Algebra of the infrared and secondary
polytopes
M. Kapranov, M. Kontsevich, Y. Soibelman
August 19, 2014

To the memory of Andrei Zelevinsky

Contents
0 Introduction

3

1 Reminder on secondary polytopes.

7

2 A commutative dg-algebra from the chain complex of Σ(A).

14

3 The L∞ -algebra.

16

4 Maurer-Cartan elements in g.

21

5 1-dimensional case: refinement to an A∞ -algebra.

23

6 Relative setting: one point “at infinity”.

25

7 Hochschild complexes and derived derivation spaces.

27

8

33

Refinement in d = 2: relative setting.
1

9 Introducing coefficients: factorizing sheaves on secondary
polytopes.
35
10 Coefficients in bimodules: d = 238

11 Analysis of the algebra R∞ and the morphism ψ for d = 2.

44

12 The universality theorem (d = 2).

49

13 Comparison with Gaiotto-Moore-Witten.

57

14 Maurer-Cartan elements for the Fukaya-Seidel category

61

15 Speculations and directions for further work

72

2

0

Introduction

The words “algebra of the infrared” in the title refer to the physical paper[GMW] by Gaiotto, Moore and Witten, to which (or, rather, to a part ofwhich) our article is a mathematical commentary.
In [GMW] the authors developed an algebraic formalism for the study ofcertain 2-dimensional massive quantum field theories with (2, 2) supersymmetry. One of the main algebraic structures introduced in the loc.cit. is
the L∞ -algebra of webs, associated with a generic finite subset A ⊂ R2 in
the plane. Physically, elements of A correspond to vacua of the theory. A
web is a plane graph with faces marked by elements of A with an additional
condition on the direction of edges, see §13 below for a review. Further, achoice of a half-plane containing A determines an A∞ -algebra (or an A∞ category, if one introduces a coefficient system). This A∞ -category has an“upper-triangular structure”, i.e. a semi-orthogonal decomposition.
Using certain “moduli spaces of ζ-instantons” the authors of loc.cit. describe a class of deformations of the above A∞ -category which describe theD-brane categories for a particular class of (2,2) supersymmetric theories:
Landau-Ginzburg models. Mathematically, the D-brane A∞ -categories corresponding to LG models are known as Fukaya-Seidel categories [HIV],[Se1].
We reinterpret and develop the algebraic structures proposed in [GMW]in a way that allows a generalization to higher dimensions (Rd , d ≥ 2 insteadof just R2 ). It turns out that using the dual language of polygons rather
than webs, one quickly uncovers certain structures well-known in toric geometry, most notably, secondary polytopes, see [GKZ]. One of the subtle andsurprising points of the GMW construction is the fact that the differential
they define, satisfies d2 = 0. In our approach this fact becomes obvious:
the cellular chain complex of any polytope, in particular, of the secondary
polytope, is indeed a complex, i.e., it has d2 = 0.
Since secondary polytopes make sense in any number of dimensions, ourdual approach leads naturally to higher-dimensional structures of “extended”
topological field theories (TFT). We postpone the study of these structures
to a later work. So the present paper is just the first step in that general
direction.
Let us discuss the contents.
Although we were strongly influenced by [GMW], we have arranged theexposition logically from scratch, implementing our dual interpretation right3

away. A detailed dictionary relating our approach and terminology withthat of [GMW], is provided in Section 13. The reader familiar with theterminology of [GMW] can start from that section and then proceed linearly
from the beginning.
Section 1 provides the general background on secondary polytopes. Roughly,
the secondary polytope of a given polytope in Rd has vertices corresponding to decompositions of the initial, “primary”, polytope into simplices (suchdecompositions are called triangulations). The usefulness of the notion of secondary polytope from the point of view of the approach of [GMW] becomes
clear in Section 2: the chain complex of the secondary polytope looks like a
dg-algebra. The main reason for that is explained earlier in Proposition 1.9:
each face of the secondary polytope is itself a product of several secondary
polytopes associated to members of a certain polyhedral subdivision of theinitial polytope.
In Section 3 we introduce an L∞ -algebra structure in the space gA spannedby polytopes with vertices in a given set A ⊂ Rd . For that we define a
differential in the symmetric algebra generated by the (shifted) dual space
to gA . The remark in the previous paragraph about the chain complex of
the secondary polytope explains why our L∞ -structure comes naturally fromsuch a chain complex. Further, one can visualize higher Lie brackets ingA geometrically. Roughly speaking, they correspond to the operation ofcomposing a convex polytope from smaller polytopes with vertices in A.
Section 4 is a reminder about Maurer-Cartan elements in L∞ -algebras.
Together with Section 7, which is devoted to a short account on deformation
theory, Hochschild complexes etc., it prepares the ground for future applications à la [GMW]: a geometrically defined Maurer-Cartan element gives rise
to a deformed L∞ -structure.
Section 5 is devoted to the most elementary case d = 1, which explains
the reason for the second construction of [GMW]: lifting of a certain L∞ algebra (related but not equal to gA ) to an A∞ -algebra. The possibility ofsuch lifting is the essential 1-dimensionality of the situation, and the purely
1-dimensional case provides a clear starting point. This is related to the factthat convex polytopes on the real line are just segments, and so one cannaturally order their vertices.
In Section 6 we study the situation when one element of A is distinguishedand referred to as an “element at infinity” (denoted ∞). This is a higherdimensional analog of the choice of a half-plane in [GMW]. In this casethe L∞ -algebra gA contains an ideal g∞ spanned by “infinite” polytopes
4

(containing ∞), and a subalgebra gfin spanned by “finite” polytopes (notcontaining ∞). We explain in Section 8 how these structures lead, in the cased = 2, to an action of gfin on R∞ , a natural A∞ -lift of the L∞ -algebra g∞ .
Combining this with the results of Sections 6, 7, we arrive at a morphism fromgfin to the Hochschild complex of R∞ (all in the case of trivial coefficients).
The case of non-trivial coefficients is the subject of Section 9. Here we
introduce and exploit an important notion of a factorizing sheaf (on the secondary polytope). This notion can be seen as a combinatorial version of factorization algebras such as used by Lurie and Costello-Gwylliam [Lur, CGw]
to describe higher-dimensional TFT. This strongly suggests that the constructions of the previous sections admit a generalization to the frameworkof higher-dimensional TFT, with, for instance, Ed−1 -algebras playing the roleof A∞ -algebras such as R∞ .
For an A∞ -algebra with an upper-triangular structure, there is a versionof the Hochschild complex which controls deformations preserving this structure. We call it the directed Hochschild complex and discuss it in Section .
After preparations in Sections 10, 11 we finally prove, in Section 12 a result
which we call the Universality Theorem. It concerns the case d = 2 and thesituation of Section 11 associated with a choice of a point ∞ (or, equivalently,of a held-plane). It says that the natural morphism from the L∞ -algebra gfinof finite polygons to the Hochschild complex of R∞ , is a quasi-isomorphism
onto the directed Hochschild complex.
This result should be compared with the Deligne conjecture, see [KoSo2],
as well as with similar results about the Swiss Cheese operad and its higherdimensional generalizations, see e.g., [Ko1]. It certainly admits a generalization to the higher-dimensional case (in which case one deals with Hochschildcomplexes of Ed−1 -algebras in the spirit of [Ko1]). This will be a subject of
separate publication.
In Section 14 we discuss possibilities for the Maurer-Cartan element giving
the Fukaya-Seidel category associated with a choice of a half-plane.
Section 15 contains a list of further applications and speculations.
We should mention here that there are several L∞ and A∞ algebras thatcan be naturally associated with a finite subset A ⊂ Rd . The biggest one
involves all marked polytopes. It has non-trivial differential corresponding
to the operation of “insertion of internal points”. On the other hand, forthe application to Fukaya-Seidel categories one needs a much smaller L∞ subalgebra which involves polytopes, each marked with the set of all its
internal points. The differential is trivial on this subalgebra, and moreover,5

we do not need to include the set of points as a separate piece of the data. In
this way we obtain a smaller L∞ -algebra, which we refer to as the geometricone. It is spanned by convex polytopes with vertices in A, with higher Liebrackets given by the operation of composing a bigger convex polytope fromsmaller ones. As we pointed out above, the geometric L∞ -algebra appears
naturally in the description of the Hochschild complex of the Fukaya-Seidel
category. The meaning of the “big” L∞ -algebra is not clear at the moment.
Acknowledgements. We thank Davide Gaiotto, Greg Moore and Edward
Witten for multiple discussions and correspondences about their deep andbeautiful paper [GMW] as well as for sending us preliminary drafts. Y.S.
is grateful to IHES for excellent research conditions. His work was partially
supported by an NSF grant. This work was supported by World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan.

61Reminder on secondary polytopes.

In this section we recall some basic material from [GKZ], Ch. 7. We refer
the reader to loc.cit. for details of constructions and proofs of statements.
Let A ⊂ Rd be a finite subset of points. We assume that:(1) The affine span of A is the whole of Rd .
(2) A is in general position, i.e., any p ≤ d + 1 points of A are the verticesof a (p − 1)-simplex.
The first assumption can always be satisfied by passing to the affine span.
The second assumption can be removed at the price of somewhat complicating the discussion and we prefer to keep it throughout the paper for simplicity.
Let Q = Conv(A) be the convex hull of A, a d-dimensional convex polytope. By a triangulation T of (Q, A) we will mean a subdivision of Q into
straight simplices of full dimension d with vertices in A so that the intersection of any two simplices is a common face (possibly empty). Note that it is
not required that each element of A appears as a vertex of a simplex of T .
A triangulation T is called regular, if there is a continuous convex function
f : Q → R such that:• f is affine-linear on each simplex of T .
• f is not affine-linear on any subset of Q which is not contained in a
simplex of T . In other words, f does indeed break along each codimension 1 simplex which is a common face of two different d-dimensional
simplices of T .
For a more systematic discussion, we need the following concept.
Definition 1.1. A marked polytope is a pair (Q, A) where Q is a convex
polytope in Rd and A ⊂ Rd is a finite subset such that Q = Conv(A), i.e.,
A contains all vertices of Q. A marked subpolytope of (Q, A) is a marked
polytope (Q0 , A0 ) such that A0 ⊂ A. Notation: (Q0 , A0 ) ⊂ (Q, A).
Definition 1.2. By a polyhedral subdivision of (Q, A) we will mean a collection P = (Qν , Aν ) of marked subpolytopes of (Q, A) which have dim Qν =dim Q such that:7(1) The Qν form a polyhedral subdivision of Q (i.e. Q = ∪ν Qν ) so thateach Qν ∩ Qν 0 is a common face of Qν and Qν 0 (possibly empty).
(2) In addition, we have Aν ∩ (Qν ∩ Qν 0 ) = Aν 0 ∩ (Qν ∩ Qν 0 ).
As for triangulations, a polyhedral subdivision P = (Qν , Aν ) is called regular,
if there is a continuous convex function f : Q → R which is affine-linear oneach Qν and is not affine linear on any subset which is not contained in some
Qν .

Qν

Qν 0

Figure 1: A triangulation and a polyhedral subdivision.
Regular polyhedral subdivisions of (Q, A) form a poset (partially orderedset) R(Q, A) with the order given by refinement. That is,P 0 = (Q0µ , A0µ ) ≤ P = (Qν , Aν ),iff P 0 induces a (necessarily regular) polyhedral subdivision of each (Qν , Aν ).
Thus, minimal elements of R(Q, A) are precisely the regular triangulations,while the unique maximal element is the subdivision consisting of (Q, A)
alone.
Remark 1.3. Note the essential role of the choices of markings Aν in thedefinition of polyhedral subdivisions and of the poset R(Q, A). We do not
require that A = ∪ν Aν . E.g. if (Q, A) is a triangle with two points in theinterior then (Q0 , A0 ) is a polyhedral subdivision provided Q0 = Q and A0 is
just one of the interior points.
More generally, if A0 ⊂ A is the set of vertices of Q, then each intermediate
subset A0 ⊂ B ⊂ A gives rise to a 1-element regular polyhedral subdivision(Q, B) ∈ R(Q, A), and(Q, B 0 ) ≤ (Q, B) iff B 0 ⊂ B.
8

R
Gψ

ψ(ω)
•

•
ω

•
Q

•
Rd

Figure 2: The polyhedron Gψ and the function fψ .
The poset R(Q, A) has an interpretation in terms of the secondary fanof A. This fan, denoted S(A), is a subdivision of RA into convex polyhedralcones CP corresponding to regular polyhedral subdivisions P ∈ R(Q, A).
To define it, we associate to each ψ : A → R a convex piecewise-affine
function fψ : Q → R as follows. We consider the unbounded polyhedronGψ ⊂ Rd+1 = R × Rd given byGψ = Conv (t, a) t ∈ R, a ∈ A ⊂ Rd , t ≥ ψ(a) .
This image of Gψ under the projection to Rd is Q, and the bottom ofGψ is the graph of a convex piecewise-affine function fψ : Q → R. The
function fψ gives rise to unique (necessarily regular) polyhedral subdivision
P(ψ) consisting of domains of affine linearity of fψ . The cone CP ⊂ RA is,by definition, the set of all ψ such that Pψ = P.
Note that each CP is invariant under translations by those ψ which comefrom global affine linear functions on Rd . We denote the space of such functions by Aff(Rd ). Then CP gives rise to the cone C P in the quotient space
RA / Aff(Rd ). The collection of these cones is called the reduced secondary
fan of A and denoted by S(A).
Proposition 1.4. The correspondence P 7→ CP is an order reversing isomorphism between the poset R(Q, A) and the poset of cones of S(A) (orderedby inclusion).

