So Φr,e(4N) is surjective since the map Φr,e(4N) is injective. This completes our claim. 10 D. CHOI AND Y. CHOIE 4. Proofs of Theorem 1 and 2 4.1. Proof of Theorem 1. First, we obtain linear relations among Fourier coefficients of modular forms of half integral weight modulo p. Let Op := {α ∈ L | α is p-integral}. M̃λ+ 1 , p(Γ0(4N)) := {H(z) = aH(n)q n ∈ Op/pOp[[q−1, q]] | H ≡ h (mod p) for some h ∈ Op[[q−1, q]] ∩Mλ+ 1 (Γ0(4N))}. S̃λ+ 1 , p(Γ0(4N)) := {H(z) = aH(n)q n ∈ Op/pOp[[q−1, q]] | H ≡ h (mod p) for some h ∈ Op[[q−1, q]] ∩ Sλ+ 1 (Γ0(4N))}. The following lemma gives the dimension of M̃λ+ 1 , p(Γ0(4N)). Lemma 4.1. Take λ ∈ N, 1 ≤ N ≤ 4 and a prime p such that p ≥ 3 if N = 1, 2, 4, p ≥ 5 if N = 3. Now take any prime ideal p ⊂ OL, p|p. Then dim M̃λ+ 1 , p(Γ0(4N)) = dimMλ+ 1 (Γ0(4N)) dim S̃λ+ 1 , p(Γ0(4N)) = dimSλ+ 1 (Γ0(4N)). Proof. Let j4N (z) = q −1 +O(q) be a meromorphic modular function with a pole only at ∞. Explicitly, these functions j4(z) = η(z)8 η(4z)8 + 8, j8(z) = η(4z)12 η(2z)4η(8z)8 j12(z) = η(4z)4η(6z)2 η(2z)2η(12z)4 , j16(z) = η2(z)η(8z) η(2z)η2(16z) Since the Fourier coefficients of η(z) and 1 are integral, the q-expansion of j4N (z) has integral coefficients. Recall that ∆4N,λ = q δλ(4N) + O(qδλ(4N)+1) is the modular form of weight λ + 1 Γ0(4N) such that the order of its zero at ∞ is higher than that of any other modular form THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 11 of the same level and weight. Denote the order of zero of ∆4N,λ at ∞ by δλ(4N). Then the basis of Mλ+ 1 (Γ0(4N)) can be chosen as (4.1) {∆4N,λ(z)j4N (z)e | 0 ≤ e ≤ δλ(4N)} . If ∆4N,λ(z) is p-integral, then {∆4N,λ(z)j4N (z)e | 0 ≤ e ≤ δλ(4N)} also forms a basis of M̃λ+ 1 ,p(Γ0(4N)). Note that δλ(4N) = dimMλ+ 1 (Γ0(4N))− 1. So from Table 1 we have (4.2) ∆4N,λ(z) = ∆4N,j(z)R4N (z) where λ ≡ j (mod 2), j ∈ {0, 1}. More precisely, one can choose ∆4N,j(z) as followings: ∆4,0(z) = θ(z), ∆4,1(z) = θ(z) ∆8,0(z) = θ(z), ∆8,1(z) = (θ(z)3 − θ(z)θ(2z)2) , ∆12,0(z) = θ(z), ∆12,1(z) = x,y,z∈Z q 3x2+2(y2+z2+yz) − x,y,z∈Z q 3x2+4y2+4z2+4yz ∆16,0(z) = (θ(z)− θ(4z)) , ∆16,1(z) = 18 (θ(z) 3 − 3θ(z)2θ(4z) + 3θ(z)θ(4z)2 − θ(4z)3) . Since θ(z) = 1+ 2 n=1 q n, the coefficients of the q-expansion of ∆4N,j(z), j ∈ {0, 1}, are p-integral. This completes the proof. � Remark 4.2. The proof of Lemma 4.1 implies that the spaces of Mλ+ 1 (Γ0(4N)) for N = 1, 2, 4 are generated by eta-quotients since θ(z) = η(2z)5 η(z)2η(4z)2 For 1 ≤ N ≤ 4 set 4N, λ+ (af(1), · · · , af(n)) ∈ Fnp | f ∈ S̃λ+ 1 (Γ0(4N)) ,Fp := Op/pOp. We define L̃S(4N, λ + ;n) to be the orthogonal complement of ṼS(4N, λ + ;n) in Fn Using Lemma 4.1, we obtain the following proposition. Proposition 4.3. Suppose that λ is a positive integer and 1 ≤ N ≤ 4. For each e ∈ N, e ≥ λ − 1, take r = 2e−λ+1. The linear map ψ̃r,e(4N) : M̃r+ 1 ,p(Γ0(4N)) → L̃S(4N, λ+ ; e · ω(4N)), defined by ψ̃r,e(4N)(g) = (b(4N, e, g; 1), · · · , b(N, e, g; e · ω(4N))) , is an isomorphism. Here b(4N, e, g; ν) is defined in (3.2). Proof. Note that dimS 3 (4N) = 0 and that dimSλ+ 1 (4N) +N + 1 + = dimMλ+ 1 (see [10]). So, from Lemma 4.1 and Table 1, it is enough to show that ψr,e(4N) is injective. If g is in the kernel of ψr,e(4N), then R4N (z) e · R4N (z)e ≡ 0 (mod p) by Sturm’s formula (see [21]). So we have g(z) ≡ 0 (mod p) since R4N(z)e 6≡ 0 (mod p). This completes the proof. � 12 D. CHOI AND Y. CHOIE Theorem 4.4. Take a prime p,N = 1, 2, 4 and f(z) := af(n)q n ∈ Sλ+ 1 (Γ0(4N)) ∩ L[[q]]. Suppose that p ⊂ OL is any prime ideal with p|p and that af (n) is p-integral for every integer n ≥ n0. If λ ≡ 2 or 2 + (mod p−1 ) or p = 2, then there exists a positive integer b such that ≡ 0 (mod p), ∀n ∈ N. Proof of Theorem 4.4. i) First, suppose that p ≥ 3: Take positive integers ℓ and b such (4.3) 3− 2α(p : λ) p2b + pm(p:λ) + ℓ(p− 1) = 2. Note that if b is large enough, that is, b > logp 3−2α(p:λ) pm(p:λ) − 2 , then there exists a positive integer ℓ satisfying (4.3). Also note that atf(0) = 0 for every cusp t of Γ0(4N) since f(z) is a cusp form. So, if r = 2e− α(p : λ) + 1, then Theorem 3.1 implies that, for g(z) ∈ M̃r+ 1 (Γ0(4N)), e·ω(4N)∑ b(4N, e, g; ν)af(νp 2b−m(p:λ)) ≡ 0 (mod p), since R4N (z)e f(z)p m(p:λ) Eℓp−1(z) e·ω(4N)∑ b(4N, e, g; ν)q−νp + a g(z) R4N (z) (0) + a g(z) R4N (z) (n)qnp af(n)q npm(p:λ) (mod p). So Proposition 4.3 implies that p2b−m(p:λ) 2p2b−m(p:λ) , · · · , a e · ω(4N)p2b−m(p:λ) ∈ ṼS 4N,α(p : λ) + 1 If α(p : λ) = 2 or 2 + , then dimSα(p:λ)+ 1 (Γ0(4N)) = dim ṼS 4N,α(p : λ) + THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 13 ii) p = 2: Note that ∆4N,1(z) R4N (z) = q−1+O(1) for N = 1, 2, 4. So, there exists a polynomial F (X) ∈ Z[X ] such that F (j4N(z)) ∆4N,1(z) R4N (z) = q−n +O(1). For an integer b, 22 > λ+ 2, let G(z) := F (j4N(z)) ∆4N,1(z) R4N(z) f(z)θ(z)2 1+2b−2λ+3. Since θ(z) ≡ 1 (mod 2), Theorem 3.1 implies that af(2b · n) ≡ 0 (mod p). � To apply Theorem 4.4, we need the following two propositions. Proposition 4.5 (Proposition 3.2 in [22]). Suppose that p is an odd prime, k and N are integers with (N, p) = 1. Let f(z) = a(n)qn ∈ Mλ+ 1 (Γ0(4N)). Suppose that ξ := cp2 d , with ac > 0. Then there exist n0, h0 ∈ N with h0|N, a sequence {a0(n)}n≥n0 and r0 ∈ {0, 1, 2, 3} such that (f |Upm|λ+ 1 ξ)(z) = 4n+r0≡0 (mod p a0(n)q 4n+r0 m , ∀m ≥ 1. Proposition 4.6 (Proposition 5.1 in [1]). Suppose that p is an odd prime such that p ∤ N and consider g(z) = a(n)qn ∈ Sλ+ 1 (Γ0(4Np j)) ∩ L[[q]], for each j ∈ N. Suppose further that p ⊂ OL is any prime ideal with p|p and that a(n) is p-integral for every integer n ≥ 1. Then there exists G(z) ∈ Sλ′+ 1 (Γ0(4N)) ∩OL[[q]] such that G(z) ≡ g(z) (mod p), where λ′ + 1 = (λ+ 1 )pj + pe(p− 1) with eN large. Remark 4.7. Proposition 4.6 was proved for p ≥ 5 in [1]. One can check that this holds also for p = 3. Now we prove Theorem 1. Proof of Theorem 1. Take Gp(z) := η(8z)48 η(16z)24 ∈M12(Γ0(16)) if p = 2, η(z)27 η(9z)3 ∈M12(Γ0(9)) if p = 3, η(4z)p η(4p2z) ∈M p2−1 (Γ0(p 2)) if p ≥ 5. 14 D. CHOI AND Y. CHOIE Using properties of eta-quotients (see [12]), note that Gp(z) vanishes at every cusp of Γ0(16) except ∞ if p = 2, and vanishes at every cusp ac of Γ0(4Np 2) with p2 ∤ N if p ≥ 3. Thus, Proposition 4.5 implies that there exist positive integers ℓ,m, k such that (f |Upm)(z)Gp(z)ℓ ∈ Sk+ 1 (Γ0(16)) if p = 2, (f |Upm)(z)Gp(z)ℓ ∈ Sk+ 1 (Γ0(4p 2N)) if p ≥ 3. Note that k ≡ λ (mod p− 1). Using Proposition 4.6, we can find F (z) ∈ Sk′+ 1 (Γ0(4N)) such that F (z) ≡ (f(z)|Upm)Gp(z)ℓ ≡ (f |Upm)(z) (mod p) and k′ ≡ k (mod p − 1). Theorem 4.4 implies that there exists a positive integer b such that (F |Up2b)(z) ≡ 0 (mod p). Thus, we have shown so far that if ρ ∈ p \ p2, all the Fourier coefficients of · F (z)|Upm+2b are p-integral. Repeat this argument to complete our claim. � 4.2. Proof of Theorem 2. Theorem 2 can be derived from Theorem 3.1 by taking a special modular form. Proof of Theorem 2. Take a positive integer ℓ and a positive even integer u such that 3− 2α(p : λ) pm(p:λ) + ℓ(p− 1) = 2. Let F (z) := ∆4N,3−α(p:λ)(z) R4N (z) and G(z) := Ep−1(z) ℓf(z)p m(p:λ) . Since Ep−1(z) ≡ 1 (mod p), we have F (z)G(z) ≡ a∆4N,3−α(p:λ)(z) R4N (z) (n)qnp af(n)q nm(p:λ) (mod p). If Fourier coefficients of f(z) at each cusp are p-integral, then ((F ·G)|2γt) (z) ≡ atF (n)q atG(n)q atf (n)q at∆4N,3−α(p:λ)(z) R4N (z) (mod p) for t ≁ ∞. Since aF (z)G(z)(0) ≡ a∆4N,3−α(p:λ)(z) R4N (z) (0)af (0) + af (p u−m(p:λ)) (mod p) , F (z)G(z) (0) ≡ at∆4N,3−α(p:λ)(z) R4N (z) (0)atf (0) (mod p) for t ≁ ∞, for large u, the Residue Theorem implies Theorem 2 by letting u = 2b. Therefore it is enough to check a p-integral property of Fourier coefficients of f(z) at each cusp: take a positive integer e such that ∆(z)ef(z) is a holomorphic modular form, where THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 15 ∆(z) := q n=1(1− qn)24. Note that the q-expansions of j4N (z) and ∆4N,12e+λ(z) at each cusp are p-integral. Thus (4.1) implies that ∆(z)ef(z) = δ12e+λ(4N)∑ cnj4N (z) n∆4N,12e+λ(z). Moreover, cn is p-integral since j4N (z) n∆4N,12e+λ(z) = q δ12e+λ(4N)−n +O qδ12e+λ(4N)−n+1 and f(z) ∈ OL[[q, q−1]]. Note that p ∤ 4N since 1 ≤ N ≤ 4 and p ≥ 5 is a prime. So Fourier coefficients of j4N (z), ∆N,12e+λ(z) and at each cusp are p-integral. This completes our claim. � 5. Proof of Theorem 3 Theorem 3 follows from Theorem 1 and Theorem 2.1. Proof of Theorem 3. Note that j(z) ∈ MH . Let g(z) := Ψ−1(j(z)) and f(z) := Ψ−1(F (z)) = af (n)q It is known (see §14 in [4]) that g(z) = (θ(z))E10(4z) 4πi∆(4z) θ(z) d (E10(4z)) 80πi∆(4z) θ(z). Since the constant terms of the q-expansions at ∞ of f(z), θ(z) and g(z) are 0, a0 (0) = and a0g(0) = · 456 , respectively, we have f(z)− kθ(z)− a0f(0) + k(1− i)/2 a0g(0) g(z) ∈ M01 (Γ0(4)). Applying Theorem 1, one obtains the result. � 6. Proofs of Theorem 4 and 5 We begin with the following proposition. Proposition 6.1. Let p be an odd prime and f(z) := af (n)q n ∈Mλ+ 1 (Γ0(4)) ∩ Zp[[q]]. If λ ≡ 2 or 3 (mod p−1 ), then p2b−m(p:λ) ≡ −(14 − 4α(p : λ))af(0) + 28 2−1 − 2−1i )pb(7−2α(p:λ)) a0f (0) (mod p) 16 D. CHOI AND Y. CHOIE for every integer b > logp 2α(p:λ)−3 pm(p:λ) + 2 Proof of Proposition 6.1. For ν ∈ Z≥0, pm(p:λ) := ν · (p− 1) + α(p : λ) + 1 For an integer b with 3− 2α(p : λ) pm(p:λ) − 2 there exists an ℓ ∈ N such that 3− 2α(p : λ) p2b + pm(p:λ) + ℓ(p− 1) = 2, since 3− 2α(p : λ) p2b + pm(p:λ) − 2 = 3− 2α(p : λ) (p2b − 1) + ν(p− 1). We have F (z) ≡ n=0 af (n)q npm(p:λ) (mod p), G(z) ≡ q−pb + 14− 4α(p : λ) + aG(1)q + · · · (mod p). Note that aG(n) is p-integral for every integer n. Moreover, we obtain F (z)G(z)|2 ( 0 −11 0 ) ≡ a0f (0) + · · · −26pb )pb(7−2α(p:λ)) + · · · (mod p), where a0f (0) is given in (1.1). Note that ∞, 0, 1 is the set of cusps of Γ0(4), so Theorem 2 implies that af (p 2b−m(p:n)) + (14− 4α(p : λ))af(0)− 28a0f (0) )pb(7−2α(p:λ)) ≡ 0 (mod p). This proves Proposition 6.1. � 6.1. Proof of Theorem 4. Now we prove Theorem 4. Proof of Theorem 4. Take f(z) := θ2λ+1(z) = 1 + r2λ+1(ℓ)q af(n)q Note that f(z) ∈Mλ+ 1 (Γ0(4)). Since (θ| 1 ( 0 −11 0 ))(z) = , we obtain af(0) = 1 and a f (0) = )2λ+1 THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 17 Since λ ≡ 2, 3 (mod p−1 ) and , we have )p2u(7−2α(p:λ)) a0f (0) pm(p:λ) )p2u(7−2α(p:λ)) ( )pm(p:λ)(2α(p:λ)+(p−1)(2[ λp−1 ]+m(p:λ))+1) )(7−2α(p:λ))(p2u−1)( )8+2(p−1)[ λp−1 ]+m(p:λ)pm(p:λ)(p−1)+(pm(p:λ)−1)(1+2α(p:λ)) )8+2[ λp−1 ](p−1)+2α(p:λ)(pm(p:λ)−1) )[ λp−1 ]+α(p:λ)m(p:λ) (mod p), for some u ∈ N. Applying Proposition 6.1, we obtain the result. � 6.2. Proof of Theorem 5. Consider the Cohen Eisenstein seriesHr+ 1 (z) := N=0H(r,N)q of weight r + 1 , where r ≥ 2 is an integer. If (−1)rN ≡ 0, 1 (mod 4), then H(r,N) = 0. If N = 0, then H(r, 0) = −B2r . If N is a positive integer and Df 2 = (−1)rN , where D is a fundamental discriminant, then (6.1) H(r,N) = L(1− r, χD) µ(d)χD(d)d r−1σ2r−1(f/d). Here µ(d) is the Möbius function. The following theorem implies that the Fourier coeffi- cients of Hr+ 1 (z) are p-integral if p−1 Theorem 6.2 ([6]). Let D be a fundamental discriminant. If D is divisible by at least two different primes, then L(1−n, χD) is an integer for every positive integer n. If D = p, p > 2, then L(1−n, χD) is an integer for every positive integer n unless gcd(p, 1−χD(g)gn) 6= 1, where g is a primitive root (mod p). Proof of Theorem 5. Note that E10(z) = E4(z)E6(z). So, E10(z)F (z), E10(z)G(z) and E10(z)W (z) are modular forms of weights, 8 · 12 , 7 · and 8 · 1 respectively. Moreover, the Fourier coefficients of those modular forms are 11-integral, since the Fourier coefficients of H 5 (z), H 7 (z) and H 9 (z) are 11-integral by Theorem 6.2. We have E10(z)F (z) = +O(q), E10(z)F (z)| 17 ( 0 −11 0 ) = (1 + i)(2i)−5 +O E10(z)G(z) = +O(q), E10(z)G(z)| 15 ( 0 −11 0 ) = (1− i)(2i)−7 +O E10(z)W (z) = +O(q), E10(z)W (z)| 17 ( 0 −11 0 ) = (1 + i)(2i)−9 +O 18 D. CHOI AND Y. CHOIE where B2r is the 2rth Bernoulli number. The conclusion now follows from Proposition 6.1. � 6.3. Proof of Theorem 6. We begin by introducing some notations (see [17]). Let V := (F2np , Q) be the quadratic space over Fp, where Q is the quadratic form obtained from a quadratic form x 7→ T [x](x ∈ Z2np ) by reducing modulo p. We denote by < x, y >:= Q(x, y)−Q(x)−Q(y), x, y ∈ F2np , the associated bilinear form and let R(V ) := {x ∈ F2np : < x, y >= 0, ∀y ∈ F2np , Q(x) = 0} be the radical of R(V ). Following [14], define a polynomial Hn,p(T ;X) := 1 if sp = 0,∏[(sp−1)/2] j=1 (1− p2j−1X2) if sp > 0, sp odd, (1 + λp(T )p (sp−1)/2X) ∏[(sp−1)/2] j=1 (1− p2j−1X2) if sp > 0, sp even, where for even sp we denote λp(T ) := 1 if W is a hyperbolic space or sp = 2n, −1 otherwise. Following [16], for a nonnegative integer µ, define ρT (p µ) by ρT (p µ)Xµ := (1−X2)Hn,p(T ;X), if p|fT , 1 otherwise. We extend the functions ρT multiplicatively to natural numbers N by defining ρT (p µ)X−µ := ((1−X2)Hn,p(T ;X)). D(T ) := GL2n(Z) \ {G ∈M2n(Z) ∩GL2n(Q) : T [G−1] half-integral}, where GL2n(Z) operates by left-multiplication and T [G −1] = T ′G−1T . Then D(T ) is finite. For a ∈ N with a|fT , let (6.2) φ(a;T ) := G∈D(T ),|det(G)|=d ρT [G−1](a/d Note that φ(a;T ) ∈ Z for all a. With these notations we state the following theorem: Theorem 6.3 ([17]). Suppose that g ≡ 0, 1 (mod 4) and let k ∈ N with g ≡ k (mod 2). A Siegel modular form F is in SMaassk+n (Γ2g) if and only if there exists a modular form f(z) = c(n)qn ∈ Sk+ 1 (Γ0(4)) THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 19 such that A(T ) = ak−1φ(a;T )c |DT | for all T . Here, DT := (−1)g · det(2T ) and DT = DT,0f T with DT,0 the corresponding fundamental discriminant and fT ∈ N. Remark 6.4. A proof of Theorem 6.3 given in [17] implies that if A(T ) ∈ Z for all T , then c(m) ∈ Z for all m ∈ N. Proof of Theorem 6. From Theorem 6.3 we can take f(z) = c(n)qn ∈ Sk+ 1 (Γ0(4)) ∩ Zp[[q]] such that F (Z) = A(T )qtr(TZ) = ak−1φ(a;T )c |DT | qtr(TZ). By Theorem 1, there exists a positive integer b such that, for every positive integer m, c(pbm) ≡ 0 (mod pj), since k ≡ 2 or 3 (mod p−1 ). Suppose that pb+2j ||DT |. If pj|a and a|fT , then ak−1φ(a;T )c |DT | ≡ 0 (mod pj). If pj ∤ a and a|fT , then pb ∣∣∣ |DT |a2 and a k−1φ(a;T )c |DT | ≡ 0 (mod pj). � Acknowledgement We thank the referee for many helpful comments which have improved our exposition. References [1] S. Ahlgren and M. Boylan Central Critical Values of Modular L-functions and Coeffients of Half Integral Weight Modular Forms Modulo ℓ, Amer. J. Math. 129 (2007), no. 2, 429–454. [2] A. Balog, H. Darmon, K. Ono, Congruences for Fourier coefficients of half-integer weight modu- lar forms and special values of L-functions, Analytic Number Theory, 105–128. Progr. Math. 138 Birkhauser, 1996. [3] B. Berndt and A. Yee, Congruences for the coefficients of quotients of Eisenstein series, Acta Arith. 104 (2002), no. 3, 297–308. [4] R. E. Borcherds, Automorphic forms on Os+2,2(R) and infinite products, Invent. Math. 120 (1995) 161–213. [5] J. H. Bruinier, K. Ono, The arithmetic of Borcherds’ exponents, Math. Ann. 327 (2003), no. 2, 293–303. [6] L. Carlitz, Arithmetic properties of generalized Bernoulli numbers, J. Reine Angew. Math. 202 1959 174–182. 20 D. CHOI AND Y. CHOIE [7] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. [8] D. Choi and Y. Choie, Linear Relations among the Fourier Coefficients of Modular Forms on Groups Γ0(N) of Genus Zero and Their Applications, to appear in J. Math. Anal. Appl. 326 (2007), no. 1, 655–666. [9] Y. Choie, W. Kohnen, K. Ono, Linear relations between modular form coefficients and non-ordinary primes, Bull. London Math. Soc. 37 (2005), no. 3, 335–341. [10] H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Lecture Notes in Mathe- matics, 627 (1977), 69–78. [11] S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004) 1623–1635. [12] B. Gordon and K. Hughes, Multiplicative properties of eta-product, Cont. Math. 143 (1993), 415-430. [13] P. Guerzhoy, The Borcherds-Zagier isomorphism and a p-adic version of the Kohnen-Shimura map, Int. Math. Res. Not. 2005, no. 13, 799–814. [14] Y. Kitaoka, Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J. 95 (1984), 73–84. [15] N. Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, 97. Springer-Verlag, New York, 1993 [16] W. Kohnen, Lifting modular forms of half-integral weight to Siegel modular forms of even genus, Math. Ann. 322 (2002), 787–809. [17] W. Kohnen and H. Kojima, A Maass space in higher genus, Compos. Math. 141 (2005), no. 2, 313–322. [18] P. Jenkins and K. Ono, Divisibility criteria for class numbers of imaginary quadratic fields, Acta Arith. 125 (2006), no. 3, 285–289. [19] T. Miyake, Modular forms, Translated from the Japanese by Yoshitaka Maeda, Springer-Verlag, Berlin, 1989 [20] J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, Lecture Notes in Math. 350, Modular Functions of One Variable III. Springer, Berlin Heidelberg, 1973, pp. 191–268. [21] J. Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985), 275–280, Lecture Notes in Math., 1240, Springer, Berlin, 1987. [22] S. Treneer, Congruences for the Coefficients of Weakly Holomorphic Modular Forms, to appear in the Proceedings of the London Mathematical Society. [23] D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002, pp.211–244. School of Liberal Arts and Sciences, Korea Aerospace University, 200-1, Hwajeon- dong, Goyang, Gyeonggi, 412-791, Korea E-mail address : choija@postech.ac.kr Department of Mathematics and Pohang Mathematical Institute, POSTECH, Pohang, 790–784, Korea E-mail address : yjc@postech.ac.kr 1. Introduction and Statement of Main Results 2. Applications: More Congruences 2.1. p-adic Limits of Borcherds Exponents 2.2. Sums of n-Squares 2.3. Quotients of Eisenstein Series 2.4. The Maass Space 3. Linear Relation among Fourier Coefficients of modular forms of Half Integral Weight 4. Proofs of Theorem ?? and ?? 4.1. Proof of Theorem ?? 4.2. Proof of Theorem ?? 5. Proof of Theorem ?? 6. Proofs of Theorem ?? and ?? 6.1. Proof of Theorem ?? 6.2. Proof of Theorem ?? 6.3. Proof of Theorem ?? Acknowledgement References
704.001
Iterated integrals and the loop product Koichi Fujii 1 Introduction The purpose of this paper is to describe string topology from the viewpoint of Chen’s iterated integrals. Let M be a compact closed oriented d-manifold and LM be the free loop space ofM , the set of unbased smooth maps from S1 toM . Let H∗(LM) be the homology of the free loop space shifted by the dimension of the manifold i.e. H∗(LM) = H∗+d(LM). Chas and Sullivan found the product on H∗(LM) which they called loop product [1]: Hp(LM)⊗Hq(LM)→ Hp+q(LM). They showed that this product makes H∗(LM) an associative, commutative algebra. Merkulov constructed a model for this product based on the theory of iter- ated integrals, especially of the formal power series connection [10]. He showed that there is an isomorphism of algebras H∗(LM) ∼= H∗(ΛM ⊗ R where ΛM is the de Rham differential graded algebra of M and R the formal completion of the free graded associative algebra generated by some noncommutative indeterminates. On the other hand, Chen showed that the cohomology of the free loop space of the simply-connected manifold is isomorphic to the cohomology of the cyclic bar complex of differential forms via Chen’s iterated integrals (see [5] or [8]): H∗(LM) ∼= H ∗(C(ΛM)). In this paper, we construct a model for the loop product based on the the- ory of the cyclic bar complex. We define a complex Hom(B(ΛM),ΛM) and its subcomplex Hom(B(ΛM),ΛM) so that the Poincaré duality induces the isomorphism of vector spaces H∗(Hom(C(ΛM),R)) ∼= H∗−d(Hom(B(ΛM),ΛM)). We can define a product on Hom(B(ΛM),ΛM) which realizes the loop product. http://arxiv.org/abs/0704.0014v1 Theorem 1.1. Let M be a compact closed oriented simply-connected manifold. Assume that H∗(M) is of finite type. Let A be a differential graded subalge- bra of ΛM such that H∗(A) ∼= H∗(ΛM) by the inclusion. Then there is an isomorphism of associative, commutative algebras H∗(LM) ∼= H∗(Hom(B(A), A)). The product defined on H∗(Hom(B(A), A)) corresponds to the loop product un- der the isomorphism. The paper is organized in the following way. In section 2, we briefly review Chen’s iterated integrals. In section 3, we give a construction of a complex Hom(B(A), A), and discuss its properties. In section 4, we give a proof of theorem 1.1. In section 5, we study the iterated integrals on the free loop space of the non-simply-connected manifolds. In section 6, we describe a relation between the product on Hom(B(A), A) and the Goldman bracket. In this paper, all the homologies have their coefficients in the field of real numbers. Acknowledgement: The author would like to thank Professor Toshitake Kohno much for helpful comments and gentle support. 2 Chen’s iterated integrals We briefly review Chen’s iterated integrals (see [5], or [8]). Let M be a finite dimensional smooth manifold and let LM be the free loop space of M , that is the space of all smooth maps from S1 to M . Let ∆k be the k-simplex {(t1, · · · , tk) ∈ R k | 0 ≤ t1 ≤ · · · ≤ tk ≤ 1}. We have an evaluation map Φk : ∆k × LM →M defined by Φk(t1, · · · , tk; γ) = (γ(t1), · · · , γ(tk)). Then define Pk to be the composition (Λ∗M)⊗k → Λ∗Mk → Λ∗(∆k × LM) → Λ∗−kLM where p∗ is the integration along the fiber of the projection p : ∆k×LM → LM . Given ω1, · · ·ωk ∈ Λ ∗M , the iterated integral ω1 · · ·ωk is a differential form on LM of total degree |ω1| + · · · |ωk| − k, defined by the formula ω1 · · ·ωk = (−1) (k−1)|ω1|+(k−2)|ω2|+···+|ωk−1|+k(k−1)/2Pk(ω1, · · · , ωk). 