UPR 1178-T
DISTA-2007
TUW-07-08
Worldsheet Instantons and Torsion Curves
Part B: Mirror Symmetry
Volker Braun1, Maximilian Kreuzer2,
Burt A. Ovrut1, and Emanuel Scheidegger3
1 Department of Physics, University of Pennsylvania,
209 S. 33rd Street, Philadelphia, PA 19104–6395, USA
2 Institute for Theoretical Physics, Vienna University of Technology,
Wiedner Hauptstr. 8-10, 1040 Vienna, Austria
3 Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale
via Bellini 25/g, 15100 Alessandria, Italy, and INFN - Sezione di Torino, Italy
Abstract
We apply mirror symmetry to the problem of counting holomorphic
rational curves in a Calabi-Yau threefold X with Z3 ⊕ Z3 Wilson lines.
As we found in Part A [1], the integral homology group H2(X,Z) = Z
Z3 ⊕ Z3 contains torsion curves. Using the B-model on the mirror of X
as well as its covering spaces, we compute the instanton numbers. We
observe that X is self-mirror even at the quantum level. Using the self-
mirror property, we derive the complete prepotential on X , going beyond
the results of Part A. In particular, this yields the first example where
the instanton number depends on the torsion part of its homology class.
Another consequence is that the threefold X provides a non-toric example
for the conjectured exchange of torsion subgroups in mirror manifolds.
Email: vbraun, ovrut@physics.upenn.edu, Maximilian.Kreuzer@tuwien.ac.at,
esche@mfn.unipmn.it
http://arxiv.org/abs/0704.0449v1
Contents
1 Introduction 1
2 Calabi-Yau Threefolds 5
2.1 The Calabi-Yau Threefold X . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Intermediate Quotient X . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Toric Geometry and Mirror Symmetry 7
3.1 Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 The Batyrev-Borisov Construction . . . . . . . . . . . . . . . . . . . . . 10
3.3 Toric Intersection Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Mori Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 B-Model Prepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Quotienting the B-Model 20
4.1 The Quotient by G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 The Quotient by G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 B-Model on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4 Instanton Numbers of X . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 B-Model on X
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.6 Instanton Numbers of X∗ . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.7 Instanton Numbers Assuming The Self-Mirror Property . . . . . . . . . 33
5 The Self-Mirror Property 34
6 Factorization vs. The (3,1,0,0,0) Curve 38
7 Towards a Closed Formula 41
8 Conclusion 43
A Triangulation of ∇̄∗ and ∇∗ 44
B The Flop of X∗ 50
Bibliography 53
1 Introduction
Counting world sheet instantons (that is, holomorphic curves) on a Calabi-Yau three-
fold has had a large number of applications in mathematics and physics, ever since it
was essentially solved by mirror symmetry several years ago [2]. The purpose of this
paper is to take into account an important subtlety that does not appear in very sim-
ple Calabi-Yau manifolds like hypersurfaces in smooth toric varieties. This subtlety
is the appearance of torsion curve classes. That is, the homology1 group
= Z3 ⊕ Z3 ⊕ Z3 (1)
contains the torsion2 subgroup Z3⊕Z3. Here, the manifold of interest X is a quotient
of one of Schoen’s Calabi-Yau manifolds [3, 4] by a freely acting symmetry group.
There are already a few known examples of such Calabi-Yau manifolds with torsion
curves [5, 6, 7, 8, 9], but the proper instanton counting has never been done before.
The prime motivation for studying these curves is that one would like to compute
the superpotential for the vector bundle moduli [10, 11, 12, 13, 14, 15, 16] in a heterotic
MSSM [17, 18, 19, 20, 21, 22, 23, 24, 25]. Our main result will be that there exist
smooth rigid rational curves in X that are alone in their homology class. This proves
that, in general, no cancellation between contributions to the superpotential W from
instantons in the same homology class can occur.
Therefore we would like to count rational curves on X . In physical terms, we need
to find the instanton correction F
X,0 to the genus zero prepotential of the (A-model)
topological string on X . This is usually written as a (convergent) power series in
h11 variables qa = e
2πita . Each summand is the contribution of an instanton, and
the (integer) coefficients are the multiplicities of instantons in each homology class.
According to [26, 27, 1] the novel feature of the 3-torsion curves on X is that for each
3-torsion generator we need an additional variable bj such that b
j = 1. The Fourier
series of the prepotential on X becomes
X,0(p, q, r, b1, b2) =
n1,n2,n3∈Z
m1,m2∈Z3
n(n1,n2,n3,m1,m2) Li3
pn1qn2rn3bm11 b
, (2)
where n(n1,n2,n3,m1,m2) is the instanton number in the curve class (n1, n2, n3, m1, m2).
For the purpose of computing the prepotential, we can either use directly the A-
model or start with the B-model and apply mirror symmetry. The A-model calculation
was carried out in the companion paper [1], entitled Part A. The results were:
• A set of powerful techniques to compute the torsion subgroups in the integral
homology and cohomology groups of X . They are spectral sequences starting
with the so-called group (co)homology of the group action on the universal cover
1In the following, Z3
= Z/3Z always denotes the integers mod 3. Similarly, we write (Z3)
⊕nZ3 = Z3 ⊕ · · · ⊕ Z3 for the Abelian group generated by n generators of order 3.
2Not to be confused with the torsion tensor of a connection.
• A closed formula for the genus zero prepotential
X,0(p, q, r, b1, b2) =
i,j=0
pbi1b
P (q)4P (r)4+O(p2) =
i,j=0
Li3(pb
2)+· · · (3)
to linear order in p, extending the one computed in [28] for the universal cover X̃ .
Here, if p(k) is the number of partitions of k ∈ Z≥, then P (q) is the generating
function for partitions,
P (q)
p(i)qi =
ln q)
. (4)
• Expanding eq. (3) as an instanton series we find that the number of rational
curves of degree (1, 0, 0, m1, m2) is:
n(1,0,0,m1,m2) = 1, ∀ m1, m2 ∈ Z3. (5)
Furthermore, these curves have normal bundle OP1(−1)⊕OP1(−1). Hence, there
are indeed 9 smooth rigid rational curves which are alone in their homology class.
Alternatively, one can start with the B-model topological string and apply mirror
symmetry, which is what we will do in this paper, entitled Part B. This will allow
us to obtain the higher order terms in p. The order in p up to which one wants to
compute the instanton numbers is only limited by computer power. We will again
find a closed formula at every order in p, however, this time by guessing it from the
instanton calculation, and hence only up to the order given by this limitation. The
way to arrive at this result is as follows:
• The universal cover X̃ admits a simple realization as a complete intersection in
a toric variety. In this situation, mirror symmetry boils down to an algorithm to
compute instanton numbers. Unfortunately, there are many non-toric divisors
which cannot be treated this way. It turns out that, after descending to X ,
precisely the torsion information is lost. In this approach one can only compute
X,0(q1, q2, q3, 1, 1).
• As a pleasant surprise we find strong evidence that the manifold X is self-mirror.
In particular, we attempt to compute the instanton numbers on the mirror X∗
by descending from the covering space X̃∗. The embedding of X̃∗ into a toric
variety is such that all 19 divisors are toric. In principle, this allows for a
complete analysis including the full Z3 ⊕Z3 torsion information, but this is too
demanding in view of current computer power.
• Although the full quotient X = X̃/(Z3 × Z3) is not toric, it turns out that a
certain partial quotient X = X̃/Z3 can be realized as a complete intersection
in a toric variety. That way, one only has to deal with h11(X) = 7 parameters,
which is manageable on a desktop computer. Assuming the self-mirror property,
we work with the mirror X
, for which again all divisors are toric, and we can
compute the expansion of F
X,0(p, q, r, 1, b2) to any desired degree. A symmetry
argument then allows one to recover the b1 dependence as well. Finally, we can
extract the instanton numbers n(n1,n2,n3,m1,m2) including the torsion information.
• As can be seen from the A-model result eq. (3), we observe that the prepotential
X,0 at order p factors into
i,j=0 b
2 times a function of p, q, r only. This
means that the instanton number does not depend on the torsion part of its
homology class. We will explain the underlying reason for this factorization
and show that it breaks down at order p3. This fits nicely with the B-model
computation at order p3, where the instanton numbers do depend on the torsion
part.
• Another consequence of the self-mirror property is that X is a non-toric example
for the conjecture of [6]. By this conjecture, certain torsion subgroups of the
integral homology groups are exchanged under mirror symmetry.
An easily readable overview and a discussion of the physical consequences of our
findings for superpotentials and moduli stabilization of heterotic models was pre-
sented in [27]. The present Part B is self-contained and can be read independently of
Part A [1]. All necessary results from Part A are reproduced in this part.
As a guide through this paper, we start in Section 2 with a brief overview of the
topology of the various spaces involved as determined in Part A [1]. This is followed by
a review of the Batyrev-Borisov construction of mirror pairs of complete intersections
in toric varieties in Section 3. We illustrate this construction by means of the covering
spaces X̃ and X̃∗ of our example. The review includes the techniques to compute the
B-model prepotential and the mirror map. These are applied in Section 4 to the partial
quotients X and X
yielding the main results stated above. This assumes that X as
well as X are self-mirror, and evidence for this property is recapitulated in Section 5.