9

Along with S(A) we will consider the dual object, the secondary polytope Σ(A) whose faces correspond to subdivisions P ∈ R(Q, A) in an orderpreserving, not reversing way. The construction of Σ(A) depends on the
choice of a translation invariant measure Vol on Rd , a different choice leading to a rescaling of Σ(A). For convenience of the reader we recall two(equivalent) definitions.
Definition 1.5. For a regular triangulation T of (Q, A) define the vector
φT ∈ RA byX
φT (ω) =Vol(∆).
∆∈Tω∈Vert(∆)

The polytope Σ(A) ⊂ RA is defined to be the convex hull of the vectors φT
for all regular triangulations T of (Q, A).
Definition 1.6. [BS] Consider the standard simplex with the set of vertices
A:


X
A
A
pω = 1 ,∆ = (pω )ω∈A ∈ R pω ≥ 0,
ωso that, by definition of the convex hull, we have a surjective projectionX
π : ∆A −→ Q = Conv(A), (pω )ω∈A 7→
pω · ω.
ωThe polytope Σ(A) is defined as the set of vector integralsZs(q) dVol ∈ RAq∈Q

for all continuous sections s : Q → ∆A of the projection π.
To explain the relation of the two definitions, note that a triangulation
A
T of (Q, A)
R defines a “tautological” piecewise linear section sT : Q → ∆ ,and φT = Q sT (q) dVol.
To each face F = FP of Σ(A) ⊂ RA we can associate its normal cone NF ⊂(RA )∗ consisting of those linear functionals l : RA → R which achieve themaximum on F and are constant on F . The cones NF form a decomposition

10

(fan) of the space (RA )∗ called the normal fan of Σ(A). Let us now identify
RA with its dual vector space by means of the standard pairingX(φ, ψ) =φ(ω)ψ(ω).
ω∈A

Under this identification we have:
Proposition 1.7. The normal fan of Σ(A) is identified with the secondary
fan S(A). In particular:(a) The face lattice of Σ(A) is isomorphic to R(Q, A) in an order-preserving
way. We denote by FP the face corresponding to a subdivision P ∈ R(Q, A).
(b) The codimension of FP is equal to the dimension of the cone C P .
(c) dim Σ(A) = |A| − d − 1 = dim(RA / Aff(Rd )).
Examples 1.8. (a) A triangulation of (Q, A) can be seen as a polyhedral
subdivision T = (Qν , Aν ) such that each Qν is a simplex and Aν consists
exactly of the vertices of Qν . In this case Σ(Aν ) is a point and so the face
FT is a vertex.
(b) Edges of Σ(A) correspond to flips (elementary modifications) of triangulations. A flip is based on a circuit, a subset Z ⊂ A which has precisely
one, up to a scalar, affine dependency
XXaω · ω = 0, aω ∈ R,aω = 0.
ω∈Z

In this case the convex hull Conv(Z) has precisely two triangulations T+ , T−with vertices in Z, so Σ(Z) is an interval. Each edge [φT , φT 0 ] of Σ(A)corresponds to a pair T , T 0 of triangulations of Q which coincide outside
Conv(Z) for some circuit Z, and restrict to T+ , T− inside Conv(Z).

Figure 3: Circuits and flips(c) Codimension 1 faces correspond to coarse subdivisions (see [GKZ]).
In the dual language used in [GMW] in case d = 2 they are called taut webs.
11

By definition, a subdivision P is coarse, if the reduced normal cone C P has
dimension 1, i.e., there exists only one, modulo scaling and adding affine
functions, convex P-piecewise-affine function f : Q → R. The following are
the most important examples of coarse subdivisions.
(c1) Let ω ∈ A be not a vertex of Q. Then (Q, A − {ω}) is a coarse
subdivision of (Q, A).
(c2) Let H is a hyperplane in Rd subdividing Q into two subpolytopes
Q1 , Q2 . Suppose that the vertices of both these polytopes lie in A. Put
Aν = A ∩ Qν . Then (Q1 , A1 ) and (Q2 , A2 ) form a coarse subdivision of(Q, A).
H
Q1
Q2

Figure 4: A 2-part coarse subdivision.
The importance of Σ(A) for our purposes comes from the following.
Proposition 1.9 (Factorization property). The face FP correspondingtoQ
a regular polyhedral subdivision P = (Qν , Aν ), has the form FP = ν Σ(Aν ),i.e., it is itself a product of several secondary polytopes.
Remark 1.10. Note that if we drop the assumption of A being in general
position, then FP will be not the full product but the fiber product of theΣ(Aν ) over the secondary polytopes of intermediate faces. This one of thecomplications we wanted to avoid in this paper.
For future convenience, we introduce the following terminology.
Definition 1.11. A marked subpolytope (Q0 , A0 ) ⊂ (Q, A) is called geometric, if A0 = A ∩ Q0 . A polyhedral subdivision P = {(Q0ν , A0ν )} of (Q, A) iscalled geometric, if each (Q0ν , A0ν ) is a geometric marked subpolytope. A face
F of Σ(A) is called geometric, if it corresponds to a geometric subdivision.

12

We will use the term subpolytope of (Q, A) for a polytope Q0 ⊂ Q withvertices from A. Notation: Q0 ⊂ (Q, A). Such subpolytopes are in bijection
with geometric marked subpolytopes (Q0 , A ∩ Q0 ).
The following property will be important for the construction of §3 tomake sense. Its proof is obvious and is left to the reader.
Proposition 1.12. Geometric faces form a right ideal in the partially ordered
set of all faces of Σ(A). That is, if F ⊂ F 0 and F is geometric, then F 0 is
geometric.
We introduce the geometric part of the secondary polytope to be theunion of the interiors of geometric faces (of all dimensions):[
Int(F ).
(1.13)
Σgeom (A) =F ⊂Σ(A)

geometric

Proposition 1.12 can be reformulated by saying that Σgeom (A) is an open
subset of Σ(A).

13

2

A commutative dg-algebra from the chain
complex of Σ(A).

The factorization property of Σ(A) implies that its chain complex looks likea (part of) a multiplicative differential in a dg-algebra.
More precisely, let k be a field of characteristic 0. For any convex polytope
P we denote by C• (P ) its cellular chain complex with coefficients in k gradedso that i-chains are positioned in degree (−i) and the differential raises thedegree by +1. Thus, as a graded vector space
MC• (P ) =or(F )[dim(F )],F ⊂Pthe direct sum over all faces. Hereor(F ) = Hcdim(F ) (Interior of F, k)
is the 1-dimensional orientation space of F .
We apply this to P = Σ(A) and form the graded vector space
M
(2.1)
V =
VA0 , VA0 = or(Σ(A0 ))[dim Σ(A0 )],(Q0 ,A0 )⊂(Q,A)

the direct sum over all marked subpolytopes of full dimension d. Here, theshift means that or(Σ(A0 )) is positioned in degree − dim Σ(A0 ). Note that if
k = R, then0or(Σ(A0 )) = Λmax RA / Aff(Rd ).
Now, take the symmetric algebra S • (V ) with the graded commutative product denoted . Each regular polyhedral subdivision P 00 = {(Q00ν , A00ν )} of eachmarked subpolytope (Q0 , A0 ) of (Q, A) produces a 1-dimensional subspaceK(2.2)
VP 00 =
VA00ν ⊂ S • (V ).
ν

The chain differentials in all the C• (Σ(A0 )) are compatible with each otherand give a differential d of degree +1(2.3)

d : S • (V ) −→ S • (V ),satisfying the Leibniz rule.
14

More precisely, let P 00 = {(Pν00 , Q00ν )} be a coarse polyhedral subdivision of(Q , A0 ), A0 ⊂ A. Using the factorization property (Proposition 1.9), we seethat the top degree part of the chain differential in C• (Σ(A0 )) gives a map
0

dP 00 : VA0 −→ VP 00 ⊂ S • (V ).
We define d to be given, on generators, by
X
d|VA0 =

dP 00 ,

P 00 coarse
subdiv. of (Q0 ,A0 )and then extend it to the whole of S • (V ) by the Leibniz rule.
Proposition 2.4. The differential d thus defined, satisfies d2 = 0 and somakes S • (V ) into a commutative dg-algebra.
Proof: In virtue of the Leibniz rule, it is enough to check that d2 = 0 on
generators. To see this, note that on VA0 the picture for d2 is embedded,
again by Proposition 1.9, into the picture for d2 in C• (Σ(A0 )). But for any
polytope P , the differential in the chain complex C• (P ) satisfies d2 = 0.
Remark 2.5. We take the symmetric algebra of V because, for a subdivision
P 00 = {(Q00ν , A00ν )} of (Q0 , A0 ), different marked polytopes (Q00ν , A00ν ) have noapparent order. But this can be refined depending on the dimension d, as
we will see later.

15

3

The L∞-algebra.

If h is a finite-dimensional dg-Lie algebra over k, then its Chevalley-Eilenberg
cochain complex is a commutative dg-algebra•
CLie(h) = S • (h∗ [−1]).

For any graded vector space V we denote by S+• (V ) ⊂ S • (V ) the maximal
ideal ⊕n≥1 V ⊗n .
As well known, a general algebra differential d on S+• (h∗ [−1]) (i.e. an
algebra derivation satisfying d2 = 0), makes h into an L∞ -algebra, see, e.g.,[KoSo1], Ch.3, Section 2.3. Equivalently we can say that L∞ -algebra on h isgiven by the algebra differential of S • (h∗ [−1]) which preserves S+• (h∗ [−1]).
Thus, in the situation of §2, we get an L∞ -structure on the vector spaceM••g = gA = V ∗ [−1] =E A0 ,(Q0 ,A0 )⊂(Q,A)

EA0 = VA∗0 [−1] = or(Σ(A0 ))∗ [−1 − dim Σ(A0 )].
In other words, the 1-dimensional k-vector space or(Σ(A))∗ (which we cancanonically identify with its dual or(Σ(A))), is now put in the positive degree
1 + dim Σ(A0 ).
•The L∞ -algebra g is, while natural, too big for our purposes. Indeed, asubpolytope Q0 ⊂ (Q, A) with vertices in A can give rise to many summands•
EA0 ⊂ g corresponding to the choice of A0 sandwiched between Vert(Q0 ) and•A ∩ Q0 . The dot in the notation g is supposed to symbolize this freedom inchoosing the set of points A0 . Taking only the “geometric summand corresponding to A0 = A ∩ Q0 , we define the subspaceMM•
g = gA =EA0 =EA∩Q; ⊂ g.
(Q0 ,A0 ) geom.

Q0 ⊂(Q,A)

Remark 3.1. We should warn the reader that in the Sections 3-8 we discuss
L∞ -algebras which are toy models of the “realistic” L∞ -algebras appearing in
the “nature” (in particular, in the study of Fukaya-Seidel categories). For thelatter situation one has to introduce a “coefficient system” on the secondary
polytope. This will be done in Section 9. Nevertheless the case of “trivial
coefficient system” discussed in Sections 3-8 has most of the features of thegeneral case, so we decided to keep it for pedagogical reasons.
16•Proposition 3.2. gA is an L∞ -subalgebra in gA , with trivial differential.
The L∞ -algebra gA (together with its various generalizations) will be theprimary object of study in this paper. We will call it the geometric L∞ algebra associated to A.
Proof of Proposition 3.2: We denote by• ⊗n

λn : g•−→ g, n = 1, 2, · · · ,the components of the L∞ -structure. Thus λ1 = d is the differential, λ2 (x ⊗y) = [x, y] is the bracket, etc.
It follows from our construction that λn corresponds to coarse subdivisions of marked subpolytopes of (Q, A) into precisely n marked subpolytopes.
More precisely, choose a generator (canonical up to sign) eA0 ∈ or(Σ(A0 )) for
each A0 ⊂ A, |A0 | = d + 1. Thus, the eA0 form a basis in g. Then eachcoarse subdivision P 00 = (Q00ν , A00ν )}nν=1 of each (Q0 , A0 ) contributes a matrix
element(3.3)

eA001 ⊗ · · · ⊗ eA00n λn eA00

= ±1,

and all non-zero matrix elements of λn are obtained in this way. Now, if each(Q00ν , A00ν ) is geometric, i.e., A00ν = A ∩ Q00ν , the A0 = A ∩ Q0 as well, i.e., (Q0 , A0 )is geometric, cf. Proposition 1.12. This means that λn (g⊗n ) ⊂ g, i.e., thatg is an L∞ -subalgebra. The fact that the differential in g vanishes, follows
from Example 3.5(a) below.
Example 3.4. Recall that we have assumed that A is in affinely general
position, i.e. each (d + 1)-element subset generates a d-simplex. Hence thelow degree part of the L∞ -algebra g looks as follows. We have g≤0 = 0,
while g1 is spanned by d-dimensional simplices, i.e., by marked subpolytopes(Q0 , A0 ) with |A0 | = d + 1. The degree 2 part g2 is spanned by circuits(|A0 | = d + 2) and so on.
•

Examples 3.5. (a) Differential in g. It corresponds to Example 1.8(c1) ofcoarse subdivisions. That is, given A0 ⊂ A and ω ∈ A0 which is not a vertex
of Q0 = Conv(A0 ), we have, specializing (3.3), thateA0 −{ω} d eA0

17

= ±1.

Writing B for A0 − {ω} (which can be arbitrary) and collecting the matrix
elements (i.e., adding, not subtracting one point in all admissible ways), wehave, cf. Fig. 5:
Xd(eB ) =±eB∪{ω} , Q0 = Conv(B).
ω∈(A∩Q0 )−B

X

d

±
•

•

•

Figure 5: Differential in g.
In other words, fixing a subpolytope Q0 ⊂ (Q, A), all eB with Conv(B) =•Q0 , span a subcomplex in (g, d). This subcomplex is isomorphic to the augmented cochain complex of the simplex ∆I , whose set of vertices is
I = (A ∩ Q0 ) − Vert(Q0 ).
In particular, it is exact, if I 6= ∅, i.e., if Q0 contains points of A other thanits vertices. This exactness is another reason why g is more important for us•
than g.
Note that if B = A ∩ Q0 , then d(eB ) = 0 (there are no more points toadd), so the differential in g vanishes.
(b) The binary bracket in g corresponds to Example 1.8(c2) of coarse subdivisions. That is, if (Q0 , A0 ) is subdivided by a hyperplane H into (Q001 , A001 )and (Q002 , A002 ), then[eA001 , eA002 ] = ±e0A(c) Consider the particular case d = 2 and assume that A ⊂ R2 is inconvex position, i.e., each element of A is a vertex of Q = Conv(A). Then•g = g and only the binary bracket is present, so g is a graded Lie algebra
in a more familiar sense (no differential, no higher λn ). As an illustration,
18

consider the further particular case when A consists of 4 points, forming
the vertices of a convex 4-gon. Then g1 has dimension 4, with the basis
vectors corresponding to the 4 triangles a, b, c, d in Fig.6. The space g2 is
1-dimensional, with the basis vector corresponding to the 4-gon itself, whileg≥3 = 0. Thus g is has five basis vectors: ea , eb , ec , ed of degree 1 and eA ofdegree 2, with the only non-zero brackets being[ea , eb ] = [ec , ed ] = eA .
a

d
b

c

Figure 6: Four points in convex position(d) Take A ⊂ R2 to be a 4-element circuit consisting of the 3 vertices ofa triangle Q and one more point ω 0 inside Q. Marked subpolytopes of (Q, A)
include 3 small triangles a, b, c in Fig. 7, one large triangle f = (Q, A−{ω 0 }),•1•and (Q, A) itself. In this case g has two non-zero components: g , with•2basis ea , eb , ec , ef and g with basis eA . The only non-trivial parts of the
L∞ -structure are the differential and the ternary bracket whose nontrivial
matrix elements are given by:d(ef ) = eA , λ3 (ea , eb , ec ) = eA ,•so the cohomology (w.r.t. d) of g has all the multiplications λn being 0. On
the other hand, g is spanned by ea , eb , ec , eA and has λ3 6= 0.

a

b

f

c

A

Figure 7: Four points in non-convex position.
Recall [Get] that an L∞ -algebra L is called nilpotent, if there exists r0 > 0such that all r-ary iterated superpositions of the λn are identically zero as
maps L⊗r → L for r > r0 .
19

•Proposition 3.6. The L∞ -algebra g (and therefore g) is nilpotent.
Proof: Since the matrix elements of λn corresponds to coarse subdivisionsof marked subpolytopes (Q0 , A0 ) ⊂ (Q, A) into n marked subpolytopes, thematrix elements of any r-ary iterated superposition of the λn correspond tosome (not necessarily coarse) subdivisions into r marked subpolytopes andtherefore vanish for r ≥ |A|.
.