3 Preliminaries In this section, we give a construction of some complexes. Let A be an arbitrary differential graded algebra in this section. Let A∨ denote the dual of A. The bar complex of A, (B(A), dB), is defined by B(A) = ⊕r≥0 ⊗ r sA, dB(ω1, · · · , ωr) = −(−1) (ω1, · · · , ωi−1, dωi, ωi+1, · · · , ωr) −(−1)εi (ω1, · · · , ωi−1, ωi ∧ ωi+1, ωi+2, · · · , ωr). Here (sA)q = Aq+1 or Aq according as 0 ≤ q or 0 < q, and εi = deg(ω1, · · · , ωi). We denote the totality of degree n elements by B(A)n. The coproductH ∗(B(A)) → H∗(B(A)) ⊗H∗(B(A)) is defined by (ω1, · · · , ωn) 7→ (ω1, · · · , ωi)⊗ (ωi+1, · · · , ωn). Chen proved the following theorem. Theorem 3.1 (Chen [5]). Let M be a simply-connected manifold and H∗(M) be of finite type. Let A be a differential graded algebra of ΛM such that A0 = R and H∗(A) ∼= H∗(ΛM) by the inclusion. Then there is an isomorphism of coalgebras H∗(B(A)) ∼= H ∗(ΩM) given by (ω1, · · · , ωn) 7→ ω1 · · ·ωn. Let F pB(A) be a filtration of B(A) such that F pB(A) = ⊕0≤r≤p ⊗ r sA. Let Hom(B(A), A∨)n = p+q=n Hom(B(A)p, A q∨) and Hom(B(A), A∨) = n Hom(B(A), A ∨)n. Its boundary is defined by δϕ(ω1, · · · , ωr)(ω) = ϕ(ω1, · · · , ωr)(dω) + (−1) |ω|ϕ(dB(ω1, · · · , ωr))(ω) − (−1)|ω|ϕ(ω2, · · · , ωr)(ω ∧ ω1) +(−1)|ω|+εr−1(|ωr|+1)ϕ(ω1, · · · , ωr−1)(ω ∧ ωr). Let us define the subcomplex of Hom(B(A), A∨), Hom(B(A), A∨), according to the Chen’s normalization of the cyclic bar complex (see [4] or [8]). We define Hom(B(A), A∨) to be the set of elements in Hom(B(A), A∨) which satisfy the following equations for any ω, ωi ∈ A >0 and f ∈ A0: −ϕ(· · ·ωi−2, fωi−1, ωi, · · · )(ω) + ϕ(· · · , ωi−1, fωi, ωi+1, · · · )(ω) +ϕ(· · · , ωi−1, df, ωi, · · · )(ω) = 0, 1 ≤ i ≤ r − 1, −ϕ(ω1, · · · , ωr)(fω) + ϕ(fω1, · · · , ωr)(ω) + ϕ(df, ω1, · · · , ωr)(ω) = 0, −ϕ(ω1, · · · , fwr)(ω) + ϕ(ω1, · · · , ωr)(fω) + ϕ(ω1, · · · , ωr, df)(ω) = 0. It can be easily seen that it is isomorphic to the dual of the normalized cyclic bar complex of A: Hom(B(A), A∨) ∼= C(A) Similarly, let Hom(B(A), A)n = p−q=n Hom(B(A)p, A q) and Hom(B(A), A) n Hom(B(A), A)n. Its boundary is defined by δϕ(ω1, · · · , ωr) = (−1)|ϕ|−εrdϕ(ω1, · · · , ωr)− (−1) |ϕ|−εrϕ(dB(ω1, · · · , ωr)) +(−1)|ϕ|−εrω1 ∧ ϕ(ω2, · · · , ωr) −(−1)(|ωr|+1)(|ϕ|+1)ϕ(ω1 · · · , ωr−1) ∧ ωr. We define Hom(B(A), A) to be the set of elements in Hom(B(A), A) which satisfy the following equations for any ω, ωi ∈ A >0 and f ∈ A0: −ϕ(· · ·ωi−2, fωi−1, ωi, · · · ) + ϕ(· · · , ωi−1, fωi, ωi+1, · · · ) +ϕ(· · · , ωi−1, df, ωi, · · · ) = 0, 1 ≤ i ≤ r − 1, −f ∧ ϕ(ω1, · · · , ωr) + ϕ(fω1, · · · , ωr) + ϕ(df, ω1, · · · , ωr) = 0, −ϕ(ω1, · · · , fwr) + ϕ(ω1, · · · , ωr) ∧ f + ϕ(ω1, · · · , ωr, df) = 0. The cup product on Hom(B(A), A) is defined by ϕ1 ∪ ϕ2(ω1, · · · , ωr) 0≤i≤r (−1)|ϕ1|(|ϕ2|+εr−εi)ϕ1(ω1, · · · , ωi) ∧ ϕ2(ωi+1, · · · , ωr). Since δ(ϕ1 ∪ ϕ2) = δϕ1 ∪ ϕ2 + (−1) |ϕ1|ϕ1 ∪ δϕ2, H∗(Hom(B(A), A)) becomes an algebra. This product can be induced on H∗(Hom(B(A), A)). The E1-term of their spectral sequences associated with the filtration F pB(A) can be calculated from the cohomology of A. Proposition 3.2. There is an isomorphism of vector spaces H∗(Hom(F pB(A)/F p−1B(A), A∨)) ∼= Hom(⊗ psH(A), H(A)∨) Proof. Let A be a differential graded subalgebra of A such that A = Ap for p > 1, A = R and A1 = dA0 ⊕A There is an isomorphism of vector spaces Hom(F qB(A)/F q−1B(A), A∨) ∼= Hom(F qB(A)/F q−1B(A), A Since A = R, there is an isomorphism H0(Hom(F qB(A)/F q−1B(A), A )) ∼= Hom(⊗sH(A), H(A) Therefore we obtain the proposition. 4 Proof of Theorem 1.1 We give the proof of theorem 1.1 in this section. There is a differential graded subalgebra of A, A, such that A = R and H(A) ∼= H(A) by the inclusion. Then we obtain the isomorphism of algebras H∗(Hom(B(A), A)) ∼= H∗(Hom(B(A), A)) by proposition 3.2. Therefore it suffices to verify the theorem in the case A0 = R. The following result is due to Chen. Theorem 4.1 (Chen [5]). H∗(LM) ∼= H∗(Hom(B(A), A Proof. We define ψ : C∗(LM)→ Hom(B(A), A ∨) by ψ(σ)(ω1, · · · , ωn)(ω) = π∗ω ∧ ω1 · · ·ωn. Let FpC∗(LM) be a filtration of C∗(LM) such that FpCr(LM) = { σ : ∆ r → LM | π ◦ σ = σ′ ◦ π′ for some σ′ ∈ Cq(M), q ≤ p, π′ : ∆r → ∆q } . Let {Erp,q} be the associated spectral sequence. Define a filtration of Hom(B(A), A FpHom(B(A), A) = {f ∈ Hom(B(A), A ∨) | f(ω1, · · · , ωn)(ω) = 0, ∀ω ∈ A ≥p+1}. It can be easily shown that ψ preserves the filtrations of C∗(LM) and Hom(B(A), A On E2-level, the map ψ : Hp(M)⊗Hq(ΩM)→ Hp(A ∨)⊗Hq(B(A) is given by σ1 ⊗ σ2 7−→ (ω1, · · · , ωn 7→ ω1 · · ·ωn) Theorem 3.1 asserts that this is an isomorphism. Therefore we obtain the theorem. Lemma 4.2. H∗(Hom(B(A), A)) ∼= H∗−d(Hom(B(A), A Proof. We define a chain map P : Hom(B(A), A)→ Hom(B(A), A∨) by P (ϕ)(ω1, · · · , ωn)(ω) = ω ∧ ϕ(ω1, · · · , ωn). Define a filtration of Hom(B(A), A) by FpHom(B(A), A) = {ϕ ∈ Hom(B(A), A) | ϕ(ω1, · · · , ωn) ∈ A ≥d−p}. The map P preserves those filtrations. On E2-level, the map P : Hd−p(A)⊗Hq(B(A) ∨)→ Hp(A ∨)⊗Hq(B(A) is given by ω ⊗ ϕ 7−→ ω ∧ τ This is isomorphic and we obtain the lemma. Proof of theorem 1.1. We can verify that H∗(LM) is isomorphic to H∗(Hom(B(A), A)) as vector spaces by composing the maps in theorem 4.1 and lemma 4.2. We can also verify that there is an isomorphism of associative, commutative algebras. Indeed, the cup product of Hom(B(A), A) on E2-level Hd−p(A)⊗Hq(B(A) ∨)⊗Hd−s(A)⊗Ht(B(A) ∨)→ H2d−p−s(A)⊗Hq+t(B(A) is given by a⊗ g ⊗ b⊗ h 7→ (−1)(d−p+q)(d−s)a ∧ b⊗ g · h, where g · h satisfies g · h(ω1, · · · , ωn) = g(ω1, · · · , ωi)h(ωi+1, · · · , ωn). Then the following theorem asserts that the loop product and the cup product coincide on E2-level. Theorem 4.3 (Cohen-Jones-Yan [6]). Let M be a simply-connected manifold. Then {Erp,q} becomes an algebra and converges to H∗(LM) as algebras. On E2-level, the product µ : Hp(M ;Hq(LM))⊗Hs(M ;Ht(LM))→ Hp+q−d(M ;Hs+t(LM)) is given by µ((a⊗ g)⊗ (b ⊗ h)) = (−1)(d−s)(p+q−d)(a · b)⊗ (gh) where a ∈ Hp(M), b ∈ Hs(M), g ∈ Hq(ΩM), h ∈ Ht(ΩM), a · b is the intersec- tion product and gh is the Pontryagin product. Therefore we obtain the theorem. 5 The conjugacy classes of fundamental groups Let π denote a fundamental group of a smooth manifold M and J denote an augmentation ideal of the group ring of π, Rπ. Chen showed that the completion of the fundamental group with respect to the powers of its augmentation ideal is isomorphic to the dual of the 0-th cohomology of the bar complex of differential forms via iterated integrals [3]: Rπ/Jp ∼= H 0(B(A))∨ where A is a differential graded subalgebra of ΛM such that A0 = R and H∗(A) ∼= H∗(M). Based on this work, we study iterated integrals on the free loop space of the non-simply-connected manifold. Let π̃ denote the set of conjugacy classes of π and J̃p denote pr(Jp) where pr is the projection of Rπ onto Rπ̃. Theorem 5.1. Let M be a smooth manifold and H∗(M) is of finite type. Let A be a differential graded subalgebra of ΛM such that the map Hq(A)→ Hq(ΛM) induced by the inclusion is isomorphic if q = 0, 1 and injective if q = 2. Then there is an isomorphism of vector spaces Rπ̃/J̃p ∼= H0(Hom(B(A), A We give the proof of this theorem in this section. Let ∗ be a fixed point in S1. In this section, let LM be a set of smooth maps from S1 to M which are constant maps near ∗. Let ΩxM be a subspace of LM whose elements send ∗ to x ∈ M . Let Diff(S1, ∗) denote diffeomorphisms of S1 which coincide with identity map near ∗. We define α, β : ∆q → LM to be equivalent by a reparameterization iff there is a smooth map τ : ∆q → Diff(S1, ∗) such that β(ξ)(t) = α(ξ)(τ(t, ξ)), ∀(t, ξ) ∈ S1 ×∆q. Let C∗(LM) be a chain complex having as a basis the totality of equiva- lence classes of smooth simplexes of LM . Let C∗(ΩxM) be a chain complex having as a basis the totality of equivalence classes of smooth simplexes of ΩxM . C∗(ΩxM) becomes a noncommutative associative algebra as follows. The prod- uct of σ1 and σ2 in C∗(ΩxM) is defined to be the path product or 0 according as degσ1+degσ2 ≤ 1 or > 1. The augmentation ε : C∗(ΩxM) → R is given by εσ = 1 or 0 according as degσ = 0 or > 0. Let σ be a smooth simplex of M . Define for each σ Cq(LM)(σ) = { niτi ∈ Cq(LM) | π♯τi = σ}. Cq(LM)(σ) becomes a noncommutative associative algebra. Let ε(σ) denote the augmentation of Cq(LM)(σ), given by niτi 7→ ni. Define a filtration of Cq(LM)(σ) by FpCq(LM) = (kerε) p ⊕ (⊕σ:∆q→M (kerε(σ)) Proposition 5.2. The map ψp : FpCq(LM) → Hom(F p−1B(A), A∨) given by (ω1, · · · , ωp) 7→ π∗ω ∧ ω1 · · ·ωp is well-defined, chain map and FpCq(LM) ⊂ kerψp. Proof. The well-definedness can be verified by the following lemma which can be verified as in proposition 1.5, proposition 4.1.1 [2], and in proposition 1.5.3 Lemma 5.3 (Chen). (1) If α and β ∈ C∗(LM) are equivalent by a reparame- terization, then ω1 · · ·ωn = β ω1 · · ·ωn. (2) If τ1, τ2 ∈ Cq(LM)(σ), then (τ1 · τ2) ω1 · · ·ωn = ω1 · · ·ωi ∧ τ ωi+1 · · ·ωn. (3) If f ∈ Λ0M , then for any i ω1 · · · fωi−1 · · ·ωn + ω1 · · · fωi · · ·ωn + ω1 · · ·ωi−1df ωi · · ·ωn = 0. To verify FpCq(LM) ⊂ kerψp, it suffices to show (kerε(σ)) p ⊂ kerψp. Let s denote the section of π, which sends points of M to the constant map. Take (σ1 − s♯σ) · (σ2 − s♯σ) · · · · ·(σp − s♯σ) ∈ (kerε(σ)) p, where σ ∈ Cq(M) and σi ∈ Cq(LM)(σ). Then (σ1 − s♯σ) · (σ2 − sσ) · · · · · (σp − s♯σ) π∗ω ∧ ω1 · · ·ωp−1 σ∗ω ∧ (σ1 − s♯σ) ω1 · · · (σk − s♯σ) ∗1 · · · ∧ (σp − s♯σ) Therefore we obtain the proposition. Let C∗(M,x) denote a set of smooth simplexes ofM neighborhood of whose vertices are at x in M . We define C ⊗ sC⊗p = C∗(M,x)⊗ sC∗(M,x) Here (sC∗(M,x))q = Cq+1(M,x) or 0 according as q > 0 or q ≤ 0. Its boundary is given by the sum of the boundary on each complex. Let us construct a chain map Φ : C ⊗ sC⊗p → FpC∗(LM)/Fp+1C∗(LM) considering the following three cases: case 1: If (σ1, · · · , σp) ∈ sC(M,x)⊗p , then Φ : (σ1, · · · , σp) 7−→ (σ1 − x) · (σ2 − x) · · · · · (σp − x) where x is regarded as a constant map. case 2: If (σ1, · · · , σp) ∈ sC(M,x)⊗p , then Φ : (σ1, · · · , σp) 7−→ (σ1 − x) · (σ2 − x) · · ·σi · · · (σp − x) where σi : ∆ 1 ∋ ξ 7→ σi(ξ)(t) ∈ ΩxM is σi(ξ)(t) σi((1 − ξ)((1 − t)v0 + tv2) + ξ(1 − 2t)v0 + 2ξtv1), if 0 ≤ t ≤ 1/2 σi((1 − ξ)((1 − t)v0 + tv2) + ξ(2 − 2t)v1 + ξ(2t− 1)v2), if 1/2 ≤ t ≤ 1 Here v0, v1, v2 are the vertices of the standard simplex ∆ case 3: If (γ, σ1, · · · , σp) ∈ C1(M,x)⊗ sC(M,x)⊗p , then Φ : (γ, σ1, · · · , σp) 7−→ γ t (σ1 − x)γt · · · γ t (σp − x)γt where γt : [0, 1] ∋ s 7→ γ(st) ∈ M , t ∈ ∆ Lemma 5.4. The following diagram commutes: C ⊗ sC⊗p −−−−→ FpC1(LM)/Fp+1C1(LM) C ⊗ sC⊗p −−−−→ FpC0(LM)/Fp+1C0(LM) Proof. For case 2, ∂′Φ(σ1, · · · , σp)− Φ∂(σ1, · · · , σp) = (σ1 − x) · · · (σ i · σ i − σ i − σ i + σ i − σ i + x) · · · (σp − x) = (σ1 − x) · · · (σ i − x) · (σ i − x) · · · (σp − x) ∈ Fp+1C0(LM) where σ i , σ i , σ i are the faces of σi. For case 3, ∂′Φ(γ, σ1, · · · , σp)− Φ∂ ′(γ, σ1, · · · , σp) = γ−1 · (σ1 − x) · γ · · · γ −1 · (σp − x) · γ − (σ1 − x) · · · (σp − x) ∈ Fp+1C0(LM). Therefore we obtain the lemma. Proposition 5.2 gives the map Hq(FpC(LM)/Fp−1C(LM))→ Hq(Hom(F pB(A)/F p−1B(A), A∨)). Lemma 5.5. For q = 0, the following map is isomorphic: H0(FpC(LM)/Fp+1C(LM)) ∼= H0(Hom(F pB(A)/F p−1B(A), A∨)). Proof. We obtain the following surjection by lemma 5.4. Φ : H0(C ⊗ sC ⊗p) ։ H0(FpC(LM)/Fp+1C(LM)). Composing with the isomorphism ⊗pH1(M) ∼= H0(C ⊗ sC ⊗p), the map ⊗pH1(M) ։ H0(FpC(LM)/Fp+1C(LM))→ Hom(⊗ pH1(A),R) is given by (σ1, · · · , σn) 7→ (ω1, · · · , ωp) 7→ ω1 · · · This is isomorphic and we obtain the lemma. Lemma 5.6. For q = 1, the following map surjective: H1(FpC(LM)/Fp+1C(LM)) ։ H1(Hom(F pB(A)/F p−1B(A), A∨)). Proof. It suffices to show that the following map obtained by lemma 5.4 is surjective. ker∂ → H1(FpC(LM)/Fp+1C(LM))→ Hom(⊗ psH(A), H(A)∨)1 If (γ, σ1, · · · , σp) ∈ ker∂ ∩ C0(M,x)⊗ sC(M,x)⊗p , then (γ, σ1, · · · , σp) 7→ (ω1, · · · , ωp) 7→ ω1 · · · ωp, if deg ω = 0 0, otherwise through the above map. If (γ, σ1, · · · , σp) ∈ ker∂ ∩ C1(M,x)⊗ sC(M,x)⊗p , then (γ, σ1, · · · , σp) 7→ (ω1, · · · , ωp) 7→ ω1 · · · when deg ω = 1. Then we can verify the surjectivity and obtain the lemma. Proof of theorem 1.1. Consider the spectral sequences ofC(LM)/FpC(LM) and Hom(F p−1B(A), A∨) associated with FqC(LM) and Hom(F qB(A), A∨), re- spectively. Lemma 5.5 asserts that ψp is isomorphic on E1-level at degree 0: H0(FqC(LM)/Fq+1C(LM)) ∼= H0(Hom(F qB(A)/F q−1B(A), A∨)). Lemma 5.6 asserts that ψp is surjective on E1-level at degree 1: H1(FqC(LM)/Fq+1C(LM)) ։ H1(Hom(F qB(A)/F q−1B(A), A∨)). Then there is an isomorphism on Er-level at degree 0 for r ≥ 1. We have Rπ̃/J̃p ∼= H0(C(LM)/FpC(LM)) ∼= H0(Hom(F pB(A), A∨)). Therefore we obtain the theorem. 6 The Goldman bracket This section is devoted to the proof of the following theorem. Theorem 6.1. Let M be a compact closed oriented surface with genus g. Then the Goldman bracket induces a Lie algebra structure on lim Rπ̃/J̃pand there is an isomorphism of Lie algebras Rπ̃/J̃p ∼= H0(Hom(B(H ∗(M)), H∗(M)∨)). Goldman showed that the vector space spanned by the free homotopy classes of closed curves on a closed oriented surface has a Lie algebra structure [9]. This work led Chas and Sullivan to the string topology. We would verify that this structure makes lim Rπ̃/J̃p a Lie algebra. On the other hand, we can construct a bracket on H0(Hom(B(H ∗(M)), H∗(M)∨)) by the cup product defined in section 3 and the Connes’s operator. Here we regard H∗(M) as a differential graded algebra with a trivial differential. Theorem 6.1 asserts that those two Lie algebras are isomorphic. First we describe a relation between this bracket and the augmentation ideal of the group ring of the surface group to induce a Lie algebra structure on Rπ̃/J̃p. Then we construct a bracket on H0(Hom(B(A), A ∨)) and verify the isomorphism of Lie algebras Rπ̃/J̃p ∼= H0(Hom(B(A), A Finally we verify the isomorphism H0(Hom(B(A), A ∨)) ∼= H0(Hom(B(H ∗(M)), H∗(M)∨). The following proposition makes lim Rπ̃/J̃p a Lie algebra. Proposition 6.2. (1) If p ≥ 1 and q ≥ 2, then [J̃p, J̃q] ⊂ J̃p+q−2. (2) If p ≥ 2 , then [J̃p,Rπ̃] ⊂ J̃p−1. Proof. We give a proof of (1). Take (σ1−x) · · · (σp−x) ∈ J̃p, (τ1−y) · · · (τq−y) ∈ J̃q, where σi ∈ ΩxM and τi ∈ ΩyM . Assume that all curves are immersions and σi τj intersect transversally for any i, j. Let {σi♯τj} denote the set of intersection points of σi and τj . Also assume that all the intersection points are distinct i.e. {σi♯τj} ∩ {σk♯τl} = φ if i 6= k or j 6= l. Then, [σ, τ ] = s∈σi♯τj {ε(s;σi, τj)γs,x · (σi − x) · · · (σp − x)(σ1 − x) · · · ·(σi−1 − x) · γ s,x · ·γs,y · (τj − y) · · · (τq − y)(τ1 − y) · · · (τj−1 − y) · γ −γs,x · (σi+1 − x) · · · (σp − x)(σ1 − x) · · · (σi−1 − x) · γ s,x · ·γs,y · (τj+1 − y) · · · (τq − y)(τ1 − y) · · · (τj−1 − y) · γ ∈ J̃p+q−2. Here γs,x is a path from s to x along σi and γs,y is a path from s to y along τj . The proof of (2) can be verified in the same way. Let A be a differential graded subalgebra of ΛM such thatH∗(A) ∼= H∗(ΛM) by the inclusion. Proposition 6.3. There is an isomorphism of vector spaces H∗(Hom(F pB(A), A)) ∼= H∗−2(Hom(F pB(A), A∨)). Proof. We define P : H∗−2(Hom(F pB(A), A))→ H∗(Hom(F pB(A), A∨)) by P (ϕ)(ω1, · · · , ωp)(ω) = ω ∧ ϕ(ω1, · · · , ωp). This map preserves the filtrations. On E1-level, the map Hom(⊗qH(A), H(A))→ Hom(⊗qH(A), H(A)∨) is isomorphic. Therefore we obtain the proposition. Now we construct a bracket on H0(Hom(B(A), A ∨)). First, we define the Connes’s operator B : H∗(Hom(F pB(A), A∨)) → H∗+1(Hom(F p−1B(A), A∨)) B(ϕ)(ω1, · · · , ωp−1)(ω) 0≤k≤p−1 (−1)(εk+1)(εp−1−εk)ϕ(ωk+1, · · · , ωp−1, ω, ω1, · · ·ωk)(1). Composing these maps and the cup product, we can define a bracket on H0(Hom(F pB(A), A∨)) by [ϕ1, ϕ2] = −P (P −1Bϕ1 ∪ P −1Bϕ2) ∈ H0(Hom(F p−1B(A), A∨)). Take 2g closed 1-forms on M , α1, · · · , αg, β1, · · ·βg, such that αi ∧ βj = δij . Let {E p.q} denote the spectral sequence of Hom(B(A), A ∨) associated with F pB(A). Notice that the cyclic group Z/pZ acts on E ∼= Hom(⊗pH1(A),R) ιϕ(ω1, · · · , ωp) = ϕ(ω2, · · · , ωp, ω1) where ι is a generator of Z/pZ. The bracket [ , ] : E p,−p⊗E q,−q → E p+q−2,−p−q+2 [ϕ1, ϕ2](ω1, · · · , ωp+q−2) i,m,n ιmϕ1(αi, ω1, · · · , ωp−1)̺ nϕ2(βi, ωp, · · · , ωp+q−2) −ιmϕ1(βi, ω1, · · · , ωp−1)̺ nϕ2(αi, ωp, · · · , ωp+q−2) where ι and ̺ are generators of Z/pZ and Z/qZ, respectively. Proposition 6.4. The following diagram commutes for p, q ≥ 1: J̃p/ ˜Jp+1 ⊗ J̃q/ ˜Jq+1 −−−−→ E ∞ ⊗ E [ , ] [ , ] J̃p+q−2/J̃p+q−1 −−−−→ E p+q−2,−p−q+2 Proof. Take σ = (σ1 − x) · · · (σp − x) ∈ FpC0(LM), τ = (τ1 − y) · · · (τq − y) ∈ FqC0(LM). Take 2g curves in M , ai, bi, as in Figure 1. Assume that σi and τj , ak, or bk, intersect transversally for any i, j, k. Also assume that τj and ak, or bk, intersect transversally for any j, k. Assume that all the intersection points are distinct. Then for any i, j, k, we can take each tubular neighborhoods of ai and bi so that it does not include some neighborhoods of intersection points of σj and τk. We fix such neighborhoods of intersection points and denote them by Up for each p. We can also take a tubular neighborhood of the diagonal map from M to M×M outside those neighborhoods of intersection points of σi and τj for any i, j i.e. S1 \ ∪pσ i (Up) S1 \ ∪pτ j (Up) = φ, ∀i, j. Here N∆ denotes the tubular neighborhood of the diagonal map. Thom class Φ of this tubular neighborhood satisfies Φ = −ε(p;σi, τj), where ε(p;σi, τj) is the intersecion number of σi and τj at p. Fig. 1 ・ ・ ・ Define e♯ : C0(LM) → C1(LM) by e♯γ(ξ)(t) = γ(ξ + t). Let ωk, 1 ≤ k ≤ n, be differential forms on M which has its support inside the tubular neighborhoods of ai and bi. Then [σi,τj] ω1 · · ·ωn p∈σi♯τj,k ε(p;σi, τj) (σi)p ω1 · · ·ωk (τj)p ωk+1 · · ·ωn p∈σi♯τj ,k ×τj | (σi)p ω1 · · ·ωk (τj)p ωk+1 · · ·ωn e♯σi×e♯τj π∗Φ ∧ p∗1 ω1 · · ·ωk ∧ p ωk+1 · · ·ωn. Here p1, p2 : LM×LM → LM are the projections. The last equality is obtained by the following lemma. Lemma 6.5. If p ∈ σi♯τj and p ′ ∈ Up ∩ σi([0, 1]), then (σi)p ω1 · · ·ωn = (σi)p′ ω1 · · ·ωn. Proof. F Let γ be the curve from p to p′ along σi inside Up. If γ and σ are in the same direction, then (σi)p′ ω1 · · ·ωn = γ·(σi)p′ ω1 · · ·ωn = (σ)p·γ ω1 · · ·ωn ω1 · · ·ωn. We can also verify the case where γ is in the direction opposite to σ in the same We have the equality e♯σ×e♯τ π∗Φ ∧ p∗1 ω1 · · ·ωk ∧ p ωk+1 · · ·ωp+q−2 e♯σ×e♯τ − p∗1(α1 ∧ β1)− p 2(α1 ∧ β1) + p 1αj ∧ p 2βj − p 1βj ∧ p ω1 · · ·ωk ∧ p ωk+1 · · ·ωp+q−2 In fact, if η ∈ Λ(M ×M) then (−1)|η|+1 e♯σ×e♯τ π∗dη ∧ p∗1 ω1 · · ·ωk ∧ p ωk+1 · · ·ωp+q−2 e♯σ×e♯τ π∗η ∧ d ω1 · · ·ωk ∧ p ωk+1 · · ·ωp+q−2 +(e♯σ) ω1 · · ·ωk (e♯τ) ωk+1 · · ·ωj ∧ ωj+1 · · ·ωp+q−2 The last equality is obtained by the following lemma. Lemma 6.6. If σ ∈ FpC0(LM), then (e♯σ) ω1 · · ·ωp−2 = 0. Proof. It suffices to show the case σ = (τ1 − x) · · · (τp − x) where x ∈ M and τi ∈ ΩxM . We define τ̄i ∈ ΩxM by τ̄i(t) = τi(pt), if (i − 1)/p ≤ t ≤ i/p 0, otherwise. Let σ̄ denote (τ̄1 − x) · · · (τ̄p − x). It can be shown that e♯σ̄ restricted on [(i − 1)/p, i/p] is contained in Fp−1C1(LM) for any i. Therefore (e♯σ) ω1 · · ·ωp−2 = (e♯σ̄) ω1 · · ·ωp−2 = 0. Jones, Geztler, and Petrack describes the map e♯ in terms of iterated inte- grals by the following theorem. Theorem 6.7 (Geztler-Jones-Petrack [8]). If σ ∈ C0(LM) and ω, ωi ∈ Λ 1 ≤ i ≤ p, then π∗ω ∧ ω1 · · ·ωp = ωk · · ·ωpωω1 · · ·ωk−1. This theorem asserts the equality e♯σ×e♯τ − p∗1(α1 ∧ β1)− p 2(α1 ∧ β1) + p 1αj ∧ p 2βj − p 1βj ∧ p ω1 · · ·ωk ∧ p ωk+1 · · ·ωn j,k,l ωk+1 · · ·ωp−1αjω1 · · ·ωk ωl+1 · · ·ωp+q−2βjωp · · ·ωl ωk+1 · · ·ωp−1βjω1 · · ·ωk ωl+1 · · ·ωp+q−2αjωp · · ·ωl Finally we obtain the equality [σ,τ ] ω1 · · ·ωp+q−2 j,k,l ωk+1 · · ·ωp−1αjω1 · · ·ωk ωl+1 · · ·ωp+q−2βjωp · · ·ωl ωk+1 · · ·ωp−1βjω1 · · ·ωk ωl+1 · · ·ωp+q−2αjωp · · ·ωl Since we can take ωi ∈ H 1(M), 1 ≤ i ≤ p + q − 2, so that their support are inside the tubular neighborhoods of aj and bj, we obtain the proposition. Proof of theorem 6.1. We obtain the following isomorphism of Lie algebras by proposition 6.4. Rπ̃/J̃p ∼= H0(Hom(B(A), A To obtain the isomorphism of Lie algebras H0(Hom(B(A), A ∨) ∼= H0(Hom(B(H ∗(M)), H∗(M)∨), we introduce the following lemma, which asserts the formality of the compact Kähler manifolds. Lemma 6.8 (ddcLemma, Deligne-Griffiths-Morgan-Sullivan [7]). Let X be a compact Kähler manifold and dc = J−1dJ where J gives the complex structure in the cotangent bundle. If α is a differential form on X such that dα = 0 and dcα = 0, and such that α = dγ, then α = ddcβ for some β. Cor. There are quasi-isomorphisms of differential graded algebras (ΛX, d)← (kerdc, d)→ (H∗dc(X), 0). Notice that a closed oriented surface endowed with a complex structure become a Kähler manifolds for the dimensional reason. Therefore the following lemma completes the proof of the theorem. Lemma 6.9. If f : A1 → A2 is a quasi-isomorphism of differential graded algebras, then the map induced by f H0(Hom(B(A1), A 1 )→ H0(Hom(B(A2), A is an isomorphism. Proof. It suffices to verify that the map induced by f f : H0(Hom(F pB(A1), A 1 )→ H0(Hom(F pB(A2), A is an isomorphism for any p. On E1-level, the map induced by f Hom(⊗sH(A1), H(A1) ∨)→ Hom(⊗sH(A2), H(A2) is an isomorphism because f is quasi-isomorphism. Therefore we obtain the lemma. Therefore we obtain the theorem. References [1] M. Chas and D. Sullivan, String topology, preprint, 1999, http://arXiv.org /abs/math.GT/9911159. [2] K.T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97(1973), 217-246. [3] K.T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83-98. [4] K.T. Chen, Reduced bar constructions on de Rham complexes, in:A.Haller and M.Tierney (eds), (Algebra, topology and category theory, 1977, pp. 19- [5] K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no.5, 831-879. [6] R.L. Cohen, J.D.S. Jones and J. Yan, The loop homology algebra of spheres and projective spaces, Categorical Decomposition Techniques in Algebraic Topology (Isle of Skye, 2001), Progr. Math., vol. 215. Birkhäuser, Basel, 2004, pp.77-92. [7] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274. [8] E. Getzler, J.D.S. Jones and S. Petrack Differential forms on loop spaces and the cyclic bar complex, Topology 30 (1991), no.3, 339-371. http://arXiv.org [9] W.M. Goldman, Invariant functions on Lie groups and Hamlitonian flows of surface group representation, Invent. Math. 85 (1986), no.2, 263-302. [10] S.A. Merkulov, De Rham Model for String Topology, International Mathe- matics Research Notices 55 (2004), 2955-2981. Introduction Chen's iterated integrals Preliminaries Proof of Theorem 1.1 The conjugacy classes of fundamental groups The Goldman bracket
In this article we discuss a relation between the string topology and differential forms based on the theory of Chen's iterated integrals and the cyclic bar complex.