Moreover, we show how the torsion subgroups are exchanged. Section 6 contains an
explanation for the breakdown of the factorization alluded to above. Putting all the
information together we try to guess a closed form for the prepotential in Section 7.
Finally, we present our conclusions in Section 8. In the course of this work we will
notice that a certain flop of X is very natural from the toric point of view, and we
will present it in Appendix B.
2 Calabi-Yau Threefolds
2.1 The Calabi-Yau Threefold X
The Calabi-Yau manifold X of interest is constructed as a free G
= Z3 × Z3 quo-
tient of its universal covering space X̃ . The latter is one of Schoen’s Calabi-Yau
threefolds [3]. It is simply connected and hence easier to study. Among its various
descriptions are the fiber product of two dP9 surfaces, a resolution of a certain T
orbifold [29], or a complete intersection in a toric variety. In the present Part B,
we will mostly use the latter viewpoint. The simplest way is to introduce the toric
ambient variety P2×P1×P2 with homogeneous coordinates
[x0 : x1 : x2], [t0 : t1], [y0 : y1 : y2]
∈ P2×P1×P2 . (6)
The embedded Calabi-Yau threefold X̃ is then obtained as the complete intersection
of a degree (0, 1, 3) and a degree (3, 1, 0) hypersurface in P2×P1×P2. We restrict
the coefficients of their defining equations Fi = 0 to a particular set of three complex
parameters λ1, λ2, λ3, such that the polynomials Fi read
x30 + x
1 + x
x0x1x2
= F1 (7a)
λ1t0 + t1
y30 + y
1 + y
λ2t0 + λ3t1
y0y1y2
= F2. (7b)
For the special complex structure parametrized by λ1, λ2, λ3 the complete intersection
is invariant under the G = Z3 × Z3 action generated by (ζ
[x0 : x1 : x2] 7→ [x0 : ζx1 : ζ
[t0 : t1] 7→ [t0 : t1] (no action)
[y0 : y1 : y2] 7→ [y0 : ζy1 : ζ
[x0 : x1 : x2] 7→ [x1 : x2 : x0]
[t0 : t1] 7→ [t0 : t1] (no action)
[y0 : y1 : y2] 7→ [y1 : y2 : y0]
One can show that the fixed points of this group action in P2 ×P1 ×P2 do not satisfy
eqns. (7a) and (7b), hence the action on X̃ is free.
2.2 The Intermediate Quotient X
The partial quotient
G1 (9)
will be of particular interest in this paper because this quotient is generated by phase
symmetries, see eq. (8a), and hence is toric. In particular, we will need a basis of
Kähler classes. As usual, we will not distinguish degree-2 cohomology and degree-4
homology classes but identify them via Poincaré duality. Part A [1] ?? shows that3
= H2(X̃,Z)G1 ⊕ Z3 = spanZ
φ, τ1, υ1, ψ1, τ2, υ2, ψ2
⊕ Z3. (10)
Hence, by abuse of notation, we can identify the free generators on X with the G1-
invariant generators on X̃ , see Part A eq. (??), via the pull back by the quotient map.
The triple intersection numbers on X = X̃/Z3 are one-third of the corresponding
intersection numbers on X̃ listed in Part A eq. (??). Hence, the intersection numbers
on X are
φτ1τ2 = 3 φτ1υ2 = 3 φτ1ψ2 = 6 φυ1τ2 = 3 φυ1υ2 = 3
φυ1ψ2 = 6 φψ1τ2 = 6 φψ1υ2 = 6 φψ1ψ2 = 12 τ
1 τ2 = 1
τ 21υ2 = 1 τ
1ψ2 = 2 τ1υ1τ2 = 3 τ1υ1υ2 = 3 τ1υ1ψ2 = 6
τ1ψ1τ2 = 3 τ1ψ1υ2 = 3 τ1ψ1ψ2 = 6 τ1τ
2 = 1 τ1τ2υ2 = 3
τ1τ2ψ2 = 3 τ1υ
2 = 3 τ1υ2ψ2 = 6 τ1ψ
2 = 6 υ
1τ2 = 3
υ21υ2 = 3 υ
1ψ2 = 6 υ1ψ1τ2 = 6 υ1ψ1υ2 = 6 υ1ψ1ψ2 = 12
2 = 1 υ1τ2υ2 = 3 υ1τ2ψ2 = 3 υ1υ
2 = 3 υ1υ2ψ2 = 6
2 = 6 ψ
1τ2 = 6 ψ
1υ2 = 6 ψ
1ψ2 = 12 ψ1τ
2 = 2
ψ1τ2υ2 = 6 ψ1τ2ψ2 = 6 ψ1υ
2 = 6 ψ1υ2ψ2 = 12 ψ1ψ
2 = 12.
Clearly, G2 acts on the partial quotient X . From Part A eq. (??) it follows that,
of the 7 non-torsion divisors above, only 3 are G2-invariant. This invariant part is
particularly manageable and will be important in the following. We find
= span
φ, τ1, τ2
with products 3τ 2i = τiφ. In particular, the triple intersection numbers on X are
τ 21 τ2 = 1, τ1φτ2 = 3, τ1τ
2 = 1, (13)
and 0 otherwise. Finally, the second Chern class of X is c2(X) = 12(τ
1 + τ
2 ). There-
fore,
· τ1 = 12, c2
· φ = 0, c2
· τ2 = 12. (14)
2.3 Variables
As we discussed in Part A ??, the instanton-generated superpotential should be
thought of as a series with one variable for each generator in H2. In particular, we
3The torsion in H2 are just the Wilson lines, that is, first Chern classes of flat line bundles. They
will play no role in the following. The torsion curves in H2, on the other hand, are the focus of this
paper.
Calabi-Yau
threefold
) Free
generators
Torsion
generators
X̃ Z19
p0, q0, . . . , q8, r0, . . . , r8
X = X̃/G1 Z
7 ⊕ Z3
P,Q1, Q2, Q3, R1, R2, R3
X = X̃/G Z3 ⊕ Z3 ⊕ Z3
p, q, r
b1, b2
Table 1: The different Calabi-Yau threefolds, curve classes, and variables used
to expand the prepotential.
will be interested in the Calabi-Yau threefolds X̃ , X , and X . For these, the degree-2
integral homology and the variables used (see Part A [1] for precise definitions) are
in summarized Table 1. Pushing down the curves by the respective quotients lets us
express the prepotential on the quotient in terms of the prepotential on the covering
space. We found in Part A that
P,Q1, Q2, Q3, R1, R2, R3, b1)
PQ51Q
3 b1, Q
2 , Q
3b1, Q3, 1, b1, Q1Q
2 , R
1, R3, 1, b
1, R1R
p, q, r, b1, b2) =
p, q, b2, b2, r, b
2, b1
. (16)
3 Toric Geometry and Mirror Symmetry
In this section we review mirror symmetry and the construction of the B-model for
the mirror of the covering space X̃ . Since X̃ is a complete intersection in a toric
variety, we can use the standard constructions. Because we expect the model to
be self-mirror, we will analyze the B-model for X̃ and its mirror X̃∗. The toric
geometry for X̃ is much simpler4 than for X̃∗, but contains less information. In this
section we will start with the simpler model in order to review the Batyrev-Borisov
construction for the mirror of a complete intersection in a toric variety. Then we will
apply this construction to the more complicated model, now without going into too
many details. We will see that, on the simpler side, not all parameters are toric and
no torsion is visible. However, on the more complicated mirror side, all parameters
are toric which will allow us, in principle, to perform the B-model computation of
4Meaning that X̃ is a complete intersection in the very simple toric variety P2 ×P1 ×P2, whereas
X̃∗ is embedded in a complicated toric ambient variety.
the complete prepotential. As X̃ ∼= X̃∗ is expected to be self-mirror, this determines
the complete prepotential F
eX∗,0
as well. In practice, however, the analysis is
computationally too involved.
Fortunately, the space X = X̃/G1 and its mirror will turn out to be both tractable
with toric methods and sufficiently informative. This quotient will be the subject
of Section 4. Finally, this is also the starting point for arguing in Section 5 that the
self-mirror property persists at the level of instanton corrections.
Recall that, in Subsection 2.1 we defined our Calabi-Yau manifold as the complete
intersection
F1 = 0, F2 = 0
⊂ P2×P1×P2 (17)
with the two polynomials F1, F2 as in eqns. (7a) and (7b), respectively. In order to
construct the mirror manifold following Batyrev and Borisov, we need to reformulate
this definition in terms of toric geometry. We review here some essential ingredients
of toric geometry, for details we refer to [30, 31] and references therein. We will give
the abstract definitions and concepts step by step, and at each step illustrate them
with the example X̃ and its mirror manifold X̃∗.
3.1 Toric Varieties
Given a lattice N of dimension d, a toric variety VΣ is defined in terms of a fan Σ
which is a collection of rational polyhedral5 cones σ ⊂ N such that it contains all
faces and intersections of its elements. VΣ is compact if the support of Σ covers all
of the real extension NR of the lattice N . The resulting d-dimensional variety VΣ is
smooth if all cones are simplicial and if all maximal cones are generated by a lattice
basis.