20

4

Maurer-Cartan elements in g.

Let L be a nilpotent L∞ -algebra over k. Recall that a Maurer-Cartan elementin L is an element γ ∈ L1 such that
(4.1)

∞
X
1
λn (γ, · · · , γ) = 0 ∈ L2 .
n!
n=0

Maurer-Cartan elements form a subset MC(L) in L1 . If dim L1 < ∞, we canconsider L1 as an affine space over k and view MC(L) as an affine subschemein L1 .
If L is a general L∞ -algebra, one can define the formal scheme of Maurerd
Cartan elements MC(L)which is the functor on local Artin k-algebras Λ
defined byΛ 7→ MC(L ⊗k m),where m is the maximal ideal of Λ.
Such a functor represents a geometric object called a formal pointed dgmanifold in [KoSo1], Ch.3.
We are interested in Maurer-Cartan elements in the finite-dimensional
nilpotent L∞ -algebra g = gA . In this section we assume that A is in general
position. As we saw, g1 is spanned by basis vectors eσ corresponding to all
possible marked d-simplices σ = (Conv(A0 ), A0 ) with A0 ⊂ A, |A0 | = d + 1.
Such a marked simplex is nothing but a straight d-simplex on vertices from
A. Thus we can viewXγ =γσ · eσ , γσ ∈ kas a “d-cochain”, i.e., a datum of numbers γσ ∈ k attached to all marked
simplices σ.
Now, g2 is spanned by basis vectors eZ corresponding to all possible circuits Z ⊂ A, |Z| = d + 2. The coefficient of the LHS of (4.1) at eZ is thenequal to the sum of two terms corresponding to the two triangulations T+ , T−
of Conv(Z), as in Example 1.8(b). More precisely, this coefficient is equal to
Y
Y
γσ −
γσ .
σ∈T+

σ∈T−

Therefore we obtain:
21

Proposition 4.2. γ is a Maurer-Cartan element in g, if and only if it is a“cocycle” in the multiplicative sense: for any circuit Z with triangulations
T+ , T− of Conv(Z) we have
Y
Y
γσ =
γσ .
σ∈T+

σ∈T−

This means that for any marked subpolytope (Q0 , A0 ) of (Q, A) we havean element γQ0 ,A0 ∈ k defined in terms of any chosen regular triangulation T 0of (Q0 , A0 ):Y
γQ0 ,A0 =
γσ ,
σ∈T 0

with the RHS being independent of the choice of T 0 .
Example 4.3. Let k = C and let Ω be any smooth d-form on Q. ThenputtingZ
γσ = eβσ ,

Ω,

βσ =
σ

we get a Maurer-Cartan element in g.
One way of interpreting Proposition 4.2 is by saying that the subscheme
MC(g) is given inside the affine space g1 by a system of binomial equations:each equation had the form of equality of exactly two monomials. This meansthat the scheme MC(g) is in fact a toric variety.
More precisely, let Sd (A) be the set of
 all d-simplices on vertices fromA
A. Thus Sd (A) is a subset in the set d+1 of all (d + 1)-element subsets inAA, and Sd (A) = d+1
when A is in affinely general position. Let ZSd (Q, A)
be the free abelian group of combinatorial d-chains in (Q, A), i.e., of integer
linear combinations of d-simplices. We write an element β ∈ ZSd (Q, A) asa system β = (βσ ) where σ ∈ Sd (A) is a d-simplex, and βσ ∈ Z. Let
Zd (Q, A) ⊂ ZSd (Q,A) be the sublattice of d-cocycles, i.e., of β as above suchthat for each circuit Z as in Proposition 4.2, we have
X
X
βσ =
βσ .
σ∈T+

σ∈T−

S (Q,A)

Let ΓA = Zd (Q, A) ∩ Z+d. This is a finitely generated abelian semigroup.
The monomial equations for MC(g) translate into the following.
Corollary 4.4. The scheme MC(g) is isomorphic to Spec k[ΓA ], the spectrum of the semigroup algebra of ΓA .
22

5

1-dimensional case: refinement to an A∞algebra.

In this section we consider the simplest case d = 1. So we assume that
A = {ω1 < · · · < ωr } ⊂ R. Then Q = Conv(A) is the interval [ω1 , ωr ]. Let
us form the vector space V as in (2.1).
If (Q0 , A0 ) is a marked subpolytope of (Q, A) and P 00 = {(Q00ν , A00ν )} is apolyhedral subdivision of (Q0 , A0 ), then each Q00ν is a sub-interval in Q0 and so
there is a natural order on the set of the intervals Q00ν , induced by the order
on R (i.e., geometrically, from left to right). This means that we can lift thedifferential d from S • (V ) (free commutative algebra generated by V ) to adifferential in T • (V ), the tensor algebra (free associative algebra) generated
by V . More precisely, we realize the vector space VP 00 from (2.2) as a subspacein T • (V ):O
VA0ν ⊂ T • (V )VP 00 =
ν(embedding given by the tensor multiplication in the order from left to right).
We then define the action of d on the generators of T • (V ) by making it act
on a summand VA0 ⊂ V as the top P
degree part of the chain differential in
0C• (Σ(A )), now considered as a map P 00 dP 00 , where P 00 runs over all coarse
subdivisions of (Q0 , A0 ) anddP 00 : VA0 −→ VP 00 ⊂ T • (V )is induced by the chain differential in C• (Σ(A0 )). Then we extend d to thewhole T • (V ) by the Leibniz rule. Similarly to Proposition 2.4, we then get:
Proposition 5.1. The differential d in T • (V ) satisfies d2 = 0 and so makesT • (V ) into an associative dg-algebra. The canonical commutativization homomorphism (T • (V ), d) → S • (V ), d) is a morphism of associative dg-algebras.
Clearly d preserves the ideal T+• (V ) := ⊕n≥1 V ⊗n ⊂ T • (V ).
•This means that the L∞ -algebra g lifts to an A∞ algebra with the sameunderlying vector space by the A∞ -analog of the formula[x, y] = xy − yx.

23•

We denote this A∞ -algebra by R. As before, we have a similar lifting of thesubalgebra g. We denote this lifting by R and it will be more fundamental•

for us than R.
Thus R has basis eij , 1 ≤ i < j ≤ r, corresponding to the geometricmarked subpolytopes
[ωi , ωj ], {ωi , ωi+1 , · · · , ωj } .
The degree of eij is j − i, and the binary multiplication is given by
eij · ejk = eik ,i < j < k,and all other binary products as well as other A∞ -data vanish. In otherwords, R is the associative algebra without unit, isomorphic to Tn (k), thealgebra of strictly upper triangular r by r matrices with zeroes on and below
the diagonal. This triangular structure is the simplest instance of the relation
between our formalism and deformations of semi-orthogonal collections, see
below.

24

6

Relative setting: one point “at infinity”.

Let (Q, A) be a marked polytope in Rd , as above. Let us add to A one moree = A ∪ {∞} and Q
e = Conv(A).
e We assume that ∞point ∞ ∈ Rd , forming Ae i.e., that it lies “far outside of A”. We also assume thatis a new vertex of Q,ethe set A is in general position. Let eg = gAe be the geometric L∞ -algebraecorresponding to A.
∞···∞···

e
Q

e
Q

Q

Q

Figure 8: ∞ as a point at the projective infinity vs. a finite point far away.
Remark 6.1. We can, if we want, imagine ∞ to be an actual “infinite”
point of the projective completion RPd = Rd ∪ RPd−1 , say the point of RPd−1
with homogeneous coordinates [0 : 0 : · · · : 1]. Note that the concepts ofhyperplanes, triangulations, convexity etc. are invariants under projective
transformations of RPd . So there is no combinatorial difference between ae on the right and the unbounded polyhedron Q
e on thebounded polytope Q
left of Fig. 8.
e A)e will be called finite, if Vert(Q0 ) ⊂ A, andA subpolytope Q0 ⊂ (Q,infinite, otherwise, i.e., if ∞ ∈ Q0 . The L∞ -algebra e
g splits, as a vector
space, into a direct sumeg = g ⊕ g∞ ,where g = gA is spanned by the summands
EA0 = or(Σ(Q0 ∩ A))∗ [−1 − dim Σ(Q0 ∩ A)]
for finite Q0 , and g∞ is spanned by similar summands for infinite Q0 .
25

Proposition 6.2. Both g and g∞ are L∞ -subalgebras of eg, and, moreover,eg∞ is an L∞ -ideal. In other words. g is the semi-direct product e
g = g n g∞ .
Proof: The L∞ -brackets are given by combining subpolytopes, say, Q001 , · · · , Q00n ,together into a coarse subdivision. If neither of the Q00i contains ∞, then the
combined marked polytope Q0 does not contain it either. This means that g
is an L∞ -subalgebra (it is nothing but gA , the geometric L∞ -algebra associated to A alone). If at least one of the Q00i does contain ∞, then so does Q0 .
This means that g∞ is a L∞ -ideal.
Remarks 6.3. (a) Let us choose a Euclidean subspace Rd−1 ⊂ Rd not passing
through ∞ and let
p : Rd − {∞} −→ Rd−1
be the projection obtained by drawing straight lines through ∞ and intersecting them with Rd−1 . In the “projective infinity” picture on the left ofFig. 8, we can think of p : Rd → Rd−1 as being the projection forgetting thelast coordinate.
We then have the point configuration A = p(A) in Rd−1 , and by ourassumption, Q = Conv(A) coincides with the image p(Q). It is instructive tocompare the L∞ -algebra g∞ with gA , the L∞ -algebra associated to A ⊂ Rd−1 .
e A)e gives a subpolytope
More precisely, each infinite subpolytope Q0 ⊂ (Q,0p(Q ) ⊂ (Q, A) via the projection p. However, not every subpolytope of(Q, A) is obtained in this way. Similarly, each coarse subdivision of Q0 gives
a coarse subdivision of p(Q0 ) but not the other way around. Nevertheless,the L∞ -algebra g∞ has, in many respects, (d − 1)-dimensional nature.
e and an element(b) The two point configurations A, A associated to A
e are precisely the two minors of Ae studied by Billera, Gelfand and∞∈A
Sturmfels [BGS]. More precisely, A is the minor by deletion and A is theminor by contraction, in the terminology of [BGS]. Among other things, ite and Σ(A) is a Minkowski
was proved in [BGS] that Σ(A) is a face of Σ(A),esummand of Σ(A).
Remark 6.4. If we assume that e
g is an ordinary dg-Lie algebra, i.e. if
λ≥3 = 0, then the Lie algebra structure on e
g gives a morphism of dg-Lie
algebrasg −→ Der(g∞ ), x 7→ ad(x) : g∞ → g∞ .
In general, what we get is a “derived” analog of such a morphism, as we
explain in the next section.
26

7

Hochschild complexes and derived derivation spaces.

In this section we recall some general principles of deformation theory andof Koszul duality.
Let P be any dg-operad (over k), and A be any P-algebra (in the
category of dg-vector spaces over k). Automorphisms of A form a group
AutP (A), and derivations of A (of all degrees) form a dg-Lie algebra DerP (A).
One can form the derived functor of A 7→ DerP (A), defininge
R DerP (A) = DerP (A),where Ae → A is a “cofibrant resolution”, i.e., a quasi-isomorphism of Palgebras with Ae being free as a P-algebra without differential. RHS of thisformula can be understood (at least in char(k) = 0 case) as an L∞ -algebra.
There is a canonical choice of a cofibrant resolution called the BoardmannVogt resolution in [KoSo2]. It was used in loc. cit. for the description ofgeneral deformation theory of algebras over operads.
Remark 7.1. If P → Q is a morphism of (dg-)operads, then any Q-algebra
A can be also considered as a P-algebra (for example, we have Ass →Com, so a commutative algebra can be considered as a particular case of an
associative algebra). In this case DerP (A) = DerQ (A), but R DerP (A) canbe different from R DerQ (A).
According to the general philosophy of deformation theory (see e.g. [KoSo1])
the aim of formal deformation theory in the case when char(k) = 0 is theconstruction of formal pointed dg-manifold (equivalently, L∞ -algebra) which
describes the local structure of the moduli space of deformations. Moreprecisely, if L is such an L∞ -algebra, then the commutative topological algebra generated by L∗ [−1] carries a structure of a dg-algebra. Its formal
spectrum (as a graded topological algebra) can be thought of as a formal
graded scheme, endowed with a (graded) vector field Q of degree +1 suchthat [Q, Q] = 0 (“homological vector field”). The formal scheme of zeros
Z(Q) admits a foliation generated by graded vector fields of the form [Q, •](all schemes are understood as functors on commutative Artin algebras). The“space of leaves” of the above foliation should be thought of as the moduli
space of the deformation problem controlled by L. The reader can look at[KoSo1] for the details and examples and at [KoSo2] for the implementation27

of this approach in the case of deformations of algebras over operads. In the
more conventional language of stacks one has the following corollary of thoseconsiderations.
Proposition 7.2 (Local structure of the moduli stack). The formal
germ of the deformation stack of A as a P-algebra is isomorphic to thequotient stackd DerP (A)) exp(R Der0 (A)).
MC(RPd DerP (A)) is the formal scheme of Maurer-Cartan elements in
Here MC(Rthe dg-Lie algebra R DerP (A), and exp(R Der0P (A)) is the formal group associated to the Lie algebra R Der0P (A), which naturally acts on the formald DerP (A)).
scheme MC(RSo we can say that R DerP (A) “governs deformations of A as a Palgebra”.
Example 7.3. Let P = Ass, and A be an associative algebra. ThenR Der(A) = R DerAss (A) is the shifted and truncated Hochschild cochain
complex of A:≥1
C ≥1 (A, A)[1] := CAss
(A, A)[1] =Homk (A, A) −→ Homk (A , A) −→ Homk (A , A) → · · · .
δ0

⊗2

δ1

⊗3

Thus Ker(δ0 ) = Der(A) is the usual space of derivations. The Lie algebra
structure on R Der(A) is given by the “brace formula”. That is, for p ∈Hom(A⊗m , A) and q ∈ Hom(A×n , A) and i = 1, ..., m we define the ith braceof p and q by p ◦i q ∈ Hom(A⊗(m+n−1) , A) by(p ◦i q)(a1 , · · · , am+n−1 ) == p a1 , · · · , ai−1 , q(ai , ai+1 , · · · , ai+n−1 ), ai+n , · · · am+n−1 .
Then[p, q] =mX±p ◦i q −

i=1

n
X

±q ◦j p.