Introduction The purpose of this paper is to describe string topology from the viewpoint of Chen’s iterated integrals. Let M be a compact closed oriented d-manifold and LM be the free loop space ofM , the set of unbased smooth maps from S1 toM . Let H∗(LM) be the homology of the free loop space shifted by the dimension of the manifold i.e. H∗(LM) = H∗+d(LM). Chas and Sullivan found the product on H∗(LM) which they called loop product [1]: Hp(LM)⊗Hq(LM)→ Hp+q(LM). They showed that this product makes H∗(LM) an associative, commutative algebra. Merkulov constructed a model for this product based on the theory of iter- ated integrals, especially of the formal power series connection [10]. He showed that there is an isomorphism of algebras H∗(LM) ∼= H∗(ΛM ⊗ R where ΛM is the de Rham differential graded algebra of M and R the formal completion of the free graded associative algebra generated by some noncommutative indeterminates. On the other hand, Chen showed that the cohomology of the free loop space of the simply-connected manifold is isomorphic to the cohomology of the cyclic bar complex of differential forms via Chen’s iterated integrals (see [5] or [8]): H∗(LM) ∼= H ∗(C(ΛM)). In this paper, we construct a model for the loop product based on the the- ory of the cyclic bar complex. We define a complex Hom(B(ΛM),ΛM) and its subcomplex Hom(B(ΛM),ΛM) so that the Poincaré duality induces the isomorphism of vector spaces H∗(Hom(C(ΛM),R)) ∼= H∗−d(Hom(B(ΛM),ΛM)). We can define a product on Hom(B(ΛM),ΛM) which realizes the loop product. http://arxiv.org/abs/0704.0014v1 Theorem 1.1. Let M be a compact closed oriented simply-connected manifold. Assume that H∗(M) is of finite type. Let A be a differential graded subalge- bra of ΛM such that H∗(A) ∼= H∗(ΛM) by the inclusion. Then there is an isomorphism of associative, commutative algebras H∗(LM) ∼= H∗(Hom(B(A), A)). The product defined on H∗(Hom(B(A), A)) corresponds to the loop product un- der the isomorphism. The paper is organized in the following way. In section 2, we briefly review Chen’s iterated integrals. In section 3, we give a construction of a complex Hom(B(A), A), and discuss its properties. In section 4, we give a proof of theorem 1.1. In section 5, we study the iterated integrals on the free loop space of the non-simply-connected manifolds. In section 6, we describe a relation between the product on Hom(B(A), A) and the Goldman bracket. In this paper, all the homologies have their coefficients in the field of real numbers. Acknowledgement: The author would like to thank Professor Toshitake Kohno much for helpful comments and gentle support. 2 Chen’s iterated integrals We briefly review Chen’s iterated integrals (see [5], or [8]). Let M be a finite dimensional smooth manifold and let LM be the free loop space of M , that is the space of all smooth maps from S1 to M . Let ∆k be the k-simplex {(t1, · · · , tk) ∈ R k | 0 ≤ t1 ≤ · · · ≤ tk ≤ 1}. We have an evaluation map Φk : ∆k × LM →M defined by Φk(t1, · · · , tk; γ) = (γ(t1), · · · , γ(tk)). Then define Pk to be the composition (Λ∗M)⊗k → Λ∗Mk → Λ∗(∆k × LM) → Λ∗−kLM where p∗ is the integration along the fiber of the projection p : ∆k×LM → LM . Given ω1, · · ·ωk ∈ Λ ∗M , the iterated integral ω1 · · ·ωk is a differential form on LM of total degree |ω1| + · · · |ωk| − k, defined by the formula ω1 · · ·ωk = (−1) (k−1)|ω1|+(k−2)|ω2|+···+|ωk−1|+k(k−1)/2Pk(ω1, · · · , ωk). 3 Preliminaries In this section, we give a construction of some complexes. Let A be an arbitrary differential graded algebra in this section. Let A∨ denote the dual of A. The bar complex of A, (B(A), dB), is defined by B(A) = ⊕r≥0 ⊗ r sA, dB(ω1, · · · , ωr) = −(−1) (ω1, · · · , ωi−1, dωi, ωi+1, · · · , ωr) −(−1)εi (ω1, · · · , ωi−1, ωi ∧ ωi+1, ωi+2, · · · , ωr). Here (sA)q = Aq+1 or Aq according as 0 ≤ q or 0 < q, and εi = deg(ω1, · · · , ωi). We denote the totality of degree n elements by B(A)n. The coproductH ∗(B(A)) → H∗(B(A)) ⊗H∗(B(A)) is defined by (ω1, · · · , ωn) 7→ (ω1, · · · , ωi)⊗ (ωi+1, · · · , ωn). Chen proved the following theorem. Theorem 3.1 (Chen [5]). Let M be a simply-connected manifold and H∗(M) be of finite type. Let A be a differential graded algebra of ΛM such that A0 = R and H∗(A) ∼= H∗(ΛM) by the inclusion. Then there is an isomorphism of coalgebras H∗(B(A)) ∼= H ∗(ΩM) given by (ω1, · · · , ωn) 7→ ω1 · · ·ωn. Let F pB(A) be a filtration of B(A) such that F pB(A) = ⊕0≤r≤p ⊗ r sA. Let Hom(B(A), A∨)n = p+q=n Hom(B(A)p, A q∨) and Hom(B(A), A∨) = n Hom(B(A), A ∨)n. Its boundary is defined by δϕ(ω1, · · · , ωr)(ω) = ϕ(ω1, · · · , ωr)(dω) + (−1) |ω|ϕ(dB(ω1, · · · , ωr))(ω) − (−1)|ω|ϕ(ω2, · · · , ωr)(ω ∧ ω1) +(−1)|ω|+εr−1(|ωr|+1)ϕ(ω1, · · · , ωr−1)(ω ∧ ωr). Let us define the subcomplex of Hom(B(A), A∨), Hom(B(A), A∨), according to the Chen’s normalization of the cyclic bar complex (see [4] or [8]). We define Hom(B(A), A∨) to be the set of elements in Hom(B(A), A∨) which satisfy the following equations for any ω, ωi ∈ A >0 and f ∈ A0: −ϕ(· · ·ωi−2, fωi−1, ωi, · · · )(ω) + ϕ(· · · , ωi−1, fωi, ωi+1, · · · )(ω) +ϕ(· · · , ωi−1, df, ωi, · · · )(ω) = 0, 1 ≤ i ≤ r − 1, −ϕ(ω1, · · · , ωr)(fω) + ϕ(fω1, · · · , ωr)(ω) + ϕ(df, ω1, · · · , ωr)(ω) = 0, −ϕ(ω1, · · · , fwr)(ω) + ϕ(ω1, · · · , ωr)(fω) + ϕ(ω1, · · · , ωr, df)(ω) = 0. It can be easily seen that it is isomorphic to the dual of the normalized cyclic bar complex of A: Hom(B(A), A∨) ∼= C(A) Similarly, let Hom(B(A), A)n = p−q=n Hom(B(A)p, A q) and Hom(B(A), A) n Hom(B(A), A)n. Its boundary is defined by δϕ(ω1, · · · , ωr) = (−1)|ϕ|−εrdϕ(ω1, · · · , ωr)− (−1) |ϕ|−εrϕ(dB(ω1, · · · , ωr)) +(−1)|ϕ|−εrω1 ∧ ϕ(ω2, · · · , ωr) −(−1)(|ωr|+1)(|ϕ|+1)ϕ(ω1 · · · , ωr−1) ∧ ωr. We define Hom(B(A), A) to be the set of elements in Hom(B(A), A) which satisfy the following equations for any ω, ωi ∈ A >0 and f ∈ A0: −ϕ(· · ·ωi−2, fωi−1, ωi, · · · ) + ϕ(· · · , ωi−1, fωi, ωi+1, · · · ) +ϕ(· · · , ωi−1, df, ωi, · · · ) = 0, 1 ≤ i ≤ r − 1, −f ∧ ϕ(ω1, · · · , ωr) + ϕ(fω1, · · · , ωr) + ϕ(df, ω1, · · · , ωr) = 0, −ϕ(ω1, · · · , fwr) + ϕ(ω1, · · · , ωr) ∧ f + ϕ(ω1, · · · , ωr, df) = 0. The cup product on Hom(B(A), A) is defined by ϕ1 ∪ ϕ2(ω1, · · · , ωr) 0≤i≤r (−1)|ϕ1|(|ϕ2|+εr−εi)ϕ1(ω1, · · · , ωi) ∧ ϕ2(ωi+1, · · · , ωr). Since δ(ϕ1 ∪ ϕ2) = δϕ1 ∪ ϕ2 + (−1) |ϕ1|ϕ1 ∪ δϕ2, H∗(Hom(B(A), A)) becomes an algebra. This product can be induced on H∗(Hom(B(A), A)). The E1-term of their spectral sequences associated with the filtration F pB(A) can be calculated from the cohomology of A. Proposition 3.2. There is an isomorphism of vector spaces H∗(Hom(F pB(A)/F p−1B(A), A∨)) ∼= Hom(⊗ psH(A), H(A)∨) Proof. Let A be a differential graded subalgebra of A such that A = Ap for p > 1, A = R and A1 = dA0 ⊕A There is an isomorphism of vector spaces Hom(F qB(A)/F q−1B(A), A∨) ∼= Hom(F qB(A)/F q−1B(A), A Since A = R, there is an isomorphism H0(Hom(F qB(A)/F q−1B(A), A )) ∼= Hom(⊗sH(A), H(A) Therefore we obtain the proposition. 4 Proof of Theorem 1.1 We give the proof of theorem 1.1 in this section. There is a differential graded subalgebra of A, A, such that A = R and H(A) ∼= H(A) by the inclusion. Then we obtain the isomorphism of algebras H∗(Hom(B(A), A)) ∼= H∗(Hom(B(A), A)) by proposition 3.2. Therefore it suffices to verify the theorem in the case A0 = R. The following result is due to Chen. Theorem 4.1 (Chen [5]). H∗(LM) ∼= H∗(Hom(B(A), A Proof. We define ψ : C∗(LM)→ Hom(B(A), A ∨) by ψ(σ)(ω1, · · · , ωn)(ω) = π∗ω ∧ ω1 · · ·ωn. Let FpC∗(LM) be a filtration of C∗(LM) such that FpCr(LM) = { σ : ∆ r → LM | π ◦ σ = σ′ ◦ π′ for some σ′ ∈ Cq(M), q ≤ p, π′ : ∆r → ∆q } . Let {Erp,q} be the associated spectral sequence. Define a filtration of Hom(B(A), A FpHom(B(A), A) = {f ∈ Hom(B(A), A ∨) | f(ω1, · · · , ωn)(ω) = 0, ∀ω ∈ A ≥p+1}. It can be easily shown that ψ preserves the filtrations of C∗(LM) and Hom(B(A), A On E2-level, the map ψ : Hp(M)⊗Hq(ΩM)→ Hp(A ∨)⊗Hq(B(A) is given by σ1 ⊗ σ2 7−→ (ω1, · · · , ωn 7→ ω1 · · ·ωn) Theorem 3.1 asserts that this is an isomorphism. Therefore we obtain the theorem. Lemma 4.2. H∗(Hom(B(A), A)) ∼= H∗−d(Hom(B(A), A Proof. We define a chain map P : Hom(B(A), A)→ Hom(B(A), A∨) by P (ϕ)(ω1, · · · , ωn)(ω) = ω ∧ ϕ(ω1, · · · , ωn). Define a filtration of Hom(B(A), A) by FpHom(B(A), A) = {ϕ ∈ Hom(B(A), A) | ϕ(ω1, · · · , ωn) ∈ A ≥d−p}. The map P preserves those filtrations. On E2-level, the map P : Hd−p(A)⊗Hq(B(A) ∨)→ Hp(A ∨)⊗Hq(B(A) is given by ω ⊗ ϕ 7−→ ω ∧ τ This is isomorphic and we obtain the lemma. Proof of theorem 1.1. We can verify that H∗(LM) is isomorphic to H∗(Hom(B(A), A)) as vector spaces by composing the maps in theorem 4.1 and lemma 4.2. We can also verify that there is an isomorphism of associative, commutative algebras. Indeed, the cup product of Hom(B(A), A) on E2-level Hd−p(A)⊗Hq(B(A) ∨)⊗Hd−s(A)⊗Ht(B(A) ∨)→ H2d−p−s(A)⊗Hq+t(B(A) is given by a⊗ g ⊗ b⊗ h 7→ (−1)(d−p+q)(d−s)a ∧ b⊗ g · h, where g · h satisfies g · h(ω1, · · · , ωn) = g(ω1, · · · , ωi)h(ωi+1, · · · , ωn). Then the following theorem asserts that the loop product and the cup product coincide on E2-level. Theorem 4.3 (Cohen-Jones-Yan [6]). Let M be a simply-connected manifold. Then {Erp,q} becomes an algebra and converges to H∗(LM) as algebras. On E2-level, the product µ : Hp(M ;Hq(LM))⊗Hs(M ;Ht(LM))→ Hp+q−d(M ;Hs+t(LM)) is given by µ((a⊗ g)⊗ (b ⊗ h)) = (−1)(d−s)(p+q−d)(a · b)⊗ (gh) where a ∈ Hp(M), b ∈ Hs(M), g ∈ Hq(ΩM), h ∈ Ht(ΩM), a · b is the intersec- tion product and gh is the Pontryagin product. Therefore we obtain the theorem. 5 The conjugacy classes of fundamental groups Let π denote a fundamental group of a smooth manifold M and J denote an augmentation ideal of the group ring of π, Rπ. Chen showed that the completion of the fundamental group with respect to the powers of its augmentation ideal is isomorphic to the dual of the 0-th cohomology of the bar complex of differential forms via iterated integrals [3]: Rπ/Jp ∼= H 0(B(A))∨ where A is a differential graded subalgebra of ΛM such that A0 = R and H∗(A) ∼= H∗(M). Based on this work, we study iterated integrals on the free loop space of the non-simply-connected manifold. Let π̃ denote the set of conjugacy classes of π and J̃p denote pr(Jp) where pr is the projection of Rπ onto Rπ̃. Theorem 5.1. Let M be a smooth manifold and H∗(M) is of finite type. Let A be a differential graded subalgebra of ΛM such that the map Hq(A)→ Hq(ΛM) induced by the inclusion is isomorphic if q = 0, 1 and injective if q = 2. Then there is an isomorphism of vector spaces Rπ̃/J̃p ∼= H0(Hom(B(A), A We give the proof of this theorem in this section. Let ∗ be a fixed point in S1. In this section, let LM be a set of smooth maps from S1 to M which are constant maps near ∗. Let ΩxM be a subspace of LM whose elements send ∗ to x ∈ M . Let Diff(S1, ∗) denote diffeomorphisms of S1 which coincide with identity map near ∗. We define α, β : ∆q → LM to be equivalent by a reparameterization iff there is a smooth map τ : ∆q → Diff(S1, ∗) such that β(ξ)(t) = α(ξ)(τ(t, ξ)), ∀(t, ξ) ∈ S1 ×∆q. Let C∗(LM) be a chain complex having as a basis the totality of equiva- lence classes of smooth simplexes of LM . Let C∗(ΩxM) be a chain complex having as a basis the totality of equivalence classes of smooth simplexes of ΩxM . C∗(ΩxM) becomes a noncommutative associative algebra as follows. The prod- uct of σ1 and σ2 in C∗(ΩxM) is defined to be the path product or 0 according as degσ1+degσ2 ≤ 1 or > 1. The augmentation ε : C∗(ΩxM) → R is given by εσ = 1 or 0 according as degσ = 0 or > 0. Let σ be a smooth simplex of M . Define for each σ Cq(LM)(σ) = { niτi ∈ Cq(LM) | π♯τi = σ}. Cq(LM)(σ) becomes a noncommutative associative algebra. Let ε(σ) denote the augmentation of Cq(LM)(σ), given by niτi 7→ ni. Define a filtration of Cq(LM)(σ) by FpCq(LM) = (kerε) p ⊕ (⊕σ:∆q→M (kerε(σ)) Proposition 5.2. The map ψp : FpCq(LM) → Hom(F p−1B(A), A∨) given by (ω1, · · · , ωp) 7→ π∗ω ∧ ω1 · · ·ωp is well-defined, chain map and FpCq(LM) ⊂ kerψp. Proof. The well-definedness can be verified by the following lemma which can be verified as in proposition 1.5, proposition 4.1.1 [2], and in proposition 1.5.3 Lemma 5.3 (Chen). (1) If α and β ∈ C∗(LM) are equivalent by a reparame- terization, then ω1 · · ·ωn = β ω1 · · ·ωn. (2) If τ1, τ2 ∈ Cq(LM)(σ), then (τ1 · τ2) ω1 · · ·ωn = ω1 · · ·ωi ∧ τ ωi+1 · · ·ωn. (3) If f ∈ Λ0M , then for any i ω1 · · · fωi−1 · · ·ωn + ω1 · · · fωi · · ·ωn + ω1 · · ·ωi−1df ωi · · ·ωn = 0. To verify FpCq(LM) ⊂ kerψp, it suffices to show (kerε(σ)) p ⊂ kerψp. Let s denote the section of π, which sends points of M to the constant map. Take (σ1 − s♯σ) · (σ2 − s♯σ) · · · · ·(σp − s♯σ) ∈ (kerε(σ)) p, where σ ∈ Cq(M) and σi ∈ Cq(LM)(σ). Then (σ1 − s♯σ) · (σ2 − sσ) · · · · · (σp − s♯σ) π∗ω ∧ ω1 · · ·ωp−1 σ∗ω ∧ (σ1 − s♯σ) ω1 · · · (σk − s♯σ) ∗1 · · · ∧ (σp − s♯σ) Therefore we obtain the proposition. Let C∗(M,x) denote a set of smooth simplexes ofM neighborhood of whose vertices are at x in M . We define C ⊗ sC⊗p = C∗(M,x)⊗ sC∗(M,x) Here (sC∗(M,x))q = Cq+1(M,x) or 0 according as q > 0 or q ≤ 0. Its boundary is given by the sum of the boundary on each complex. Let us construct a chain map Φ : C ⊗ sC⊗p → FpC∗(LM)/Fp+1C∗(LM) considering the following three cases: case 1: If (σ1, · · · , σp) ∈ sC(M,x)⊗p , then Φ : (σ1, · · · , σp) 7−→ (σ1 − x) · (σ2 − x) · · · · · (σp − x) where x is regarded as a constant map. case 2: If (σ1, · · · , σp) ∈ sC(M,x)⊗p , then Φ : (σ1, · · · , σp) 7−→ (σ1 − x) · (σ2 − x) · · ·σi · · · (σp − x) where σi : ∆ 1 ∋ ξ 7→ σi(ξ)(t) ∈ ΩxM is σi(ξ)(t) σi((1 − ξ)((1 − t)v0 + tv2) + ξ(1 − 2t)v0 + 2ξtv1), if 0 ≤ t ≤ 1/2 σi((1 − ξ)((1 − t)v0 + tv2) + ξ(2 − 2t)v1 + ξ(2t− 1)v2), if 1/2 ≤ t ≤ 1 Here v0, v1, v2 are the vertices of the standard simplex ∆ case 3: If (γ, σ1, · · · , σp) ∈ C1(M,x)⊗ sC(M,x)⊗p , then Φ : (γ, σ1, · · · , σp) 7−→ γ t (σ1 − x)γt · · · γ t (σp − x)γt where γt : [0, 1] ∋ s 7→ γ(st) ∈ M , t ∈ ∆ Lemma 5.4. The following diagram commutes: C ⊗ sC⊗p −−−−→ FpC1(LM)/Fp+1C1(LM) C ⊗ sC⊗p −−−−→ FpC0(LM)/Fp+1C0(LM) Proof. For case 2, ∂′Φ(σ1, · · · , σp)− Φ∂(σ1, · · · , σp) = (σ1 − x) · · · (σ i · σ i − σ i − σ i + σ i − σ i + x) · · · (σp − x) = (σ1 − x) · · · (σ i − x) · (σ i − x) · · · (σp − x) ∈ Fp+1C0(LM) where σ i , σ i , σ i are the faces of σi. For case 3, ∂′Φ(γ, σ1, · · · , σp)− Φ∂ ′(γ, σ1, · · · , σp) = γ−1 · (σ1 − x) · γ · · · γ −1 · (σp − x) · γ − (σ1 − x) · · · (σp − x) ∈ Fp+1C0(LM). Therefore we obtain the lemma. Proposition 5.2 gives the map Hq(FpC(LM)/Fp−1C(LM))→ Hq(Hom(F pB(A)/F p−1B(A), A∨)). Lemma 5.5. For q = 0, the following map is isomorphic: H0(FpC(LM)/Fp+1C(LM)) ∼= H0(Hom(F pB(A)/F p−1B(A), A∨)). Proof. We obtain the following surjection by lemma 5.4. Φ : H0(C ⊗ sC ⊗p) ։ H0(FpC(LM)/Fp+1C(LM)). Composing with the isomorphism ⊗pH1(M) ∼= H0(C ⊗ sC ⊗p), the map ⊗pH1(M) ։ H0(FpC(LM)/Fp+1C(LM))→ Hom(⊗ pH1(A),R) is given by (σ1, · · · , σn) 7→ (ω1, · · · , ωp) 7→ ω1 · · · This is isomorphic and we obtain the lemma. Lemma 5.6. For q = 1, the following map surjective: H1(FpC(LM)/Fp+1C(LM)) ։ H1(Hom(F pB(A)/F p−1B(A), A∨)). Proof. It suffices to show that the following map obtained by lemma 5.4 is surjective. ker∂ → H1(FpC(LM)/Fp+1C(LM))→ Hom(⊗ psH(A), H(A)∨)1 If (γ, σ1, · · · , σp) ∈ ker∂ ∩ C0(M,x)⊗ sC(M,x)⊗p , then (γ, σ1, · · · , σp) 7→ (ω1, · · · , ωp) 7→ ω1 · · · ωp, if deg ω = 0 0, otherwise through the above map. If (γ, σ1, · · · , σp) ∈ ker∂ ∩ C1(M,x)⊗ sC(M,x)⊗p , then (γ, σ1, · · · , σp) 7→ (ω1, · · · , ωp) 7→ ω1 · · · when deg ω = 1. Then we can verify the surjectivity and obtain the lemma. Proof of theorem 1.1. Consider the spectral sequences ofC(LM)/FpC(LM) and Hom(F p−1B(A), A∨) associated with FqC(LM) and Hom(F qB(A), A∨), re- spectively. Lemma 5.5 asserts that ψp is isomorphic on E1-level at degree 0: H0(FqC(LM)/Fq+1C(LM)) ∼= H0(Hom(F qB(A)/F q−1B(A), A∨)). Lemma 5.6 asserts that ψp is surjective on E1-level at degree 1: H1(FqC(LM)/Fq+1C(LM)) ։ H1(Hom(F qB(A)/F q−1B(A), A∨)). Then there is an isomorphism on Er-level at degree 0 for r ≥ 1. We have Rπ̃/J̃p ∼= H0(C(LM)/FpC(LM)) ∼= H0(Hom(F pB(A), A∨)). Therefore we obtain the theorem. 6 The Goldman bracket This section is devoted to the proof of the following theorem. Theorem 6.1. Let M be a compact closed oriented surface with genus g. Then the Goldman bracket induces a Lie algebra structure on lim Rπ̃/J̃pand there is an isomorphism of Lie algebras Rπ̃/J̃p ∼= H0(Hom(B(H ∗(M)), H∗(M)∨)). Goldman showed that the vector space spanned by the free homotopy classes of closed curves on a closed oriented surface has a Lie algebra structure [9]. This work led Chas and Sullivan to the string topology. We would verify that this structure makes lim Rπ̃/J̃p a Lie algebra. On the other hand, we can construct a bracket on H0(Hom(B(H ∗(M)), H∗(M)∨)) by the cup product defined in section 3 and the Connes’s operator. Here we regard H∗(M) as a differential graded algebra with a trivial differential. Theorem 6.1 asserts that those two Lie algebras are isomorphic. First we describe a relation between this bracket and the augmentation ideal of the group ring of the surface group to induce a Lie algebra structure on Rπ̃/J̃p. Then we construct a bracket on H0(Hom(B(A), A ∨)) and verify the isomorphism of Lie algebras Rπ̃/J̃p ∼= H0(Hom(B(A), A Finally we verify the isomorphism H0(Hom(B(A), A ∨)) ∼= H0(Hom(B(H ∗(M)), H∗(M)∨). The following proposition makes lim Rπ̃/J̃p a Lie algebra. Proposition 6.2. (1) If p ≥ 1 and q ≥ 2, then [J̃p, J̃q] ⊂ J̃p+q−2. (2) If p ≥ 2 , then [J̃p,Rπ̃] ⊂ J̃p−1. Proof. We give a proof of (1). Take (σ1−x) · · · (σp−x) ∈ J̃p, (τ1−y) · · · (τq−y) ∈ J̃q, where σi ∈ ΩxM and τi ∈ ΩyM . Assume that all curves are immersions and σi τj intersect transversally for any i, j. Let {σi♯τj} denote the set of intersection points of σi and τj . Also assume that all the intersection points are distinct i.e. {σi♯τj} ∩ {σk♯τl} = φ if i 6= k or j 6= l. Then, [σ, τ ] = s∈σi♯τj {ε(s;σi, τj)γs,x · (σi − x) · · · (σp − x)(σ1 − x) · · · ·(σi−1 − x) · γ s,x · ·γs,y · (τj − y) · · · (τq − y)(τ1 − y) · · · (τj−1 − y) · γ −γs,x · (σi+1 − x) · · · (σp − x)(σ1 − x) · · · (σi−1 − x) · γ s,x · ·γs,y · (τj+1 − y) · · · (τq − y)(τ1 − y) · · · (τj−1 − y) · γ ∈ J̃p+q−2. Here γs,x is a path from s to x along σi and γs,y is a path from s to y along τj . The proof of (2) can be verified in the same way. Let A be a differential graded subalgebra of ΛM such thatH∗(A) ∼= H∗(ΛM) by the inclusion. Proposition 6.3. There is an isomorphism of vector spaces H∗(Hom(F pB(A), A)) ∼= H∗−2(Hom(F pB(A), A∨)). Proof. We define P : H∗−2(Hom(F pB(A), A))→ H∗(Hom(F pB(A), A∨)) by P (ϕ)(ω1, · · · , ωp)(ω) = ω ∧ ϕ(ω1, · · · , ωp). This map preserves the filtrations. On E1-level, the map Hom(⊗qH(A), H(A))→ Hom(⊗qH(A), H(A)∨) is isomorphic. Therefore we obtain the proposition. Now we construct a bracket on H0(Hom(B(A), A ∨)). First, we define the Connes’s operator B : H∗(Hom(F pB(A), A∨)) → H∗+1(Hom(F p−1B(A), A∨)) B(ϕ)(ω1, · · · , ωp−1)(ω) 0≤k≤p−1 (−1)(εk+1)(εp−1−εk)ϕ(ωk+1, · · · , ωp−1, ω, ω1, · · ·ωk)(1). Composing these maps and the cup product, we can define a bracket on H0(Hom(F pB(A), A∨)) by [ϕ1, ϕ2] = −P (P −1Bϕ1 ∪ P −1Bϕ2) ∈ H0(Hom(F p−1B(A), A∨)). Take 2g closed 1-forms on M , α1, · · · , αg, β1, · · ·βg, such that αi ∧ βj = δij . Let {E p.q} denote the spectral sequence of Hom(B(A), A ∨) associated with F pB(A). Notice that the cyclic group Z/pZ acts on E ∼= Hom(⊗pH1(A),R) ιϕ(ω1, · · · , ωp) = ϕ(ω2, · · · , ωp, ω1) where ι is a generator of Z/pZ. The bracket [ , ] : E p,−p⊗E q,−q → E p+q−2,−p−q+2 [ϕ1, ϕ2](ω1, · · · , ωp+q−2) i,m,n ιmϕ1(αi, ω1, · · · , ωp−1)̺ nϕ2(βi, ωp, · · · , ωp+q−2) −ιmϕ1(βi, ω1, · · · , ωp−1)̺ nϕ2(αi, ωp, · · · , ωp+q−2) where ι and ̺ are generators of Z/pZ and Z/qZ, respectively. Proposition 6.4. The following diagram commutes for p, q ≥ 1: J̃p/ ˜Jp+1 ⊗ J̃q/ ˜Jq+1 −−−−→ E ∞ ⊗ E [ , ] [ , ] J̃p+q−2/J̃p+q−1 −−−−→ E p+q−2,−p−q+2 Proof. Take σ = (σ1 − x) · · · (σp − x) ∈ FpC0(LM), τ = (τ1 − y) · · · (τq − y) ∈ FqC0(LM). Take 2g curves in M , ai, bi, as in Figure 1. Assume that σi and τj , ak, or bk, intersect transversally for any i, j, k. Also assume that τj and ak, or bk, intersect transversally for any j, k. Assume that all the intersection points are distinct. Then for any i, j, k, we can take each tubular neighborhoods of ai and bi so that it does not include some neighborhoods of intersection points of σj and τk. We fix such neighborhoods of intersection points and denote them by Up for each p. We can also take a tubular neighborhood of the diagonal map from M to M×M outside those neighborhoods of intersection points of σi and τj for any i, j i.e. S1 \ ∪pσ i (Up) S1 \ ∪pτ j (Up) = φ, ∀i, j. Here N∆ denotes the tubular neighborhood of the diagonal map. Thom class Φ of this tubular neighborhood satisfies Φ = −ε(p;σi, τj), where ε(p;σi, τj) is the intersecion number of σi and τj at p. Fig. 1 ・ ・ ・ Define e♯ : C0(LM) → C1(LM) by e♯γ(ξ)(t) = γ(ξ + t). Let ωk, 1 ≤ k ≤ n, be differential forms on M which has its support inside the tubular neighborhoods of ai and bi. Then [σi,τj] ω1 · · ·ωn p∈σi♯τj,k ε(p;σi, τj) (σi)p ω1 · · ·ωk (τj)p ωk+1 · · ·ωn p∈σi♯τj ,k ×τj | (σi)p ω1 · · ·ωk (τj)p ωk+1 · · ·ωn e♯σi×e♯τj π∗Φ ∧ p∗1 ω1 · · ·ωk ∧ p ωk+1 · · ·ωn. Here p1, p2 : LM×LM → LM are the projections. The last equality is obtained by the following lemma. Lemma 6.5. If p ∈ σi♯τj and p ′ ∈ Up ∩ σi([0, 1]), then (σi)p ω1 · · ·ωn = (σi)p′ ω1 · · ·ωn. Proof. F Let γ be the curve from p to p′ along σi inside Up. If γ and σ are in the same direction, then (σi)p′ ω1 · · ·ωn = γ·(σi)p′ ω1 · · ·ωn = (σ)p·γ ω1 · · ·ωn ω1 · · ·ωn. We can also verify the case where γ is in the direction opposite to σ in the same We have the equality e♯σ×e♯τ π∗Φ ∧ p∗1 ω1 · · ·ωk ∧ p ωk+1 · · ·ωp+q−2 e♯σ×e♯τ − p∗1(α1 ∧ β1)− p 2(α1 ∧ β1) + p 1αj ∧ p 2βj − p 1βj ∧ p ω1 · · ·ωk ∧ p ωk+1 · · ·ωp+q−2 In fact, if η ∈ Λ(M ×M) then (−1)|η|+1 e♯σ×e♯τ π∗dη ∧ p∗1 ω1 · · ·ωk ∧ p ωk+1 · · ·ωp+q−2 e♯σ×e♯τ π∗η ∧ d ω1 · · ·ωk ∧ p ωk+1 · · ·ωp+q−2 +(e♯σ) ω1 · · ·ωk (e♯τ) ωk+1 · · ·ωj ∧ ωj+1 · · ·ωp+q−2 The last equality is obtained by the following lemma. Lemma 6.6. If σ ∈ FpC0(LM), then (e♯σ) ω1 · · ·ωp−2 = 0. Proof. It suffices to show the case σ = (τ1 − x) · · · (τp − x) where x ∈ M and τi ∈ ΩxM . We define τ̄i ∈ ΩxM by τ̄i(t) = τi(pt), if (i − 1)/p ≤ t ≤ i/p 0, otherwise. Let σ̄ denote (τ̄1 − x) · · · (τ̄p − x). It can be shown that e♯σ̄ restricted on [(i − 1)/p, i/p] is contained in Fp−1C1(LM) for any i. Therefore (e♯σ) ω1 · · ·ωp−2 = (e♯σ̄) ω1 · · ·ωp−2 = 0. Jones, Geztler, and Petrack describes the map e♯ in terms of iterated inte- grals by the following theorem. Theorem 6.7 (Geztler-Jones-Petrack [8]). If σ ∈ C0(LM) and ω, ωi ∈ Λ 1 ≤ i ≤ p, then π∗ω ∧ ω1 · · ·ωp = ωk · · ·ωpωω1 · · ·ωk−1. This theorem asserts the equality e♯σ×e♯τ − p∗1(α1 ∧ β1)− p 2(α1 ∧ β1) + p 1αj ∧ p 2βj − p 1βj ∧ p ω1 · · ·ωk ∧ p ωk+1 · · ·ωn j,k,l ωk+1 · · ·ωp−1αjω1 · · ·ωk ωl+1 · · ·ωp+q−2βjωp · · ·ωl ωk+1 · · ·ωp−1βjω1 · · ·ωk ωl+1 · · ·ωp+q−2αjωp · · ·ωl Finally we obtain the equality [σ,τ ] ω1 · · ·ωp+q−2 j,k,l ωk+1 · · ·ωp−1αjω1 · · ·ωk ωl+1 · · ·ωp+q−2βjωp · · ·ωl ωk+1 · · ·ωp−1βjω1 · · ·ωk ωl+1 · · ·ωp+q−2αjωp · · ·ωl Since we can take ωi ∈ H 1(M), 1 ≤ i ≤ p + q − 2, so that their support are inside the tubular neighborhoods of aj and bj, we obtain the proposition. Proof of theorem 6.1. We obtain the following isomorphism of Lie algebras by proposition 6.4. Rπ̃/J̃p ∼= H0(Hom(B(A), A To obtain the isomorphism of Lie algebras H0(Hom(B(A), A ∨) ∼= H0(Hom(B(H ∗(M)), H∗(M)∨), we introduce the following lemma, which asserts the formality of the compact Kähler manifolds. Lemma 6.8 (ddcLemma, Deligne-Griffiths-Morgan-Sullivan [7]). Let X be a compact Kähler manifold and dc = J−1dJ where J gives the complex structure in the cotangent bundle. If α is a differential form on X such that dα = 0 and dcα = 0, and such that α = dγ, then α = ddcβ for some β. Cor. There are quasi-isomorphisms of differential graded algebras (ΛX, d)← (kerdc, d)→ (H∗dc(X), 0). Notice that a closed oriented surface endowed with a complex structure become a Kähler manifolds for the dimensional reason. Therefore the following lemma completes the proof of the theorem. Lemma 6.9. If f : A1 → A2 is a quasi-isomorphism of differential graded algebras, then the map induced by f H0(Hom(B(A1), A 1 )→ H0(Hom(B(A2), A is an isomorphism. Proof. It suffices to verify that the map induced by f f : H0(Hom(F pB(A1), A 1 )→ H0(Hom(F pB(A2), A is an isomorphism for any p. On E1-level, the map induced by f Hom(⊗sH(A1), H(A1) ∨)→ Hom(⊗sH(A2), H(A2) is an isomorphism because f is quasi-isomorphism. Therefore we obtain the lemma. Therefore we obtain the theorem. References [1] M. Chas and D. Sullivan, String topology, preprint, 1999, http://arXiv.org /abs/math.GT/9911159. [2] K.T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97(1973), 217-246. [3] K.T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83-98. [4] K.T. Chen, Reduced bar constructions on de Rham complexes, in:A.Haller and M.Tierney (eds), (Algebra, topology and category theory, 1977, pp. 19- [5] K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no.5, 831-879. [6] R.L. Cohen, J.D.S. Jones and J. Yan, The loop homology algebra of spheres and projective spaces, Categorical Decomposition Techniques in Algebraic Topology (Isle of Skye, 2001), Progr. Math., vol. 215. Birkhäuser, Basel, 2004, pp.77-92. [7] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274. [8] E. Getzler, J.D.S. Jones and S. Petrack Differential forms on loop spaces and the cyclic bar complex, Topology 30 (1991), no.3, 339-371. http://arXiv.org [9] W.M. Goldman, Invariant functions on Lie groups and Hamlitonian flows of surface group representation, Invent. Math. 85 (1986), no.2, 263-302. [10] S.A. Merkulov, De Rham Model for String Topology, International Mathe- matics Research Notices 55 (2004), 2955-2981. Introduction Chen's iterated integrals Preliminaries Proof of Theorem 1.1 The conjugacy classes of fundamental groups The Goldman bracket