Let Σ(1) denote the set of one-dimensional cones (rays) with primitive generators ρi,
i = 1, . . . , n. The simplest description of VΣ introduces n homogeneous coordinates zi
corresponding to the generators ρi of the rays in Σ
(1). These homogeneous coordinates
are then subjected to weighted projective identifications
z1 : · · · : zn
1 z1 : · · · : λ
a = 1, . . . , h (18)
for any nonzero complex number λ ∈ C×, where the integer n-vectors q
i are genera-
tors of the linear relations
i ρi = 0 among the primitive lattice vectors
6 ρi. In or-
der to obtain a well-behaved quotient, we must exclude an exceptional set Z(Σ) ⊂ Cn
5Here, the tip of the cone is always the origin of N . A cone is rational if it is spanned by rays
which pass through lattice points (other than the origin), that is, have rational slopes. A cone is
polyhedral if it is the cone over an (d − 1)-dimensional polytope. In other words, curved faces are
not allowed.
6We will use the same symbol ρ∗ to denote the generators in Σ
(1) and the corresponding primitive
lattice vectors in N .
that is defined in terms of the fan, as will be explained below. Hence, the quotient is
Cn − Z(Σ)
, (19)
where Γ ≃ N/ span{ρi} is a finite abelian group. There are h = n−d independent C
identifications, therefore the complex dimension of VΣ equals the rank d of the lattice
N . The identifications by Γ are only non-trivial if the ρi do not span the lattice N .
Refinements of the lattice N with fixed ρi can hence be used to construct quotients
of toric varieties VΣ by discrete phase symmetries such as Z3. Such quotients will be
discussed in Section 4. Note that the rays ρi are in 1–to–1 correspondence with the
(C×)-invariant divisors Di on VΣ, which are defined as
zi = 0
⊂ VΣ. (20)
Conversely, the homogeneous coordinate zi is a section of the line bundle O(Di).
For example, consider the simplest compact toric variety, the projective space Pd.
Its fan Σ = Σ(∆) is generated by the n = d+ 1 vectors
ρ1 = e1, ρ2 = e2, . . . , ρn−1 = ed, ρn = −
ei (21)
of a d-dimensional simplex ∆. They satisfy a single linear relation,
i=1 ρi = 0.
Therefore qi = 1 for all i, and the homogeneous coordinates in eq. (18) are the usual
homogeneous coordinates on Pd.
For products of toric varieties we simply extend the relations for any single factor
by zeros and take the union of them. Hence, the fan of the polyhedron ∆∗ describing
the 5-dimensional toric variety P2×P1×P2 in eq. (17) is generated by the n = 5+3 = 8
vectors
ρ1 = e1, ρ2 = e2, ρ3 = −e1 − e2, ρ4 = e3,
ρ5 = −e3, ρ6 = e4, ρ7 = e5, ρ8 = −e4 − e5
satisfying the linear relations
ρi = 0. (23)
Except for the origin, there are no other lattice points in the interior of ∆∗. The
corresponding homogeneous coordinates will be denoted by
z1 = x0, z2 = x1, z3 = x2,
z4 = t0, z5 = t1,
z6 = y0, z7 = y1, z8 = y2.
In more general situations, given a polytope ∆∗ ⊂ N we will denote the resulting toric
variety by P∆∗ = VΣ(∆∗).
3.2 The Batyrev-Borisov Construction
Batyrev showed that a generic section ofK−1
, the anticanonical bundle of P∆∗, defines
a Calabi-Yau hypersurface if ∆∗ is reflexive, which means, by definition, that ∆∗ and
its dual
x ∈MR
∣∣∣ (x, y) ≥ −1 ∀y ∈ ∆∗
are both lattice polytopes. Here, M = Hom(N,Z) is the lattice dual to N and MR is
its real extension. Mirror symmetry corresponds to the exchange of ∆ and ∆∗ [32].
The generalization of this construction to complete intersections of codimension r > 1
is due to Batyrev and Borisov [33, 34]. For that purpose, they introduced the notion
of a nef partition. Consider a dual pair of d-dimensional reflexive polytopes ∆ ⊂
∗ ⊂ NR. In that context, a partition E = E1 ∪ · · · ∪ Er of the set of vertices
of ∆∗ into disjoint subsets E1, . . . , Er is called a nef-partition if there exist r integral
upper convex Σ(∆∗)-piecewise linear support functions φl : NR → R, l = 1, . . . , r such
φl(ρ) =
1 if ρ ∈ El,
0 otherwise.
Each φl corresponds to a divisor
D0,l =
Dρ (27)
on P∆∗ , and their intersection
Y = D0,1 ∩ · · · ∩D0,r (28)
defines a family Y of Calabi-Yau complete intersections of codimension r. Moreover,
each φl corresponds to a lattice polyhedron ∆l defined as
x ∈MR
∣∣∣ (x, y) ≥ −φl(y) ∀y ∈ NR
. (29)
The lattice points m ∈ ∆l correspond to monomials
〈m,ρi〉
i ∈ Γ (P∆∗ ,O(D0,l)) . (30)
One can show that the sum of the functions φl is equal to the support function of K
and, therefore, the corresponding Minkowski sum is ∆1 + · · · + ∆r = ∆. Moreover,
the knowledge of the decomposition E = E1 ∪ · · · ∪Er is equivalent to that of the set
of supporting polyhedra Π(∆) = {∆1, . . . ,∆r}, and therefore this data is often also
called a nef partition.
It can be shown that given a nef partition Π(∆) the polytopes7
{0} ∪ El
⊂ NR (31)
define again a nef partition Π∗(∇) = {∇1, . . . ,∇r} such that the Minkowski sum
∇ = ∇1 + · · · + ∇r is a reflexive polytope. This is the combinatorial manifestation
of mirror symmetry in terms of dual pairs of nef partitions of ∆∗ and ∇∗, which we
summarize in the diagram
∆ = ∆1 + . . .+∆r ∆
∇1, . . . ,∇r
Mirror
Symmetry
uujjj
MR NR
∆1, . . . ,∆r
(∆l,∇l′) ≥ −δl l′ ∇ = ∇1 + . . .+∇r
. (32)
In the horizontal direction, we have the duality between the lattices M and N and
mirror symmetry goes from the upper right to the lower left. The other diagonal has
also a meaning in terms of mirror symmetry as we will explain below. The complete
intersections Y ⊂ P∆∗ and Y
∗ ⊂ P∇∗ associated to the dual nef partitions are then
mirror Calabi-Yau varieties.
Let us now apply the Batyrev-Borisov construction to the complete intersection
eq. (17), hence r = 2. There exist several nef-partitions of ∆∗. The one which has
the correct degrees (3, 1, 0) and (0, 1, 3) is, up to exchange of t0 and t1, E1 = {ρi |i =
1, . . . , 4} and E2 = {ρi |i = 5, . . . , 8}. Adding the origin and taking the convex hull
yields the polytopes
ρ1, . . . , ρ4, 0
, ∇2 =
ρ5, . . . , ρ8, 0
, (33)
where the ρi are defined in eq. (22). The two divisors cutting out the Calabi-Yau
threefold are, according to eq. (27),
D0,1 =
Di, D0,2 =
Di ⇒ X̃ = D0,1 ∩D0,2 ⊂ P∆∗ (34)
Note that, while ∆∗ has no further lattice points, its dual ∆ has 18 vertices and 300
lattice points. Using the computer package PALP [35], we determine the associated
polytopes ∆1 and ∆2 of the global sections of O(D0,1) and O(D0,2), respectively. In
an appropriate lattice basis there is, up to symmetry, a unique nef partition consisting
ν1, . . . , ν6, 0
, ∆2 =
ν7, . . . , ν12, 0
, (35)
7The brackets
· · ·
denote the convex hull.
where
ν1 = 2e1 − e2, ν2 = −e1 + 2e2, ν3 = −e1 − e2,
ν4 = 2e1 − e2 − e3, ν5 = −e1 + 2e2 − e3, ν6 = −e1 − e2 − e3,
ν7 = 2e4 − e5, ν8 = −e4 + 2e5, ν9 = −e4 − e5,
ν10 = e3 + 2e4 − e5, ν11 = e3 − e4 + 2e5, ν12 = e3 − e4 − e5.
Among these 12 vectors there are the 7 independent linear relations
3ν3 + ν4 + ν5 − 2ν6 = 0, 3ν9 + ν10 + ν11 − 2ν12 = 0,
ν1 − ν3 − ν4 + ν6 = 0, −ν1 + ν2 + ν4 − ν5 = 0,
ν7 − ν9 − ν10 + ν12 = 0, −ν7 + ν8 + ν10 − ν11 = 0,
−ν2 + ν5 − ν8 + ν11 = 0.