j=1

Example 7.4. Let P = Lie and A = L be a Lie algebra. Then R DerLie (L) =is the shifted and truncated Chevalley-Eilenberg cochain complex of L withcoefficients in the adjoint representation:


δ0
δ0
≥1
∗
2 ∗
3 ∗
CLie (L, L)[1] = L ⊗ L −→ Λ L ⊗ L −→ Λ L ⊗ L → · · ·28

General principles of Koszul duality: Let P be a quadratic Koszul
operad with Koszul dual operad P ! (see e.g. [GiKa] about basics on Koszul
duality for operads). A weak P-algebra structure (also called a P∞ -algebra
structure) on a graded vector space A is the same as a P ! -algebra differential
d on the free P ! -algebra FP ! (A∗ [−1]), satisfying d2 = 0.
We remark that we define the free P ! -algebra FP ! (A∗ [−1]) in such a waythat it does not have 0-ary operations.
In particular:(a) An L∞ -structure on a graded vector space L is a multiplicative differential
on the (completed) symmetric algebra S+• (L∗ [−1]).
(b) An A∞ -structure on a graded vector space R is a multiplicative differential on the (completed) tensor algebra T+• (R∗ [−1]).
Although technically one should consider completed tensor products (sincecorresponding structures are defined in dual terms of the differential cofree
coalgebras), we will often skip the word “completed”, since our examples
are nilpotent in the sense that only finitely many “higher” operations are
non-trivial.
Most important for us will be the following principle.
Proposition 7.5. A datum consisting of:(1) A P∞ -algebra A.
(2) An L∞ -algebra L.
(3) An L∞ -morphism α : L → R DerP∞ (A)is the same as a datum consisting of:(i) A differential dL in the (completed) algebra S • (L∗ [−1]) which preserves
the maximal ideal S+• (L∗ [−1]).
(ii) A differential d in the (completed) graded spaceS • (L∗ [−1]) ⊗k FP ! (A∗ [−1])which is a P ! -algebra derivation, satisfies d2 = 0 as well as the following Leibniz formula:d(a⊗x) = dL∗ [−1] (a)⊗x+(−1)|a| a⊗d(1⊗x),
29

a ∈ S • (L∗ [−1]), x ∈ FP ! (A∗ [−1]).

Proof: We explain a more conceptual argument and then spell out additional
algebraic details in the examples we will be interested in. Suppose data (i)-(ii)are given. Then (i) implies that L is an L∞ -algebra, thus giving the datum(2). Further, considering the restriction d|A∗ [−1] and projecting the image on
1 ⊗ FP ! (A∗ [−1]), we get a P ! -algebra differential in FP ! (A∗ [−1]), i.e., we
get a P∞ -algebra structure on A which we will call the “initial” structure.
This gives the datum (1). Now, by general principles of deformation theory, Spec S • (L∗ [−1]) is a formal commutative dg-manifold (functor on local
artinian commutative dg-algebras) whose underlying ordinary (non-dg) ford
mal commutative manifold is MC(L),the formal scheme of Maurer-Cartan
elements in L. So we denoted
RMC(L)= Spec S • (L∗ [−1]).
Thus our differential d in the tensor product is the same as a family of P∞ dstructures on A parametrized by RMC(L)which gives the datum (3) by
Proposition 7.2. The argument in the opposite direction is similar.
Example 7.6 (Particular case P = Ass). A datum consisting of:(1) An A∞ -algebra R.
(2) An L∞ -algebra L.
(3) An L∞ -morphism α : L → R Der(R) = C ≥1 (R, R)[1]
is the same as an algebra differential d in the (completed) tensor productS • (L∗ [−1]) ⊗ T • (R∗ [−1]) which preserves S+• (L∗ [−1]) ⊗ 1 as well as 1 ⊗T+• (R∗ [−1]).
The relations of the data (1) and (2) to the differential d has alreadybeen explained in the general proof of Proposition 7.5. Let us explain how
to obtain the datum (3) explicitly.
We recall, first of all, the general meaning of the concept of an L∞ morphism β : L → L0 between two L∞ -algebras L and L0 whose L∞ operations we denote by λn , λ0n respectively. By definition, β is the sameas a morphism of commutative dg-algebras(7.7)

β ∗ : S • (L0∗ [1]) −→ S • (L∗ [1]),the datum which spells out to a collection of morphisms of graded vector
spaces(7.8)

βn : S n (L[−1]) −→ L0 [−1],30n≥0satisfying a chain of compatibility conditions. More precisely, any collectionof the βn in (7.8) defines a unique morphisms of commutative graded algebras
β ∗ as in (7.7) and the conditions on the βn say that β ∗ thus defined, actuallycommutes with the differentials. Note, in particular, that β1 can be seen,
after shift, as a morphism of graded vector spaces β1 [1] : L −→ L0 and thefirst in the chain of conditions on the βn says that β1 [1] commutes with thedifferentials λ1 and λ01 in L and L0 . This is the “underlying morphism” of
the L∞ -morphism β, and higher βn can be seen as higher homotopies for its
compatibility with the brackets.
We now specialize to our situation: L0 = C ≥1 (R, R)[1]. Suppose we are
given a differential d in S • (L∗ [−1])⊗T • (R∗ [−1]) which preserves S • (L∗ [−1]).
We describe the corresponding L∞ -morphism α in terms of its componentsMαn : S n (L[−1]) −→ C ≥1 (R, R)[1] =Homk (R⊗m , R)[1 − m].
m≥1More precisely, the matrix element
αn,m : S n (L[−1]) → Homk (R⊗m , R)[1 − m]of αn with the respect to the above direct sum decomposition of the RHS is
obtained as the dual of the matrix element of the restriction d|R∗ [−1] which
we denotedn,m : R∗ [−1] −→ S n (L∗ [−1]) ⊗ T m (R∗ [−1]),

T m (R∗ [−1]) = (R∗ )⊗m [−m].

Example 7.9 (Particular case P = Lie). A datum consisting of:(1) An L∞ -algebra L∞ .
(2) An L∞ -algebra L.
≥1(3) An L∞ -morphism α : L → R DerLie (R) = CLie
(L, L)[1]

is the same as an algebra differential d in the (completed) symmetric algebraS • (L∗ [−1]) ⊗ S • (L∗∞ [−1]) = S • (L ⊕ L∞ )∗ [−1] ,preserving S+• (L∗ [−1]) ⊗ 1 as well as 1 ⊗ S+• (L∗∞ [−1]). The correspondence
between the matrix elements of d and the matrix elements of the components
αn is completely analogous to Example 7.6, and we leave it to the reader.
31Note that the data of a differential d as above, is the same as an L∞ structure on L ⊕ L∞ such that L is an L∞ -subalgebra and L∞ is an L∞ ideal. This is precisely the kind of situation we had in §6. So we obtain thefollowing.
Corollary 7.10. In the situation of §6, we have a natural L∞ -moprhism
α : g −→ R DerLie (g∞ ).

32

8

Refinement in d = 2: relative setting.

e =Consider the relative setting of §6 of which we keep the notation: AA ∪ {∞} etc.
In this case the L∞ -algebra g∞ can be upgraded to an A∞ -algebra R∞ ,similar to §5. This is because, for each purely infinite coarse subdivision P 00 ={(Q00ν , A00ν )} of an infinite marked subpolygon (Q0 , A0 ), ∞ ∈ A0 , the infinite
marked polygons Q00ν , are naturally ordered using the chosen orientation ofR2 and the projection p : R2 − {∞} → R as in Remark 6.3(a).
∞

1

2

3

Figure 9: Ordering of infinite polygons in R2 .
So if we put
V∞ =

M

VA0 ,

(Q0 ,A0 ) infinite

then the formulas for the chain differential in the C• (Σ(A0 )) define an algebra•

differential d in T • (V∞ ) with d2 = 0. This means that R∞ = V∞∗ [−1] is an
A∞ -algebra.
••

Similarly to §5, R∞ = g∞ as a graded vector space. Further, the only
A∞ -operations in R∞ are the differential and the binary bracket, so R∞ is
an associative dg-algebra.
•

As before, wee define the geometric subalgebra R∞ ⊂ R∞ spanned bygeometric infinite marked polygons (Q0 , A ∩ Q0 ). Thus R∞ = g∞ as a vector
space and the differential on R∞ is trivial, so R∞ is a graded associative
algebra.
The product of two infinite marked subpolygons a, b is zero unless theirunion is again an infinite marked subpolygon (in particular, the union is33

convex), with a being on the left of b and meeting b along an edge, see Fig.
10.
∞

∞
mult.

a

b

ab

Figure 10: Multiplication on R∞ .
e A),eNow, looking at all coarse subdivisions of all marked subpylgons of (Q,both finite and infinite, we get an algebra differential inM
S • (V ) ⊗ T • (V∞ ), V =VA0 .
(Q0 ,A0 ) finite

This differential preserves S • (V ) and gives there the differential defining g∞ .
So we obtain, by Proposition 7.5 and, more particularly, Example 7.6, L∞ morphisms
•

(8.1)

•

•

φ : g∞ −→ R DerAss (R∞ ),φ : g∞ −→ R DerAss (R∞ ),of which the second one will be more important for us. This is the settingof Gaiotto-Moore-Witten [GMW] in the “case of trivial coefficients”. In thenext sections we explain how can one generalize our setting to fully include
that of [GMW].

34

9Introducing coefficients: factorizing sheaves
on secondary polytopes.

Since we assumed that A ⊂ Rd is in general position, all proper faces of Q are
simplices. Each subset of d points {ω1 , · · · , ωd } ⊂ A gives a (d − 1)-simplex
σ = Conv{ω1 , · · · , ωd }.
By an oriented simplex we will mean a simplex together with a numbering of vertices considered up to an even permutation. We will use thenotation σ = hω1 , · · · , ωd i to denote the oriented (d − 1)-simplex given by
the numbering of the verices ω1 , · · · , ωd . We write σ for “σ with the opposite
orientation”, i.e, σ = hωs(1) , · · · , ωs(d) i, where s is any odd permutation.
Definition 9.1. A system of coefficients for A is a datum N = (Nσ ) of
cochain complexes, one for each oriented (d − 1)-simplex σ with vertices inA, so that Nσ = Nσ∗ .
We fix an orientation of Rd . Then for each marked subpolytope (Q0 , A0 ) ⊂(Q, A), each codimension 1 face of Q0 is naturally an oriented (d−1)-simplex.
We denote
O
Nσ .
(9.2)
NA0 =
σ⊂∂Q0

Let P = {(Qν , Aν )} be a regular polyhedral subdivision of (Q, A). We defineONA0ν .
(9.3)
NP =
ν

If P 0 is a refinement of P, then we have morphisms of cochain complexes
γP 0 P : NP 0 → NP
obtained by taking traces Nσ ⊗ Nσ → k over those (d − 1)-faces of P 0 which
are internal, i.e., are not faces of P and thus belong to exactly two polytopes
from P 0 .
Proposition 9.4. The maps γP 0 P are transitive for chains of refinements:
if P 00 refines P 0 and P 0 refines P, thenγP 00 P = γP 0 P ◦ γP 00 P 0 .
35

We now interpret this proposition in terms of constructible complexes ofsheaves on secondary polytopes. Let X be a CW-complex with a cellular
stratification (Xα )α∈I . In other words, (I, ≤) is a poset together with astrictly monotone function d : I → Z+ , each Xα is a locally closed subspaceof X homeomorphic to the Euclidean space Rd(α) , and the condition α ≤ β
is equivalent to the Xα being contained in the closure of Xβ . A sheaf F of
k-vector spaces on X constant along each Xα is, as well known [Cur][GM]described by the following data:(1) Vector spaces Fα = H 0 (Xα , F) (stalk of F).
(2) Generalization maps γαβ : Fα → Fβ defined for any α ≤ β and satisfying the transitivity conditions, i.e., forming a representation of theposet I in k-vector spaces.
Proposition 9.4 implies that the data (NP , γP 0 ,P ) give rise to a complex ofsheaves NA on the secondary polytope Σ(A) which is constant on each open
face FP◦ with stalk NP and with the generalization maps being the γP 0 P . We
then have the following counterpart of Proposition 1.9.
Proposition 9.5. The complex NA satisfies
Q the factorization property: therestriction of NA to any closed face FP = (Q0ν ,A0ν )∈P Σ(A0i ) is identified withthe exterior tensor product ν NA0ν .
The fact that the generalization maps for a cellular sheaf go from thestalks at smaller cells to the stalks at bigger cells, is a part of the general
phenomenon that sheaves have cohomology rather than homology. In particular, the cellular cochain complex of a sheaf F given by (Fα , γαβ ) has the
form
 M

M
•
or(Xα ) ⊗ Fα −→or(Xα ) ⊗ Fα −→ · · ·Ccell (X, F) =dim Xα =0

dim Xα =1d(α)

where or(Xα ) = Hc (Xα , k) is the 1-dimensional orientation space of thecell Xα and the differentials are induced by the γαβ .
So sticking to the cohomological picture, we form the graded vector space
M
VAcoh
VAcoh
= NA0 ⊗ or(Σ(A0 ))[− dim Σ(A0 )].
VNcoh =
0 ,
0
A0 ⊂A

|A0 |≥d+1

36

Then the cellular cochain complexes of the factorizing sheaves NA0 unite togive a coalgebra differential d, d2 = 0, in the symmetric coalgebra S • (VNcoh ).
This is an alternative way of saying that:
Proposition 9.6. (a) The cellular cochain differentials of the NA0 give riseto an L∞ -algebra structure onM••
gN = gA,N =EA0 , EA0 = NA0 ⊗ or(Σ(A0 ))[− dim Σ(A0 ) − 1].
(Q0 ,A0 )⊂(Q,A)•

The L∞ -algebra gN is nilpotent.
(b) The subspaceM

gN = gA,N =EA∩Q0

Q0 ⊂(Q,A)•

is an L∞ -subalgebra in gN .
•As before gN will be more important for us than gN .
Thus, the component g1N (where Maurer-Cartan elements live) isM− dim Σ(A0 )
g1N =
NA0
⊗ or(Σ(A0 )), A0 = A ∩ Q0 .
Q0 ⊂(Q,A)In other words, an element of g1N is a rule associating to each marked dsimplex (Q0 , A0 ) ⊂ (Q, A) an element of NA0 0 , to each circuit (Q0 , A0 ) anelement of NA−10 and so on.
e = A ∪ {∞}, as in §6. Let N
e be
Relative case. Suppose A is extended to A
e We then generalize Propositions 6.2 and Corollarya coefficient system for A.
7.10 to:Proposition 9.7. (a) We have a semi-direct product decompositiongNe = gN n gNe ,∞where gN,∞ is the direct sum of summands EA0 corresponding to infinite
marked subpolytopes (Q0 , A0 ). Therefore we have an L∞ -morphism α : gNe →R DerLie (gNe ,∞ ).
(b) If d = 2, then gNe ,∞ lifts to an A∞ -algebra ANe ,∞ and we have an
L∞ -morphism ψ : gNe → R DerAss (RNe ,∞ ).
37

10

Coefficients in bimodules: d = 2

The systems of coefficients considered in §9 consisted in data associated tocodimension 1 simplices only. It would be natural to consider more generalcoefficient systems which would associate some data to simplices of all dimensions from 0 up to (d − 1). In this section we explain how to do this in
the case d = 2.
Let us denote elements of A as i, j, k and so on.
Definition 10.1. An extended system of coefficients for A is a following system of data:(1) For each i ∈ A, an associative dg-algebra Si .
(2) For each oriented edge (1-simplex) (i, j), i, j ∈ A, a (differential graded)(Si , Sj )-bimodule Nij which we assume projective, of finite rank, overthe graded algebra underlying Si ⊗k Sjop .
(3) For each (i, j), a pairing
βij : Nij ⊗k Nji −→ Si ⊗k Sj ,which is a morphism of (Si ⊗k Sj , Si ⊗k Sj )-bimodules. It is required
that:(4) The pairing βij is non-degenerate, i.e., the morphism of (Si , Sj )-bimodules
βijt : Nij → HomSj ⊗k Siop (Nji , Sj ⊗k Siop )induced by βij , is an isomorphism.
(5) The diagram
Nij ⊗k Nji

βij/ Si ⊗k Sj

perm.

perm.