704.002
arXiv:0704.0015v2 [hep-th] 10 Mar 2008 Preprint typeset in JHEP style - HYPER VERSION Fermionic superstring loop amplitudes in the pure spinor formalism Christian Stahn Department of Physics, University of North Carolina Chapel Hill, NC 27599–3255, USA E-mail: stahn@physics.unc.edu Abstract: The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor super- space. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states. Keywords: Superstrings, Pure Spinors. http://arxiv.org/abs/0704.0015v2 mailto:stahn@physics.unc.edu http://jhep.sissa.it/stdsearch Contents 1. Introduction 1 2. Zero mode integration 2 2.1 Symmetry considerations and tensorial formulae 3 2.2 A spinorial formula 5 2.3 Component-based approach 7 3. One-loop amplitudes 7 3.1 Review: four bosons 8 3.2 Four fermions 10 3.3 Two bosons, two fermions 10 4. Two-loop amplitudes 12 4.1 Review: four bosons 13 4.2 Four fermions 14 4.3 Two bosons, two fermions 15 5. Discussion 16 A. Reduction to kinematic bases 17 A.1 Four bosons 17 A.2 Four fermions 18 A.3 Two bosons, two fermions 20 B. A gamma matrix representation 21 1. Introduction The quantisation of the ten-dimensional superstring using pure spinors as world-sheet ghosts [1] has overcome many difficulties encountered in the Green-Schwarz (GS) and Ramond-Neveu-Schwarz (RNS) formalisms. Most notably, by maintaining manifest space- time supersymmetry, the pure spinor formalism has yielded super-Poincaré covariant multi- loop amplitudes, leading to new insights into perturbative finiteness of superstring theory [2, 3]. Counting fermionic zero modes is a powerful technique in the computation of loop amplitudes in the pure spinor formalism and has for example been used to show that at least four external states are needed for a non-vanishing massless loop amplitude [2]. Furthermore, the structure of massless four-point amplitudes is relatively simple because all – 1 – fermionic worldsheet variables contribute only through their zero modes. In the expressions derived for the one-loop [2] and two-loop [4] amplitudes, supersymmetry was kept manifest by expressing the kinematic factors as integrals over pure spinor superspace [5] involving three pure spinors λ and five fermionic superspace coordinates θ, K1-loop = (λA)(λγmW )(λγnW )Fmn K2-loop = (λγmnpqrλ)(λγsW )FmnFpqFrs (1.1) where the pure spinor superspace integration is denoted by 〈. . . 〉, and Aα(x, θ), Wα(x, θ) and Fmn(x, θ) are the superfields of ten-dimensional Yang-Mills theory. The kinematic factors in (1.1) have been explicitly evaluated for Neveu-Schwarz states at two loops [6] and one loop [7], and were found to match the amplitudes derived in the RNS formalism [8]. This provided important consistency checks in establishing the validity of the pure spinor amplitude prescriptions. (Related one-loop calculations had been reported in [9].) In this paper, it will be shown how to compute the kinematic factors in (1.1) when the superfields are allowed to contribute fermionic fields, as is relevant for the scattering of fermionic closed string states as well as Ramond/Ramond bosons. It turns out that the calculation of fermionic amplitudes presents no additional difficulties, making (1.1) a good practical starting point for the computation of four-point loop amplitudes in a unified fashion. This practical aspect of the supersymmetric pure spinor amplitudes was also emphasised in [10], where the tree-level amplitudes were used to construct the fermion and Ramond/Ramond form contributions to the four-point effective action of the type II theories. This paper is organised as follows. In section 2, different methods to compute pure spinor superspace integrals are explored. These methods are then applied to the explicit evaluation of the kinematic factors of massless four-point amplitudes at the one-loop level in section 3, and at the two-loop level in section 4. In both these sections, the bosonic calcu- lations are briefly reviewed before separately considering the cases of two and four Ramond states. Particular attention will be paid to the constraints imposed by simple exchange symmetries. An appendix contains algorithms which were used to reduce intermediate expressions encountered in the amplitude calculations to a canonical form. 2. Zero mode integration The calculation of scattering amplitudes in the pure spinor formalism leads to integrals over zero modes of the fermionic worldsheet variables λ and θ. Both θ and λ are 16-component Weyl spinors, the λ are commuting and the θ anticommuting, and λ is subject to the pure spinor constraint (λγmλ) = 0. The amplitude prescriptions [1, 2] require three zero modes of λ and five zero modes of θ to be present, and a Lorentz covariant object T̄ αβγ,δ1...δ5 ≡ λαλβλγθδ1 . . . θδ5 = T̄ (αβγ),[δ1...δ5] (2.1) was constructed such that the Yang-Mills antighost vertex operator V = (λγmθ)(λγnθ)(λγpθ)(θγmnpθ) has = 1 . (2.2) – 2 – In this section, different methods of computing such “pure superspace integrals” are ex- plored. As an example, a typical correlator encountered in the two-loop calculations of section 4 is considered: F (ki, ui) = k (λγmnpq[rλ)(λγs]u1)(θγn abθ)(θγbu2)(θγqu3)(θγsu4) (2.3) Here, ki and ui are the momenta and spinor wavefunctions of the four external particles. 2.1 Symmetry considerations and tensorial formulae One systematic approach to evaluate the zero mode integrals is to find expressions for all tensors that can be formed from (2.1). By Fierz transformations, one can always write the product of two θ spinors as (θγ[3]θ), where γ[k] denotes the antisymmetrised product of k gamma matrices. Due to the pure spinor constraint, the only bilinear in λ is (λγ[5]λ), and it is thus sufficient to consider the three cases (λγ[5]λ)(λ{γ[1] or γ[3] or γ[5]}θ)(θγ[3]θ)(θγ[3]θ) . (2.4) Lorentz invariance then implies that it must be possible to express these tensors as sums of suitably symmetrised products of metric tensors, resulting in a parity-even expression, plus a parity-odd part made up from terms which in addition contain an epsilon tensor. The parity-even parts may be constructed [6] starting from the most general ansatz compatible with the symmetries of the correlator and then using spinor identities along with the normalisation (2.2) to determine all coefficients in the ansatz. Duality properties of the spinor bilinears can be used to determine the parity-odd part [7]. An extensive (and almost exhaustive) list of correlators is found in [11], including the (λγ[1]θ) and (λγ[3]θ) cases of the above list: (λγmnpqrλ)(λγuθ)(θγfghθ)(θγjklθ) = − 4 mnpqr m̄n̄p̄q̄r̄ + εmnpqrm̄n̄p̄q̄r̄ δm̄n̄fg δ (δr̄l δ u + δ u − δr̄uδhl ) [fgh][jkl] (2.5) (λγmnpqrλ)(λγstuθ)(θγfghθ)(θγjklθ) = −24 mnpqr m̄n̄p̄q̄r̄ + εmnpqrm̄n̄p̄q̄r̄ δm̄j δ δq̄sδ u − δkhδr̄u) [fgh][jkl](fgh↔jkl) (2.6) (Here, the brackets (fgh↔ jkl) denote symmetrisation under simultaneous interchange of fgh with ijk, with weight one.) The remaining correlator with the (λγ[5]θ) factor can be derived in the same way, using an ansatz consisting of six parity-even structures. Taking a trace between the two γ[5] factors and noting that (λγmnpqrλ)(λγabcdeθ) . . . (λγmnpq[bλ)(λγcde]θ) . . . one finds a relation to (2.6). This is sufficient to determine all coefficients in the ansatz, and the result is (λγmnpqrλ)(λγabcdeθ)(θγfghθ)(θγjklθ) mnpqr m̄n̄p̄q̄r̄ + εmnpqrm̄n̄p̄q̄r̄ m̄n̄p̄ (−δehδr̄l + 2δel δr̄h) + δm̄n̄ab δcdfgδ (δehδ l − 3δel δr̄h) [abcde][fgh][jkl](fgh↔jkl) (2.7) – 3 – One may find it surprising that the derivation of these tensorial expressions only made use of properties of (pure) spinors, and of the normalisation condition (2.2). However, it can be seen from representation theory that the correlator (2.1) is uniquely characterised, up to normalisation, by its symmetry. To see this, note that [12] the spinor products λ3 and θ5 transform in λ(αλβλγ) : Sym3 S+ = [00003] ⊕ [10001] θ[δ1 . . . θδ5] : Alt5 S+ = [00030] ⊕ [11010] . (2.8) (Here, λ and θ are taken to be in the S+ irrep of SO(1,9), with Dynkin label [00001].) The tensor product of these contains only one copy of the trivial representation. This applies to any spinors λ, which means that the pure spinor property cannot be essential to the derivation of the tensorial identities. The use of the pure spinor constraint merely allows for simpler derivations of the same identities. As an illustration of this approach, consider the correlator of eq. (2.3). Leaving the momenta aside for the moment by setting F = k2ak r F̃ , the task is to compute (λγmnpq[rλ)(λγs]u1)(θγn abθ)(θγbu2)(θγqu3)(θγsu4) After applying two Fierz transformations, (λγmnpq[r|λ)(λγcθ)(θγn abθ)(θγjklθ) |s]γcγbu2) 3!·16 (λγmnpq[r|λ)(λγcdeθ)(θγn abθ)(θγjklθ) |s]γcdeγbu2) 2·5!·16 (λγmnpq[r|λ)(λγcdefgθ)(θγn abθ)(θγjklθ) |s]γcdefgγbu2) 3!·16(u3γqγjklγsu4) , one obtains a combination of the fundamental correlators listed in (2.5), (2.6) and (2.7). A reliable evaluation of the numerous index symmetrisations is made possible by the use of a computer algebra program. In doing these calculations with Mathematica, an essential tool is the GAMMA package [13], expanding the products of gamma matrices in a γ[k] basis. The result consists of two parts, F̃ = F̃ (δ) + F̃ (ε), with F̃ (δ) = 1 mpru2)(u3γ au4) + r (u1γ iu2)(u3γiu4) + . . . ai1i2u2)(u3γ i1i2u4) (92 terms) (2.9) F̃ (ε) = − 1 1209600 εi1...i7 mpr(u1γ i1...i7u2)(u3γ au4) + . . . 604800 εampri1...i6(u1γ i3...i9u2)(u3γ i7i8i9u4) (34 terms) (2.10) The epsilon tensors in the second part can be eliminated using the fact that the ui are chiral spinors: If all the indices on γ[k]ui are contracted into an epsilon tensor, one uses εi1...ik′j1...jkγ j1...jkγ11 = (−) k(k+1) k! γi1...ik′ , (2.11) where γ11 = 1 εi0...i9γ i0...i9 . More generally, if all but r indices of γ[k]ui are contracted, εi1...ik′ j1...jkγ p1...prj1...jkγ11 = (−) k(k+1) (k′ − r)! pr ...p1 [i1...ir γir+1...i′k] . (2.12) – 4 – The result of these manipulations is F̃ (ε) =− 1 mpru2)(u3γ au4)− 1280δ r (u1γ amiu2)(u3γiu4) + . . . 11200 i1i2i3u2)(u3γ i1i2i3u4) (53 terms) (2.13) (Note that while the epsilon terms in the basic correlator formulae were easily obtained from the delta terms by using Poincaré duality, this cannot be done here in any obvious way.) The last step in the evaluation of (2.3) is to contract with the momenta, F = k2ak r F̃ , and to simplify the expressions using the on-shell identities i ki = 0, k i = 0, /kiui = 0. It is shown in appendix A.2 that there are only ten independent scalars, denoted by B1 . . . B10, that can be formed from four momenta and the four spinors u1 . . . u4. With respect to this basis, the result is F (δ) = 1 48·10080 695s12(u1/k3u2)(u3/k1u4) + · · ·+ 233s213(u1γau2)(u3γau4) (7 terms) 48·10080 (695, 775, 0,−80, 356, 356, 0, 233, 233, 0)B1 ...B10 , F (ε) = 1 48·10080 (−23,−7, 0,−16, 28, 28, 0, 7, 7, 0)B1 ...B10 , F = 1 10080 (14, 16, 0,−2, 8, 8, 0, 5, 5, 0)B1 ...B10 , (2.14) where sij = ki · kj . 2.2 A spinorial formula While the derivation of tensorial identities for correlators of the form (2.4) is relatively straightforward and elegant, it may be a tedious task to transform the expressions encoun- tered in amplitude calculations to match this pattern. As seen in the example calculated above, this is particularly true if additional spinors are involved, making it necessary to ap- ply Fierz transformations. It is therefore desirable to use a covariant correlator expression with open spinor indices. Such an expression was given in [1, 2]: T̄αβγ,δ1...δ5 = N−1 (γm)αδ1(γn)βδ2(γp)γδ3(γmnp) (αβγ)[δ1...δ5] , (2.15) where N is a normalisation constant and the brackets ()[] denote (anti-)symmetrisation with weight one. (Note that the right hand side is automatically gamma-matrix traceless: any gamma-trace (γr)αβ × (γm)α[δ1|(γn)β|δ2|(γp)γ|δ3(γmnp)δ4δ5] = −(γmnr)[δ1δ2(γmnp)δ3δ4(γp)δ5]γ = 0 vanishes due to the double-trace identity (γabθ) α(θγabcθ) = 0, which follows from the fact that the tensor product (Alt3 S+)⊗ S− does not contain a vector representation and therefore the vector (ψγabθ)(θγ abcθ) has to vanish for all spinors ψ, and can also be shown by applying a Fierz transformation.) This prescription was originally motivated [2] by the fermionic expansion of the Yang-Mills antighost vertex operator V , V = Tαβγ,δ1...δ5λ αλβλγθδ1 . . . θδ5 (2.16) Tαβγ,δ1...δ5 = (γm)αδ1(γ n)βδ2(γ p)γδ3(γmnp)δ4δ5 (αβγ)[δ1...δ5] – 5 – where T is related to T̄ by a parity transformation, up to the overall constant N . (Since T̄ is uniquely determined by its symmetries, any covariant expression will be proportional to T̄ , after symmetrisation of the spinor indices, and this is merely the simplest choice.) Equation (2.15) immediately yields an algorithm to convert any correlator into traces of gamma matrices or, if additional spinors are involved, bilinears in those spinors. It is, however, already very tiresome to determine the normalisation constant N by hand. The main advantage of this approach is that it clearly lends itself to implementation on a computer algebra system, which can easily carry out the spinor index symmetrisations, simplify the gamma products (again using the GAMMA package), and compute the traces. For example, N〈V 〉 = (γm)αδ1(γn)βδ2(γp)γδ3(γmnp) (αβγ)[δ1...δ5] (γx)αδ1(γy)βδ2(γz)γδ3(γ xyz)δ4δ5 = − 1 Tr(γxγ m)Tr(γyγ n)Tr(γzγ p)Tr(γxyzγpnm) + . . . Tr(γzγpnmγ zyxγnγxγ p) (60 terms) = 5160960 . The correct normalisation is therefore obtained by setting N = 5160960. Returning to the example correlator (2.3), one finds that the calculation is by far simpler than with the previous method. After carrying out the symmetrisations (αβγ)[δi], one obtains NF̃ = 1 Tr(γxγ mnpq[r|)(u3γqγ xyzγsu4)(u1γ |s]γzγbu2) + . . . (u2γbγ xyzγqu3)(u1γsγyγ mnpq[rγzγ s]u4) , (24 terms) where elementary index re-sorting has reduced the number of terms from 60 to 24. Ex- panding the gamma products leads to NF̃ = 476 δpr (u1γ mu4)(u2γ au3) + · · ·+ 815(u1γ ai1i2i3i4u2)(u3γ i1i2i3i4u4) , (294 terms) which, in contrast to (2.10), contains no epsilon terms as there are not enough free indices present. Note that this intermediate result contains terms with with u1 paired with u3 or u4, so it is not possible to directly compare to eqs. (2.9) and (2.13). However, after contracting with the momenta k2ak r and decomposing the result in the basis B1 . . . B10, one again obtains F = 1 10080 (14, 16, 0,−2, 8, 8, 0, 5, 5, 0)B1 ...B10 , (2.17) in agreement with (2.14). The algorithm just outlined will be the method of choice for all correlator calculations in the later sections of this paper and can easily be applied to a wider range of problems. The only limitation is that the larger the number of gamma matrices and open indices of the correlator, the slower the computer evaluation will be. For example, the correlator considered in eq. (5.2) of [11], mnm1n1...m4n4 (λγpγm1n1θ)(λγqγm2n2θ)(λγrγm3n3θ)(θγmγnγpqrγ m4n4θ) = − 2 m1n1...m4n4 εmnm1n1...m4n4 , (2.18) can still be verified with this method but this already requires substantial runtime. – 6 – 2.3 Component-based approach A third method to evaluate the zero mode integrals consists of choosing a gamma matrix representation, expanding the integrand as a polynomial in spinor components, and then applying (2.15) to the individual monomials. This procedure seems particularly appealing if at some stage of the calculation one works with a matrix representation anyhow, in order to reduce the results to a canonical form (e.g. as outlined in appendix A). An efficient decomposition algorithm (of k4u1u2u3u4 scalars, say) only needs a few non-zero momentum and spinor wavefunction components to distinguish all independent scalars, and therefore k and u can be replaced by sparse vectors. Furthermore, a trivial observation allows for a much quicker numeric evaluation of correlator components than a naive use of (2.15): In view of (2.16), one can equivalently compute the components of the parity- transformed expression V̄ = (λ̄γmθ̄)(λ̄γnθ̄)(λ̄γpθ̄)(θ̄γmnpθ̄), where λ̄ and θ̄ are spinors of chirality opposite to that of λ, θ. In the representation given in appendix B, V̄ coincides with V |λ→λ̄,θ→θ̄, and V = 192λ9λ9λ9θ1θ2θ3θ4θ9 + · · ·+ 480λ1λ2λ3θ1θ9θ10θ13θ15 + . . . (100352 terms) The monomials in the fermionic expansion of V̄ then correspond to the arguments of non-zero correlators, and the coefficients of those monomials are, up to normalisation and symmetry factors, the correlator values. Unfortunately, it turns out that the complexity of typical correlators (e.g. the one given in (2.3)) makes it difficult to carry out the expansion in fermionic components in any straightforward way and limits this method to special applications. For example, the coefficients in (2.18) can be checked relatively easily by choosing particular index values, such as (λγpγ12θ)(λγqγ21θ)(λγrγ34θ)(θγ0γ0γpqrγ 12λ1λ1λ1θ1θ9θ10θ11θ12 + · · ·+ 12λ16λ16λ16θ5θ6θ7θ8θ16 (For fixed values of pqr, one gets no more than about 105 monomials of the form λ3θ5). This approach may thus still be helpful in situations where the result has been narrowed down to a simple ansatz. 3. One-loop amplitudes The amplitude for the scattering of four massless states of the type IIB superstring was computed [2] in the pure spinor formalism as A = KK̄ (Im τ)5 G(zi, zj) ki·kj , (3.1) where G(zi, zj) is the scalar Green’s function, and the kinematic factor is given by the product KK̄ of left- and right-moving open superstring expressions, K1-loop = (λA1)(λγ mW2)(λγ nW3)F4,mn cycl(234) . (3.2) – 7 – Here the indices 1 . . . 4 label the external states and “· · ·+ cycl(234) ” denotes the addition of two other terms obtained by cyclic permutation of the indices 234. The spinor super- field Aα and its supercovariant derivatives, the vector gauge superfield Am = m DαAβ as well as the spinor and vector field strengths Wα = 1 (γm)αβ(DβAm − ∂mAβ) and Fmn = 18(γmn) β = 2∂[mAn], describe ten-dimensional super-Yang-Mills theory. The physical fields of this theory, a gauge boson and a gaugino, are found in the leading components Am| = ζm and Wα| = ûα and correspond to the Neveu-Schwarz and Ramond superstring states. The superfields Aα and W α as well as the gaugino field ûα are anticommuting.1 To facilitate computer calculations involving polynomials in the spinor components, and for easier comparison with the literature, it will be more convenient to work with commuting fermion wavefunctions uα. Fortunately, as the kinematic factors with fermionic external states are multilinear functions of the distinctly labelled spinors ûi, it is straightforward to translate between the two conventions: Any monomial expression in û1 . . . û4 (and possibly fermionic coordinates θ) corresponds to the same expression in u1 . . . u4, multiplied by the signature of the permutation sorting the ûi (and any θ variables) into some fixed order, such as (θ · · · θ)ûα11 û Choosing a gauge where θαAα = 0, the on-shell identities 2D(αAβ) = γ αβAm , DαW β = 1 (γmn)α have been used to derive recursive relations [10, 14, 15] for the fermionic expansion A(n)α = (γmθ)αA (n−1) m , A (θγmW (n−1)) , Wα(n) = − 1 (γmnθ)α∂mA (n−1) where f (n) = 1 θαn · · · θα1(Dα1 · · ·Dαnf)|. These recursion relations were explicitly solved in [10], reducing the fermionic expansion to a simple repeated application of the derivative operator Omq = 12 (θγm qpθ)∂p: A(2k)m = (2k)! [Ok]mqζq , A(2k+1)m = (2k+1)! [Ok]mq(θγqû) . (3.3) With this solution at hand, one has all ingredients to evaluate the kinematic factor (3.2) for the three cases of zero, two, or four fermionic states. 3.1 Review: four bosons The kinematic factor involving four bosons was considered in [7] and this calculation will now be reviewed briefly. First, note that the outcome is not fixed by symmetry: The result must be gauge invariant [2] and therefore expressible in terms of the field strengths F1 . . . F4. The cyclic symmetrisation in (3.2) yields expressions symmetric in F2, F3, F4, and acting on scalars constructed from the Fi only, the (234) symmetrisation is equivalent to complete symmetrisation in all labels (1234). Thus the result must be a linear combination of the 1Thanks to Carlos Mafra for pointing this out. – 8 – two gauge invariant symmetric F 4 scalars, namely the single trace Tr(F(1F2F3F4)) and double trace Tr(F(1F2)Tr(F3F4)), leaving one relative coefficient to be determined. Since all four states are of the same kind, one may first evaluate the correlator for one labelling and then carry out the cyclic symmetrisation: 1-loop = (λA1)(λγ mW2)(λγ nW3)F4,mn cycl (234) The different ways to saturate θ5 result in a sum of terms of the form XABCD = 1 )(λγ 2 )(λγ (3.4) with A+B +C +D = 5 and A, B, C odd, D even: (λA1)(λγ mW2)(λγ nW3)F4,mn = X3110 +X1310 +X1130 +X1112 . Note that X1310 and X1130 are related by exchange of the labels 2 and 3. This exchange can be carried out after computing the correlator, an operation which will in the following be denoted by π23. Using (3.3) for the superfield expansions and replacing ∂m → ikm, one obtains X3110 = − 1512F tuX̃3110 , X̃3110 = (λγ[t|γpqθ)(λγ|u]γrsθ)(λγaθ)(θγ amnθ) X1112 = − 1128 ik tuX̃1112 , X̃1112 = (λγ[m|γpqθ)(λγ|a]γrsθ)(λγnθ)(θγa X1310 = − 1384 ik tuX̃1310 , X̃1310 = (λγ[t|γmaθ)(λγ|u]γrsθ)(λγnθ)(θγa The method outlined in section 2.2 is readily applicable to these correlators. For example, for X3111, the trace evaluation yields X̃3110 = N Tr(γaγ z)Tr(γxyzγ anm)Tr(γxγqpγ [t|)Tr(γyγsrγ |u]) + · · · · · ·+ 1 Tr(γ[u|γrsγzyxγqpγ |t]γxγaγ yγmnaγz) (60 terms) δmprs δ tu − 1315δ rs − 145δ δmnpr δ [mn][pq][rs][tu](pq↔rs) Upon contracting with the field strengths, momenta and polarisations, and symmetrising over the cyclic permutations (234) (with weight 3), one finds that all three contributions are separately gauge invariant: X3110 + cycl(234) = − 11 13440 Tr(F(1F2F3F4)) + Tr(F(1F2)Tr(F3F4)) X1112 + cycl(234) = − 19 53760 Tr(F(1F2F3F4)) + 215040 Tr(F(1F2)Tr(F3F4)) (1 + π23)X1310 + cycl(234) = − 1 10240 4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4)) The sum X3110 +X1112 has the right ratio of single- and double-trace terms to be propor- tional to the well-known result t8F 4, and the last line exhibits the right ratio by itself. The overall kinematic factor is therefore K4B1-loop = − 12560 4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4)) = − 1 15360 4 , (3.5) in agreement with the expressions derived in the RNS [16] and Green-Schwarz [17] for- malisms. – 9 – 3.2 Four fermions The four-fermion kinematic factor could be evaluated in the same way as in the four-boson case by summing up all terms XABCD, A + B + C + D = 5, now with A, B, C even and D odd. Note however that this time, the outcome is fixed by symmetry: The cyclic symmetrisation in (3.2) leads to a completely symmetric dependence on û2, û3, û4, and therefore to a completely antisymmetric dependence on u2, u3, u4. Acting on scalars of the form k2u1u2u3u4, antisymmetrising over [234] is equivalent to antisymmetrising over [1234], and there is only one completely antisymmetric k2u1u2u3u4 scalar. Without further calculation, one can infer that the kinematic factor is proportional to that scalar, K4F1-loop = const · (u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3) which of course agrees with the RNS amplitude (see e.g. [16], eq. (3.67)). 