The convex hull ∇∗ = 〈∆1,∆2〉 yields the fan Σ(∇
∗) and, consequently, the toric
variety P∇∗ . Let D
i , i = 1, . . . , 12 be the divisors associated to the vertices νi. Then,
by eq. (27), the nef partition eq. (35) defines the divisors
D∗0,1 =
D∗i , D
0,2 =
D∗i , ⇒ X̃
∗ = D∗0,1 ∩D
0,2 ⊂ P∇∗ (38)
cutting out the mirror complete intersection X̃∗. In contrast to ∆∗, the polytope ∇∗
contains extra integral points. We find that it contains, in addition to the origin and
the vertices in eq. (36), the 26 points
ν13 =
(ν4 + ν5 + ν6) = −e3, ν12+6k+i+j =
(ν3k+i + 2ν3k+j),
ν14 =
(ν10 + ν11 + ν12) = e3, ν15+6k+i+j =
(ν3k+j + 2ν3k+i)
∀ k ∈ {0, . . . , 3}, (i, j) ∈
(1, 2), (1, 3), (2, 3)
For completeness, note that the dual polytope ∇ has 15 vertices and 24 lattice points.
Running PALP to compute the Hodge numbers using the formula of [36], we obtain
= h1,2
= h1,1
= h1,2
= 19, (40)
in agreement with Part A [1], eq. (??).
So far, we have mainly focused on the information contained in the reflexive poly-
topes ∆∗ and ∇∗ and ignored their duals. We have already mentioned that in the
reflexive case a generic section of K−1
defines a Calabi-Yau manifold, and that such
sections are provided by the lattice points of ∆. In other words, ∆ and ∇ are the
Newton polytopes of Y and Y ∗, respectively. That is, the complete intersection Y
(Y ∗) is defined by r polynomial equations, and the exponents of the monomials in
each equation are the lattice points in ∆ (∇). More precisely, the Minkowski sum for,
say, ∆ = ∆1 + · · ·+∆r defines r homogeneous polynomials
Fl(z) =
∇l′∩N
〈m,ρi〉+δl l′
i , l = 1, . . . , r (41)
with coefficients al,m ∈ C. The simultaneous vanishing of F1, . . . , Fr then defines
the complete intersection Calabi-Yau manifold Y ⊂ P∆∗ . Exchanging ∆l and ∇l′ in
eq. (41) yields the equations F ∗l defining the mirror manifold Y
∗. It is in this sense
that the map from the upper left to the lower right in eq. (32) is also a manifestation
of mirror symmetry. Since we will not need the actual polynomials for X̃ and X̃∗, we
refrain from writing them explicitly. Instead, we refer the reader to Section 4, where
we determine the equations in a simpler situation.
3.3 Toric Intersection Ring
Up to now we have only considered one of the ingredients in the fan Σ, namely, the
generators ρ ∈ Σ(1) which defined the C× action in eq. (19). The second ingredient
is the exceptional set Z(Σ). It corresponds to fixed loci of a continuous subgroup of
for which the quotient eq. (19) is not well defined. Therefore, these loci have
to be removed. In terms of the homogeneous coordinates zi, this happens precisely
when a subset {zi |i ∈ I}, I ⊆ {1, . . . , n}, of the coordinates vanishes simultaneously
such that there is no cone σ ∈ Σ containing all of the ρi ⊆ σ, i ∈ I. Hence, the set
Z(Σ) is the union of the sets ZI = {[z1 : · · · : zn] |zi = 0 ∀i ∈ I}. Minimal index
sets I with this property are called primitive collections [37]. In order to determine
the index sets I we need a coherent8 triangulation T = T (∆∗) of the polytope ∆∗
for which all simplices contain the origin. Different triangulations will yield different
exceptional sets and, hence, different toric varieties. However, for simplicity, we will
mostly suppress the choice of a triangulation in the notation. In the case of complete
intersections, only those triangulations of ∆∗ are compatible with a given nef partition
that can be lifted to a triangulation of the corresponding Gorenstein cone, see [38].
The polytope defining projective space Pd admits a unique triangulation with
the required properties, and this triangulation consists of n = d + 1 simplices. The
only primitive collection is I = {1, . . . , n}. This is well-known from the definition
of projective space, where we have to remove the origin z1 = · · · = zd+1 = 0 from
Cd+1. Similarly, the polyhedron ∆∗ for the ambient space P∆∗ of X̃ admits a unique
triangulation, and the primitive collections are those of its factors, that is,
I1 = {1, 2, 3}, I2 = {4, 5}, I3 = {6, 7, 8}. (42)
8Coherent triangulations, sometimes also called regular triangulations, satisfy some technical
property that is equivalent to the associated toric quotient being Kähler.
The mirror polyhedron∇∗, on the other hand, admits a huge number of triangulations.
We will discuss particularly interesting triangulations of the mirror polyhedron at the
end of Appendix A.
The primitive collections determine the cohomology ring of toric varieties and,
together with the nef partition, complete intersections. Recall that if the collection
ρi1 , . . . , ρik of rays is not contained in at least one cone, then the corresponding ho-
mogeneous coordinates zil are not allowed to vanish simultaneously. Therefore, the
corresponding divisors Dil have no common intersection. Hence, we obtain non-linear
relations RI = Di1 · . . . ·Dik = 0 in the intersection ring. It can be shown that all such
relations are generated by the primitive collections I = {i1, . . . , ik} defined above.
The ideal generated by these RI is called Stanley-Reisner ideal
ISR =
RI , I primitive collection
⊂ Z[D1, . . . , Dn], (43)
and Z[D1, . . . , Dn]/ISR is the Stanley-Reisner ring. The intersection ring of a non-
singular compact toric variety PΣ is [39]
= Z [D1, . . . , Dn]
(m, ρi)Di
. (44)
In other words, the intersection ring can be obtained from the Stanley-Reisner ring by
adding the linear relations
(m, ρi)Di = 0, where it is sufficient to take a set of basis
vectors for m ∈ M . In particular, the intersection number of the divisors spanning a
maximal-dimensional simplicial cone σ = span
R≥{ρi1 , . . . , ρid} is
Di1 · . . . ·Did =
Vol(σ)
, (45)
where Vol(σ) is the lattice-volume, that is, the geometric volume divided by the
volume 1
of a basic simplex. For practical purposes it is sufficient to compute one
of these volumes, the remaining intersections can be obtained using the linear and
non-linear relations.
Having found the intersection ring of the ambient toric variety, we now turn to
the complete intersection Y ⊂ P∆∗ . The toric part of its even-degree intersection ring
is [40]
Hevtoric
= Q [D1, . . . , Dn]
IY , (46)
where IY is the ideal quotient
(m, ρi)Di
D0,l. (47)
Note that it can happen that some of the Di appear as generators of IY . This means
that they can be set to zero in the intersection ring. Geometrically, this means that
these divisors do not intersect a generic complete intersection Y . While the inter-
section ring depends on the triangulation T (∆∗) through the primitive collections
defining the Stanley-Reisner ideal, we conjecture that the divisors Di not intersecting
Y are independent of the choice of triangulation. This conjecture is proven for r = 1
and supported by a large amount of empirical evidence for r > 1. We conclude that
the dimension dimH2toric(Y ) is in general smaller than h
1,1(Y ) for the following two
reasons: Only h = n − d = dimH2(P∆∗,Z) divisors are realized in the ambient toric
variety P∆∗ , and some of them may not descend to the complete intersection Y . Using
the adjunction formula we can compute the the Chern classes of Y by expanding
c(Y ) =
(1 +Di)
(1 +D0,l)
. (48)
The intersection ring together with the second Chern class determine the diffeomor-
phism type of a simply-connected Calabi-Yau manifold [41]. If we consider the coho-
mology with integral coefficients there can be torsion and, in fact, this is what this
paper is all about. Unfortunately, a combinatorial formula in terms of the fan Σ(∆)
for the torsion in the integral cohomology of a toric Calabi-Yau manifold is only known
in the hypersurface case [6].
We now illustrate these concepts in the example of the complete intersection X̃ ⊂
P∆∗ = P
2 × P1 × P2 and its mirror manifold X̃∗. In eq. (42) we already determined
the primitive collections, hence the corresponding Stanley-Reisner ideal is
ISR =
D1D2D3, D4D5, D6D7D8
. (49)
The linear equivalences are D1 = D2, D1 = D3, D4 = D5, D6 = D7, D6 = D8 and,
hence, we can choose K1 = D4, K2 = D1, K3 = D6 as a basis for H
2(P∆∗). In terms
of this basis, we obtain D0,1 = K1+3K2 and D0,2 = K1+3K3, see eq. (27). Therefore,
the ideal I
in eq. (34) is
2K2 −K2
2K3, K1K2 − 3K2
2, K1K3 − 3K3
2, K1
2, K2
3, K3
. (50)
Next, we define the restriction of the Ki to X̃ to be the divisors
J̃i = Ki · X̃ = Ki(K1 + 3K2)(K1 + 3K3). (51)
We need to compute one of the intersection numbers directly from the volume of a
cone, say, J̃1J̃2J̃3 = K1K2K3(K1 + 3K2)(K1 + 3K3) = 9K1K
3 , where we made use
of the relations in I
. Using eq. (45), this intersection can be evaluated to be
3 = 9D1D2D4D6D7 = 9/Vol
〈ρ1, ρ2, ρ4, ρ6, ρ7〉
= 9/Vol
〈e1, e2, e3, e4, e5〉
Then, again using eq. (50), we see that the only non-vanishing intersection numbers
and the second Chern class are
J̃22 J̃3 = 3, J̃1J̃2J̃3 = 9, J̃2J̃
3 = 3,
· J̃1 = 0, c2
· J̃2 = 36, c2
· J̃3 = 36.