Nji ⊗k Nijβji/ Sj ⊗k Si

is commutative.
Thus the setting of Definition 9.1, specialized to d = 2, corresponds tothe case when all Si = k.
38

Remark 10.2. The data of an extended system of coefficients can be interpreted as follows. The datum (1) means that we have a (pre-)triangulated
dg-category Ci . More precisely, let us assume that Si is smooth and proper[TV], in particular, that dimk Si < ∞. Then we take Ci = D Mod<∞toSi
be the derived category formed by finite-dimensional right dg-modules overSi . Note that Ci has a Serre functor [BK], i.e., a covariant exact functor
Si : Ci → Ci equipped with natural isomorphisms
HomCi (M, M 0 )∗ ' HomCi (M 0 , Si (M )).
Here and elsewhere ∗ means dualization over k. Explicitly, Si is given by
the Nakayama formula∗(10.3)Si (M ) = R HomSi (M, Si ) .
The datum (2) means that we have an exact functor Fij : Ci → Cj (givenby the tensor product with Nij ). Finally, the datum (3) means that Fij isleft adjoint to Si Fji (or, equivalently, that Fji is left adjoint to Sj Fij ). This
follows from (10.3).
Note that for a pair of exact functors F : C → D, G : D → C between
triangulated categories C, D with Serre functors SC , SD the condition “F isleft adjoint to SC G” is symmetric in F and G (unlike the usual conditionof adjointness). Indeed, such “twisted adjointness” is expressed via natural
isomorphisms
HomD (F (X), Y )∗ ' HomC (G(Y ), X),

X ∈ C, Y ∈ D.

We fix an orientation of R2 . Suppose given an extended system of coefficients (Si , Nij ). Let Q0 ⊂ (Q, A) be a subpolygon. Let us label the vertices ofQ0 cyclically, in the counterclockwise order as i0 , · · · , im . We can then form
the cyclic tensor product
NQ0 = Ni0 i1 ⊗Si1 Ni1 i2 ⊗Si2 · · · ⊗Sim−1 Nim−1 im ⊗Siop ⊗Sim Nim i0 ,0which can be pictorially represented in Figure 11, which makes evident its
cyclic symmetry (independence on the choice of the starting point i0 ). Notethat NQ0 is just a cochain complex: all module structures have been used in
the formation of the tensor products.
39

N i0i1 ⊗Si1 N

i1

i0

i2

⊗ Si 2

···

⊗

n−
1

Ni

3

Si

1
n−

in

Ni2 i3 ⊗⊗Sin Nini0⊗S· · · ⊗Si

Figure 11: The cyclic tensor product.
Remark 10.4. More generally, we can define a cyclic tensor product Nσ forany closed oriented edge path σ = (i0 , i1 , · · · , in , i0 ), not nevessarily comingfrom the boundary of a convex polygon. As before, Nσ depends only on σ asa closed path and not on the choice of a starting point.
Let
σ1 = (· · · , h, i, j, k, · · · ), σ2 = (· · · , p, j, i, q, · · · )be two closed oriented edge paths having one edge [i, j] in common, with
opposite orientations. Then we can concatenate σ1 and σ2 along this edge,
erasing it and obtaining a new closed oriented edge path
σ1 ∗[i,j] σ2 = (· · · , p, j, k, · · · , h, i, q, · · · ),see Fig. 12.

40

k
p
j

σ1
h

σ2

i
q

Figure 12: Concatenation.
In this situation, the pairing βij gives rise to the concatenation map
γσ[i,j]: Nσ1 ⊗k Nσ2 −→ Nσ1 ∗[i,j] σ2 .
1 ,σ2(10.5)

To define it, consider two decomposable tensors
nσ1 = · · · ⊗ nhi ⊗ nij ⊗ njk ⊗ · · · ∈ Nσ1 ,nσ2 = · · · ⊗ npj ⊗ nji ⊗ niq ⊗ · · · ∈ Nσ2 ,where nij ∈ Nij etc. Suppose
βij (nij ⊗ nji ) =Xs0ν ⊗ s00ν ,s0ν ∈ Si , s00ν ∈ Sj .

νThen we define(nσ1 ⊗ nσ2 ) =γσ[i,j]1 ,σ2X· · · ⊗ npj s00ν ⊗ njk ⊗ · · · ⊗ nhi s0ν ⊗ niq ⊗ · · · .

νIt is immediate to check that this gives a well defined map of cyclic tensor
products.
If now P is a polygonal subdivision of Q0 = Conv(A0 ) into {Q00ν }, we define
O
NP =
NQ00ν
ν

41

(tensor product over k). Note that the concatenation maps (10.5) corresponding to various intermediate edges of P, commute with each other, so
the result of applying them all in any order, is independent on the order. We
call this result the composition map and denote it(10.6)

γP : NP −→ NA0 .

Further, if P 0 is a subdivision refining P, the composition maps (10.6) corresponding to all polygons of P, give rise to a map(10.7)

γP 0 P : NP 0 −→ NP

which we call the generalization map to emphasize that we want to use it toconstruct cellular sheaves on secondary polytopes. More precisely, we havethe following statement whose proof is straightforward.
Proposition 10.8. The data (NP , γP 0 P ) define a factorizing system of complexes of sheaves NA0 on the Σ(A0 ), with the stalk of NA0 on the face FP ⊂
Σ(A0 ) being NP . Denote
EA0 = NA0 ⊗ or(Σ(A0 ))[− dim Σ(A0 ) − 1].
Then the differentials in the cellular cochain complexes of the NA0 unite toL•make the graded vector space g =(Q0 ,A0 )⊂(Q,A) EA0 into a nilpotent L∞ L•
algebra. The subspace g = Q0 ⊂(Q,A) EA∩Q0 is an L∞ -subalgebra in g.
e = A∪{∞}, Q
e = Conv(A).
e Let (Si , Nij )Relative situation (as in §6). A
e For any infinite marked
be an extended coefficient system for A (not A)00polytope (Q , A ) with vertices ∞, i0 , · · · , im in counterclockwise order, we
define the linear tensor product
LA0 = Ni0 i1 ⊗Si1 Ni1 i2 ⊗Si2 · · · ⊗Sim−1 Nim−1 im .
This is an (Si0 , Sim )-bimodule. If P is a subdivision of a (possibly infinite)
subpolygon (Q0 , A0 ) into (Q00ν , A00ν ), then we defineOONP =NA00ν ⊗
LA00ν .
A00
ν 63∞

A00
ν 3∞

and these complexes, again, give a factorizable system of complexes of sheaves
on the Σ(A0 ). Restricting to the geometric summands as before, we obtain:
42

Proposition 10.9. Denoting for A0 3 ∞,FA0 = LA0 ⊗ or(Σ(A0 ))[− dim Σ(A0 ) − 1],
we get the following data:(1) An L∞ -algebra g =L(2) An A∞ -algebra R∞ =Q0 63∞ EA∩Q0 .

L

Q0 3∞ FA∩Q0 .

(3) An L∞ -morphism ψ : g → R Der(R∞ ).

Remark 10.10. Study of Fukaya-Seidel categories (see §14 below for morebackground) leads naturally to a structure somewhat more general and symmetric than an extended system of coefficients. In that structure, for eachpoint i ∈ A we have not just a category, but a local system of triangulated
categories (together with pre-triangulated dg-enhancements) Ci over a small
circle Si1 centered at i. It is required that:(a) Each fiber category Ci,θ , θ ∈ Si1 has Serre functor Si .
(b) The monodromy around the circle Si1 gives rise to the functor isomorphic to Si ◦ [d], where [d] denotes the functor of shift of the grading byd ∈ Z (for the Fukaya-Seidel category associated with the pair (X, W )the number d is the dimension of the complex manifold X).
Further, for each pair of different points i, j ∈ A we have functors Fij :Cj,θj → Ci,θi and Fji : Ci,θi → Cj,θj between the fiber categories over the points
θi ∈ Si1 and θj ∈ Sj1 which are intersection points of the straight segment ij
with circles Si1 and Sj1 . These functors are required to be “twisted adjoint”
to each other in the sense of the Remark 10.2.
A choice of a “generic” point at infinity (equivalently, of a generic halfplane) breaks the local rotational symmetry, since each circle Si1 contains theintersection point with the ray from i to infinity. This allows us to replaceeach Si1 by a segment. After this, the more general structure described above
reduces effectively to an extended system of coefficients.

43

11

Analysis of the algebra R∞ and the morphism ψ for d = 2.

e = A ∪ {∞} is in general position, A has a total
Since we assumed that A
order by the slope of the line (i∞) from left to right. We get a decomposition
M
R∞ =
Rij ,
i,j∈A

i<j

where Rij is the direct sum of the summands FA∩Q0 corresponding to infinite
polygons Q0 whose edges through ∞ are (i∞) and (j∞). Note that Rij isan (Si , Sj )-bimodule.
Proposition 11.1. (a) The higher compositions mn , n ≥ 3, in R∞ , aretrivial. The only non-trivial part of the binary compositions m2 are the maps
µijk : Rij ⊗ Rjk → Rik ,i < j < k.

Thus R∞ is a strictly upper-triangular dg-algebra (without unit).
(b) The multiplication µij is Sj -bilinear with respect to the right Sj -module
structure on Rij and the left Sj -module structure on Rjk . Therefore, putting
Rii = Si , we obtain a non-strictly triangular associative dg-algebra with unit
M
R=
Rij .
i≤j

Remark 11.2. Note that part (b) means that ModR has a semi-orthogonal
decomposition with quotients
ModSi1 , ModSi2 , · · · , ModSir ,where i1 < i2 < · · · ir are all the elements of A in our order.
Proof of Proposition 11.1: (a) The only coarse subdivisions of an infinite
polygon into all infinite subpolygons contain either 2 parts (and so accountfor m2 ), or 1 part (and so account for m1 = d, as in Examples 1.8 (c2) and(c1). The subdivisions into 1 part are excluded in the geometric subalgebra•

R∞ ⊂ R∞ . This shows part (a). Part (b) is clear.
44

We now describe the morphismψ : g −→ R Der(R∞ ) = C ≥1 (R∞ , R∞ )[1]from Proposition 10.5 explicitly. Because of the direct sum decompositions
MMg=
EA0 , R∞ =FB , |A0 |, |B| ≥ 3,
A0 ⊂A geom.

B3∞ geom.

ψ is given by its matrix elements with respect to the induced direct sum decompositions of the source and target. These matrix elements are morphismsof dg-vector spaces(11.3)(B ,··· ,Bm |C)ψA0 1: EA0 −→ Homk (FB1 ⊗k · · · ⊗k FBm , FC ),given for all finite geometric A0 ⊂ A and all infinite geometric B1 , · · · , Bm , C ⊂e of cardinality ≥ 3. By the analysis of Example 7.6, ψ (B0 1 ,··· ,Bm |C) 6= 0 onlyAA
if Q0 = Conv(A0 ) together with the Pν = Conv(Bν ), forms a coarse regularsubdivision of P = Conv(C). In other words, nonzero matrix elements of ψe A)ecorrespond to coarse regular subdivisions of infinite subpolygons in (Q,which contain exactly one finite polygon (the rest being infinite). Let us
call such subdivisions 1-finite. We will now describe all 1-finite subdivisions
explicitly, starting with the following class of examples.
Example 11.4. Let Q0 be a finite subpolygon of (Q, A), and let
e0 = Conv(A0 ∪ {∞}), A
e0 = Ae∩Q
e0 .
A0 = A ∩ Q0 , Q
e lie on a line, the boundary ∂Q0 splits
Since we assumed that no 3 points of A
into two distinct parts meeting at two points:• The positive boundary, i.e., the union of sides facing ∞, drawn thick in
Figure 13 and denoted ∂+ Q0 .
• The negative boundary, denoted ∂− Q0 formed by the sides which lie on
the opposite side of ∞.