3.3 Two bosons, two fermions In evaluating (3.2) for two bosons and two fermions, the cyclic symmetrisations affect whether the W and F superfields contribute bosons or fermions. Only the label of the Aα superfield stays unaffected, and one has to choose whether it should contribute a boson or a fermion. Since its fermionic expansion starts with the bosonic polarisation vector, A1,α ∼ (/ζ1θ)α, the calculation can be simplified by choosing a labelling where particle 1 is a fermion. (Of course, the final result must be independent of this choice.) The assignment of the other three labels is then irrelevant and will be chosen as f1f2b3b4. Writing out the cyclic permutations, two of the three terms are essentially the same because they are related by interchange of the labels 3 and 4. The kinematic factor is then K2B2F1-loop(f1f2b3b4) = (1 + π34) (even) 1 )(λγ (even) 2 )(λγ (odd) (even) (even) 1 )(λγ (odd) 3 )(λγ (odd) (odd) Unlike in the four-fermion calculation, the result is not fixed by symmetry. There are five independent ku1u2F3F4 scalars (see appendix A, eq. (A.6)), denoted by C1 . . . C5, and there are two independent combinations of these scalars with the required [12](34) symmetry. Expanding the superfields and collecting terms with θ5, the first line yields a combination of terms XABCD with A, B, D odd and C even. There is only one θ 5 combination coming from the second line, which will be denoted by X ′2111 ≡ (−π24)X2111: K2B2F1-loop = (1 + π34) (X4010 +X2210 +X2030 +X2012) +X 2111 , with the correlators X4010 = ζ3c k nX̃4010 , X̃4010 = (λγaθ)(θγa pqθ)(θγpu1)(λγ [mu2)(λγ n]γbcθ) X2210 = − i12k nX̃2210 , X̃2210 = (λγaθ)(θγau1)(λγ [m|γbcθ)(θγcu2)(λγ |n]γdeθ) X2030 = − i36k nX̃2030 , X̃2030 = (λγaθ)(θγau1)(λγ [mu2)(λγ n]γbcθ)(θγc X2012 = − i12k ζ3c k ζ4e X̃2012 , X̃2012 = (λγaθ)(θγau1)(λγ [mu2)(λγ n]γbcθ)(θγn X ′2111 = ζ3c k 2111 , X̃ 2111 = (λγaθ)(θγau1)(λγ [m|γbcθ)(λγ|n]γdeθ)(θγnu2) – 10 – (The numerical coefficient in X ′2111 includes a sign coming from the θ, û ordering: there is an odd number of θs between u1 and u2.) Evaluating these expressions as outlined in section 2.2, the spinor wavefunctions ui present no complication. The last part takes the simplest form: One finds (λγaθ)(θγau1)(λγ mγbcθ)(λγnγdeθ)(θγnu2) = − 1 (2δbcm[d(u1γe]u2) + δ m(u1γ c]deu2)) and therefore X̃ ′2111 = − 1480 δ[bm(u1γ c]γdeu2) + δ m(u1γ e]γbcu2) The result for X̃4010 is X̃4010 = δbqmn(u1γ cu2)− 190δ mq(u1γ nu2) + δbcmn(u1γ qu2)− 12520δ q (u1γ bcnu2) δbq(u1γ cmnu2) + δbm(u1γ cnqu2) + bcmnqu2) [bc][mn] For the evaluation of X̃2210, it is useful to consider the more general correlator (λγaθ)(θγau1)(λγ [m|γbcθ)(λγ|n]γdeθ)(θγxu2) mn(u1γ cu2) + . . . 201600 δmx (u1γ bcdenu2) + · · · − 11403200 (u1γ bcdemnxu2) [mn][bc][de] (27 terms) 9676800 εbcdemni1i2i3i4(u1γ i1i2i3i4xu2)− 12419200εbcdemnxi1i2i3(u1γ i1i2i3u2) . This time, even using the method of section 2.2, there are sufficiently many open indices and long enough traces for epsilon tensors to appear. Using eqs. (2.11) and (2.12), they can be re-written into γ[5,7] terms: (λγaθ)(θγau1)(λγ [m|γbcθ)(λγ|n]γdeθ)(θγxu2) mn(u1γ cu2) + . . . 16800 δmx (u1γ bcdenu2) + · · · − 133600 (u1γ bcdemnxu2) [mn][bc][de] (27 terms) A good check on the sign of the epsilon contributions is that X̃ ′2111 is recovered when contracting with ηnx, involving a cancellation of all γ [5] terms. To obtain X̃2210, one multiplies by −ηcx: X̃2210 = δdemn(u1γ bu2) + δbdmn(u1γ eu2) + δbmde (u1γ nu2) + 20160 δdm(u1γ benu2) δbm(u1γ denu2) + 20160 δbd(u1γ emnu2) + bdemnu2) [de][mn] For the calculation of X2030 and X2012, one may first evaluate a more general correlator 〈(λγaθ)(θγau1)(λγ[mu2)(λγn]γbcθ)(θγxγdeθ)〉 and then contract with ηcx and ηnx, respec- tively. The results are X̃2030 = δdemn(u1γ bu2) + δbdmn(u1γ eu2)− 11440δ de (u1γ nu2)− 1710080δ m(u1γ benu2) 10080 δbm(u1γ denu2)− 11440δ d(u1γ emnu2) + bdemnu2) [mn][de] X̃2012 = δdebm(u1γ cu2) + δbcdm(u1γ eu2)− 11440δ de(u1γ mu2) + δdm(u1γ bceu2) 10080 δbm(u1γ cdeu2) + 10080 δbd(u1γ cemu2)− 13360 (u1γ bcdemu2) [bc][de] – 11 – After multiplication with the momenta and polarisations, all individual contributions are gauge invariant and can be expanded in the basis C1 . . . C5 listed in (A.6): (1 + π34)X4010 = 483840 (−6,−16,−40, 6, 0)C1 ...C5 (1 + π34)X2210 = 483840 (−18,−104,−176, 18, 0)C1 ...C5 (1 + π34)X2030 = 483840 (−21, 42,−42, 21, 0)C1 ...C5 (1 + π34)X2012 = 483840 (−39, 78,−78, 39, 0)C1 ...C5 X ′2111 = − i11520 (1, 0, 4,−1, 0)C1 ...C5 The sum can be written as K2B2F1-loop = X 2111 = − i3840 (1, 0, 4,−1, 0)C1 ...C5 = − i s13(u2/ζ3(/k2 + /k3)/ζ4u1) + s23(u2/ζ4(/k2 + /k4)/ζ3u1) (3.6) and again agrees with the amplitude computed in the RNS result, see [16] eq. (3.37). 4. Two-loop amplitudes The pure spinor formalism was used in [4, 2] to compute the two-loop type-IIB amplitude involving four massless states, d2Ω11d 2Ω12d i,j ki · kj G(zi, zj) (det ImΩ)5 K2-loop(ki, zi) , where Ω is the genus-two period matrix, and the integration over fermionic zero modes is encapsulated in K2-loop = ∆12∆34 (λγmnpqrλ)(λγsW1)F2,mnF3,pqF4,rs perm(1234) (4.1) ≡ ∆12∆34K12 +∆13∆24K13 +∆14∆23K14 . (4.2) The kinematic factors K12, K13, K14 are accompanied by the basic antisymmetric biholo- morphic 1-form ∆, which is related to a canonical basis ω1, ω2 of holomorphic differentials via ∆ij = ∆(zi, zj) = ω1(zi)ω2(zj) − ω2(zi)ω1(zj). The superfields Wαi and Fi,mn are the spinor and vector field strengths of the i-th external state, as in section 3. One encounters superspace integrals of the form Y (abcd) = (λγmnpqrλ)(λγsWa)Fb,mnFc,pqFd,rs . (4.3) The symmetries of the λ3 combination [4] in this correlator include the obvious symmetry under mn↔ pq, and also (λγ[mnpqrλ)(λγs])α = 0 (this holds for pure spinors λ and can be seen by dualising, and holds for unconstrained spinors λ as part of a λ3θ5 scalar, as seen from the representation content (2.8)), and allow one to shuffle the F factors: Y (abcd) = Y (acbd) , Y (abcd) + Y (acdb) + Y (adbc) = 0 . (4.4) – 12 – 4.1 Review: four bosons The case of four Neveu-Schwarz states was considered in [6] and will be briefly reviewed here. As all three kinematic factors K12, K13 and K14 are equivalent, it is sufficient to consider K12 in detail. With all external states being identical, the symmetrisations of (4.1) can be carried out at the end of the calculation: K4B12 = 4 W[1F2]F[3F4] W[3F4]F[1F2] = (1− π12)(1− π34)(1 + π13π24) W1F2F3F4 Expanding the superfields and adopting the notation YABCD(abcd) = (λγmnpqrλ)(λγsW (A)a )F F (C)c,pqF the Neveu-Schwarz states come from terms of the form YABCD ≡ YABCD(1234) with A odd and B, C, D even. Using the shuffling identities (4.4) to simplify, one obtains W1F2F3F4 = Y5000 + Y1400 + Y1040 + Y1004 + Y3200 + Y3020 + Y3002 + Y1220 + Y1202 + Y1022 = (1 + π23)(1− π24) Y5000 + Y1400 + Y3200 + Y1022 and therefore K4B12 can be written as the image of a symmetrisation operator S4B: K4B12 = S4B Y5000 + Y1400 + Y3200 + Y1022 S4B = (1− π12)(1− π34)(1 + π13π24)(1 + π23)(1− π24) It is worth noting at this point that, on the sixteen-dimensional space of Lorentz scalars built from the four field strengths Fi and two momenta, the symmetriser S4B has rank four. The correlators were computed in [6], using the method outlined in section 2.1. Two are zero, Y5000 = Y1400 = 0, and the remaining ones are Y3200 = (λγmnpqrλ)(λγsγabθ)(θγb cdθ)(θγn Y1022 = F 1abF (λγmnpq[rλ)(λγs]γabθ)(θγq cdθ)(θγs In reducing those two contributions to a set of independent scalars, one finds that they both are not just sums of (k · k)F 4 terms but also contain terms of the form k · F terms. The latter are projected out by the symmetriser S4B, and the result is K4B12 = S4B(Y3200 + Y1022) = 1120 (s13 − s23) 4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4)) (s13 − s23)t8F 4 . By trivial index exchange, one obtains K13 and K14, and the total is K4B2-loop = (s13 − s23)∆12∆34 + (s12 − s23)∆13∆24 + (s12 − s13)∆14∆23 4 , (4.5) a product of the completely symmetric one-loop kinematic factor t8F 4 and a completely symmetric combination of the momenta and the ∆ij. – 13 – 4.2 Four fermions The calculation involving four Ramond states is very similar to the bosonic one. Focussing on the K12 part, the symmetrisations in (4.1) can again be rewritten as action of sym- metrisation operators on the correlator of superfields with one particular labelling: K4F12 (ûi) = (1− π12)(1 − π34)(1 + π13π24) W1F2F3F4 û1û2û3û4 = 4(1− π12) W1F2F3F4 û1û2û3û4 The last step follows from the fact that all scalars of the form k4u4 (see appendix A.2), and therefore all k4û4 scalars, are invariant under π13π24 and have π12 = π34. This time, on expanding the superfields, one collects the terms YABCD with A even and B, C, D odd. After using (4.4) to simplify, W1F2F3F4 û1û2û3û4 = Y2111 + Y0311 + Y0131 + Y0113 = (1 + π23)(1− π24) Y2111 + Y0311 and after translating to commuting wavefunctions ui, which multiplies every permutation operator with its signature, one obtains K4F12 (ui) = S4F Y2111(ui) + Y0311(ui) , S4F = 4(1 + π12)(1− π23)(1 + π24) . This symmetriser has rank three, and the result is again not determined by symmetry. Two correlators have to be computed: Y2111(ui) = (−2)k1ak2mk3pk4r (λγmnpq[rλ)(λγs]γabθ)(θγbu1)(θγnu2)(θγqu3)(θγsu4) Y0311(ui) = (−23)k (λγmnpq[rλ)(λγs]u1)(θγn abθ)(θγbu2)(θγqu3)(θγsu4) With four fermions present, the method of section 2.2 is preferred as it does not involve re- arranging the fermions using Fierz identities. The first correlator was covered as an example in that section, and the second one can be evaluated in the same fashion. Expressed in the basis listed in (A.5), the results are Y2111(ui) = (−19,−21, 21, 19,−17,−17, 0, 0, 0, 0)B1 ...B10 , Y0311(ui) = 15120 (−14,−16, 0, 2,−8,−8, 0,−5,−5, 0)B1 ...B10 . After acting with the symmetriser S4F, one obtains the same u4 scalar encountered in the one-loop amplitude, K4F12 (ui) = S4F(13Y2111(ui) + Y0311(ui)) = (−1,−2, 1, 2,−1,−2, 0, 0, 0, 0)B1 ...B10 (s23 − s13) (u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3) The K13 and K14 parts again follow by index exchange, and the total result K4F2-loop(ui) = (s23 − s13)∆12∆34 + (s23 − s12)∆13∆24 + (s13 − s12)∆14∆23 (u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3) (4.6) is again a simple product of the one-loop kinematic factor and a combination of the ∆ij and momenta. – 14 – 4.3 Two bosons, two fermions As in the one-loop calculation of section 3.3, in the mixed case one has to pay some attention to the permutations in (4.1) since they affect which superfields contribute fermionic fields. The complete symmetrisation makes it irrelevant which labels are assigned to the two fermions, and the convention f1f2b3b4 will be used here. The kinematic factor K 12 is then distinguished from the other two, K2B2F13 and K 14 . Carrying out the symmetrisations in (4.1) and using the identities (4.4), one finds K12(û1, û2, ζ3, ζ4) = (1− π12)(1− π34)K̃ , K13(û1, û2, ζ3, ζ4) = (2 · 1+ π12 + π34 + 2π12π34)K̃ , K14(û1, û2, ζ3, ζ4) = (1+ 2π12 + 2π34 + π12π34)K̃ , where, schematically, (even) (odd) (even) (even) (odd) (even) (odd) (odd) . (4.7) In translating to commuting variables u1 and u2, the permutation operator π12 changes sign, and therefore2 K12(u1, u2, ζ3, ζ4) = (1+ π12)(1− π34)K̃ , K13(u1, u2, ζ3, ζ4) = (2 · 1− π12 + π34 − 2π12π34)K̃ , K14(u1, u2, ζ3, ζ4) = (1− 2π12 + 2π34 − π12π34)K̃ . Expanding the superfields, the contributions to K̃ are: Y4100 = − i48k (λγmnpqrλ)(λγsγabθ)(θγbγ cdθ)(θγcu1)(θγnu2) Y0500 = (λγmnpqrλ)(λγsu1)(θγn abθ)(θγb cdθ)(θγdu2) Y0140 = (λγmnpqrλ)(λγsu1)(θγnu2)(θγq abθ)(θγb Y0104 = (λγmnpqrλ)(λγsu1)(θγnu2)(θγ|s] abθ)(θγb Y2300 = (λγmnpqrλ)(λγsγabθ)(θγbu1)(θγn cdθ)(θγeu2) Y2120 = (λγmnpqrλ)(λγsγabθ)(θγbu1)(θγnu2)(θγq Y2102 = (λγmnpqrλ)(λγsγabθ)(θγbu1)(θγnu2)(θγ|s] Y0320 = (λγmnpqrλ)(λγsu1)(θγn abθ)(θγbu2)(θγq Y0302 = (λγmnpqrλ)(λγsu1)(θγn abθ)(θγbu2)(θγ|s] Y0122 = (λγmnpqrλ)(λγsu1)(θγnu2)(θγq abθ)(θγs] Y3011 = (λγmnpqrλ)(λγsγabθ)(θγb cdθ)(θγcu1)(θγnu2) Y1211 = F 3abk (λγmnpqrλ)(λγsγabθ)(θγn cdθ)(θγqu1)(θγ|s]u2) Y1031 = F 3abF (λγmnpqrλ)(λγsγabθ)(θγq cdθ)(θγdu1)(θγ|s]u2) Y1013 = F 3abF (λγmnpqrλ)(λγsγabθ)(θγqu1)(θγ|s] cdθ)(θγdu2) 2This sign change is crucial to avoid the erroneous conclusion that the two-boson, two-fermion kinematic factor cannot be of the same product form as in the four-boson or four-fermion cases, which would be in contradiction to the supersymmetric identities derived in [18]. – 15 – These correlators can be evaluated exactly as described in section 3.3. One finds that Y0500 = Y0140 = Y0104 = 0, and the sum of the remaining terms reduces to K̃ = Y4100 + Y2300 + Y2120 + Y2102 + Y0320 + Y0302 + Y0122 + Y3011 + Y1211 + Y1031 + Y1013 (s12 + s13)× (1, 0, 4,−1, 0)C1 ...C5 . After applying the symmetrisation operators, (1+ π12)(1− π34)K̃ = i180 (s12 + 2s13)× (1, 0, 4,−1, 0)C1 ...C5 , (2 · 1− π12 + π34 − 2π12π34)K̃ = i180 (2s12 + s13)× (1, 0, 4,−1, 0)C1 ...C5 , (1− 2π12 + 2π34 − π12π34)K̃ = i180 (s12 − s13)× (1, 0, 4,−1, 0)C1 ...C5 , the total kinematic factor is seen to be K2-loop(u1, u2, ζ3, ζ4) = − i180 (s23−s13)∆12∆34+(s23−s12)∆12∆34+(s13−s12)∆12∆34 × (1, 0, 4,−1, 0)C1 ...C5 (4.8) and displays the same simple product form as in the four-boson and four-fermion case. 5. Discussion In this paper, different methods were discussed to efficiently evaluate the superspace inte- grals appearing in multiloop amplitudes derived in the pure spinor formalism. Extending previous calculations [6, 7] restricted to Neveu-Schwarz states, it was then shown how the treatment of Ramond states poses no additional difficulties. While the bosonic calculations of [6, 7] have, in conjunction with supersymmetry, already established the equivalence of the massless four-point amplitudes derived in the pure spinor and RNS formalisms, it would be interesting to make contact between the results of sections 4.2 / 4.3 and two-loop amplitudes involving Ramond states as computed in the RNS formalism (see for example [19]). The assistance of a computer algebra system seems indispensible in explicitly evaluat- ing pure spinor superspace integrals. To avoid excessive use of custom-made algorithms, it would be desirable to implement these calculations in a wider computational framework particular adapted to field theory calculations [20]. The methods outlined in this paper should be easily applicable to future higher-loop amplitude expressions derived from the pure spinor formalism, and, it is hoped, to other superspace integrals. Acknowledgements The author would like to thank Louise Dolan for discussions, and Carlos Mafra for valuable correspondence. This work is supported by the U.S. Department of Energy, grant no. DE- FG01-06ER06-01, Task A. – 16 – A. Reduction to kinematic bases In calculating scattering amplitudes one encounters kinematic factors which are Lorentz invariant polynomials in the momenta, polarisations and/or spinor wavefunctions of the scattered particles. It can be a non-trivial task to simplify such expressions, taking into account the on-shell identities i ki = 0, k i = 0, ki · ζi = 0, /kiui = 0, and, in the case of fermions, re-arrangements stemming from Fierz identities. More generally, one would like to know how many independent combinations of some given fields (subject to on-shell identitites) there are, and how to reduce an arbitrary expres- sion with respect to some chosen basis. This appendix outlines methods to address these problems, with an emphasis on algorithms which can easily be transferred to a computer algebra system. These methods are not limited to dealing with pure spinor calculations but the scope will be restricted to amplitudes of four massless vector or spinor particles in ten dimensions. A.1 Four bosons It is not difficult to reduce polynomials in the momenta and polarisations to a canonical form. The momentum conservation constraint i ki = 0 is solved by eliminating one momentum (for example k4), all k i are set to zero, and one of the two remaining quadratic combinations of momenta is eliminated (for example s23 → −s12− s13, where sij ≡ ki ·kj). Then all products ki · ζi are set to zero, and one extra k · ζ product is replaced (when eliminating k4, the replacement is k3 · ζ4 → (−k1 − k2) · ζ4). The remaining monomials are then independent. (This is at least the case with the low powers of momenta encountered in the calculations of sections 3 and 4, where there are enough spatial directions for all momenta/polarisations to be linearly independent.) The implementation of these reduction rules on a computer is straightforward. The easiest way to obtain scalars which are also invariant under the gauge symmetry ki → ζi is to start with expressions constructed from the field strengths F abi = 2∂ i . For the one-loop calculations of section 3.1, the relevant basis consists of gauge invariant scalars containing only the four field strengths F1 . . . F4. One finds six independent combinations, Tr(F1F2F3F4) Tr(F1F2F4F3) Tr(F1F3F2F4) Tr(F1F2)Tr(F3F4) Tr(F1F3)Tr(F2F4) Tr(F1F4)Tr(F2F3) In the two-loop calculations of section 4.1, all monomials have two more momenta. There are sixteen independent gauge invariant scalars of the form kkF1F2F3F4, and twelve of them may be constructed from the previous basis by multiplication with s12 and s13: A1 = s12 Tr(F1F2F3F4), A2 = s13 Tr(F1F2F3F4), etc. One choice for the additional four is A13 = k3 · F1 · F2 · k3 Tr(F3F4) A15 = k3 · F1 · F4 · k2 Tr(F2F3) A14 = k4 · F1 · F3 · k2 Tr(F2F4) A16 = k4 · F2 · F3 · k4 Tr(F1F4) . – 17 – As an example application of the computer algorithms, one may check that the symmetri- sation operator of section 4.1, S4B = (1− π12)(1− π34)(1 + π13π24)(1 + π23)(1− π24) , acts as S4BA1 = 8A1 + 4A2 − 4A3 + 4A4 + 8A5 + 16A6 . . . S4BA16 = −6A1 + 6A3 − 6A5 − 12A6 + 32A7 + 3A8 + A9 + 3A10 + A11 + 3A12 and has rank four. A.2 Four fermions In dealing with the spinor wavefunctions ui one has to face two issues: Fierz identities, and the Dirac equation. Fierz identities not only allow one to change the order of the spinors but also give rise to relations between different expressions in one spinor order. The Dirac equation often simplifies terms with momenta contracted into (uiγ [n]uj) bilinears. In this section it is shown how to construct bases for terms of the form (k2 or k4) × u1u2u3u4. A significant simplification comes from noting that the Dirac equation allows one to rewrite (uiγ [n]uj) bilinears into terms with lower n if more than one momentum is contracted into the γ[n]. A good first step is therefore to disregard the momenta temporarily and find all independent scalars and two-index tensors built from u1, . . . , u4. From the SO(10) representation content, (S+)⊗4 = 2 · 1+ 6 · + 3 ·˜+ (tensors with rank > 2) , one expects two scalars and nine 2-tensors. The scalars are easily found by considering, as in [21], T1(1234) = (u1γ au2)(u3γau4) , T3(1234) = (u1γ abcu2)(u3γabcu4) . and similarly for the other two inequivalent orders of the four spinors. (Note there is no T5 because of self-duality of the γ[5].) From Fierz transformations, one learns that all T3 terms can be reduced to T1 by T3(1234) = −12T1(1234)− 24T1(1324) and permutations, and the identity (γa)(αβ(γ a)γ)δ = 0 implies that T1(1234) + T1(1324) + T1(1423) = 0, leaving for example T1(1234) and T1(1324) as independent scalars. Generalising this approach to two-index tensors, it turns out that it is sufficient to start with T11(1234) = (u1γ mu2)(u3γ nu4) , T31(1234) = (u1γ aγmγnu2)(u3γau4) , T33(1234) = (u1γ abγmu2)(u3γabγ nu4) , – 18 – and permutations of the spinor labels. It would be very tiresome to systematically apply a variety of Fierz transformations by hand and to find an independent set. Fortunately, by choosing a gamma matrix representation (such as the one listed in appendix B) and reducing all expressions to polynomials in the independent spinor components u1i , . . . , u this problem can be solved with computer help. As expected, one finds that the Tij(abcd) span a nine-dimensional space, and a basis can be chosen as T11(1234), T11(1324), T11(1423), T11(3412), T11(2413), T11(2314), T31(1234), T31(1324), T31(2314) . (A.1) A typical relation reducing the other Tij(abcd) to this basis is T31(3412) = 2T11(1234) − 2T11(3412) + T31(1324) + T31(2314) + 2ηmnT1(1234) . (A.2) Having solved the first step, it is now easy to include the two or four momenta, taking the Dirac equation into account. Consider first the case of two momenta. Starting from the two-tensors in (A.1), one gets the three independent scalars (u1/k3u2)(u3/k1u4) , (u1/k2u3)(u2/k1u4) , (u1/k2u4)(u2/k1u3) . In addition, there are four products of the two independent scalars T1(1234) and T1(1324) with the two independent momentum invariants s12 and s13. By contracting (A.2) with momenta, one can show that s12T1(1324) − s13T1(1234) = −(u1/k3u2)(u3/k1u4) + (u1/k2u3)(u2/k1u4)− (u1/k2u4)(u2/k1u3) , (A.3) and this relation can be used to eliminate s12T1(1324). (It will become clear later that there are no independent other relations like this one.) There are thus six independent k2u1 · · · u4 scalars: (u1/k3u2)(u3/k1u4) s12 T1(1234) (u1/k2u3)(u2/k1u4) s13 T1(1234) (A.4) (u1/k2u4)(u2/k1u3) s13 T1(1324) Note that there is only one completely antisymmetric combination of those, given by the right hand side of (A.3). Similarly, in the case of four momenta, one finds ten independent k4u1 · · · u4 scalars: B1 = s12 (u1/k3u2)(u3/k1u4) B2 = s13 (u1/k3u2)(u3/k1u4) B3 = s12 (u1/k2u3)(u2/k1u4) B4 = s13 (u1/k2u3)(u2/k1u4) B5 = s12 (u1/k2u4)(u2/k1u3) B6 = s13 (u1/k2u4)(u2/k1u3) (A.5) B7 = s 12 T1(1234) B8 = s12s13 T1(1234) B9 = s 13 T1(1234) B10 = s 13 T1(1324) – 19 – Working in a gamma matrix representation, it is again simple to construct a computer algorithm which reduces any given k2u1 · · · u4 or k4u1 · · · u4 scalar into polynomials of the spinor and momentum components. The Dirac equation can then be solved by breaking up the sixteen-component spinors ui into eight-dimensional chiral spinors u i and u i , as in eq. (B.1). One obtains polynomials in the momentum components kai and the independent spinor components (uci ) 1...8. However, a great disadvantage of this procedure is that it breaks manifest Lorentz invariance. For example, one encounters expressions which contain subsets of terms proportional to the square of a single momentum and are therefore equal to zero, but it is difficult to recognise this with a simple algorithm. The easiest solution is to choose several sets of particular vectors ki satisfying k i = 0 and i ki = 0 and to evaluate all expressions on these vectors. (By choosing integer arithmetic, one easily avoids issues of numerical accuracy.) Substituting these sets of momentum vectors in the bases (A.4) and (A.5) gives full rank six and ten respectively, showing they are indeed linearly independent. Equipped with a computer algorithm for these basis decompositions, one finds, for example, that the symmetriser S4F of section 4.2, S4F = 4(1 + π12)(1 − π23)(1 + π24) , acts on the B1 . . . B10 basis as S4FB1 = −12B4 + 12B5 + 12B6 , . . . S4FB10 = 8B1 + 16B2 − 8B3 − 16B4 + 8B5 + 16B6 − 24B7 − 24B8 − 24B9 and has rank three. A.3 Two bosons, two fermions The combined methods of the last two sections can easily be extended to the mixed case of two bosons and two fermions. In the one-loop calculation of section 3.3, one encounters scalars of the form ku1u2F3F4. A basis of such objects is given by C1 = (u1γ au2)k C2 = (u1γ au2)F C3 = (u1γ au2)F c (A.6) C4 = (u1γ abcu2)F C5 = (u1γ abcu2)F There are two combinations antisymmetric in [12] and symmetric in (34): −C1 + 4C2 +C4 and C2 + C3 . Finally, there are ten independent scalars of the form k3u1u2F3F4 (relevant to the two-loop calculation of section 4.3), and they can all be obtained by multiplication of C1 . . . C5 with the two momentum invariants s12 and s13. – 20 – B. A gamma matrix representation A convenient representation of the SO(1,9) gamma matrices is given by the 32×32 matrices 0 (γa)αβ (γa)αβ 0 where (γ0)αβ = 116 = (γ 0)αβ , (γ9)αβ = −18 0 = −(γ9)αβ , and (γa)αβ = −(γa)αβ, a = 1 . . . 8, is a real, symmetric 16×16 representation for the SO(8) Clifford algebra, (γa)αβ = (σa)T 0 , a = 1 . . . 8 , as given in appendix 5.B of [21]. The matrices Γa satisfy the SO(1,9) Clifford algebra relations, {Γa,Γb} = 2ηab132 , ηab = (+−− · · · −) , and bilinears of chiral spinors (with, say, positive chirality) are constructed as (uΓ[a1...ak]v) = (uγ[a1...ak ]v) = uα(γ[a1)αβ(γ a2)βγ . . . 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The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states.