Note that only h
toric(X̃) = 3 of the h
1,1(X̃) = 19 parameters are realized torically.
Comparing the triple intersection numbers with eq. (13), it is clear that these 3 toric
divisors are precisely the G-invariant divisors on X̃ .
A similar, though much more complicated, calculation can be done for X̃∗ ⊂ P∇∗ .
Using the results of Appendix A one can show that, among the points in eq. (39), the
14 divisors D∗13, D
14, D
12+6k+i+j , D
15+6k+i+j, k = 0, 2 appear as generators of eq. (47)
and, therefore, do not intersect X̃∗. Subtracting from the remaining 24 divisors in
eqns. (36) and (39) the remaining 5 linear relations in eq. (37), we find that all
toric(X̃
∗) = h1,1(X̃∗) = 19 moduli are realized torically.
3.4 Mori Cone
As we have just seen, the cohomology classes Di span H
2(PΣ,R) = H
1,1(PΣ). The
Kähler classes of a smooth projective toric variety PΣ form an open cone in H
1,1(PΣ)
called the Kähler cone K(PΣ). This cone has a combinatorial description in terms of
the fan Σ, which we now review.
First, define a support function to be a continuous function ψ : NR → R given on
each cone σ ∈ Σ by an mσ ∈MR via
ψ(ρ) = (mσ, ρ) ∀ρ ∈ σ ⊂ NR. (54)
A support function determines a divisor D =
ψ(ρi)Di. We say that D is convex if
ψ is a convex function on NR. The convex classes form a non-empty strongly convex
polyhedral cone in H1,1(PΣ) whose interior is the Kähler cone K(PΣ). Such a support
function is strictly convex if and only if
ψ(ρi1 + · · ·+ ρik) > ψ(ρi1) + · · ·+ ψ(ρik) (55)
for every primitive collection I = {i1, . . . , ik} [40]. The dual of the Kähler cone
K(PΣ) is called the Mori cone or the cone of numerically effective curves NE(PΣ).
Its generators can be described by vectors l(a) of the corresponding linear relations∑
i ρi = 0. Each face of the Kähler cone K(PΣ) is dual to an edge of NE(PΣ).
These edges are generated by curves c(a), and the entries of the vector l(a) are
= c(a) ·Di. (56)
A practical algorithm to find the generators for l(a) in terms of the triangulation T (∆∗)
is described in [42]. Of course, we are not interested in the ambient space but in a
complete intersection Y ⊂ P∆∗ . The restriction of a Kähler class on the ambient space
yields a Kähler class on Y , but not every Kähler class on Y arises that way. We define
the toric part of the Kähler cone on Y as the restriction [43]
K(Y )toric = K(PΣ)
⊂ K(Y ). (57)
In the simplicial case, we can always take the basis Ji of H
toric(Y,Q) to be edges of the
Kähler cone. The dual of the toric Kähler cone of Y is the (toric) Mori cone NE(Y )toric.
This is sufficient for mirror symmetry purposes, however, it can be larger than the
actual cone of effective curves. Once the generators l(a) of NE(P∆∗) are determined,
we need to add the information about the nef partition. For this purpose, we define
= −D0,m · c
(a) m = 1, . . . , r. (58)
Finally, it is customary to write the generators of the Mori cone NE(Y )toric as
l(a) =
0,1 , . . . , l
0,r ; l
1 , . . . , l
, (59)
which are, by abuse of notation, again denoted by l(a). The knowledge of the (toric)
Mori cone is important for several reasons. It defines the local coordinates on the
complex structure moduli space of the mirror Y ∗ near the point of maximal unipotent
monodromy. Moreover, the generators enter the coefficients of the fundamental period
which is a solution of the Picard-Fuchs equations as we will review in Subsection 3.5.
For example, using the unique primitive collections in eq. (42), the Mori cone for
P∆∗ is generated
l(1) =(0, 0, 0, 1, 1, 0, 0, 0)
l(2) =(1, 1, 1, 0, 0, 0, 0, 0)
l(3) =(0, 0, 0, 0, 0, 1, 1, 1).
Recalling the nef partition D0,1 = D1 + · · ·+D4, D0,2 = D5 + · · ·+D8, we prepend
(−D0,1 · c
(a),−D0,2 · c
(a)) = (−3, 0), (−1,−1), (0,−3), a = 1, 2, 3, to obtain the gener-
ators
l(1) =(−1,−1; 0, 0, 0, 1, 1, 0, 0, 0)
l(2) =(−3, 0; 1, 1, 1, 0, 0, 0, 0, 0)
l(3) =( 0,−3; 0, 0, 0, 0, 0, 1, 1, 1)
of the Mori cone NE(X̃)toric. Due to the large number of toric moduli, the calculation
for the Mori cone NE(P∇∗) of the ambient toric variety of the mirror X̃
∗ is much more
complex.
9We sort the Mori cone generators such that the first one corresponds to the P1 of the ambient
space, and the second and third generator are the hyperplane sections of the two P2. In other words,
we have J̃a · c
(b) = δb
. This is the basis of curves that we used for the A-model computation.
3.5 B-Model Prepotential
Mirror symmetry identifies the quantum corrected Kähler moduli space of Y with
the classical complex structure moduli space of Y ∗, see the excellent treatise in [43]
for details. The deformations of the complex structure of Y ∗ are encoded in the
periods ̟ =
Ω and the latter can be computed from the equations F ∗l that cut
out Y ∗ ⊂ P∇∗ . Given the Mori cone eq. (59) and the classical intersections numbers
κabc = Ja · Jb · Jc we follow [44, 45, 38, 43] to write down a local expansion of the
periods, convergent near the large complex structure point, which is characterized
by its maximal unipotent monodromy. In the following, we will review just the bare
essentials.
The coefficients ai in the polynomial constraints F
l of the complete intersection Y
see eq. (41), define the complex structure of Y ∗. A particular set of local coordinates
ua on the complex structure moduli space on Y
∗ is defined by
i b = 1, . . . , h (62)
where h
toric(Y ) and am,0 is the coefficient in (41) corresponding to the origin in
∇l. In these coordinates, the point of maximal unipotent monodromy is at ub = 0.
We define the cohomology-valued period
̟(u, J) =
{na≥=0}
0,mJa
a=1 l
0,mna
a=1 l
una+Jaa . (63)
where (x)n = Γ(x + n)/Γ(x) is the Pochhammer symbol. Note that the choice of
triangulation is implicit in the generators l(a) of the Mori cone. Expanding ̟(u, J)
by cohomology degree yields
̟(u, J) = ̟(0)(u) +
̟(1)a (u)Ja +
̟(2)a (u)κabcJbJc −̟
(3)(u) dVol, (64)
where dVol is the volume form. The coefficients in eq. (64) are the fundamental period
̟(0)(u), that is, the unique solution to the Picard-Fuchs equations holomorphic at
ua = 0, and
̟(1)a (u) = ∂Ja̟(u, J)|J=0, ̟
a (u) =
κabc∂Jb∂Jc̟(u, J)|J=0,
̟(3)(u) = −
κabc∂Ja∂Jb∂Jc̟(u, J)|J=0.
These coefficients coincide with the basis of solutions of the Picard- Fuchs equations
obtained from the Frobenius method in [46, 31]. The B-model prepotential FBY ∗,0 is
Y ∗,0(u) =
2̟(0)(u)2
̟(0)(u)̟(3)(u) +
̟(1)a (u)̟
a (u)
. (66)
At the large complex structure point the mirror map defines natural flat coordinates
on the Kähler moduli space of the original manifold Y , which are
i (u)
̟0(u)
, i = 1, . . . , h. (67)
We also define qj = e
2πitj = uj + O(u
2). One way to obtain the prepotential is to
compute its third derivatives
C∗abc = DaDbDcF
Y ∗,0 =
Ω ∧ ∂a∂b∂cΩ, (68)
and apply the Picard-Fuchs operators. This leads to linear differential equations,
which determine C∗abc up to a common constant, see again [46, 43] for details. The
quantum corrected three point function Cijk(q) on Y follows from C
abc(u) using the
inverse mirror map eq. (67) u = u(t), and one obtains
Cijk(q) =
̟(0)(u(q))2
C∗abc(u(q)). (69)
In practice, we use the formula
Cijk(q) = ∂ti∂tj
k (u(q))
̟(0)(u(q))
. (70)
Integrating three times with respect to ti yields the prepotential F
Y ∗,0(t) up to a
polynomial of degree three in ti which can be determined partially by the topological
data of Y .