45∞Π1 Π2 ... Πm
η1

η2 · · ·

ηm

∂+ Q0

Q0

∂− Q0
Figure 13: The positive and negative boundary.
We denote the sides of ∂+ Q0 by η1 , · · · , ηm in the natural order from left
to right (counter-clockwise rotation around ∞), and let Π1 , · · · , Πm be thetriangles with one vertex at ∞ and the opposite sides being η1 , · · · , ηm . Lete ν . Then (Πν , Dν ) is an infinite marked subpolygon of (Q0 , A0 ),also Dν = A∩Πand we have a polygonal subdivision (see Fig. 13)(11.5)

e0 , Ae0 ) = (Q0 , A0 ) ∪ (Π1 , D1 ) ∪ · · · ∪ (Πm , Dm ).
(Q

It is easy to see that (11.5) is a 1-finite subdivision. Indeed, the unique (upto scalar factors and adding global affine-linear functions) piecewise-affine
e0 → R is determined by putting f (∞) = 1 and f |A0 = 0.
function f : Q
However, Example 11.4 does not give the most general form of a 1-finite
subdivision. To obtain the general form, considering the following situation,
extending the above one.
e A)
e be an infinite subpolygon and Q0 ⊂ P be a
Example 11.6. Let P ⊂ (Q,finite subpolygon such that ∂− Q0 is contained in ∂P and is the union of edges
from ∂P . The two remaining finite parts of ∂P (not in ∂− Q0 ) will be called
handles, see Fig. 14. We denote the left handle by λ and edges constituting
it by λ1 , · · · , λp . Similarly, we denote the right handle by ρ and the edges
constituting it by ρ1 , · · · , ρq , in our order from left to right. The handles
may be empty, i.e., it may happen that p = 0 or q = 0 or both.
46

∞
P

P1
Left handle
λ = (λ1 , · · · , λp )here p = 1

η1

P2 ...
η2 · · ·Pm

Right handle
ρ = (ρ1 , · · · , ρq )here q = 2

ηm

Q0

∂− Q0
Figure 14: General 1-finite subdivision.
e0 of Q0 and ∞, we get the situation
Note that by taking the convex hull Q
described in Example 11.4, from which we retain the notation. We then havea decomposition of P into Q0 and polygons P1 , · · · Pm , where:• Pν = Πν for ν = 2, · · · , m − 1.
• The finite part of ∂P1 is composed out of λ and η1 .
• The finite part of ∂Pm is composed out of ηm and ρ.
Proposition 11.7. The subdivision described in Example 11.6, is 1-finite,and each 1-finite subdivision is obtained in this way.
Proof: By construction, the subdivision contains exactly one finite subpolygon. Let us show that it is regular. For this, note that, as in Example 11.4,
the conditions f (∞) = 1 and fQ0 = 0 determine a unique continuous fonction f : P → R which is affine on each polygon and breaks exactly along
the subdivision. Further, this function is unique with these properties up toscalar factors and adding global affine functions. This follows from analyzing
e0 = Conv(Q ∪ {∞}). Therefore our subdivision is coarse
the restriction to Q
and therefore is 1-finite.
Conversely, note that the construction of Example 11.6 by its very nature
produces all subdivisions of any infinite polygon P which contain exactly one
finite subpolygon.
47

In the situation of Example 11.6, we put
e ∩ Pν , C = A
e ∩ P.
A0 = A ∩ Q0 , Bν = A(B ,··· ,Bm |C)

In order to describe the corresponding matrix element ψA0 1
it in the transposed way, as a map, we writeψAC0 ;B1 ,··· ,Bm : EA0 ⊗k FB1 ⊗k · · · ⊗k FBm −→ FC .

(11.8)Let us denote

Nλ = Nλ1 ⊗S · · · ⊗S Nλp ,Nρ = Nρ1 ⊗S · · · ⊗S Nρq

the linear tensor products corresponding to the handles λ and ρ considered
as oriented edge paths. Here and in the sequel we write ⊗S for the tensor
product over the algebra Si corresponding to the vertex common to the twoedges in an edge path. Then
FB1 = Nλ ⊗S Eη1 ,

FBν = Eην , 2 ≤ ν ≤ m − 1,
FC = Nλ ⊗S EA0 ⊗S Nρ .

FBm = Eηm ⊗S Nρ ,Let
(11.9)

γQ0 : EA0 ⊗k FD1 ⊗k · · · ⊗k FDm −→ FAe0

be the composition map obtained, similarly to (10.6), by substituting thepairings βij over all the intermediate edges [i, j] in the coarse subdivision(11.5). Define(11.10)

ψAC0 ;B1 ,··· ,Bm = IdNλ ⊗S γQ0 ⊗S IdNρ .

Then the source and target of ψAC0 ;B1 ,··· ,Bm are as specified in (11.8).
Proposition 11.11. The maps ψAC0 ;B1 ,··· ,Bm corresponding to all pairs (P, Q0 )e A)e and a finite subpolygon Q0
consisting of an infinite subpolygon P ⊂ (Q,with ∂− Q0 contained in the finite part of ∂P , are precisely all the matrix
elements of the L∞ -morphism ψ : g → C ≥1 (R∞ , R∞ )[1] with respect to thedirect sum decompositions of the source and target described above.
Proof: This is a consequence of the analysis of Example 7.6 applied to our
case.
48

12

The universality theorem (d = 2).

We work in the setting of §10 and 11. So we have the triangular dg-algebra
M
R=
Rij , Rii = Si .
i≤j

We are interested in deformations of R which, first, preserve the triangularstructure and, second, do not change the quotients of the semi-orthogonal
decomposition. The second condition means that the algebras Si are notdeformed. Such deformations are governed by the ordered Hochschild complex→−•
C (R, R), with→−n
C (R, R) =
=

M


Hom

Si0 ⊗Siopn
Ri0 i1 ⊗Si1 Ri1 i2 ⊗Si2 · · · ⊗Sin−1 Rin−1 , Rin , Ri0 in

i0 <i1 <···<in(strict inequalities under the direct sum sign).
→−
Remark 12.1. Note that C • (R, R) is a subcomplex in the ordinary Hochschildcomplex C • (R, R), specified by the multi-linearity conditions. Moreover, thissubcomplex is closed with respect to the Lie bracket and is thus a dg-Lie subalgebra.
Theorem 12.2. The L∞ -moprhism ψ factors through an L∞ -morphism→−Ψ : g −→ C ≥1 (R, R)[1].
The morphism Ψ is a quasi-isomorphism.
Remarks 12.3. (a) Note that the differential in g is trivial, while that in→− ≥1
C (R, R)[1] is not.
(b) Theorem 12.2 suggests an alternative way to think about the L∞ algebra g of convex polygons. Namely, we can start with the dg-algebra R anddefine the L∞ -structure on g by transferring, along the quasi-isomorphism→−
Ψ, the DGLA structure from C • (R, R) to a L∞ -structure on g. Explicit
formulas for the transfer in terms of sums over planar trees can be written
similarly to those from [KoSo3], Section 4. Note that the transfered structure
a priori depends on a choice of ∞ and hence is different from the one described49

in Section 9. Because of our theorem, these two L∞ structures are equivalent.
In particular, the transfered structure does not, up to equivalence, depend
on a choice of ∞.
On the other hand, the transfered structure has the advantage that ite i.e.,
can be defined in terms of the oriented matroid generated by the set A,e
by the knowledge of which pairs consisting of a point ω ∈ A and a subset
e satisfy ω ∈ Conv(A0 ), see [OM]. This is a much weaker and more
A0 ⊂ A,
combinatorial datum, in particular, it cannot be used to recover the conceptsof a regular decomposition or a convex piecewise affine function.
→−→−
Proof of the theorem: For simplicity, let us write C • for C ≥1 (R, R)[1] in thesequel.
Let us prove the first statement of the theorem. By Proposition 11.11, the(B ,··· ,Bm |C)components of ψ are the maps ψA0 1
from (11.3) and (11.10) and wekeep the corresponding notation. These are non-zero only if FB1 , · · · , FBm
are summands of Ri0 i1 , Ri1 i2 , ..., Rim−1 im and FC is a summand of Ri0 im .
They are also bilinear with respect to the intermediate algebras. This proves→−that ψ factors through a morphism Ψ with values in C • .
Let us now prove that Ψ is a quasi-isomorphism. It will be convenient toorganize the proof into several steps.
L→−
A. Interpretation of C • via closed paths. Recall that Rij = Q0 FA∩Q0
where Q0 runs over infinite subpolygons with edges [∞, i] and [j, ∞].
Definition 12.4. A sequence P0 , P1 , · · · , Pn of marked infinite polygons iscalled admissible, if its left and right infinite edges match as in Fig. 15, thatis, there are i0 < · · · < in such that P0 has edges [∞, i0 ] and [in , ∞] which Pν
has edges [∞, iν−1 ] and [iν , ∞]. Note that it is not required that P0 actuallycontains any of the Pν .

50

∞

∞

P0

i0

P1

P2

in

Pn

i1

is thisbig polygon

i0

in

i2
P0

Figure 15: An admissible sequence.
e ∩ Pν . For an admissible sequence P0 , P1 , · · · , Pn we
We denote Bν = A
defineP0FP1 ,··· ,Pn = HomSi0 ⊗Siop FB1 ⊗Si1 FB2 ⊗Si2 · · · ⊗Sin−1 FBn , FB0 .
n

Then

→
−n
C =

M

FPP10,··· ,Pn .

P0 ,P1 ,··· ,Pn admiss.

We note further that FPP10,··· ,Pn is the same as the cyclic tensor product Nσ (inthe sense of Remark 10.4) over the closed edge path σ obtained by runningover the negative boundaries ∂− P1 , ∂− P2 , · · · , ∂− Pn and then over ∂− P0 in
the opposite direction, see Fig. 16.

σ=

i0

∂− P1

∂− P2

i1

∂− Pn

in

i2
∂− P0

Figure 16: FPP10,··· ,Pn as a cyclic tensor product Nγ .
→−B. Filtration of C • by handle length. Note that closed edge paths σ→−labeling summands in C • , can retrace parts of themselves on the left or51

on the right. Similarly to §11, we will call such retraced parts handles anddenote them by λ and ρ, see Fig. 17.

ρ

λ
η1

η2 · · ·ηm

Figure 17: A closed path with handles.
Lemma 12.5. Put

MM→−
Gl C • =FPP10,··· ,Pn ,nthe second sum running over admissible sequences P0 , P1 , · · · , Pn with thesum of edge lengths of the two “handles” on the left and right being ≥ l.
→−
Then G is a decreasing filtration of C • by subcomplexes.
→−
Proof: The differential in C • is the sum d + δ, where δ is the Hochschilddifferential, induced by the multiplication in R, and d is induced by thedifferential in R itself. We recall the formula forδ : Hom(R⊗n , R) −→ Hom(R⊗(n+1) , R)
(in our case, Hom and ⊗ are taken over intermediate algebras, as explained
above):(δf )(a1 , · · · , an+1 ) = a1 f (a2 , · · · , an+1 ) ++nX(−1)ν f (a1 , · · · , aν aν+1 , · · · , an+1 ) + (−1)n f (a1 , · · · , an )an+1 .

ν=1Recall also the formula for the multiplication m2 in R:
The canonical map FB1 ⊗ FB2 → FB1 ∪B2 , if
m2 |FB1 ⊗FB2 = Q1 ∪ Q2 is convex and Q1 is on the left of Q2 ;0, otherwise.
52

e ∩ Qν .
Here Q1 and Q2 are two infinite subpolygons, and Bν = A→−
Consider a summand FPP10,··· ,Pn ⊂ C n (R, R). The first and last terms in
the formula for δ, applied to this summand, consist in adding an infinite
polygon P to the picture on the left and on the right so that ∂− P together
with ∂− P0 form a convex polygon (if the union is not convex, the contributionis 0). See Fig. 18.
∞

∞

P

P1

P

···

Pn

P1

P0

···

Pn

P0

a1 f (a2 , · · · , an+1 )f (a1 , · · · , an )an+1

Figure 18: The first and last terms in δ.
The new summands in which these terms lie, correspond to closed edge
paths with the length of the left or right handle increased. The remaining
summands (−1)ν f (a1 , · · · , aν aν+1 , · · · , an+1 ) of δ send FPP10,··· ,Pn into summands obtained by all possible splittings of each of the polygons Pν , ν =1, · · · , n, into two subpolygons along some edge of the form [∞, p] where p
is some intermediate vertex. If Pν is a triangle and cannot be split, thenthe corresponding contribution is 0. See Fig. 19. These summands have thesame lengths of handles. Therefore δ preserves the filtration G, and so doesd + δ.

53∞pf (a1 , · · · , aν aν+1 , · · · , an+1 )Figure 19: Intermediate terms in δ.
→−
C. Ψ : g → gr0 C • is an embedding with exact quotient. Consider themorphism of complexes
→
−
→
−
→
−
Ψ : g −→ gr0G C • = C • /G1 C • ,→−induced by Ψ. We can think of gr0G C • as the direct sum of cyclic tensor
products Nσ corresponding to closed paths σ without handles (and comingfrom admissible sequences).
On the other hand, g is the direct sum of cyclic tensor products takenover the boundaries of (convex) finite subpolygons Q0 . These boundaries are→−
particular cases of the paths we obtain for gr0G C • . Further, it follows from
Proposition 11.11 that the only matrix element of Ψ on EQ0 not landing in→−
G1 C • , is the transpose of the map γQ0 from (11.9). This transpose is in fact
an isomorphism (identity)t

γQ0 : EA0 −→ N∂Q0 = EA0 .
This establishes the following.
Lemma 12.6. Ψ is an embedding of complexes, whose image is the direct sumof summands FPP10,··· ,Pn such that the negative boundaries ∂− Pν , ν = 1, · · · , n,are segments which, together with ∂− P0 , form a convex polygon.
54

So it is enough to prove that Coker(Ψ) is exact. As a vector space,
Coker(Ψ), is simply the direct sum of all the summands FPP10,··· ,Pn without
handles, other than those forming a convex polygon as above.
Now, in grF the first and last terms of the formula for δ disappear (theyincrease the sum of the lengths of the handles). The remaining intermediate
terms correspond to splitting each Pν , ν = 1, ..., n along possible intermediate
vertices p of Pν as in Fig. 19:
PνPν0 ∪ Pν00 .

The summand corresponding to the split configuration is identical with theoriginal one:
FPP10,··· ,Pν−1 ,Pν0 ,Pν00 ,Pν+1 ,··· ,Pn = FPP10,··· ,Pn .
→−Thus grF C • is split into a direct sum of complexes of the form
FPP10,··· ,Pn ⊗k C • (∆r−1 ),where:(a) P0 , P1 , · · · , Pn is an admissible sequence such that none of the intermediate adjacent pairs Pν , Pν+1 , i + 1 ≤ n, can be combined together toform a convex polygon (from which they could be obtained by splitting).
(b) r is the number of intermediate vertices of P1 , · · · , Pn .
(c) ∆r−1 is the (r − 1)-dimensional combinatorial simplex whose vertices
correspond to the intermediate vertices (the ways of splitting) in (b),and C • (∆r−1 ) is the augmented simplicial cochain complex of ∆r−1
with coefficients in k. Note that C • (∆r−1 ) is exact if r ≥ 0.
So the remaining summands FPP10,··· ,Pn which will not be tensored with anexact complex, correspond to admissible sequences P0 , P1 , · · · , Pn which:(1) Cannot be split anywhere, i.e., each Pν , ν = 1, · · · , n, is a triangle, andso ∂− Pν is a segment.
(2) Cannot appear in the splitting of something else. This means that thesegments ∂− P1 , · · · , ∂− Pn form an upwardly convex broken line from i0
to in , which, together with ∂− P0 (a downwardly convex broken line)
55

form a convex finite polygon Q0 representing a summand in Ψ(g), seeFig. 20, where ∂− P0 depicted curved to emphasize that it can consistof many segments.
∞∂− P 1∂− P2∂− P n∂− P0

Figure 20: A convex polygon appearing.
These are precisely the summands forming Ψ(g). This proves that Ψ is aquasi-isomorphism.
→−
D. All gr≥1 C • are exact. To finish the proof of Theorem 12.2, it will now→−
be sufficient to show that each grlG C • , l ≥ 1, is an exact complex.
This is done similarly to Step C. Since the first and last summands in δ→−
disappear in grG , the differential induced by δ on grlG C • consists in splittings→−of the polygons Pν into two subpolygons, as in Fig. 19. So grlG C • splits intoa direct sum of sub complexes corresponding to different total shapes of thepictures, and the differential in each such summand consists of splittings (theδ part) plus the differential d induced from dR .
Now, if l ≥ 1, the picture representing each summand has at least one
handle, λ or ρ, nontrivial. If λ is nontrivial, then the edge path σ representing
the picture allows for nontrivial splittings of the union of λ and η1 , see Fig.
17. As in part C, his will exhibit a nontrivial (i.e., contractible) C • (∆r−1 ) asa tensor factor of our summand. Similarly if ρ is nontrivial.
Theorem 12.2 is proved.
56

13

Comparison with Gaiotto-Moore-Witten.