Introduction 1 2. Zero mode integration 2 2.1 Symmetry considerations and tensorial formulae 3 2.2 A spinorial formula 5 2.3 Component-based approach 7 3. One-loop amplitudes 7 3.1 Review: four bosons 8 3.2 Four fermions 10 3.3 Two bosons, two fermions 10 4. Two-loop amplitudes 12 4.1 Review: four bosons 13 4.2 Four fermions 14 4.3 Two bosons, two fermions 15 5. Discussion 16 A. Reduction to kinematic bases 17 A.1 Four bosons 17 A.2 Four fermions 18 A.3 Two bosons, two fermions 20 B. A gamma matrix representation 21 1. Introduction The quantisation of the ten-dimensional superstring using pure spinors as world-sheet ghosts [1] has overcome many difficulties encountered in the Green-Schwarz (GS) and Ramond-Neveu-Schwarz (RNS) formalisms. Most notably, by maintaining manifest space- time supersymmetry, the pure spinor formalism has yielded super-Poincaré covariant multi- loop amplitudes, leading to new insights into perturbative finiteness of superstring theory [2, 3]. Counting fermionic zero modes is a powerful technique in the computation of loop amplitudes in the pure spinor formalism and has for example been used to show that at least four external states are needed for a non-vanishing massless loop amplitude [2]. Furthermore, the structure of massless four-point amplitudes is relatively simple because all – 1 – fermionic worldsheet variables contribute only through their zero modes. In the expressions derived for the one-loop [2] and two-loop [4] amplitudes, supersymmetry was kept manifest by expressing the kinematic factors as integrals over pure spinor superspace [5] involving three pure spinors λ and five fermionic superspace coordinates θ, K1-loop = (λA)(λγmW )(λγnW )Fmn K2-loop = (λγmnpqrλ)(λγsW )FmnFpqFrs (1.1) where the pure spinor superspace integration is denoted by 〈. . . 〉, and Aα(x, θ), Wα(x, θ) and Fmn(x, θ) are the superfields of ten-dimensional Yang-Mills theory. The kinematic factors in (1.1) have been explicitly evaluated for Neveu-Schwarz states at two loops [6] and one loop [7], and were found to match the amplitudes derived in the RNS formalism [8]. This provided important consistency checks in establishing the validity of the pure spinor amplitude prescriptions. (Related one-loop calculations had been reported in [9].) In this paper, it will be shown how to compute the kinematic factors in (1.1) when the superfields are allowed to contribute fermionic fields, as is relevant for the scattering of fermionic closed string states as well as Ramond/Ramond bosons. It turns out that the calculation of fermionic amplitudes presents no additional difficulties, making (1.1) a good practical starting point for the computation of four-point loop amplitudes in a unified fashion. This practical aspect of the supersymmetric pure spinor amplitudes was also emphasised in [10], where the tree-level amplitudes were used to construct the fermion and Ramond/Ramond form contributions to the four-point effective action of the type II theories. This paper is organised as follows. In section 2, different methods to compute pure spinor superspace integrals are explored. These methods are then applied to the explicit evaluation of the kinematic factors of massless four-point amplitudes at the one-loop level in section 3, and at the two-loop level in section 4. In both these sections, the bosonic calcu- lations are briefly reviewed before separately considering the cases of two and four Ramond states. Particular attention will be paid to the constraints imposed by simple exchange symmetries. An appendix contains algorithms which were used to reduce intermediate expressions encountered in the amplitude calculations to a canonical form. 2. Zero mode integration The calculation of scattering amplitudes in the pure spinor formalism leads to integrals over zero modes of the fermionic worldsheet variables λ and θ. Both θ and λ are 16-component Weyl spinors, the λ are commuting and the θ anticommuting, and λ is subject to the pure spinor constraint (λγmλ) = 0. The amplitude prescriptions [1, 2] require three zero modes of λ and five zero modes of θ to be present, and a Lorentz covariant object T̄ αβγ,δ1...δ5 ≡ λαλβλγθδ1 . . . θδ5 = T̄ (αβγ),[δ1...δ5] (2.1) was constructed such that the Yang-Mills antighost vertex operator V = (λγmθ)(λγnθ)(λγpθ)(θγmnpθ) has = 1 . (2.2) – 2 – In this section, different methods of computing such “pure superspace integrals” are ex- plored. As an example, a typical correlator encountered in the two-loop calculations of section 4 is considered: F (ki, ui) = k (λγmnpq[rλ)(λγs]u1)(θγn abθ)(θγbu2)(θγqu3)(θγsu4) (2.3) Here, ki and ui are the momenta and spinor wavefunctions of the four external particles. 2.1 Symmetry considerations and tensorial formulae One systematic approach to evaluate the zero mode integrals is to find expressions for all tensors that can be formed from (2.1). By Fierz transformations, one can always write the product of two θ spinors as (θγ[3]θ), where γ[k] denotes the antisymmetrised product of k gamma matrices. Due to the pure spinor constraint, the only bilinear in λ is (λγ[5]λ), and it is thus sufficient to consider the three cases (λγ[5]λ)(λ{γ[1] or γ[3] or γ[5]}θ)(θγ[3]θ)(θγ[3]θ) . (2.4) Lorentz invariance then implies that it must be possible to express these tensors as sums of suitably symmetrised products of metric tensors, resulting in a parity-even expression, plus a parity-odd part made up from terms which in addition contain an epsilon tensor. The parity-even parts may be constructed [6] starting from the most general ansatz compatible with the symmetries of the correlator and then using spinor identities along with the normalisation (2.2) to determine all coefficients in the ansatz. Duality properties of the spinor bilinears can be used to determine the parity-odd part [7]. An extensive (and almost exhaustive) list of correlators is found in [11], including the (λγ[1]θ) and (λγ[3]θ) cases of the above list: (λγmnpqrλ)(λγuθ)(θγfghθ)(θγjklθ) = − 4 mnpqr m̄n̄p̄q̄r̄ + εmnpqrm̄n̄p̄q̄r̄ δm̄n̄fg δ (δr̄l δ u + δ u − δr̄uδhl ) [fgh][jkl] (2.5) (λγmnpqrλ)(λγstuθ)(θγfghθ)(θγjklθ) = −24 mnpqr m̄n̄p̄q̄r̄ + εmnpqrm̄n̄p̄q̄r̄ δm̄j δ δq̄sδ u − δkhδr̄u) [fgh][jkl](fgh↔jkl) (2.6) (Here, the brackets (fgh↔ jkl) denote symmetrisation under simultaneous interchange of fgh with ijk, with weight one.) The remaining correlator with the (λγ[5]θ) factor can be derived in the same way, using an ansatz consisting of six parity-even structures. Taking a trace between the two γ[5] factors and noting that (λγmnpqrλ)(λγabcdeθ) . . . (λγmnpq[bλ)(λγcde]θ) . . . one finds a relation to (2.6). This is sufficient to determine all coefficients in the ansatz, and the result is (λγmnpqrλ)(λγabcdeθ)(θγfghθ)(θγjklθ) mnpqr m̄n̄p̄q̄r̄ + εmnpqrm̄n̄p̄q̄r̄ m̄n̄p̄ (−δehδr̄l + 2δel δr̄h) + δm̄n̄ab δcdfgδ (δehδ l − 3δel δr̄h) [abcde][fgh][jkl](fgh↔jkl) (2.7) – 3 – One may find it surprising that the derivation of these tensorial expressions only made use of properties of (pure) spinors, and of the normalisation condition (2.2). However, it can be seen from representation theory that the correlator (2.1) is uniquely characterised, up to normalisation, by its symmetry. To see this, note that [12] the spinor products λ3 and θ5 transform in λ(αλβλγ) : Sym3 S+ = [00003] ⊕ [10001] θ[δ1 . . . θδ5] : Alt5 S+ = [00030] ⊕ [11010] . (2.8) (Here, λ and θ are taken to be in the S+ irrep of SO(1,9), with Dynkin label [00001].) The tensor product of these contains only one copy of the trivial representation. This applies to any spinors λ, which means that the pure spinor property cannot be essential to the derivation of the tensorial identities. The use of the pure spinor constraint merely allows for simpler derivations of the same identities. As an illustration of this approach, consider the correlator of eq. (2.3). Leaving the momenta aside for the moment by setting F = k2ak r F̃ , the task is to compute (λγmnpq[rλ)(λγs]u1)(θγn abθ)(θγbu2)(θγqu3)(θγsu4) After applying two Fierz transformations, (λγmnpq[r|λ)(λγcθ)(θγn abθ)(θγjklθ) |s]γcγbu2) 3!·16 (λγmnpq[r|λ)(λγcdeθ)(θγn abθ)(θγjklθ) |s]γcdeγbu2) 2·5!·16 (λγmnpq[r|λ)(λγcdefgθ)(θγn abθ)(θγjklθ) |s]γcdefgγbu2) 3!·16(u3γqγjklγsu4) , one obtains a combination of the fundamental correlators listed in (2.5), (2.6) and (2.7). A reliable evaluation of the numerous index symmetrisations is made possible by the use of a computer algebra program. In doing these calculations with Mathematica, an essential tool is the GAMMA package [13], expanding the products of gamma matrices in a γ[k] basis. The result consists of two parts, F̃ = F̃ (δ) + F̃ (ε), with F̃ (δ) = 1 mpru2)(u3γ au4) + r (u1γ iu2)(u3γiu4) + . . . ai1i2u2)(u3γ i1i2u4) (92 terms) (2.9) F̃ (ε) = − 1 1209600 εi1...i7 mpr(u1γ i1...i7u2)(u3γ au4) + . . . 604800 εampri1...i6(u1γ i3...i9u2)(u3γ i7i8i9u4) (34 terms) (2.10) The epsilon tensors in the second part can be eliminated using the fact that the ui are chiral spinors: If all the indices on γ[k]ui are contracted into an epsilon tensor, one uses εi1...ik′j1...jkγ j1...jkγ11 = (−) k(k+1) k! γi1...ik′ , (2.11) where γ11 = 1 εi0...i9γ i0...i9 . More generally, if all but r indices of γ[k]ui are contracted, εi1...ik′ j1...jkγ p1...prj1...jkγ11 = (−) k(k+1) (k′ − r)! pr ...p1 [i1...ir γir+1...i′k] . (2.12) – 4 – The result of these manipulations is F̃ (ε) =− 1 mpru2)(u3γ au4)− 1280δ r (u1γ amiu2)(u3γiu4) + . . . 11200 i1i2i3u2)(u3γ i1i2i3u4) (53 terms) (2.13) (Note that while the epsilon terms in the basic correlator formulae were easily obtained from the delta terms by using Poincaré duality, this cannot be done here in any obvious way.) The last step in the evaluation of (2.3) is to contract with the momenta, F = k2ak r F̃ , and to simplify the expressions using the on-shell identities i ki = 0, k i = 0, /kiui = 0. It is shown in appendix A.2 that there are only ten independent scalars, denoted by B1 . . . B10, that can be formed from four momenta and the four spinors u1 . . . u4. With respect to this basis, the result is F (δ) = 1 48·10080 695s12(u1/k3u2)(u3/k1u4) + · · ·+ 233s213(u1γau2)(u3γau4) (7 terms) 48·10080 (695, 775, 0,−80, 356, 356, 0, 233, 233, 0)B1 ...B10 , F (ε) = 1 48·10080 (−23,−7, 0,−16, 28, 28, 0, 7, 7, 0)B1 ...B10 , F = 1 10080 (14, 16, 0,−2, 8, 8, 0, 5, 5, 0)B1 ...B10 , (2.14) where sij = ki · kj . 2.2 A spinorial formula While the derivation of tensorial identities for correlators of the form (2.4) is relatively straightforward and elegant, it may be a tedious task to transform the expressions encoun- tered in amplitude calculations to match this pattern. As seen in the example calculated above, this is particularly true if additional spinors are involved, making it necessary to ap- ply Fierz transformations. It is therefore desirable to use a covariant correlator expression with open spinor indices. Such an expression was given in [1, 2]: T̄αβγ,δ1...δ5 = N−1 (γm)αδ1(γn)βδ2(γp)γδ3(γmnp) (αβγ)[δ1...δ5] , (2.15) where N is a normalisation constant and the brackets ()[] denote (anti-)symmetrisation with weight one. (Note that the right hand side is automatically gamma-matrix traceless: any gamma-trace (γr)αβ × (γm)α[δ1|(γn)β|δ2|(γp)γ|δ3(γmnp)δ4δ5] = −(γmnr)[δ1δ2(γmnp)δ3δ4(γp)δ5]γ = 0 vanishes due to the double-trace identity (γabθ) α(θγabcθ) = 0, which follows from the fact that the tensor product (Alt3 S+)⊗ S− does not contain a vector representation and therefore the vector (ψγabθ)(θγ abcθ) has to vanish for all spinors ψ, and can also be shown by applying a Fierz transformation.) This prescription was originally motivated [2] by the fermionic expansion of the Yang-Mills antighost vertex operator V , V = Tαβγ,δ1...δ5λ αλβλγθδ1 . . . θδ5 (2.16) Tαβγ,δ1...δ5 = (γm)αδ1(γ n)βδ2(γ p)γδ3(γmnp)δ4δ5 (αβγ)[δ1...δ5] – 5 – where T is related to T̄ by a parity transformation, up to the overall constant N . (Since T̄ is uniquely determined by its symmetries, any covariant expression will be proportional to T̄ , after symmetrisation of the spinor indices, and this is merely the simplest choice.) Equation (2.15) immediately yields an algorithm to convert any correlator into traces of gamma matrices or, if additional spinors are involved, bilinears in those spinors. It is, however, already very tiresome to determine the normalisation constant N by hand. The main advantage of this approach is that it clearly lends itself to implementation on a computer algebra system, which can easily carry out the spinor index symmetrisations, simplify the gamma products (again using the GAMMA package), and compute the traces. For example, N〈V 〉 = (γm)αδ1(γn)βδ2(γp)γδ3(γmnp) (αβγ)[δ1...δ5] (γx)αδ1(γy)βδ2(γz)γδ3(γ xyz)δ4δ5 = − 1 Tr(γxγ m)Tr(γyγ n)Tr(γzγ p)Tr(γxyzγpnm) + . . . Tr(γzγpnmγ zyxγnγxγ p) (60 terms) = 5160960 . The correct normalisation is therefore obtained by setting N = 5160960. Returning to the example correlator (2.3), one finds that the calculation is by far simpler than with the previous method. After carrying out the symmetrisations (αβγ)[δi], one obtains NF̃ = 1 Tr(γxγ mnpq[r|)(u3γqγ xyzγsu4)(u1γ |s]γzγbu2) + . . . (u2γbγ xyzγqu3)(u1γsγyγ mnpq[rγzγ s]u4) , (24 terms) where elementary index re-sorting has reduced the number of terms from 60 to 24. Ex- panding the gamma products leads to NF̃ = 476 δpr (u1γ mu4)(u2γ au3) + · · ·+ 815(u1γ ai1i2i3i4u2)(u3γ i1i2i3i4u4) , (294 terms) which, in contrast to (2.10), contains no epsilon terms as there are not enough free indices present. Note that this intermediate result contains terms with with u1 paired with u3 or u4, so it is not possible to directly compare to eqs. (2.9) and (2.13). However, after contracting with the momenta k2ak r and decomposing the result in the basis B1 . . . B10, one again obtains F = 1 10080 (14, 16, 0,−2, 8, 8, 0, 5, 5, 0)B1 ...B10 , (2.17) in agreement with (2.14). The algorithm just outlined will be the method of choice for all correlator calculations in the later sections of this paper and can easily be applied to a wider range of problems. The only limitation is that the larger the number of gamma matrices and open indices of the correlator, the slower the computer evaluation will be. For example, the correlator considered in eq. (5.2) of [11], mnm1n1...m4n4 (λγpγm1n1θ)(λγqγm2n2θ)(λγrγm3n3θ)(θγmγnγpqrγ m4n4θ) = − 2 m1n1...m4n4 εmnm1n1...m4n4 , (2.18) can still be verified with this method but this already requires substantial runtime. – 6 – 2.3 Component-based approach A third method to evaluate the zero mode integrals consists of choosing a gamma matrix representation, expanding the integrand as a polynomial in spinor components, and then applying (2.15) to the individual monomials. This procedure seems particularly appealing if at some stage of the calculation one works with a matrix representation anyhow, in order to reduce the results to a canonical form (e.g. as outlined in appendix A). An efficient decomposition algorithm (of k4u1u2u3u4 scalars, say) only needs a few non-zero momentum and spinor wavefunction components to distinguish all independent scalars, and therefore k and u can be replaced by sparse vectors. Furthermore, a trivial observation allows for a much quicker numeric evaluation of correlator components than a naive use of (2.15): In view of (2.16), one can equivalently compute the components of the parity- transformed expression V̄ = (λ̄γmθ̄)(λ̄γnθ̄)(λ̄γpθ̄)(θ̄γmnpθ̄), where λ̄ and θ̄ are spinors of chirality opposite to that of λ, θ. In the representation given in appendix B, V̄ coincides with V |λ→λ̄,θ→θ̄, and V = 192λ9λ9λ9θ1θ2θ3θ4θ9 + · · ·+ 480λ1λ2λ3θ1θ9θ10θ13θ15 + . . . (100352 terms) The monomials in the fermionic expansion of V̄ then correspond to the arguments of non-zero correlators, and the coefficients of those monomials are, up to normalisation and symmetry factors, the correlator values. Unfortunately, it turns out that the complexity of typical correlators (e.g. the one given in (2.3)) makes it difficult to carry out the expansion in fermionic components in any straightforward way and limits this method to special applications. For example, the coefficients in (2.18) can be checked relatively easily by choosing particular index values, such as (λγpγ12θ)(λγqγ21θ)(λγrγ34θ)(θγ0γ0γpqrγ 12λ1λ1λ1θ1θ9θ10θ11θ12 + · · ·+ 12λ16λ16λ16θ5θ6θ7θ8θ16 (For fixed values of pqr, one gets no more than about 105 monomials of the form λ3θ5). This approach may thus still be helpful in situations where the result has been narrowed down to a simple ansatz. 3. One-loop amplitudes The amplitude for the scattering of four massless states of the type IIB superstring was computed [2] in the pure spinor formalism as A = KK̄ (Im τ)5 G(zi, zj) ki·kj , (3.1) where G(zi, zj) is the scalar Green’s function, and the kinematic factor is given by the product KK̄ of left- and right-moving open superstring expressions, K1-loop = (λA1)(λγ mW2)(λγ nW3)F4,mn cycl(234) . (3.2) – 7 – Here the indices 1 . . . 4 label the external states and “· · ·+ cycl(234) ” denotes the addition of two other terms obtained by cyclic permutation of the indices 234. The spinor super- field Aα and its supercovariant derivatives, the vector gauge superfield Am = m DαAβ as well as the spinor and vector field strengths Wα = 1 (γm)αβ(DβAm − ∂mAβ) and Fmn = 18(γmn) β = 2∂[mAn], describe ten-dimensional super-Yang-Mills theory. The physical fields of this theory, a gauge boson and a gaugino, are found in the leading components Am| = ζm and Wα| = ûα and correspond to the Neveu-Schwarz and Ramond superstring states. The superfields Aα and W α as well as the gaugino field ûα are anticommuting.1 To facilitate computer calculations involving polynomials in the spinor components, and for easier comparison with the literature, it will be more convenient to work with commuting fermion wavefunctions uα. Fortunately, as the kinematic factors with fermionic external states are multilinear functions of the distinctly labelled spinors ûi, it is straightforward to translate between the two conventions: Any monomial expression in û1 . . . û4 (and possibly fermionic coordinates θ) corresponds to the same expression in u1 . . . u4, multiplied by the signature of the permutation sorting the ûi (and any θ variables) into some fixed order, such as (θ · · · θ)ûα11 û Choosing a gauge where θαAα = 0, the on-shell identities 2D(αAβ) = γ αβAm , DαW β = 1 (γmn)α have been used to derive recursive relations [10, 14, 15] for the fermionic expansion A(n)α = (γmθ)αA (n−1) m , A (θγmW (n−1)) , Wα(n) = − 1 (γmnθ)α∂mA (n−1) where f (n) = 1 θαn · · · θα1(Dα1 · · ·Dαnf)|. These recursion relations were explicitly solved in [10], reducing the fermionic expansion to a simple repeated application of the derivative operator Omq = 12 (θγm qpθ)∂p: A(2k)m = (2k)! [Ok]mqζq , A(2k+1)m = (2k+1)! [Ok]mq(θγqû) . (3.3) With this solution at hand, one has all ingredients to evaluate the kinematic factor (3.2) for the three cases of zero, two, or four fermionic states. 3.1 Review: four bosons The kinematic factor involving four bosons was considered in [7] and this calculation will now be reviewed briefly. First, note that the outcome is not fixed by symmetry: The result must be gauge invariant [2] and therefore expressible in terms of the field strengths F1 . . . F4. The cyclic symmetrisation in (3.2) yields expressions symmetric in F2, F3, F4, and acting on scalars constructed from the Fi only, the (234) symmetrisation is equivalent to complete symmetrisation in all labels (1234). Thus the result must be a linear combination of the 1Thanks to Carlos Mafra for pointing this out. – 8 – two gauge invariant symmetric F 4 scalars, namely the single trace Tr(F(1F2F3F4)) and double trace Tr(F(1F2)Tr(F3F4)), leaving one relative coefficient to be determined. Since all four states are of the same kind, one may first evaluate the correlator for one labelling and then carry out the cyclic symmetrisation: 1-loop = (λA1)(λγ mW2)(λγ nW3)F4,mn cycl (234) The different ways to saturate θ5 result in a sum of terms of the form XABCD = 1 )(λγ 2 )(λγ (3.4) with A+B +C +D = 5 and A, B, C odd, D even: (λA1)(λγ mW2)(λγ nW3)F4,mn = X3110 +X1310 +X1130 +X1112 . Note that X1310 and X1130 are related by exchange of the labels 2 and 3. This exchange can be carried out after computing the correlator, an operation which will in the following be denoted by π23. Using (3.3) for the superfield expansions and replacing ∂m → ikm, one obtains X3110 = − 1512F tuX̃3110 , X̃3110 = (λγ[t|γpqθ)(λγ|u]γrsθ)(λγaθ)(θγ amnθ) X1112 = − 1128 ik tuX̃1112 , X̃1112 = (λγ[m|γpqθ)(λγ|a]γrsθ)(λγnθ)(θγa X1310 = − 1384 ik tuX̃1310 , X̃1310 = (λγ[t|γmaθ)(λγ|u]γrsθ)(λγnθ)(θγa The method outlined in section 2.2 is readily applicable to these correlators. For example, for X3111, the trace evaluation yields X̃3110 = N Tr(γaγ z)Tr(γxyzγ anm)Tr(γxγqpγ [t|)Tr(γyγsrγ |u]) + · · · · · ·+ 1 Tr(γ[u|γrsγzyxγqpγ |t]γxγaγ yγmnaγz) (60 terms) δmprs δ tu − 1315δ rs − 145δ δmnpr δ [mn][pq][rs][tu](pq↔rs) Upon contracting with the field strengths, momenta and polarisations, and symmetrising over the cyclic permutations (234) (with weight 3), one finds that all three contributions are separately gauge invariant: X3110 + cycl(234) = − 11 13440 Tr(F(1F2F3F4)) + Tr(F(1F2)Tr(F3F4)) X1112 + cycl(234) = − 19 53760 Tr(F(1F2F3F4)) + 215040 Tr(F(1F2)Tr(F3F4)) (1 + π23)X1310 + cycl(234) = − 1 10240 4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4)) The sum X3110 +X1112 has the right ratio of single- and double-trace terms to be propor- tional to the well-known result t8F 4, and the last line exhibits the right ratio by itself. The overall kinematic factor is therefore K4B1-loop = − 12560 4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4)) = − 1 15360 4 , (3.5) in agreement with the expressions derived in the RNS [16] and Green-Schwarz [17] for- malisms. – 9 – 3.2 Four fermions The four-fermion kinematic factor could be evaluated in the same way as in the four-boson case by summing up all terms XABCD, A + B + C + D = 5, now with A, B, C even and D odd. Note however that this time, the outcome is fixed by symmetry: The cyclic symmetrisation in (3.2) leads to a completely symmetric dependence on û2, û3, û4, and therefore to a completely antisymmetric dependence on u2, u3, u4. Acting on scalars of the form k2u1u2u3u4, antisymmetrising over [234] is equivalent to antisymmetrising over [1234], and there is only one completely antisymmetric k2u1u2u3u4 scalar. Without further calculation, one can infer that the kinematic factor is proportional to that scalar, K4F1-loop = const · (u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3) which of course agrees with the RNS amplitude (see e.g. [16], eq. (3.67)). 3.3 Two bosons, two fermions In evaluating (3.2) for two bosons and two fermions, the cyclic symmetrisations affect whether the W and F superfields contribute bosons or fermions. Only the label of the Aα superfield stays unaffected, and one has to choose whether it should contribute a boson or a fermion. Since its fermionic expansion starts with the bosonic polarisation vector, A1,α ∼ (/ζ1θ)α, the calculation can be simplified by choosing a labelling where particle 1 is a fermion. (Of course, the final result must be independent of this choice.) The assignment of the other three labels is then irrelevant and will be chosen as f1f2b3b4. Writing out the cyclic permutations, two of the three terms are essentially the same because they are related by interchange of the labels 3 and 4. The kinematic factor is then K2B2F1-loop(f1f2b3b4) = (1 + π34) (even) 1 )(λγ (even) 2 )(λγ (odd) (even) (even) 1 )(λγ (odd) 3 )(λγ (odd) (odd) Unlike in the four-fermion calculation, the result is not fixed by symmetry. There are five independent ku1u2F3F4 scalars (see appendix A, eq. (A.6)), denoted by C1 . . . C5, and there are two independent combinations of these scalars with the required [12](34) symmetry. Expanding the superfields and collecting terms with θ5, the first line yields a combination of terms XABCD with A, B, D odd and C even. There is only one θ 5 combination coming from the second line, which will be denoted by X ′2111 ≡ (−π24)X2111: K2B2F1-loop = (1 + π34) (X4010 +X2210 +X2030 +X2012) +X 2111 , with the correlators X4010 = ζ3c k nX̃4010 , X̃4010 = (λγaθ)(θγa pqθ)(θγpu1)(λγ [mu2)(λγ n]γbcθ) X2210 = − i12k nX̃2210 , X̃2210 = (λγaθ)(θγau1)(λγ [m|γbcθ)(θγcu2)(λγ |n]γdeθ) X2030 = − i36k nX̃2030 , X̃2030 = (λγaθ)(θγau1)(λγ [mu2)(λγ n]γbcθ)(θγc X2012 = − i12k ζ3c k ζ4e X̃2012 , X̃2012 = (λγaθ)(θγau1)(λγ [mu2)(λγ n]γbcθ)(θγn X ′2111 = ζ3c k 2111 , X̃ 2111 = (λγaθ)(θγau1)(λγ [m|γbcθ)(λγ|n]γdeθ)(θγnu2) – 10 – (The numerical coefficient in X ′2111 includes a sign coming from the θ, û ordering: there is an odd number of θs between u1 and u2.) Evaluating these expressions as outlined in section 2.2, the spinor wavefunctions ui present no complication. The last part takes the simplest form: One finds (λγaθ)(θγau1)(λγ mγbcθ)(λγnγdeθ)(θγnu2) = − 1 (2δbcm[d(u1γe]u2) + δ m(u1γ c]deu2)) and therefore X̃ ′2111 = − 1480 δ[bm(u1γ c]γdeu2) + δ m(u1γ e]γbcu2) The result for X̃4010 is X̃4010 = δbqmn(u1γ cu2)− 190δ mq(u1γ nu2) + δbcmn(u1γ qu2)− 12520δ q (u1γ bcnu2) δbq(u1γ cmnu2) + δbm(u1γ cnqu2) + bcmnqu2) [bc][mn] For the evaluation of X̃2210, it is useful to consider the more general correlator (λγaθ)(θγau1)(λγ [m|γbcθ)(λγ|n]γdeθ)(θγxu2) mn(u1γ cu2) + . . . 