Mirror symmetry then ensures that the B-model prepotential, eq. (66), is equal to
the A-model prepotential. That is,
FY,0(q) = F
Y ∗,0(u(q)). (71)
This allows us to compute the instanton numbers nd. For the case of interest,
X̃ ∈ P∆∗ = P
2 × P1 × P2, (72)
we refer to [28] where this program been carried out in detail. The same calculation
can in principle be done on the mirror X∗, but the large number of toric moduli
again makes it highly extensive. Instead, we refer to the next section where a suitable
quotient of X̃∗ will be treated in detail for which the computations are reasonably
simple.
4 Quotienting the B-Model
In this section we consider the quotientX = X̃/G in terms of toric geometry and study
the mirror of X in this context. In order to achieve this, we first analyze the partial
quotient X = X̃/G1. Using the techniques introduced in Section 3, we construct the
mirror X
. Using their toric realization, we perform the B-model computation for the
non-perturbative prepotentials F
and F
, respectively. Finally, we explain how
one can implement the quotient by G2 on both sides in order to obtain X and X
4.1 The Quotient by G1
We start with a review of the general discussion of free quotients of complete intersec-
tions in toric geometry in [31]. Consider a fan Σ ⊂ NR and pick a lattice refinement
N̄ such that Γ = N̄/N is a finite abelian group. Such a lattice refinement consists of
a finite sequence of lattice refinements of the form N → N +wpZ which are described
by a vector wp =
αpiρi with αpi ∈ Z. The group Γ is then isomorphic to
Zkp .
Let Σ̄ be the fan obtained from Σ by relating everything to the lattice N̄ . In this
context, we make some additional identifications in the toric quotient eq. (19) [47].
One finds that VΣ̄ = VΣ/Γ is the quotient of VΣ by the finite abelian group Γ. Its
action on the homogeneous coordinates is by multiplication by phases
z1 : · · · : zn
ξα1z1 : · · · : ξ
, ξ = e
k , (73)
for every cyclic subgroup of order k. We will denote such group actions by Zk :
(α1, . . . , αn). If VΣ is a compact toric variety, then the quotient VΣ̄ is never free [39].
However, a hypersurface or complete intersection in VΣ need not intersect the set
of fixed points, and in that case we get a smooth quotient manifold with nontrivial
fundamental group.
We now apply this to P∆∗ = P
2 × P1 × P2 defined in eq. (22). The first step in
performing the quotient of P∆∗ by G1 thus amounts to a refinement N̄ = wZ+N of
the lattice N with index |G1| = 3. From the definition eq. (8a) of the action of G1 on
P∆∗ and eq. (24) we read off that the refinement is by a vector
ρ2 + 2ρ3 + ρ7 + 2ρ8
+ Z5. (74)
The resulting polytope ∆̄∗ admits the same nef partition as ∆∗ in eq. (33),
∇̄1 = 〈ρ̄1, . . . , ρ̄4, 0〉, ∇̄2 = 〈ρ̄5, . . . , ρ̄8, 0〉. (75)
where we express the generators ρ̄ in terms of ρ as
ρ̄i = ρi, i = 1, . . . , 6 ,
ρ̄7 = ρ7 + e1 + 2e2 + e4 + 2e5, ρ̄8 = ρ8 − e1 − 2e2 − e4 − 2e5.
It is easy to check that the ρ̄i satisfy the same linear relations eq. (23) as the ρi,
and that w = 1
(ρ̄1 − ρ̄2 + ρ̄6 − ρ̄7) = −e2 − e5. The ρ̄i together with w therefore
indeed generate the lattice N̄ . Note that, while all 8 non-zero lattice points of ∆̄∗
are vertices, the dual polytope ∆̄ has 18 vertices and 102 points. Using PALP [35]
again, we compute the lattice points of the polytope ∇̄∗ = 〈∆̄1, ∆̄2〉 ⊂MR, which will
describe the ambient space of the mirror X
of X . We find
∆̄1 = 〈ν̄1, . . . , ν̄6, 0〉, ∆̄2 = 〈ν̄7, . . . , ν̄12, 0〉, (77)
where we express the vertices ν̄i in terms of the vertices νi of ∇
ν̄3k+1 = ν3k+1, ν̄3k+2 = ν3k+2 − e5, ν̄3k+3 = ν3k+3 + e5, k = 0, . . . , 3. (78)
Again, it is easy to check that the ν̄i satisfy the same linear relations eq. (37) as the
νi. It turns out that the lattice points of ∇̄
∗ generate a sublattice M̄ of index 3 in M ,
and the lattice refinement is generated by
ν̄1 + 2ν̄2 + 2ν̄7 + ν̄8
= e2 + e4 − e5. (79)
Among the points of ∇∗ listed in eq. (39) only ν13 and ν14 are also lattice points of
the sublattice M̄ . In fact, we have ν̄13 = ν13 and ν̄14 = ν14. Hence, ∇̄
∗ has 12 vertices
and 15 lattice points; its dual ∇̄ = ∇̄1 + ∇̄2 has 42 lattice points among which 15 are
vertices10.
Once we have the polytopes ∆̄∗ and ∇̄∗, we can construct X and X
as complete
intersections entirely analogous to X̃ and X̃∗, see Section 3. That is, using eq. (27),
we define
X = D̄0,1 ∩ D̄0,2, X
= D̄∗0,1 ∩ D̄
0,2 (81)
in terms of the nef partitions eq. (75) and (77), respectively. Here, D̄i and D̄
i denote
the divisors associated to the generators ρ̄i and ν̄i, respectively. The absence of fixed
points of the G1 action on the complete intersection X̃ is guaranteed by the fact
that the resulting polytope ∆̄∗ ⊂ N̄R has no additional lattice points [31]. Hence,
X = X̃/G1 has a non-trivial fundamental group π1(X) = Z3. Surprisingly, it turns
out that the mirror X
is a free quotient as well. To see this recall that, as noticed
above, the lattice points of ∇̄∗ generate a sublattice M̄ of index 3 inM . Furthermore,
∇̄∗ also has no additional lattice points with respect to ∇∗. Therefore, there is a
10Note that all of our polytopes differ from the non-free Z3 × Z3 quotient of ∆
∗ defined in [28],
Proposition 7.1. In the notation of [31] their quotient is
∇∗ 6= P
1 1 1 0 0 0 0 0
0 0 0 1 1 1 0 0
0 0 0 0 0 0 1 1
Z3 : 0 1 2 0 0 0 0 0
Z3 : 0 0 0 0 1 2 0 0
and has 21 points and 8 vertices in the lattice N .
group G∗1 ≃ Z3 acting torically on P∇∗ . On the homogeneous coordinates this action
g∗1 :
z1 : · · · : z12
ζz1 : ζ
2z2 : z3 : · · · : z6 : ζ
2z7 : ζz8 : z9 : · · · : z12
. (82)
Hence, X
= X̃∗/G∗1 also has a non-trivial fundamental group π1(X
) = Z3. Note
that this never happens for hypersurfaces in toric varieties [6]. Having the toric
representation of X and X
, we can now compute their Hodge numbers. It turns out
= h1,2
= h1,1
= h1,2
= 7, (83)
in agreement with Part A [1], eq. (??).
4.2 The Quotient by G2
We now turn to the G2 action, which does not act torically. Hence, we cannot, in
principle, find a toric variety containing X = X/G2 as we did for the G1 quotient
above. However, at least we have to ensure that X and X
are G2-symmetric. This
can be achieved via suitable symmetries in the toric data.
The easy part of the toric data for X is the polytope ∆̄∗. The G2 action on the
ambient space permutes the homogeneous coordinates, see eq. (8b). In terms of toric
geometry, this means that it permutes the corresponding points of the polytope. That
is11,
g2 : ρ̄i 7→ ρ̄1+(i mod 3) ∀i ∈ {1, 2, 3},
g2 : ρ̄4 7→ ρ̄4, ρ̄5 7→ ρ̄5,
g2 : ρ̄5+i 7→ ρ̄6+(i mod 3) ∀i ∈ {1, 2, 3}.
It induces a mirror group actionG∗2 onX
which is geometrical, rather than a quantum
symmetry as discussed in [48]. The action of G∗2 is obviously the dual group action
on the dual lattice M , which again must be a symmetry of the relevant polytope ∇̄∗.
We find that
g∗2 : ν̄3k+i 7→ ν̄3k+1+(i mod 3) ∀k = 0, . . . , 3, i ∈ {1, 2, 3}. (85)
As a check on the mirror group action, note that the matrix of scalar products, see
eq. (87) below, is invariant. That is,
g2(ρ̄l), g
2(ν̄l′)
ρ̄l, ν̄l′
∀ l, l′. (86)
By abuse of notation, we denote the corresponding cyclic permutation of homogeneous
coordinates by g∗2 as well. Using this action, we define the mirror of X to be X
11We define the modulus operation such that (i mod 3) ∈ {0, 1, 2}.
/G∗2. This idea has already been used for the construction of mirrors of orbifolds
of the quintic [49] soon after the discovery of the first mirror construction by Greene
and Plesser.