In this section we provide a dictionary between the approach and terminologyof [GMW] and the ones we use in the main body of the paper.
Vacua = elements of A. The construction of [GMW] starts with a finite
set V of “vacua” whose elements are denoted by i, j, k, etc., and a “weightmap” z : V → C (which we assume injective). Our set A is the image of thismap. So in this section we denote elements of A by zi , i ∈ V.
Webs=polygonal decompositions. We recall:Definition 13.1. ([GMW]) A plane web is a graph w in the plane R2 = C
together with a labeling of faces (“countries”) by vacua so that:(1) Different faces have different labels.
(2) If an edge is oriented so that i is on the left and j on the right, thenthis edge is parallel to the vector zi − zj (i.e., has the same oriented
direction but possibly different nonzero length).
Remark 13.2. More precisely, in [GMW] the labels are allowed to be repeated, with (1) replaced by a weaker condition: labels differ across edges.
Such repeating patterns are accounted for by taking the symmetric algebra
as in §2, and we do not consider them here.

j

i

k

n
m

l

Figure 21: A plane web. Edges with arrows go to infinity.
We call the set of vacua that appear as labels of the faces of a web w, thesupport of w. We further recall that two webs w and w0 are said to be of the57

same deformation type, if they are topologically equivalent in the following
restricted sense: w0 can obtained from w by translation and stretching (butnot rotating) of the edges. Following [GMW], we denote by D(w) the set
formed by all webs of the same deformation type as w, considered modulo
overall translations. This set is called the moduli space of webs in the fixeddeformation type.
Proposition 13.3. Let A0 ⊂ A and Q0 = Conv(A0 ). Then:(a) Deformation types of webs with support A0 are in bijection with regular
geometric polygonal decompositions P 0 of (Q0 , A0 ), i.e., with geometricfaces FP 0 of the secondary polytope Σ(A0 ).
(b) If w is a web whose deformation type corresponds to a decomposition
P 0 , then the moduli space D(w) is identified with the reduced normal
cone C P 0 to the face FP 0 ⊂ Σ(A0 ), see §1.
Proof: Given a polygonal decomposition P 0 = {(Q00ν , A00ν )} of (Q0 , A0 ), we form
the dual graph by putting one vertex in each polygon Q00ν and joining themas in Fig. 22:

zj

zi
zn

zk

zm

zl

Figure 22: Webs are dual to decompositions.
This gives a plane graph Γ with faces labelled by elements of A0 but notnecessarily satisfying the condition (2) of Definition 13.1. We now show thatpossible ways of deforming Γ to a web, modulo translations, are in bijection
with C P 0 .
We first recall that C P 0 consists of convex P 0 -piecewise affine functions
Q0 → R, considered modulo adding global affine functions on Q0 , i.e., C P 0 =58

CP 0 / Aff(R2 ), where CP 0 consists of actual convex P 0 -piecewise affine functions.
Next, we note that rotation by 90◦ transforms the condition (2) of Definition 13.1 into:(20 ) An edge having i on the left and j on the right, is perpendicular tothe vector zi − zj (i.e., forms, together with this vector, a positively
oriented orthogonal frame).
So we will work with (20 ) instead of (2). Let us identify (R2 )∗ with R2 bymeans of the standard scalar product.
Suppose given a piecewise-linear f ∈ CP 0 (an actual function, not a classmodulo adding global affine functions). For each polygon Q00ν ∈ P 0 let pν ∈(R2 )∗ be the slope of the affine function f |Q00ν . By the above, we can regardpν as a point of R2 . If two polygons Q00µ , Q00ν ∈ P 0 have a common edge,
the slopes of f must match on this edge, This simply means that pµ − pν isperpendicular to the edge.
So we obtain a plane graph w topologically equivalent to Γ by joining pµand pν by a straight line internal whenever Q00µ and Q00ν have a common edge
By the above, w satisfies (20 ), i.e., it is a (90◦ rotation of a) web. In this way
the condition (20 ) expresses the existence of a global piecewise affine function
f with the slope pν on each Q00ν . The function f is defined by the knowledgeof the pν up to an additive constant. Further, the convexity of f is expressedby saying that the edges of w (with respect to our choice of orientations)have positive length (a concave break would result in the length of an edge
counted as negative). Translating w as a whole amounts to adding a global
linear function to f . We leave the remaining details to the reader.
We now briefly indicate the meaning, in our terms, of some further terminology of [GMW].
Cyclic fans of vacua = subpolygons Q0 = Conv(A0 ), A0 ⊂ A. Indeed,according to loc. cit., a cyclic fan of vacua is a cyclically ordered set I ={i1 , · · · , in } such that the rays (zik − zik+1 )R+ are oriented clockwise. This
simply corresponds to zi1 , · · · zin being the vertices of a convex n-gon, in thiscyclic order.
Further, if a web w corresponds to a regular polygonal subdivision P 0 of
0
(Q , A0 ) then vertices of w are labeled, by the above, by polygons Q00ν of P 0 ,and the local fan of w at a vertex v, denoted Iv (w), corresponds the polygons
Q00ν labeling v. The fan at infinity, denoted I∞ (w), corresponds to Q0 itself.
59

Rigid, taut and sliding webs. They correspond to faces of a secondary
polytope Σ(A0 ), A0 ⊂ A of codimension 0, 1, and 2. This is a consequence of
Proposition 13.3.
Convolution of webs. It corresponds to further subdivision of one polygon
of a polygonal decomposition:∗=

Figure 23: Convolution of webs in dual language.
Half-plane webs. They correspond, in the dual language, to polygonale = Conv(A)e where A
e = A ∪ {∞},
subdivisions of the extended polytope Q
see Figs. 8 and 24. The choice of the direction towards ∞ corresponds to the
choice of a a half-plane H in [GMW]. In Fig. 24, H is positioned horizontally.
···∞···

H

Figure 24: A half-plane web in the dual language.

60

14

Maurer-Cartan elements for the FukayaSeidel category

In this section we sketch, using our approach, an application of the previous
considerations to the Fukaya-Seidel (FS) categories. We assume familiarity
with the conventional approach to the subject [Se1], so our reminder in that
respect will be minimal.
A. Setup. Let X be a complex Kähler manifold X of complex dimension k.
The Kähler metric, denoted ω 1,1 , can be understood as a family of complex
valued Hermitian scalar products in the C-vector spaces Tx X. Globally, it
gives a smooth isomorphism of C-vector bundles(14.1)

ω 1,1 : T X −→ T ∗ X,

where T X is the conjugate bundle to the C-vector bundle√ T X. Separating
the real and imaginary parts, we write ω 1,1 = g + ω −1. Thus g is a
Riemannian metric and ω is a symplectic form on X. We also assume that
X is equipped with a nowhere vanishing holomorphic volume form Ωk,0 (tobe used later to obtain Z-gradings of various Floer complexes).
Let W : X → C be a holomorphic function with only non-degenerate(Morse) critical points. These points correspond to vacua of the LandauGinzburg theory associated to W . We denote the set of critical points by
V and assume it finite: V = {x1 , · · · , xr }. We denote by wi = W (xi ) the
critical value at xi and put A = {w1 , · · · , wr } ⊂ C. We assume that A isin (affine) general position: no three points lie on a line. By identifying C
with R2 , we can apply the formalism of previous sections to this choice of A.
In particular, we have the polygon Q = Conv(A) and can speak about its
triangulations, its secondary polytope etc.
The real part Re(W ) : X → R is a Morse function on X (as a real 2kdimensional manifold), with the same critical points x1 , · · · , xr . Each xi isMorse with the same signature (k, k). As usual in Morse theory, by using the
Riemannian metric g, we associate to xi the unstable manifold (also known
in this context as the Lefschetz thimble) Ti = Ti (W ) ' Rk ⊂ X. That is, Ti
is the union of downward gradient trajectories φ(t) originating from xi , i.e.,such that limt→−∞ φ(t) = xi . We recall the following well known
Proposition 14.2. (a) The gradient flow of Re(W ) with respect to g is equal
to the Hamiltonian flow of Im(W ) with respect to ω.
61(b) Each Ti is Lagrangian with respect to ω.
Since every Hamiltonian flow preserves the Hamiltonian, part (a) implies
that the images W (Ti ) lie on horizontal half-lines wi + R≤0 ⊂ C. See Fig. 25,
where we also indicated the hyperplane H describing the “direction towards
infinity”, cf. Fig. 24. Its boundary is now positioned vertically,
H

wi
wj
...
wl

Figure 25: Thimbles project to half-lines.
Definition 14.3. The set A is called horizontally generic, if no two points of
A havethe same imaginary part 1 . For an angle θ we say that A is θ-generic,√−1θ
A is horizontally generic.
if e
We assume that A is horizontally
generic (which can be always assured by√−1θW ). Then the half-lines wi + R≤0 are disjoint,
replacing W with some eand so the thimbles Ti are disjoint closed Lagrangian submanifolds of X,
which we assume numbered in order of increasing imaginary parts: Im(w1 ) <· · · < Im(wr ). We recall that k is our fixed base field of characteristic 0. The
Fukaya-Seidel category FS = FS(X, W ) is a k-linear A∞ -category in which
T1 , · · · , Tr are objects, forming an exceptional collection:
Hom•F S (Ti , Ti ) = k,

Hom•F S (Ti , Tj ) = 0, i > j.

We denote
RS = RS(X, W ) =M

Hom•F S (Ti , Tj )i≤jthe total A∞ -algebra associated to FS. The goal is to obtain RS by deforming an appropriate algebra R as in §11 with respect to an appropriate
Maurer-Cartan element.
1

e = A ∪ {−∞} is in general position, see §6 andIn other words, the extended set A
Fig.8.

62

Remark 14.4. There is no doubt that the formalism of Fukaya-Seidel categories generalizes to a situation much more general than the one describedabove: we must be able to allow W to have non-Morse (and even nonisolated) critical points. In this case, each critical value wi (their number will
still be finite in any algebraic situation) would give rise to a “local FukayaSeidel category” Ci . These local categories would then embed into a global
category FS and form there a semi-orthogonal family. We expect that thismore general setting to be still analyzable by the methods of this paper. In
particular, the generality of extended systems of coefficients in §10 is (intentionally) geared to this future context, the category Ci corresponding to thedg-algebra Si , see Remark 10.2.
B. The system of coefficients. The algebra R comes from an extended
coefficient system (Si , Nij• ) as in §10. We put Si = k (so the setting of §9 is
in fact sufficient for this section). This corresponds to the fact that each xi
is a Morse singular point for W . The graded vector spaces Nij• are obtainedby analyzing those angles θ for which W is not θ-generic. These are precisely
θij = − arg(wi − wj ) ∈ S 1 = R/2πZ,i 6= j.

We associate to them the unit vectors√

ζij = e

−1θij


=

wi − wj|wi − wj |

−1
.

√

Note that for any θ ∈ S 1 the function Re(e −1θ W ) : X → R is again a Morsefunction with critical points x1 , · · · , xn√of signature (k, k) and so gives riseto the “θ-rotated” thimbles Tiθ = Ti (e −1θ W ). The image of Tiθ under W
lies now on the half-line originating from wi and having angle −θ with thereal axis. Following [GMW] we introduce the following
Definition 14.5. (a) By a ζij -soliton we will mean a gradient trajectory
φ = φ(t) of Re(ζij W ) originating (at t = −∞) from xi and terminating (att = +∞) at xj . Thus the image of a ζij -soliton under W is the straightinterval [wi , wj ].
(b) By a gradient polygon we mean a sequence φ = (φi0 i1 , φi1 i2 , · · · , φim i0 ),where φiν iν+1 is a ζiν iν+1 -soliton. A gradient polygon is called convex, or acyclic fan of solitons, if the intervals [wiν , wiν+1 ] form the boundary of aconvex polygon, in the counter-clockwise order.
63θThus the union of all ζij -solitons is the intersection of the thimbles Tijijθand Tj ji Since existence of gradient trajectories between critical points of thesame index is a codimension 1 phenomenon in the space of Morse functions,see, e.g., [KS], the number of ζij -solitons is “typically” finite. We assume
this to be the case and denote this number by nij = nji . Further, let Nij be
the k-vector space having, as basis vectors, symbols eφ corresponding to theζij -solitons φ. So dim(Nij ) = nij .
We now explain how to introduce a Z-grading on each Nij . As usualwith Floer complexes (of which the Nij are particular cases, see below), thegrading is not fully canonical. More canonical are the following two types of
constraints, which admit a solution but not uniquely.
Duality: Nji = Nij∗ as a graded space. Cf. the requirement (4) of Definition
10.1 .
Maslov index grading of cyclic products: The grading in each tensor
product of the form Ni0 i1 ⊗ · · · Nim−1 im ⊗ Nim i0 , m ≥ 1 (cf. Fig. 11, where all
Si = k), is given topologically as follows. By definition, this tensor producthas a basis {eφ } labelled by gradient polygons φ, and we define the degreeof each eφ . This is done by an instance of the Maslov index construction,
which we review here in the generality that we need, referring to [Se1], §11
for the general case.
For a topological space Y and x, y ∈ Y we denote by Πx,y Y the set of
homotopy classes of paths in Y beginning at x an ending at y, i.e., the set ofmorphisms from x to y in the fundamental groupoid of X.
For a symplectic R-vector bundle E over some base B we denote byLG(E) the bundle of the Lagrangian Grassmannians of the fibers of E overB.
Consider first the case when B = pt, so E is a symplectic vector space.
Let L1 , L2 ∈ LG(E) be two Lagrangian subspaces meeting transversely, i.e.,
E = L1 ⊕ L2 . In this case there is a canonical homotopy class κL1 ,L2 ∈ΠL1 ,L2 LG(E), defined uniquely by three properties:(1) Naturality with respect to symplectic isomorphisms E → E 0 .
(2) Additivity in symplectic direct sums. That is, if E = E 0 ⊕ E 00 , andLν = L0ν ⊕ L00ν , then κL1 ,L2 is the image of (κL01 ,L02 , κL001 ,L002 ) under themorphism of fundamental groupoids induced by LG(E 0 ) × LG(E 00 ) ,→LG(E).
64

(3) Normalization in dimension 2. If dimR (E) = 2, then LG(E) ' R1 = S 1
is a circle, equipped with orientation. In this case κL1 ,L2 is represented
by the circle arc going from L1 to L2 in the positive direction.
Let us call κL1 ,L2 the symplectic bridge between L1 and L2 .
Returning now to our situation, each gradient polygon φ gives a piecewise continuous closed path $φ in LG(T X) obtained by associating to any
internal point p of the segment φiν iν+1 the tangent space at p to the thimble Tiν (ζiν iν+1 W ). At each point iν , the path $φ has a simple discontinuity(jump), with the two limit values before and after the jump forming a pairof transversal Lagrangian subspaces in Txν X. Joining each such pair by
the symplectic bridge, we obtain a closed continuous path $φ∧ in LG(T X),
defined uniquely up to homotopy.
Taking the exterior power and applying the holomorphic volume formk,0
Ω gives a continuous map (“phase”)
Λk

Ωk,0

R
LG(T X) −→LG(ΛkC T X) −→ LG(C) = RP 1 = S 1 .