201600 δmx (u1γ bcdenu2) + · · · − 11403200 (u1γ bcdemnxu2) [mn][bc][de] (27 terms) 9676800 εbcdemni1i2i3i4(u1γ i1i2i3i4xu2)− 12419200εbcdemnxi1i2i3(u1γ i1i2i3u2) . This time, even using the method of section 2.2, there are sufficiently many open indices and long enough traces for epsilon tensors to appear. Using eqs. (2.11) and (2.12), they can be re-written into γ[5,7] terms: (λγaθ)(θγau1)(λγ [m|γbcθ)(λγ|n]γdeθ)(θγxu2) mn(u1γ cu2) + . . . 16800 δmx (u1γ bcdenu2) + · · · − 133600 (u1γ bcdemnxu2) [mn][bc][de] (27 terms) A good check on the sign of the epsilon contributions is that X̃ ′2111 is recovered when contracting with ηnx, involving a cancellation of all γ [5] terms. To obtain X̃2210, one multiplies by −ηcx: X̃2210 = δdemn(u1γ bu2) + δbdmn(u1γ eu2) + δbmde (u1γ nu2) + 20160 δdm(u1γ benu2) δbm(u1γ denu2) + 20160 δbd(u1γ emnu2) + bdemnu2) [de][mn] For the calculation of X2030 and X2012, one may first evaluate a more general correlator 〈(λγaθ)(θγau1)(λγ[mu2)(λγn]γbcθ)(θγxγdeθ)〉 and then contract with ηcx and ηnx, respec- tively. The results are X̃2030 = δdemn(u1γ bu2) + δbdmn(u1γ eu2)− 11440δ de (u1γ nu2)− 1710080δ m(u1γ benu2) 10080 δbm(u1γ denu2)− 11440δ d(u1γ emnu2) + bdemnu2) [mn][de] X̃2012 = δdebm(u1γ cu2) + δbcdm(u1γ eu2)− 11440δ de(u1γ mu2) + δdm(u1γ bceu2) 10080 δbm(u1γ cdeu2) + 10080 δbd(u1γ cemu2)− 13360 (u1γ bcdemu2) [bc][de] – 11 – After multiplication with the momenta and polarisations, all individual contributions are gauge invariant and can be expanded in the basis C1 . . . C5 listed in (A.6): (1 + π34)X4010 = 483840 (−6,−16,−40, 6, 0)C1 ...C5 (1 + π34)X2210 = 483840 (−18,−104,−176, 18, 0)C1 ...C5 (1 + π34)X2030 = 483840 (−21, 42,−42, 21, 0)C1 ...C5 (1 + π34)X2012 = 483840 (−39, 78,−78, 39, 0)C1 ...C5 X ′2111 = − i11520 (1, 0, 4,−1, 0)C1 ...C5 The sum can be written as K2B2F1-loop = X 2111 = − i3840 (1, 0, 4,−1, 0)C1 ...C5 = − i s13(u2/ζ3(/k2 + /k3)/ζ4u1) + s23(u2/ζ4(/k2 + /k4)/ζ3u1) (3.6) and again agrees with the amplitude computed in the RNS result, see [16] eq. (3.37). 4. Two-loop amplitudes The pure spinor formalism was used in [4, 2] to compute the two-loop type-IIB amplitude involving four massless states, d2Ω11d 2Ω12d i,j ki · kj G(zi, zj) (det ImΩ)5 K2-loop(ki, zi) , where Ω is the genus-two period matrix, and the integration over fermionic zero modes is encapsulated in K2-loop = ∆12∆34 (λγmnpqrλ)(λγsW1)F2,mnF3,pqF4,rs perm(1234) (4.1) ≡ ∆12∆34K12 +∆13∆24K13 +∆14∆23K14 . (4.2) The kinematic factors K12, K13, K14 are accompanied by the basic antisymmetric biholo- morphic 1-form ∆, which is related to a canonical basis ω1, ω2 of holomorphic differentials via ∆ij = ∆(zi, zj) = ω1(zi)ω2(zj) − ω2(zi)ω1(zj). The superfields Wαi and Fi,mn are the spinor and vector field strengths of the i-th external state, as in section 3. One encounters superspace integrals of the form Y (abcd) = (λγmnpqrλ)(λγsWa)Fb,mnFc,pqFd,rs . (4.3) The symmetries of the λ3 combination [4] in this correlator include the obvious symmetry under mn↔ pq, and also (λγ[mnpqrλ)(λγs])α = 0 (this holds for pure spinors λ and can be seen by dualising, and holds for unconstrained spinors λ as part of a λ3θ5 scalar, as seen from the representation content (2.8)), and allow one to shuffle the F factors: Y (abcd) = Y (acbd) , Y (abcd) + Y (acdb) + Y (adbc) = 0 . (4.4) – 12 – 4.1 Review: four bosons The case of four Neveu-Schwarz states was considered in [6] and will be briefly reviewed here. As all three kinematic factors K12, K13 and K14 are equivalent, it is sufficient to consider K12 in detail. With all external states being identical, the symmetrisations of (4.1) can be carried out at the end of the calculation: K4B12 = 4 W[1F2]F[3F4] W[3F4]F[1F2] = (1− π12)(1− π34)(1 + π13π24) W1F2F3F4 Expanding the superfields and adopting the notation YABCD(abcd) = (λγmnpqrλ)(λγsW (A)a )F F (C)c,pqF the Neveu-Schwarz states come from terms of the form YABCD ≡ YABCD(1234) with A odd and B, C, D even. Using the shuffling identities (4.4) to simplify, one obtains W1F2F3F4 = Y5000 + Y1400 + Y1040 + Y1004 + Y3200 + Y3020 + Y3002 + Y1220 + Y1202 + Y1022 = (1 + π23)(1− π24) Y5000 + Y1400 + Y3200 + Y1022 and therefore K4B12 can be written as the image of a symmetrisation operator S4B: K4B12 = S4B Y5000 + Y1400 + Y3200 + Y1022 S4B = (1− π12)(1− π34)(1 + π13π24)(1 + π23)(1− π24) It is worth noting at this point that, on the sixteen-dimensional space of Lorentz scalars built from the four field strengths Fi and two momenta, the symmetriser S4B has rank four. The correlators were computed in [6], using the method outlined in section 2.1. Two are zero, Y5000 = Y1400 = 0, and the remaining ones are Y3200 = (λγmnpqrλ)(λγsγabθ)(θγb cdθ)(θγn Y1022 = F 1abF (λγmnpq[rλ)(λγs]γabθ)(θγq cdθ)(θγs In reducing those two contributions to a set of independent scalars, one finds that they both are not just sums of (k · k)F 4 terms but also contain terms of the form k · F terms. The latter are projected out by the symmetriser S4B, and the result is K4B12 = S4B(Y3200 + Y1022) = 1120 (s13 − s23) 4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4)) (s13 − s23)t8F 4 . By trivial index exchange, one obtains K13 and K14, and the total is K4B2-loop = (s13 − s23)∆12∆34 + (s12 − s23)∆13∆24 + (s12 − s13)∆14∆23 4 , (4.5) a product of the completely symmetric one-loop kinematic factor t8F 4 and a completely symmetric combination of the momenta and the ∆ij. – 13 – 4.2 Four fermions The calculation involving four Ramond states is very similar to the bosonic one. Focussing on the K12 part, the symmetrisations in (4.1) can again be rewritten as action of sym- metrisation operators on the correlator of superfields with one particular labelling: K4F12 (ûi) = (1− π12)(1 − π34)(1 + π13π24) W1F2F3F4 û1û2û3û4 = 4(1− π12) W1F2F3F4 û1û2û3û4 The last step follows from the fact that all scalars of the form k4u4 (see appendix A.2), and therefore all k4û4 scalars, are invariant under π13π24 and have π12 = π34. This time, on expanding the superfields, one collects the terms YABCD with A even and B, C, D odd. After using (4.4) to simplify, W1F2F3F4 û1û2û3û4 = Y2111 + Y0311 + Y0131 + Y0113 = (1 + π23)(1− π24) Y2111 + Y0311 and after translating to commuting wavefunctions ui, which multiplies every permutation operator with its signature, one obtains K4F12 (ui) = S4F Y2111(ui) + Y0311(ui) , S4F = 4(1 + π12)(1− π23)(1 + π24) . This symmetriser has rank three, and the result is again not determined by symmetry. Two correlators have to be computed: Y2111(ui) = (−2)k1ak2mk3pk4r (λγmnpq[rλ)(λγs]γabθ)(θγbu1)(θγnu2)(θγqu3)(θγsu4) Y0311(ui) = (−23)k (λγmnpq[rλ)(λγs]u1)(θγn abθ)(θγbu2)(θγqu3)(θγsu4) With four fermions present, the method of section 2.2 is preferred as it does not involve re- arranging the fermions using Fierz identities. The first correlator was covered as an example in that section, and the second one can be evaluated in the same fashion. Expressed in the basis listed in (A.5), the results are Y2111(ui) = (−19,−21, 21, 19,−17,−17, 0, 0, 0, 0)B1 ...B10 , Y0311(ui) = 15120 (−14,−16, 0, 2,−8,−8, 0,−5,−5, 0)B1 ...B10 . After acting with the symmetriser S4F, one obtains the same u4 scalar encountered in the one-loop amplitude, K4F12 (ui) = S4F(13Y2111(ui) + Y0311(ui)) = (−1,−2, 1, 2,−1,−2, 0, 0, 0, 0)B1 ...B10 (s23 − s13) (u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3) The K13 and K14 parts again follow by index exchange, and the total result K4F2-loop(ui) = (s23 − s13)∆12∆34 + (s23 − s12)∆13∆24 + (s13 − s12)∆14∆23 (u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3) (4.6) is again a simple product of the one-loop kinematic factor and a combination of the ∆ij and momenta. – 14 – 4.3 Two bosons, two fermions As in the one-loop calculation of section 3.3, in the mixed case one has to pay some attention to the permutations in (4.1) since they affect which superfields contribute fermionic fields. The complete symmetrisation makes it irrelevant which labels are assigned to the two fermions, and the convention f1f2b3b4 will be used here. The kinematic factor K 12 is then distinguished from the other two, K2B2F13 and K 14 . Carrying out the symmetrisations in (4.1) and using the identities (4.4), one finds K12(û1, û2, ζ3, ζ4) = (1− π12)(1− π34)K̃ , K13(û1, û2, ζ3, ζ4) = (2 · 1+ π12 + π34 + 2π12π34)K̃ , K14(û1, û2, ζ3, ζ4) = (1+ 2π12 + 2π34 + π12π34)K̃ , where, schematically, (even) (odd) (even) (even) (odd) (even) (odd) (odd) . (4.7) In translating to commuting variables u1 and u2, the permutation operator π12 changes sign, and therefore2 K12(u1, u2, ζ3, ζ4) = (1+ π12)(1− π34)K̃ , K13(u1, u2, ζ3, ζ4) = (2 · 1− π12 + π34 − 2π12π34)K̃ , K14(u1, u2, ζ3, ζ4) = (1− 2π12 + 2π34 − π12π34)K̃ . Expanding the superfields, the contributions to K̃ are: Y4100 = − i48k (λγmnpqrλ)(λγsγabθ)(θγbγ cdθ)(θγcu1)(θγnu2) Y0500 = (λγmnpqrλ)(λγsu1)(θγn abθ)(θγb cdθ)(θγdu2) Y0140 = (λγmnpqrλ)(λγsu1)(θγnu2)(θγq abθ)(θγb Y0104 = (λγmnpqrλ)(λγsu1)(θγnu2)(θγ|s] abθ)(θγb Y2300 = (λγmnpqrλ)(λγsγabθ)(θγbu1)(θγn cdθ)(θγeu2) Y2120 = (λγmnpqrλ)(λγsγabθ)(θγbu1)(θγnu2)(θγq Y2102 = (λγmnpqrλ)(λγsγabθ)(θγbu1)(θγnu2)(θγ|s] Y0320 = (λγmnpqrλ)(λγsu1)(θγn abθ)(θγbu2)(θγq Y0302 = (λγmnpqrλ)(λγsu1)(θγn abθ)(θγbu2)(θγ|s] Y0122 = (λγmnpqrλ)(λγsu1)(θγnu2)(θγq abθ)(θγs] Y3011 = (λγmnpqrλ)(λγsγabθ)(θγb cdθ)(θγcu1)(θγnu2) Y1211 = F 3abk (λγmnpqrλ)(λγsγabθ)(θγn cdθ)(θγqu1)(θγ|s]u2) Y1031 = F 3abF (λγmnpqrλ)(λγsγabθ)(θγq cdθ)(θγdu1)(θγ|s]u2) Y1013 = F 3abF (λγmnpqrλ)(λγsγabθ)(θγqu1)(θγ|s] cdθ)(θγdu2) 2This sign change is crucial to avoid the erroneous conclusion that the two-boson, two-fermion kinematic factor cannot be of the same product form as in the four-boson or four-fermion cases, which would be in contradiction to the supersymmetric identities derived in [18]. – 15 – These correlators can be evaluated exactly as described in section 3.3. One finds that Y0500 = Y0140 = Y0104 = 0, and the sum of the remaining terms reduces to K̃ = Y4100 + Y2300 + Y2120 + Y2102 + Y0320 + Y0302 + Y0122 + Y3011 + Y1211 + Y1031 + Y1013 (s12 + s13)× (1, 0, 4,−1, 0)C1 ...C5 . After applying the symmetrisation operators, (1+ π12)(1− π34)K̃ = i180 (s12 + 2s13)× (1, 0, 4,−1, 0)C1 ...C5 , (2 · 1− π12 + π34 − 2π12π34)K̃ = i180 (2s12 + s13)× (1, 0, 4,−1, 0)C1 ...C5 , (1− 2π12 + 2π34 − π12π34)K̃ = i180 (s12 − s13)× (1, 0, 4,−1, 0)C1 ...C5 , the total kinematic factor is seen to be K2-loop(u1, u2, ζ3, ζ4) = − i180 (s23−s13)∆12∆34+(s23−s12)∆12∆34+(s13−s12)∆12∆34 × (1, 0, 4,−1, 0)C1 ...C5 (4.8) and displays the same simple product form as in the four-boson and four-fermion case. 5. Discussion In this paper, different methods were discussed to efficiently evaluate the superspace inte- grals appearing in multiloop amplitudes derived in the pure spinor formalism. Extending previous calculations [6, 7] restricted to Neveu-Schwarz states, it was then shown how the treatment of Ramond states poses no additional difficulties. While the bosonic calculations of [6, 7] have, in conjunction with supersymmetry, already established the equivalence of the massless four-point amplitudes derived in the pure spinor and RNS formalisms, it would be interesting to make contact between the results of sections 4.2 / 4.3 and two-loop amplitudes involving Ramond states as computed in the RNS formalism (see for example [19]). The assistance of a computer algebra system seems indispensible in explicitly evaluat- ing pure spinor superspace integrals. To avoid excessive use of custom-made algorithms, it would be desirable to implement these calculations in a wider computational framework particular adapted to field theory calculations [20]. The methods outlined in this paper should be easily applicable to future higher-loop amplitude expressions derived from the pure spinor formalism, and, it is hoped, to other superspace integrals. Acknowledgements The author would like to thank Louise Dolan for discussions, and Carlos Mafra for valuable correspondence. This work is supported by the U.S. Department of Energy, grant no. DE- FG01-06ER06-01, Task A. – 16 – A. Reduction to kinematic bases In calculating scattering amplitudes one encounters kinematic factors which are Lorentz invariant polynomials in the momenta, polarisations and/or spinor wavefunctions of the scattered particles. It can be a non-trivial task to simplify such expressions, taking into account the on-shell identities i ki = 0, k i = 0, ki · ζi = 0, /kiui = 0, and, in the case of fermions, re-arrangements stemming from Fierz identities. More generally, one would like to know how many independent combinations of some given fields (subject to on-shell identitites) there are, and how to reduce an arbitrary expres- sion with respect to some chosen basis. This appendix outlines methods to address these problems, with an emphasis on algorithms which can easily be transferred to a computer algebra system. These methods are not limited to dealing with pure spinor calculations but the scope will be restricted to amplitudes of four massless vector or spinor particles in ten dimensions. A.1 Four bosons It is not difficult to reduce polynomials in the momenta and polarisations to a canonical form. The momentum conservation constraint i ki = 0 is solved by eliminating one momentum (for example k4), all k i are set to zero, and one of the two remaining quadratic combinations of momenta is eliminated (for example s23 → −s12− s13, where sij ≡ ki ·kj). Then all products ki · ζi are set to zero, and one extra k · ζ product is replaced (when eliminating k4, the replacement is k3 · ζ4 → (−k1 − k2) · ζ4). The remaining monomials are then independent. (This is at least the case with the low powers of momenta encountered in the calculations of sections 3 and 4, where there are enough spatial directions for all momenta/polarisations to be linearly independent.) The implementation of these reduction rules on a computer is straightforward. The easiest way to obtain scalars which are also invariant under the gauge symmetry ki → ζi is to start with expressions constructed from the field strengths F abi = 2∂ i . For the one-loop calculations of section 3.1, the relevant basis consists of gauge invariant scalars containing only the four field strengths F1 . . . F4. One finds six independent combinations, Tr(F1F2F3F4) Tr(F1F2F4F3) Tr(F1F3F2F4) Tr(F1F2)Tr(F3F4) Tr(F1F3)Tr(F2F4) Tr(F1F4)Tr(F2F3) In the two-loop calculations of section 4.1, all monomials have two more momenta. There are sixteen independent gauge invariant scalars of the form kkF1F2F3F4, and twelve of them may be constructed from the previous basis by multiplication with s12 and s13: A1 = s12 Tr(F1F2F3F4), A2 = s13 Tr(F1F2F3F4), etc. One choice for the additional four is A13 = k3 · F1 · F2 · k3 Tr(F3F4) A15 = k3 · F1 · F4 · k2 Tr(F2F3) A14 = k4 · F1 · F3 · k2 Tr(F2F4) A16 = k4 · F2 · F3 · k4 Tr(F1F4) . – 17 – As an example application of the computer algorithms, one may check that the symmetri- sation operator of section 4.1, S4B = (1− π12)(1− π34)(1 + π13π24)(1 + π23)(1− π24) , acts as S4BA1 = 8A1 + 4A2 − 4A3 + 4A4 + 8A5 + 16A6 . . . S4BA16 = −6A1 + 6A3 − 6A5 − 12A6 + 32A7 + 3A8 + A9 + 3A10 + A11 + 3A12 and has rank four. A.2 Four fermions In dealing with the spinor wavefunctions ui one has to face two issues: Fierz identities, and the Dirac equation. Fierz identities not only allow one to change the order of the spinors but also give rise to relations between different expressions in one spinor order. The Dirac equation often simplifies terms with momenta contracted into (uiγ [n]uj) bilinears. In this section it is shown how to construct bases for terms of the form (k2 or k4) × u1u2u3u4. A significant simplification comes from noting that the Dirac equation allows one to rewrite (uiγ [n]uj) bilinears into terms with lower n if more than one momentum is contracted into the γ[n]. A good first step is therefore to disregard the momenta temporarily and find all independent scalars and two-index tensors built from u1, . . . , u4. From the SO(10) representation content, (S+)⊗4 = 2 · 1+ 6 · + 3 ·˜+ (tensors with rank > 2) , one expects two scalars and nine 2-tensors. The scalars are easily found by considering, as in [21], T1(1234) = (u1γ au2)(u3γau4) , T3(1234) = (u1γ abcu2)(u3γabcu4) . and similarly for the other two inequivalent orders of the four spinors. (Note there is no T5 because of self-duality of the γ[5].) From Fierz transformations, one learns that all T3 terms can be reduced to T1 by T3(1234) = −12T1(1234)− 24T1(1324) and permutations, and the identity (γa)(αβ(γ a)γ)δ = 0 implies that T1(1234) + T1(1324) + T1(1423) = 0, leaving for example T1(1234) and T1(1324) as independent scalars. Generalising this approach to two-index tensors, it turns out that it is sufficient to start with T11(1234) = (u1γ mu2)(u3γ nu4) , T31(1234) = (u1γ aγmγnu2)(u3γau4) , T33(1234) = (u1γ abγmu2)(u3γabγ nu4) , – 18 – and permutations of the spinor labels. It would be very tiresome to systematically apply a variety of Fierz transformations by hand and to find an independent set. Fortunately, by choosing a gamma matrix representation (such as the one listed in appendix B) and reducing all expressions to polynomials in the independent spinor components u1i , . . . , u this problem can be solved with computer help. As expected, one finds that the Tij(abcd) span a nine-dimensional space, and a basis can be chosen as T11(1234), T11(1324), T11(1423), T11(3412), T11(2413), T11(2314), T31(1234), T31(1324), T31(2314) . (A.1) A typical relation reducing the other Tij(abcd) to this basis is T31(3412) = 2T11(1234) − 2T11(3412) + T31(1324) + T31(2314) + 2ηmnT1(1234) . (A.2) Having solved the first step, it is now easy to include the two or four momenta, taking the Dirac equation into account. Consider first the case of two momenta. Starting from the two-tensors in (A.1), one gets the three independent scalars (u1/k3u2)(u3/k1u4) , (u1/k2u3)(u2/k1u4) , (u1/k2u4)(u2/k1u3) . In addition, there are four products of the two independent scalars T1(1234) and T1(1324) with the two independent momentum invariants s12 and s13. By contracting (A.2) with momenta, one can show that s12T1(1324) − s13T1(1234) = −(u1/k3u2)(u3/k1u4) + (u1/k2u3)(u2/k1u4)− (u1/k2u4)(u2/k1u3) , (A.3) and this relation can be used to eliminate s12T1(1324). (It will become clear later that there are no independent other relations like this one.) There are thus six independent k2u1 · · · u4 scalars: (u1/k3u2)(u3/k1u4) s12 T1(1234) (u1/k2u3)(u2/k1u4) s13 T1(1234) (A.4) (u1/k2u4)(u2/k1u3) s13 T1(1324) Note that there is only one completely antisymmetric combination of those, given by the right hand side of (A.3). Similarly, in the case of four momenta, one finds ten independent k4u1 · · · u4 scalars: B1 = s12 (u1/k3u2)(u3/k1u4) B2 = s13 (u1/k3u2)(u3/k1u4) B3 = s12 (u1/k2u3)(u2/k1u4) B4 = s13 (u1/k2u3)(u2/k1u4) B5 = s12 (u1/k2u4)(u2/k1u3) B6 = s13 (u1/k2u4)(u2/k1u3) (A.5) B7 = s 12 T1(1234) B8 = s12s13 T1(1234) B9 = s 13 T1(1234) B10 = s 13 T1(1324) – 19 – Working in a gamma matrix representation, it is again simple to construct a computer algorithm which reduces any given k2u1 · · · u4 or k4u1 · · · u4 scalar into polynomials of the spinor and momentum components. The Dirac equation can then be solved by breaking up the sixteen-component spinors ui into eight-dimensional chiral spinors u i and u i , as in eq. (B.1). One obtains polynomials in the momentum components kai and the independent spinor components (uci ) 1...8. However, a great disadvantage of this procedure is that it breaks manifest Lorentz invariance. For example, one encounters expressions which contain subsets of terms proportional to the square of a single momentum and are therefore equal to zero, but it is difficult to recognise this with a simple algorithm. The easiest solution is to choose several sets of particular vectors ki satisfying k i = 0 and i ki = 0 and to evaluate all expressions on these vectors. (By choosing integer arithmetic, one easily avoids issues of numerical accuracy.) Substituting these sets of momentum vectors in the bases (A.4) and (A.5) gives full rank six and ten respectively, showing they are indeed linearly independent. Equipped with a computer algorithm for these basis decompositions, one finds, for example, that the symmetriser S4F of section 4.2, S4F = 4(1 + π12)(1 − π23)(1 + π24) , acts on the B1 . . . B10 basis as S4FB1 = −12B4 + 12B5 + 12B6 , . . . S4FB10 = 8B1 + 16B2 − 8B3 − 16B4 + 8B5 + 16B6 − 24B7 − 24B8 − 24B9 and has rank three. A.3 Two bosons, two fermions The combined methods of the last two sections can easily be extended to the mixed case of two bosons and two fermions. In the one-loop calculation of section 3.3, one encounters scalars of the form ku1u2F3F4. A basis of such objects is given by C1 = (u1γ au2)k C2 = (u1γ au2)F C3 = (u1γ au2)F c (A.6) C4 = (u1γ abcu2)F C5 = (u1γ abcu2)F There are two combinations antisymmetric in [12] and symmetric in (34): −C1 + 4C2 +C4 and C2 + C3 . Finally, there are ten independent scalars of the form k3u1u2F3F4 (relevant to the two-loop calculation of section 4.3), and they can all be obtained by multiplication of C1 . . . C5 with the two momentum invariants s12 and s13. – 20 – B. A gamma matrix representation A convenient representation of the SO(1,9) gamma matrices is given by the 32×32 matrices 0 (γa)αβ (γa)αβ 0 where (γ0)αβ = 116 = (γ 0)αβ , (γ9)αβ = −18 0 = −(γ9)αβ , and (γa)αβ = −(γa)αβ, a = 1 . . . 8, is a real, symmetric 16×16 representation for the SO(8) Clifford algebra, (γa)αβ = (σa)T 0 , a = 1 . . . 8 , as given in appendix 5.B of [21]. The matrices Γa satisfy the SO(1,9) Clifford algebra relations, {Γa,Γb} = 2ηab132 , ηab = (+−− · · · −) , and bilinears of chiral spinors (with, say, positive chirality) are constructed as (uΓ[a1...ak]v) = (uγ[a1...ak ]v) = uα(γ[a1)αβ(γ a2)βγ . . . (γak ])γδv This representation is particularly suitable for the calculations outlined in appendix A because it allows a simple decomposition of SO(1,9) spinors into SO(8) spinors due to its block structure: Γ0 · · ·Γ9 = 116 0 0 −116 , Γ1 · · ·Γ8 = 18 0 0 0 0 −18 0 0 0 0 18 0 0 0 0 −18 Therefore, the Dirac equation for a chiral 16-component spinor u, (γa)αβ∂au α = 0 , can be solved by splitting u into two chiral eight-component spinors of SO(8), with γ1...8 One obtains the coupled equations (∂0 + ∂9)u s − (σ · ∂)uc = 0 (∂0 − ∂9)uc − (σT · ∂)us = 0 (with eight-dimensional dot products). These can be solved for us in terms of uc: (σ · ∂)uc = (σ · k)uc , (B.1) where k+ = −i∂+ = −i√ (∂0 + ∂9). – 21 – References [1] N. Berkovits, Super-Poincaré covariant quantization of the superstring, J. High Energy Phys. 04 (2000) 018 [hep-th/0001035]. [2] N. Berkovits, Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, J. High Energy Phys. 09 (2004) 047 [hep-th/0406055]. [3] N. Berkovits, New higher-derivative R4 theorems [hep-th/0609006]. [4] N. Berkovits, Super-Poincaré covariant two-loop superstring amplitudes, J. High Energy Phys. 01 (2006) 005 [hep-th/0503197]. [5] N. Berkovits, Explaining pure spinor superspace [hep-th/0612021]. [6] N. Berkovits and C.R. Mafra, Equivalence of two-loop superstring amplitudes in the pure spinor and RNS formalisms, Phys. Rev. Lett. 96 (2006) 011602 [hep-th/0509234]. [7] C.R. Mafra, Four-point one-loop amplitude computation in the pure spinor formalism, J. High Energy Phys. 01 (2006) 075 [hep-th/0512052]. [8] E. D’Hoker and D.H. Phong, Two loop superstrings, 1. Main formulas, Phys. Lett. B 529 (2002) 241, [hep-th/0110247]. [9] L. Anguelova, P.A. Grassi and P. Vanhove, Covariant one-loop amplitudes in D = 11, Nucl. Phys. B 702 (2004) 269 [hep-th/0408171]. [10] G. Policastro and D. Tsimpis, R4, purified, Class. and Quant. Grav. 23 (2006) 4753 [hep-th/0603165]. [11] N. Berkovits and C.R. Mafra, Some superstring amplitude computations with the non-minimal pure spinor formalism, J. High Energy Phys. 11 (2006) 079 [hep-th/0607187]. [12] A. Cohen, M. van Leeuwen and B. Lisser, LiE: A Computer algebra package for Lie group computations, v. 2.2 (1998), http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/ [13] U. Gran, GAMMA: A Mathematica package for performing gamma-matrix algebra and Fierz transformations in arbitrary dimensions [hep-th/010508]. [14] H. Ooguri, J. Rahmfeld, H. Robins, J. Tannenhauser, Holography in superspace, J. High Energy Phys. 07 (2000) 045 [hep-th/0007104]. [15] P.A. Grassi, L. Tamassia, Vertex operators for closed superstrings, J. High Energy Phys. 07 (2004) 071 [hep-th/0405072]. [16] J.J. Atick and A. Sen, Covariant one loop fermion emission amplitudes in closed string theories, Nucl. Phys. B 293 (1987) 317. [17] M. B. Green and J. H. Schwarz, Supersymmetrical dual string theory. 3. Loops and renormalisation, Nucl. Phys. 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704.002