Following eq. (41), the equations for the Calabi-Yau complete intersections X and
are defined by evaluating the matrix of scalar products 〈ρ̄i, ν̄j〉+ δl l′ , which are
〈 , 〉+ δl l′ ν̄1 ν̄2 ν̄3 ν̄4 ν̄5 ν̄6 ν̄13 ν̄7 ν̄8 ν̄9 ν̄10 ν̄11 ν̄12 ν̄14
ρ̄1 3 0 0 3 0 0 1 0 0 0 0 0 0 0
ρ̄2 0 3 0 0 3 0 1 0 0 0 0 0 0 0
ρ̄3 0 0 3 0 0 3 1 0 0 0 0 0 0 0
ρ̄4 1 1 1 0 0 0 0 0 0 0 1 1 1 1
ρ̄5 0 0 0 1 1 1 1 1 1 1 0 0 0 0
ρ̄6 0 0 0 0 0 0 0 3 0 0 3 0 0 1
ρ̄7 0 0 0 0 0 0 0 0 3 0 0 3 0 1
ρ̄8 0 0 0 0 0 0 0 0 0 3 0 0 3 1
The equations of X can now be read off from the columns of eq. (87), and one finds
F1 = (λ5t0 + λ6t1)(x
0 + x
1 + x
2) + (λ7t0 + λ8t1)x0x1x2, (88a)
F2 = (λ1t0 + λ4t1)(y
0 + y
1 + y
2) + (λ2t0 + λ3t1)y0y1y2, (88b)
where the G2-symmetry has been imposed. Note that the last monomial in each
equation corresponds to the vector 0 ∈ ∆̄l, l = 1, 2. Two of the eight coefficients λm
can be fixed by normalizing the equations, say λ4 = λ5 = 1, and three correspond to
the symmetries of P1, that is, SL(2) transformations of [t0 : t1]. Hence, we can, for
example, set λ6 = λ7 = λ8 = 0. This leaves us with 3 complex structure deformations
λ1, λ2, and λ3, see eqns. (7a) and (7b).
The equations defining X∗ correspond to the rows of eq. (87), that is,
F ∗1 = a1(z
4 + z
5 + z
6)z13 + (a2z10z11z12z14 + a3z4z5z6z13)z1z2z3, (89a)
F ∗2 = a4(z
10 + z
11 + z
12)z14 + (a5z4z5z6z13 + a6z10z11z12z14)z7z8z9, (89b)
where, again, invariance under G∗2 has been imposed and the last monomial of each
equation comes from the lattice point 0 ∈ ∇̄l, , l = 1, 2. Both equations are homo-
geneous with respect to all seven scaling degrees that follow from the linear relations
eq. (37). Among the twelve scalings of the coordinates zi, six are compatible with the
cyclic permutations g∗2, see eq. (85). Subtracting the three G2 symmetric indepen-
dent scalings among the relations eq. (37), there remains one torus action that acts
effectively on the parameters plus two normalizations of the equations. As expected,
the six parameters am of the equations of X
∗ thus become the 3 complex structure
moduli.
So far, we only considered the polytopes ∆̄∗ and ∇̄∗. However, this is only part of
the toric data defining the manifolds X and X
, respectively. In addition, we need the
triangulations and the corresponding exceptional sets. A change in the triangulation
corresponds to a flop of the toric variety. The very real danger is that not all, and
perhaps none, of the flopped Calabi-Yau manifolds are G2-symmetric. For X ⊂ P∆̄∗
this turns out to be unproblematic, but for X
⊂ P∇̄∗ we will find a condition for the
choice of a triangulation.
4.3 B-Model on X
We now return to the discussion of the triangulations and the intersection ring of X .
The analogous, but technically much more involved discussion of X
will be presented
in Subsection 4.5.
For X everything is straightforward since the G1-quotient did not introduce ad-
ditional lattice points in the associated polytope ∆̄∗. Therefore, just like for the
polytope ∆∗ of the covering space X̃ , there exists a unique triangulation. In particu-
lar the primitive collections, the Stanley-Reisner ideal, and the ideal IX are identical
to the ones in eqns. (42), (49), and (50) since they are derived from the same triangu-
lation. Moreover, one can easily see that this triangulation is G2-invariant and, hence,
X is G2 symmetric.
The only change is in the normalization of the intersection ring in eq. (52), since
the total volume has to be divided by 3 = |G1|. This can also be seen in eq. (76),
where the volume of the cone is now 3 instead of 1. Hence, on X the intersection ring
and the second Chern class are
J̄22 J̄3 = 1, J̄1J̄2J̄3 = 3, J̄2J̄
3 = 1,
· J̄1 = 0, c2
· J̄2 = 12, c2
· J̄3 = 12.
Comparing these intersection numbers with eq. (13), it is clear that the toric divisors
should be identified with the G1-invariant divisors on X as
J̄1 = φ, J̄2 = τ1, J̄3 = τ2. (91)
The curves spanning the Mori cone on the cover turn out to be G1-invariant as well.
Therefore, the Mori cones NE(P∆̄∗) and NE(X)toric are identical to those in eqns. (60)
and (61), respectively.
Following the steps given in Section 3 we now want to compute the B-model pre-
potential FB
, plug in the mirror map, and obtain the prepotential on X
(P,Q1, Q2, Q3, R1, R2, R3, b1). (92)
We immediately realize the following two caveats:
• We do not know how to incorporate the torsion curves H2(X,Z)tors = Z3 into
the toric mirror symmetry calculation.
• Of the 7 Kähler classes on X , only 3 are toric.
This means that only 3 out of the 7 + 1 variables in the prepotential are accessible,
and the remaining ones are set to one. Looking at the intersection numbers eq. (90),
it is clear that the 3 divisors are precisely the G2-invariant divisors on X , see eq. (13).
Therefore, these 3 variables must be those that map to the variables p, q, and r on
X . By comparing with eq. (16), we see that the corresponding variables on X are P ,
Q1, and R1. Hence, we actually only compute
(P,Q1, 1, 1, R1, 1, 1, 1) =
n1,n2,n3
nX(n1,n2,n3) Li3
P n1Qn21 R
. (93)
In effect, this means that the resulting instanton numbers are not just the instantons
in a single integral homology class, but the instanton numbers in a whole set of integral
homology classes. The instanton numbers sum over all curve classes that cannot be
distinguished by P,Q1, R1 ∈ Hom
H2(X,Z),C
. Up to total degree 4 and the
symmetry
nX(n1,n2,n3) = n
(n1,n3,n2)
, (94)
the resulting instanton numbers are
nX(1,0,0) = 27 n
(1,0,1) = 108 n
(1,0,2) = 378 n
(1,0,3) = 1080
nX(1,1,1) = 432 n
(1,1,2) = 1512 n
(2,0,1) = −54 n
(2,0,2) = −756
nX(2,1,1) = 864 n
(3,0,1) = 9.
4.4 Instanton Numbers of X
Knowing the prepotential on X , we now want to divide out the free G2 action and
arrive at the prepotential on X . Since we do not know the complete expansion but
only eq. (93), we have to set b1 = b2 = 1 in the descent equation (16). This yields
p, q, r, 1, 1) =
p, q, 1, 1, r, 1, 1, 1
n1,n2,n3
nX(n1,n2,n3) Li3
pn1qn2rn3
Up to the symmetry nX
(n1,n2,n3)
(n1,n3,n2)
, the non-vanishing instanton numbers for
X up to total degree 5 are
nX(1,0,0) = 9 n
(1,0,1) = 36 n
(1,0,2) = 126 n
(1,0,3) = 360
nX(1,0,4) = 945 n
(1,1,1) = 144 n
(1,1,2) = 504 n
(1,1,3) = 1440
nX(1,2,2) = 1764 n
(2,0,1) = −18 n
(2,0,2) = −252 n
(2,0,3) = −1728
nX(2,1,1) = 288 n
(2,1,2) = 3960 n
(3,0,1) = 3 n
(3,0,2) = 252
nX(3,1,1) = 756,
Unfortunately, this direct calculation misses the torsion information and only yields
the expansion F
X,0(p, q, r, 1, 1). The b1 dependence was lost because the toric methods
do not yield this part, and the b2 dependence was lost because the relevant divisor on
X was not toric. Comparing with the full expansion of the prepotential
p, q, r, b1, b2) =
n1,n2,n3
m1,m2
nX(n1,n2,n3,m1,m2) Li3
pn1qn2rn3bm11 b
, (98)
see Part A eq. (??), this means we only obtain the sum of the instanton numbers over
all torsion classes
nX(n1,n2,n3) =
m1,m2=0
nX(n1,n2,n3,m1,m2). (99)
Clearly, this destroys the torsion information, that is, the instanton numbers nX
(n1,n2,n3)
do not depend on the torsion part of the integral homology. For comparison purposes,
we list the instanton numbers nX(n1,n2,n3) for 0 ≤ n1, n2, n3 ≤ 5 in Table 2.
4.5 B-Model on X
We now study the mirror X
, which sits in a more complicated ambient toric variety.
Consequently, the analysis is more involved. The big advantage, however, will turn
out to be that all h11(X
) = 7 Kähler moduli are toric, which will enable us to obtain
the full instanton expansion.