Denote this composite map Θ. Note that LG(C) = S 1 has a canonicalcounterclockwise orientation.
Definition 14.6. The Maslov index d(φ) of a gradient polygon φ is thewinding number of the continuous closed path Θ($φ∧ ) in S 1 = LG(C). We
assign to the basis vector eφ the degree d(φ).
C. Transversality assumptions. Note that the intersection of the Laθθgrangian manifolds Tijij and Tj ji is never transversal: if non-empty, it has
dimension at least one. There are two ways of realizing nij as the number of intersection points of two Lagrangian manifolds which can be (and,“typically”, are) transversal.
(1) Take an intermediate point p on the interval [wi , wj ] and consider the(smooth) Kähler manifold W −1 (p) of dimension k−1. The intersectionsθVi (p) = Ti ij ∩ W −1 (p),θVj (p) = Tj ji ∩ W −1 (p)are Lagrangian spheres (“vanishing cycles”) in W −1 (p) whose intersecθ
θtion points are in bijection with trajectories constituting Tijij and Tj ji .
65See Fig. 26. We assume that Vi (p) and Vj (p) intersect transversally in
W −1 (p).

W −1 (p)Vi (p)

θ

Ti ij

θ

Ti ij

xi

xj
Vj (p)

wj

wip

Figure 26: Vanishing cycles over a midpoint.
(2) For small  > 0 put θ0 = θij −  and θ00 = θji + . Then the thimbles
000Tiθ and Tjθ project by W onto two half lines which intersect in a point
q somewhere near the interval [wi , wj ], see Fig. 27. The intersection
000
points of Tiθ and Tjθ , all lying over q, are again in bijection withθθ
trajectories constituting Tijij ∩ Tj ji . We assume (this assumption is000equivalent to that in (1)) that Tiθ and Tjθ intersect transversally in X.
000

W (Tjθ )W (Tiθ )qwj

wi

Figure 27: Perturbed thimbles in projection to C.

D. The instanton equation. Suppose that the Kähler manifold X is exact,i.e., there exists a 1-form λ primitive for ω, i.e., dλ = ω. It is convenient66

to write λ symbolically as λ = p dq which refers to a particular choice of aprimitive in local Darboux coordinates p = (p1 , · · · , pk ), q = (q1 , · · · , qk ).
The ζij -solitons, being Hamiltonian flow curves for H = Im(ζij W ) are,according to elementary Hamiltonian mechanics, critical points of the “phasespace action” which is the functionalZZhζij (φ) =p dq + Hdt :=φ∗ λ + Im(ζij W )dt.
φR

It is defined on the space Xij of smooth maps φ = φ(t) : R → X with theboundary conditions
lim φ(t) = xi ,t→−∞

lim φ(t) = xj .

t→+∞This interpretation is used in [GMW] for constructing both L∞ and A∞
structures in terms of supersymmetric quantum mechanics. That is, ourtransversality assumptions imply that hζij is itself a Morse function on Xij ,i.e., the ζij -solitons are non-degenerate critical points. The grading on Nijis, up to shift, given by the relative Morse (Floer) index of hζij .
The Riemannian metric g = Re(ω 1,1 ) on X defines the L2 -Riemannian
metric on Xij :Z(δ1 φ, δ2 φ) =
δ1 φ(t), δ2 φ(t) g dt,
R

and so we can speak about the gradient flow of hζij on Xij . We will view aparametrized curve in Xij with parameter s as a map Φ = Φ(t, s) : R2 → X
or, equivalently,
as an X-valued function Φ(τ ) of one complex variable τ =√t + −1s ∈ C.
Definition 14.7. (a) The Kähler gradient of W (with respect to ω 1,1 ) is thevector field gradω1,1 (W ) on X (i.e., a smooth section of T X) obtained fromthe section dW of T ∗ X by applying the composite isomorphism(ω 1,1 )−1v→v

T ∗ X −→ T X −→ T X,
where (ω 1,1 )−1 is the inverse to (14.1), and v → v is the canonical C-antilinear
identification between T X and T X.
(b) For ζ ∈ C∗ , the ζ-instanton equation, or the Witten equation [FJR],
is the following condition on an X-valued function Φ = Φ(τ ) : C → X:∂Φ= −ζ gradω1,1 (W ).
∂τ67To explain the meaning of the equation, at any τ0 ∈ C√∂∂∂
Φ(τ0 ) =Φ(τ0 ) + −1 Φ(τ0 )∂τ∂t∂s
is a C-linear combination of two tangent vectors to X at Φ(τ0 ), so itself anelement of TΦ(τ0 ) X. It is required that this element is equal to the value of−ζ gradω1,1 (W ) at Φ(τ0 ).
The ζ-instanton equation can be seen as a complex ∂-analog of the ordinary differential equation of downward gradient flow for Re(ζW ). In fact, it
reduces to that equation, if we assume that Φ(t, s) = φ(t) is independent ons.
Proposition 14.8 ([GMW]). A curve Φ(s) in Xij represented by a mapΦ = Φ(t, s) : R2 → C, is a gradient trajectory for hζij , if and only if itsatisfies the ζij -instanton equation.
Examples 14.9. (a) If X = C with coordinate z and ω 1,1 = dz dz is thestandard flat Kähler metric, then the equation has the form∂Φ= −ζ W 0 (Φ).
∂τ(b) More generally,system on
P if1,1(z1 , · · · , zk ) is a local holomorphic coordinate
1
1,1
X, and ω =ωa,b dza dz b , then we can write Φ(τ ) as (Φ (τ ), · · · , Φk (τ ))and the equation has the form


X
√
∂Φa
∂
∂
∂W
=
+ −1
Φa = −ζ
η a,b
.
b
∂τ
∂t
∂s
∂Φ
b
1,1
k.
Here kη a,b k is the inverse matrix to kωa,b

E. The Maurer-Cartan element. Let g = gA,N be the L∞ -algebra corresponding to our set A and the coefficient system given by the Nij defined in§B. Thus, see §10,Mg =NQ0 ⊗ or(Σ(A0 )), A0 = Q0 ∩ A,
Q0

where Q0 runs over all convex subpolygons with vertices on A.
68

Fix one such Q0 and suppose that it has vertices i0 , · · · , im (counterclockwise). Then the basis of NQ0 is formed by the vectors eφ for all gradientpolygons φ = (φi0 i1 , · · · , φim i0 ) that project (under W ) onto the boundary∂Q0 . Let w0 be the web dual to Q0 , i.e., w0 is the normal fan of Q0 , see §13
and Fig. 28. Thus w0 has one vertex and the sectors Cν of w are in bijection
with the vertices iν of Q0 .
Fix further a gradient polygon φ covering Q0 and let Mζ (φ) be the moduli
space of solutions of the ζ-instanton equations Φ : C → X with the following
asymptotic conditions:(1) When τ approaches infinity deep inside Cν , then Φ(τ ) approaches xiν .
(2) Consider an edge e of w0 separating some Cν and Cν+1 , see Fig. 28.
For a point τ ∈ e let l(τ ) be the line orthogonal to e passing throughτ . Then, as τ → ∞ on e, the restriction Φ|l(τ ) approaches the soliton
φiν ,iν+1 .

w0
l(τ )

e
τ

Cν

iν
Cν−1

iν+1

iν−1
Q0

Cν+1

Figure 28: The asymptotic condition for ζ-instantons.
The proposal of [GMW] can be mathematically summarized as follows.
Conjecture 14.10. (a) Under sufficient genericity assumptions, Mζ (φ) is amanifold of dimension d(φ) − 1, equipped with a natural orientation.
(b) In particular, for φ such that d(φ) = 1, i.e., eφ is a basis vector of g1 ,we have a well defined signed cardinality γζ (φ) ∈ Z of Mζ (φ) (signs come69

from the orientation of the 0-dimensional manifold). The element
Xγ = γζ =γζ (φ)eφ ∈ g1
d(φ)=1is a Maurer-Cartan element.
(c) The deformation of the A∞ -algebra R∞ with respect to γζ is identified
with RS(ζW ), the A∞ -algebra associated to the Fukaya-Seidel category ofζW .
Remarks 14.11. (a) Since Nij is spanned by the critical points of hζij on
Xij , it carries a natural Floer differential. This differential is in fact a part ofthe datum provided by α, corresponding to m = 1, i.e., to gradient polygons
with 2 edges, so that m = 1 and i0 = i, i1 = j.
(b) Suppose that m ≥ 2 and let φ be a gradient polygon lying over theboundary of a convex sub polygon Q0 . In this case it seems likely that forany ζ-instanton Φ ∈ Mζ (φ), the compositionΦWC −→ X −→ Cmaps C bijectively onto the interior of Q0 . If this is so, then one can think
about ζ-instantons as “sections” of W over Q0 .
(c) The argument justifying the Maurer-Cartan condition for γ proposed
in [GMW] is based on “counting the ends” of the 1-dimensional manifoldsMζ (φ) for eφ ∈ g2 . More generally, this type of argument suggests theexistence of a compactification Mζ (φ) of Mζ (φ) by adding strata which areproducts of Mζ (ψ) for “smaller” gradient polygons ψ. More precisely, strata
are labelled by data P consisting of:(1) A regular polygonal subdivision P = {Q00ν } of the polygon Q0 (image of
φ under W ).
(2) An assignment, to each intermediate edge of the subdivision, of an
instanton projecting onto this edge.
Such a datum can be thought of as a lifting of P into X and is equivalent
to a “colored web” in the terminology of [GMW]. Each polygon Q00ν is thenlifted to a gradient polygon ψ ν , and the stratum Mζ (P) is the product of
the Mζ (ψ ν ). In other words, the compactification Mζ (φ) has the strata
70structure and the factorization property, completely parallel to those of thesecondary polytope Σ(A ∩ Q0 ).
Further, for a sequence of instantons (Φn ) in Mζ (φ), the condition of
approaching the stratum Mζ (P), i.e., the condition that lim Φn ∈ Mζ (P),can be seen as a kind of tropical degeneration. That is, fixing ε > 0, consider
the sets
Un (ε) := {τ ∈ C : kdτ Φn k2 ≥ ε}.
Then, as ε → 0 and n → ∞ simultaneously in a compatible way (∀ ∃n), thesets Un (ε) converge, in the Gromov-Hausdorff sense, to the web w dual toP. The shape of each individual Un (ε) can be compared to the amoeba of aplane algebraic curve.

71

15

Speculations and directions for further work

1. From Stasheff polytopes to a categorical structure. A ζij -instantonin §14 is the m = 2 case of the following “catastrophe” (non-generic occurrence) for a Morse function: a chain of m critical points of the same index
connected by gradient trajectories in a sequential way. It was shown in [KS]that generic deformations of such a catastrophe are governed by the Stasheff
polytope Km whose vertices correspond to bracketings of m + 1 factors. As
well known, combinatorics of the Km is at the basis of the formalism of A∞ algebras and categories. It is therefore interesting to relate this appearanceof the Km to other categorical structures and to wall crossing formulas in
2 and more dimensions. This seems especially appealing since the Km also
describe [KS] higher syzygies among the Steinberg relations for elementary
matrices eij (λ) ∈ GLN :eij (λ)ejk (µ) = ejk (µ)eik (λ + µ)eij (λ).
Such relations, each having the shape of a pentagon K3 , appear naturally inthe wall-crossing formulas.
2.Higher TFT structures and Ed -algebras. It seems certain that in ddimensions the L∞ -algebra g from §3 can be refined to an Ed -algebra, and g∞
to an Ed−1 -algebra. We plan to address this in a subsequent paper, currently
in preparation. The possibility of such refinement is in agreement with thehigher analog of the Deligne conjecture [KoSo2, Lur, Gin] which says thatthe deformation complex of an Ed−1 -algebra is naturally an Ed -algebra. The
quasi-isomorphism Φ in the Universality Theorem should then be a morphismof Ed -algebras. Further, the full higher-dimensional analog of the concept ofa coefficient system from §10 should associate to any simplex of codimension
p with vertices in A, an Ep -algebra.
3. Curvilinear theory. It would be interesting to develop the analog of
the formalism of Fukaya-Seidel categories and of [GMW] for the case when
the potential W takes values not in C but in some Riemann surface S. In
this case the set A of critical values makes S into a “marked surface” in thesense of Teichmüller theory and one can use isotopy classes of curvilinear
triangulations of S with vertices in A in order to analyze various categoricalstructures. In particular, geodesics of quadratic differentials as well as spectral networks of Gaiotto-Moore-Neitzke [GMN] seem like natural objects toappear in such a theory.
72

4. Infinite-dimensional case. Much of the recent “physical” work on
Picard-Lefschetz theory and its generalizations was motivated by the infinitedimensional example of the complexified Chern-Simons functional [Wit]. Here
X is the (universal cover of the) space of all connections in a principal bundle on a 3-manifold M , whose structure group is a complex semi simple Liegroup. The set A of critical values of the Chern-Simons functional consists ofcomplex numbers of substantial arithmetic importance (involving volumes ofhyperbolic manifolds, regulators of elements of K3 (Q) and the like). The nature of polygons and triangulations that can be formed out of these numbers
remains mysterious. It would be very interesting if Picard-Lefschetz theory
imposed some constraints on the convex geometry of these numbers.
The case of the holomorphic Chern-Simons functional (M is a 3-dimensional
complex Calabi-Yau) is even more mysterious.

73

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Continuation

Of

Chern-Simons

Theory,M.Ka.: Kavli Institute for Physics and Mathematics of the Universe (WPI), 5-1-5
Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan. Email: mikhail.kapranov@ipmu.jp
M.Ko.: Institute des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Buressur-Yvette, France. Email: maxim@ihes.fr
Y. So.: Department of Mathematics, Kansas State University, Cardwell Hall, Manhattan, KS 66506 USA. Email: soibel@math.ksu.edu

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