Lifetime of doubly charmed baryons Chao-Hsi Chang1,2 ∗, Tong Li3†, Xue-Qian Li3‡ and Yu-Ming Wang4§ 1CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, P.R. China 2Institute of Theoretical Physics, Chinese Academy of Sciences, P.O.Box 2735, Beijing 100080, P.R. China 3 Department of Physics, Nankai University, Tianjin, 300071, P.R. China 4Institute of High Energy Physics, Chinese Academy of Sciences, P.O.Box 918, Beijing 100049, P.R. China Abstract In this work, we evaluate the lifetimes of the doubly charmed baryons Ξ+cc, Ξ cc and Ω cc. We care- fully calculate the non-spectator contributions at the quark level where the Cabibbo-suppressed di- agrams are also included. The hadronic matrix elements are evaluated in the simple non-relativistic harmonic oscillator model. Our numerical results are generally consistent with that obtained by other authors who used the diquark model. However, all the theoretical predictions on the lifetimes are one order larger than the upper limit set by the recent SELEX measurement. This discrepancy would be clarified by the future experiment, if more accurate experiment still confirms the value of the SELEX collaboration, there must be some unknown mechanism to be explored. ∗ email: zhangzx@itp.ac.cn † email: allongde@mail.nankai.edu.cn ‡ email: lixq@nankai.edu.cn § email: wangym@mail.ihep.ac.cn http://arxiv.org/abs/0704.0016v1 I. INTRODUCTION The quite large difference of the lifetimes between D± and D0 and the lifetimes close to each other for B± and B0 are well explained by taking into account the non-spectator effects[1]. This success implies that the mechanism which governs the reactions at quark level is well understood. When we apply the mechanism to the heavy baryon case, some problems emerge. The famous puzzle in the heavy-flavor field that the lifetime of Λb is remarkably shorter than that of B meson is much alleviated recently when the operators of higher dimensions are taken into account[2, 3]. The more recent experimental value of the ratio τ(Λb)/τ(B 0) = 1.041± 0.057[4] is close to the theoretical evaluation[3]. However, in the theoretical works, one can notice that the evaluation of hadronic matrix elements is still very rough and based on some approximations. The possible errors brought up by the uncertainties in the hadronic matrix elements are still uncontrollable. In our recent work[5], we find that the short-distance contributions to the branching ratio of Λb → Λγ which is evaluated in the PQCD approach, are much smaller than that from long-distance effects. Therefore, even though one has a full reason to believe that the low-energy QCD should solve the discrepancy if it exists, he must find a proper way to deal with the hadronic matrix elements. The observation of doubly charmed baryon Ξ+cc by the SELEX Collaboration at FERMILAB[6] provides an opportunity to investigate the hidden problems. Hopefully the study may shed some lights on the unknown non-perturbative QCD effects which result in obvious difference between baryons and mesons. Because Ξ+cc contains two heavy quarks, by the heavy quark effective theory (HQET) the situation may become relatively simple and clear compared to the case of Λb or Λc which possesses only one heavy quark. Thus a careful study on the Ξ+cc is necessary and interesting. Several groups already investigated the two-heavy-flavor baryons a long time ago[7, 8]. In their work, the evaluation of the hadronic matrix elements is based on the quark-diquark structure of the baryons. This is definitely reasonable, it is believed that two heavy quarks can constitute a more stable and compact color-anti-triplet diquark[9]. However, since charm quark, even b-quark, is not so heavy that the degree of freedom of the light flavor can be ignored, the diquark scenario may bring up certain errors, especially when evaluating lifetimes of baryons, because only inclusive processes are concerned. In this work, we do not use the diquark picture, but instead, adopt a simpler non-relativistic model for the baryon and re-evaluate the hadronic matrix elements. As a by-product, one can compare the results by the diquark picture with that by the three valence-quark picture. It may help us to better understand the diquark picture and its application range. The advantage is obvious, that we only concern the inclusive processes in terms of the optical theorem when calculating the lifetime. Therefore, we do not need to deal with the hadronization to light hadrons. The only non-perturbative effects come from the wave function of the heavy baryon. Moreover, since there are two heavy quarks in the baryon, the relativistic effects are not so significant and the framework of non-relativistic harmonic oscillator model might lead to a reasonable result. Moreover, at the quark level, we carry out similar calculations as that in the literature, but we keep some new operators which are CKM suppressed and contribute to the lifetime. They appear at the non-spectator scattering at order of 1 in heavy quark expansion(HQE). Later, our numerical results show that their contributions are indeed very tiny to make any substantial contributions. All the concerned parameters in the model are obtained by fitting data, therefore we avoid some theoretical uncertainties and obtain reasonable results. Comparing these results with data, we may gain information about the the whole picture. Our paper is organized as follows. In Section.II we derive the formulation for the lifetimes of Ξ+cc, Ξ cc and Ω cc which include the non-spectator effects. In Section.III, we use a simple model, i.e. the harmonic oscillator, to estimate the hadronic matrix elements. In Section.IV we present our numerical results along with the values of all the input parameters. The last section is devoted to our conclusion and discussion. II. FORMULATION FOR LIFETIMES OF Ξ+cc, Ξ cc , Ω A. Spectator Contribution to Lifetimes of Ξ+cc, Ξ cc , Ω The lifetime is determined by the inclusive decays. Thus one can use the optical theorem to obtain the total width (lifetime) of the heavy hadron by calculating the absorptive part of the forward-scattering amplitude. The total width is then written as Γ(HQ → X) = d4x〈HQ|T̂ |HQ〉 = 〈HQ|Γ̂|HQ〉, (1) where T̂ = T{iLeff(x),Leff(0)} (2) and Leff is the relevant effective Lagrangian. 1/mQ is the expansion parameter, and the non-local operator T̂ is expanded as a sum of local operators and the corresponding Wilson coefficients include terms with increasing powers of 1/mQ. Definitely, the lowest dimen- sional term dominates in the limit mQ → ∞ and it is the dimension-three operator c̄c. The total width of a charmed hadron Hc is determined by Im〈Hc|T̂ |Hc〉[10] with a proper normalization[11]. Γ(Hc → f) = 192π3 |VCKM |2{c3(f)〈Hc|c̄c|Hc〉 +c5(f) 〈Hc|c̄iσµνGµνc|Hc〉 6 (f) 〈Hc|(c̄Γiq)(q̄Γic)|Hc〉 )}, (3) where the coefficients ci(f) depend on the masses of the internal quarks in the loop. The coefficient c3(f) has been calculated to one-loop order[12, 13, 14] whereas the coefficient c5(f) is evaluated at the tree level[15, 16]. VCKM is the Cabibbo-Kabayashi-Maskawa mixing matrix elements and Gµυ is the gluonic field strength tensor. Since the third term involves light quarks, it can be different for charmed hadrons with various light flavors. Thus, the difference appears at the 1/m3c order and in the hadronic matrix elements of four-quark operators. The contributions at orders higher than 1/m3c are neglected. To the lowest order, the main contribution comes from the heavy quark(charm quark) decays, while the light flavors are treated as spectators. The contributions are due to the semileptonic and the nonleptonic decays as follows: Γ(c→ s) = l=e,µ Γc→sl̄υ + q(q′)=u,d,s Γc→sq̄q′ (4) The semileptonic and nonleptonic decay rates of the c quark up to order 1/m2c has been evaluated by many authors[17], and here we would directly use their results. B. Non-spectator Contributions to Inclusive Decays of Ξ+cc, Ξ cc , Ω The total width of hadrons which involve at least one charm quark c can be decomposed into two parts Γ(HQ → f) = Γspectator + Γnonspectator. (5) For the spectator scenario, the contribution to the total width of the (ccd)-baryon ground state Ξ+cc, the (ccu)-baryon ground state Ξ cc and the (ccs)-baryon ground state Ω cc should be a sum of decays rates of two c−quarks individuallynamely Γspecccq ≃ 2Γspecc , q = u, d, s. (6) To derive the non-spectator contributions for decays of Ξ+cc, Ξ cc and Ω cc, we need the relevant effective Lagrangian:[18] L(∆c=1)eff (µ = mc) = − {VcsV ∗ud[C1(µ)s̄γµLcūγµLd+ C2(µ)ūγµLcs̄γµLd] +VcdV ud[C1(µ)d̄γ µLcūγµLd+ C2(µ)ūγ µLcd̄γµLd] +VcsV us[C1(µ)s̄γ µLcūγµLs+ C2(µ)ūγ µLcs̄γµLs] l=e,µ s̄γµLcν̄lγ µLl}+ h.c. (7) where L denotes 1−γ5 (i) The inclusive decays of Ξ+cc: There are four diagrams which contribute to the the width of Ξ+cc, as shown in Fig.1. Here we also include the Feynman diagrams which are CKM suppressed. Fig 1.(a),(c) are the W-exchange diagrams (WE), while Fig 1.(b),(d) are the pauli-interference diagrams (PI). Here Fig 1.(d) is arisen from the semi-leptonic decay of the charm quarks with the d−quark in Ξ+cc. For the WE-type diagrams, we derive the contribution to the width as (|Vcs|2|Vud|2C(zs+, zu+) + |Vcd|2|Vud|2C(zu+, zd+))P 2+ {[C21 (µ) + C22 (µ)]c̄γµLcd̄γµLd+ 2C1(µ)C2(µ)c̄γµLdd̄γµLc}, (8) where P+ = pc + pd, zq+ = (q = u, d, s). The definition of the function C(z1, z2) is C(z1, z2) = −[−2(x32 − x31)− (x22 − x21)(3 + 2z1 − 2z2) + 4z1(x2 − x1)], (9) where x1,2 = (1+z1−z2)∓ (1+z1−z2)2−4z1 . In the expressions q and q̄ are free field opearotors of quark and antiquark, and we will show in next section that all the non-perturbative QCD effects are included in the wavefunctions. Their explicit expressions are given as (2π)3 α=1,2 bqα(k)u q (k)e −ikx + d+qα(k)υ q (k)e (2π)3 α=1,2 b+qα(k)ū q (k)e ikx + dqα(k)ῡ q (k)e . (11) For Ξ+cc, q=c, u. The contributions from the Pauli-interference(PI) non-spectator diagrams to the width of Ξ+cc are: PI = − {|Vud|2|Vcd|2Fµν(zu−, zd−)[NC21 (µ)c̄γµLdd̄γνLc+ C22 (µ)c̄iγµLdj d̄jγνLci +2C1(µ)C2(µ)c̄γ µLdd̄γνLc] + 2|Vcd|2Fµν(0, zl−)c̄γµLdd̄γνLc}, (12) where zq− = (q = u, d, e, µ) and P− = pc−pd. The definition of the function Fµν(z1, z2) is Fµν(z1, z2) = −[2(x32 − x31)− (2 + z1 − z2)(x22 − x21) + 3(x2 − x1)]P 2−gµν +[2(x32 − x31)− 3(x22 − x21)]P−µP−ν , (13) where the definitions of z1 and z2 are the same as before. (ii) The inclusive decays of Ξ++cc : The non-spectator contribution to the width of Ξ++cc come from the diagrams shown in Fig.2. That is caused by an interference of the produced u−quark from decay of one of the charm quarks with the u−quark in Ξ++cc . Here we also include the CKM suppressed Feynman diagrams. The contribution is PI = − {|Vcs|2|Vud|2Fµν(zs−, zd−) + |Vcs|2|Vus|2Fµν(zs−, zs−) +|Vcd|2|Vud|2Fµν(zd−, zd−)} {C21(µ)c̄iγµLuj ūjγνLci +NC22 (µ)ūγµLcc̄γνLu+ 2C1(µ)C2(µ)ūγµLcc̄Lνu}, where z− = (q = s, d), P− = pc − pu. (iii) For the inclusive decays of Ω+cc: The non-spectator contributions for Ω+cc not only come from the Pauli interference of the s− quark produced in the non-leptonic, but also from the semi-leptonic decay of the charm quarks with the s−quark in Ω+cc, the later one is suggested by Voloshin et al.[19]. As above, here we include the CKM suppressed WE non-spectator diagrams. The WE non-spectator contribution to the width Ω+cc is |Vus|2|Vcs|2C(zu+, zs+)P 2+ {[C21 (µ) + C22 (µ)]c̄γµLcs̄γµLs+ 2C1(µ)C2(µ)c̄γµLss̄γµLc}, where zq+ = , q = u, s and P+ = pc + ps. The PI non-spectator contribution to the width of Ω+cc is PI = − {|Vcs|2|Vud|2Fµν(zu−, zd−) + |Vcs|2|Vus|2Fµν(zu−, zs−)} {NC21(µ)c̄γµLss̄γνLc + C22(µ)c̄iγµLsj s̄jγνLci + 2C1(µ)C2(µ)c̄γµLss̄γνLc} |Vcs|2Fµν(0, zl−)c̄γµLss̄γνLc, (16) where zq− = , q = u, d, s, e, µ and P− = pc − ps. Sandwiching the operators between initial and final Ξ+cc, Ξ cc , Ω cc states, we obtain the hadronic matrix elements: WE/PI = 〈Ξ+cc(P = 0, s)|Γ̂ WE/PI |Ξ+cc(P = 0, s)〉 PI = 〈Ξ cc (P = 0, s)|Γ̂ PI |Ξ cc (P = 0, s)〉 WE/PI = 〈Ω+cc(P = 0, s)|Γ̂ WE/PI |Ω+cc(P = 0, s)〉. (17) III. THE HADRONIC MATRIX ELEMENTS Because the hadronic matrix elements are fully determined by the non-perturbative QCD effects which cannot be reliably evaluated at present yet, we need to invoke concrete phenomenological models to carry out the computations. In this work, we adopt a simple non-relativistic model, i.e. the harmonic oscillator[20]. This model has been widely em- ployed in similar researches[21, 22, 23, 24, 25, 26]. In fact, an advantage of the calculations of the lifetimes of heavy hadrons is that one does not need to deal with the hadronization process of lighter products (quarks or even gluons) and the heavy hadrons can be well described by such simple non-relativistic models, and the results are relatively reliable than for light hadron decays. (i)The inclusive decays of Ξ+cc: In the harmonic oscillator model, the wavefunction of Ξ+cc is expressed as |Ξ+cc〉 and |Ξ+cc(P, s)〉 = AB color,spin χspin,flavorϕcolor d3pρd 3pλΨΞ+cc(pρ,pλ)|ci(pq1 , sq1), cj(pq2, sq2), dk(pq3, sq3)〉. The normalization condition for |Ξ+cc(P, s)〉 is 〈Ξ+cc(P, s)|Ξ+cc(P′, s′)〉 = (2π)3 δ3(P−P′)δs,s′, (19) where χspin,flavorϕcolor are the spin-flavor and color wavefunctions respectively. Their explicit expressions are χs= 1 ,flavor = (2|c↑c↑d↓〉 − |c↑c↓d↑〉 − |c↓c↑d↑〉) (20) ϕcolor = ǫijk. (21) AB is the normalization constant. The spatial wavefunction ΨΞ+cc is a three-body harmonic oscillator wavefunction and expressed as ΨΞ+cc = exp(− ). (22) Here aρ and aλ parameters reflecting the non-perturbative effects. In the above expressions, the Jacobi transformations of p1, p2, p3 which are the momenta of the three valence quarks ccd, and variables pρ, pλ, P are p1 − p2√ ,pλ = p1 + p2 − 2mcmd p3 22mc+md ,P = p1 + p2 + p3. (23) We choose the center-of-mass frame of Ξ+cc, i.e. (P=0) to calculate the hadronic matrix elements. Substituting the four-quark operators into the expressions, we obtain the non- spectator WE contributions to the width of Ξ+cc as WE = 64π 2G2FP +(|Vcs|2|Vud|2C(zs+, zu+) + |Vcd|2|Vud|2C(zu+, zd+))(C1(µ)− C2(µ))2 |AB|2[2(1 + )]3/2 d3pρd exp[− ]exp[− (pλ + 1 + 2mc (pρ − p′ρ))2 ]ūcγµLucūdγ µLud, and the PI contribution is PI = − π2G2F{|Vcd|2|Vud|2Fµν(zu−, zd−)[−NC21 (µ) + C22 (µ)− 2C1(µ)C2(µ)] −2|Vcd|2Fµν(0, zl−)}|AB|2[2(1 + )]3/2 d3pρd exp[− ]exp[− (pλ + 1 + 2mc (pρ − p′ρ))2 ]ūcγ µLud ūdγ νLuc, where the sum over spin means a sum over the polarizations of the three valence quarks of Ξ+cc with their corresponding C-G coefficients in the spin-flavor wavefunction. uq, ūq denote the Dirac spinors of free quarks q and the expression is Eq +mq Eq+mq χ (26) ūq = Eq +mq 1 − σ·p Eq+mq in our case q denotes c and d quarks. (ii)The inclusive decays of Ξ++cc : The contribution from the PI non-spectator diagrams to the width of Ξ++cc is Ξ++cc PI = − π2G2F{|Vcs|2|Vud|2Fµν(zs−, zd−) + |Vcs|2|Vus|2Fµν(zs−, zs−) +|Vcd|2|Vud|2Fµν(zd−, zd−)}(C21(µ)−NC22 (µ)− 2C1(µ)C2(µ))|AB|2[2(1 + )]3/2 d3pρd ρexp[− ]exp[− (pλ + 1 + 2mc (pρ − p′ρ))2 µLuu ūuγ νLuc. (28) Similar to the case of Ξ+cc, the sum over spin means a sum of the polarizations of the three valence quarks of Ξ++cc with their C-G coefficients. One only needs to replace u by d in pρ, pλ and other expressions are similar to that for Ξ (iii)The inclusive decays of Ω+cc: The contribution from the W-boson exchange(WE) non-spectator diagrams to the width of Ω+cc is WE = 64π 2G2FP +|Vus|2|Vcs|2C(zu+, zs+)(C1(µ)− C2(µ))2 |AB|2[2(1 + )]3/2 d3pρd exp[− ]exp[− (pλ + 1 + 2mc (pρ − p′ρ))2 ]ūcγµLucūsγ µLus, whereas that from the Pauli-interference(PI) non-spectator diagrams is PI = − π2G2F{[|Vcs|2|Vud|2Fµν(zu−, zd−) + |Vcs|2|Vus|2Fµν(zu−, zs−)] [−NC21 (µ) + C22 (µ)− 2C1(µ)C2(µ)]− 2|Vcs|2Fµν(0, zl−)}|AB|2[2(1 + )]3/2 d3pρd ρexp[− ]exp[− (pλ + 1 + 2mc (pρ − p′ρ))2 µLusūsγ νLuc. (30) The sum over polarizations is similar to that for Ξ+cc and Ξ IV. INPUT PARAMETERS AND NUMERICAL RESULTS To obtain the decay amplitudes, we adopt the input parameters as follows[7, 27]: GF = 1.166×10−5GeV−2, |Vcs| = 0.9737, |Vud| = 0.9745, C1(mc) = 1.3, C2(mc) = −0.57,mc = 1.60 GeV, ms = 0.45 GeV, mu = md = 0.3 GeV, m s = 0.2GeV, m u = m d = 0, MΞ+cc = MΞ++cc = 3.519 GeV, MΩ+cc = 3.578 GeV, MΞ+∗cc −MΞ+cc = MΞ++∗cc −MΞ++cc = MΩ+∗cc −MΩ+cc = 0.132 GeV. Here mq∗ denotes the current quark mass of flavor q. The non-perturbative parameters aρ, aλ in the harmonic oscillator wavefunctions are selected as follows: for J/ψ, in ref.[20], a2ρ = 0.33GeV 2, for D−mesons, a2ρ = 0.25GeV2. For TABLE I: The numerical results about the contributions from the different components and the evaluated lifetime for the doubly charmed baryons. For a comparison, in the following table, we list the corresponding lifetimes predicted by the authors of ref.[7] where the diquark picture was employed. It is noted that in ref.[7], the authors used various input parameters and obtained slightly diverse results, we take average values of the numbers in the table. There is only one datum for the lifetimes on τ given by the SELEX collaboration which is also listed the table. Ξ+cc Γspec(10 −12GeV) ΓWEnon (10 −13GeV) ΓPInon(10 −15GeV) τ (ps) τ (ps) in ref.[7] exp(ps) 2.01 6.43 -3.36 0.25 0.19 0.033 Ξ++cc Γspec(10 −12GeV) ΓPInon(10 −12GeV) τ (ps) τ (ps) in ref.[7] 2.01 -1.02 0.67 0.52 − Ω+cc Γspec(10 −12GeV) ΓWEnon (10 −14GeV) ΓPInon(10 −12GeV) τ (ps) τ (ps) in ref.[7] 2.01 4.25 1.10 0.21 0.22 − the doubly charmed baryons, because aρ reflects the coupling between two charm quarks, we set it to be the same as that for J/ψ. aλ reflects the coupling of the light quark with these two charm quark, thus we can reasonably set it to be the same as aρ in D-mesons. With these parameters as input, the lifetimes of the doubly charmed baryons can be evaluated out (see TABLE.I), if the non-spectator effects are taken into account. V. CONCLUSION AND DISCUSSION In this work, we evaluate the lifetimes of doubly charmed baryons with the non-spectator effects being properly taken into account. As argued in the introduction, to evaluate the lifetimes (the total widths), only the inclusive processes are concerned, and then the non- perturbative effects are all from the wavefunctions of the doubly charmed baryons. Due to existence of the two heavy charm quarks, the non-relativistic harmonic oscillator model should apply in this case. Mainly, we carefully calculate the contribution of non-perturbative effects to the lifetimes in the model, which are closely related to the bound states of the baryons. Our numerical results indicate that the non-spectator contributions to the lifetimes of Ξ+cc, Ξ cc and Ω cc are substantial. The non-spectator contributions to the width of Ξ cc are mainly from the WE diagrams (the PI diagrams which contribute are CKM suppressed), since the WE contribution is constructive, therefore the lifetime of Ξ+cc is much suppressed. By contraries, for Ξ++c and Ω cc, the non-spectator contributions are mainly from the PI diagrams and the net effect is destructive. It is noted that for Ω+cc there still are Cabibbo- suppressed WE diagrams, but for Ξ++cc there are only PI diagrams. Therefore the predicted lifetime of Ξ++cc is larger than that of other two baryons. We also employ other values for parameters aρ, aλ and find that the resultant values can vary within 20% uncertainty. Our results are τ(Ξ+cc) = 0.25 ps τ(Ξ cc ) = 0.67 ps and τ(Ω cc) = 0.21 ps. These are generally consistent with the results obtained by Kiselev et al.[7] and Guberina et al.[8], even though they used different models for calculating the hadronic matrix elements. Concretely, they used the diquark picture and attributed the non-perturbative effects into the wavefunction of the diaquark at origin. Kiselev et al. gave τ(Ξ+cc) ∼ 0.16 − 0.22 ps τ(Ξ++cc ) ∼ 0.40− 0.65 ps and τ(Ω+cc) ∼ 0.24− 0.28. Although all the theoretical predictions based on different models agree with each other, they are obviously one order larger than the upper limit of the measured value on the lifetime of Ξ+cc (0.033 ps) by the SELEX collaboration[6]. This deviation, as suggested by some authors, may come from experiments[28]. So far the difference between theoretical predictions and experimental data may imply some unknown physics mechanisms which drastically change the value, if the future experiment, say at LHCb, confirms the measure- ment of the SELEX. 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D72 (2005) 034021. http://arxiv.org/abs/hep-ph/0005102 http://arxiv.org/abs/hep-ph/0208231 http://arxiv.org/abs/hep-ph/0702054 http://arxiv.org/abs/hep-ph/0610205 http://arxiv.org/abs/hep-ph/0701212 http://arxiv.org/abs/hep-ph/0609237 FIG. 1: non-spectator effects contribution to lifetime of Ξccd FIG. 2: non-spectator effects contribution to lifetime of Ξccu FIG. 3: non-spectator effects contribution to lifetime of Ωccs Introduction Formulation for Lifetimes of cc+, cc++, cc+ Spectator Contribution to Lifetimes of cc+, cc++, cc+ Non-spectator Contributions to Inclusive Decays of cc+, cc++, cc+ The hadronic matrix elements Input parameters and Numerical results Conclusion and Discussion References
In this work, we evaluate the lifetimes of the doubly charmed baryons $\Xi_{cc}^{+}$, $\Xi_{cc}^{++}$ and $\Omega_{cc}^{+}$. We carefully calculate the non-spectator contributions at the quark level where the Cabibbo-suppressed diagrams are also included. The hadronic matrix elements are evaluated in the simple non-relativistic harmonic oscillator model. Our numerical results are generally consistent with that obtained by other authors who used the diquark model. However, all the theoretical predictions on the lifetimes are one order larger than the upper limit set by the recent SELEX measurement. This discrepancy would be clarified by the future experiment, if more accurate experiment still confirms the value of the SELEX collaboration, there must be some unknown mechanism to be explored.
Introduction Formulation for Lifetimes of cc+, cc++, cc+ Spectator Contribution to Lifetimes of cc+, cc++, cc+ Non-spectator Contributions to Inclusive Decays of cc+, cc++, cc+ The hadronic matrix elements Input parameters and Numerical results Conclusion and Discussion References

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