Since the polytope ∇̄∗ in eq. (78) is not simplicial, we have to specify a resolution of
the singularities, that is, a triangulation T (∇̄∗). Moreover, not any triangulation will
do, but we have to make sure that it is compatible with the action of the permutation
group G∗2. While a tedious technicality, the existence of such a resolution has to
be shown in order to establish the existence of a geometrical mirror family of X .
In particular, we show in Appendix A that there is no projective resolution of the
ambient space among the 720 coherent star triangulations of ∇̄∗ that respects the
permutation symmetry eq. (85). In other words, if one demands G∗2 symmetry then
the ambient toric variety cannot be chosen to be Kähler, but only a complex manifold.
Clearly, in that case there is no Kähler cone and the usual toric mirror symmetry
algorithm does not work. What comes to the rescue is that there are two classes of
non-symmetric projective resolutions for which the symmetry-violating exceptional
sets do not intersect X
. Hence the complete intersection is G2-symmetric, even
though the ambient space is not.
We conclude that the extended Kähler moduli space of X
contains two sym-
metric phases. We will denote these two classes of triangulations by T± = T±(∇̄
see Appendix A. In fact, the two phases are topologically distinct, and only the tri-
angulation T+ describes the threefold X
that we are interested in. In Appendix B,
we will investigate the other triangulation T− which describes a flop of X
(0,n2,n3)
(3,n2,n3)
0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 0 0 0 0 0
2 0 0 0 0 0 0
3 0 0 0 0 0 0
4 0 0 0 0 0 0
5 0 0 0 0 0 0
0 1 2 3 4 5
0 0 3 252 4158 40173 287415
1 3 756 15390 164280 1259685 7763364
2 252 15390 426708 5427684 46537092 310465062
3 4158 164280 5427684 73971360 657552966 4487097816
4 40173 1259685 46537092 657552966 5948103483 41016575313
5 287415 7763364 310465062 4487097816 41016575313 284581389204
nX(1,n2,n3) n
(4,n2,n3)
0 1 2 3 4 5
0 9 36 126 360 945 2268
1 36 144 504 1440 3780 9072
2 126 504 1764 5040 13230 31752
3 360 1440 5040 14400 37800 90720
4 945 3780 13230 37800 99225 238140
5 2268 9072 31752 90720 238140 571536
0 1 2 3 4 5
0 0 0 −144 −6048 −107280 −1235520
1 0 −306 −12348 −207000 −2273400 −19066500
2 −144 −12348 348480 14609520 235219680 2505155400
3 −6048 −207000 14609520 520226784 8245864800 87989812560
4 −107280 −2273400 235219680 8245864800 131759049600 1417949658000
5 −1235520 −19066500 2505155400 87989812560 1417949658000 15365394415800
(2,n2,n3)
(5,n2,n3)
0 1 2 3 4 5
0 0 −18 −252 −1728 −9000 −38808
1 −18 288 3960 27648 143748 620928
2 −252 3960 54432 380160 1976472 8537760
3 −1728 27648 380160 2654208 13799808 59609088
4 −9000 143748 1976472 13799808 71748000 309920688
5 −38808 620928 8537760 59609088 309920688 1338720768
0 1 2 3 4 5
0 0 0 45 5670 189990 3508920
1 0 36 13140 474840 8793648 111499020
2 45 13140 1112886 38961252 777759975 10723515300
3 5670 474840 38961252 1952428464 47357606430 732897531720
4 189990 8793648 777759975 47357606430 1237373786439 19911043749420
5 3508920 111499020 10723515300 732897531720 19911043749420 327006066948660
Table 2: Summed instanton numbers nX
(n1,n2,n3)
m1,m2
(n1,n2,n3,m1,m2)
(hence not distinguishing torsion) com-
puted by mirror symmetry. The table contains all non-vanishing instanton numbers for 0 ≤ n1, n2, n3 ≤ 6.
Following Subsection 3.3, given the triangulation T+, we can determine the prim-
itive collections. This immediately yields the Stanley-Reisner ideal
ISR =
D̄1D̄13, D̄2D̄4, D̄2D̄13, D̄3D̄4, D̄3D̄5, D̄3D̄13, D̄4D̄10, D̄4D̄11, D̄4D̄12,
D4D̄14, D̄5D̄10, D̄5D̄11, D̄5D̄12, D̄5D̄14, D̄6D̄10, D̄6D̄11, D̄6D̄12,
D̄6D̄14, D̄13D̄10, D̄13D̄11, D̄13D̄12, D̄13D̄14, D̄7D̄14, D̄8D̄10, D̄8D̄12,
D̄8D̄14, D̄9D10, D̄9D̄14, D̄1D̄2D̄3, D̄1D̄2D̄6, D1D̄5D̄6, D̄4D̄5D̄6,
D̄7D̄8D̄9, D̄7D̄9D̄11, D̄7D̄11D̄12, D̄10D̄11D̄12
(100)
where we dropped the superscript ∗ on D̄ for ease of notation. From this, in turn, we
obtain the generators l̄
+ of the Mori cone NE(P∇̄∗):
+ =( 0, 0, 0, 0, 0, 0, 1, 0, 0,−1, 0, 0, 0, 1)
+ =( 1, 0, 0,−1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)
+ =(−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
+ =( 0,−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0, 0)
+ =( 0, 0,−1, 0, 0, 1, 0,−1, 0, 0, 1, 0, 0, 0)
+ =( 0, 0, 0, 0, 0, 0,−1, 0, 1, 1, 0,−1, 0, 0)
+ =( 0, 0, 0, 0, 0, 0, 0, 1,−1, 0,−1, 1, 0, 0)
+ =( 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0,−3, 0)
+ =( 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0,−3).
(101)
A dual basis for the generators of the Kähler cone K(P∇̄∗) is
K̄1 = D̄13 + 2 D̄1 − D̄2 − D̄3 + D̄9 + D̄7 + D̄8 + 3 D̄4,
K̄2 = 3 D̄1 + D̄13 + 3 D̄4,
K̄3 = D̄13 + 2 D̄1 + 3 D̄4,
K̄4 = D̄13 + 2 D̄1 − D̄2 + 3 D̄4,
K̄5 = D̄13 + 2 D̄1 − D̄2 − D̄3 + 3 D̄4,
K̄6 = D̄13 + 2 D̄1 − D̄2 − D̄3 + D̄9 + D̄8 + 3 D̄4,
K̄7 = D̄8 + D̄13 + 2 D̄1 − D̄2 − D̄3 + 3 D̄4,
K̄8 = D̄4 + D̄1,
K̄9 = D̄10 + D̄7.
(102)
The Calabi-Yau complete intersection X
is then defined by X
= K̄1K̄2. It turns
out that the divisors D̄13, D̄14 do not intersect X
. Therefore, all
toric
= h1,1
= 7 (103)
Kähler moduli are realized torically. Since there are two divisors that do not intersect,
finding the Mori cone is somewhat subtle. First, we have to restrict the lattice of linear
relations to the sublattice orthogonal to these two directions. For the generators of
the toric Mori cone NE(X
)toric, this means that l̄
+ → 3l̄
+ + l̄
+ , l̄
+ → 3l̄
+ + l̄
and that we drop l̄
+ , l̄
+ as well as the entries corresponding to intersections with
D̄13, D̄14. In addition, we prepend the intersection numbers with D̄0,1 and D̄0,2. This
yields
+ =(−3, 0; 0, 0, 0, 0, 0, 0, 3, 0, 0,−2, 1, 1)
+ =( 0,−3; 3, 0, 0,−2, 1, 1, 0, 0, 0, 0, 0, 0)
+ =( 0, 0;−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0)
+ =( 0, 0; 0,−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0)
+ =( 0, 0; 0, 0,−1, 0, 0, 1, 0,−1, 0, 0, 1, 0)
+ =( 0, 0; 0, 0, 0, 0, 0, 0,−1, 0, 1, 1, 0,−1)
+ =( 0, 0; 0, 0, 0, 0, 0, 0, 0, 1,−1, 0,−1, 1).
(104)
The dual basis of divisors is
J̄∗1 =
K̄21K̄2, J̄
K̄1K̄
2 , J̄
5 = K̄1K̄2K̄5,
J̄∗3 = K̄1K̄2K̄3, J̄
4 = K̄1K̄2K̄4,
J̄∗6 = K̄1K̄2K̄6, J̄
7 = K̄1K̄2K̄7.
(105)
We now try to identify this basis J̄∗1 , . . . , J̄
7 of divisors onX
with the basis {φ, τ1, υ1, ψ1, τ2, υ2, ψ2}
of divisors on X in eq. (10). It turns out that there is more than one way to identify
the bases if one only wants to preserve the triple intersection numbers. To obtain a
unique answer, we also need to identify the actions by G∗2 and G2 as well. First, the
G∗2 action on H
2(P∇̄∗ ,Z) is defined by eq. (85). Using the linear equivalence relations
2D̄1 − D̄2 − D̄3 + 2D̄4 − D̄5 − D̄6 = 0
−D̄1 + 2D̄2 − D̄3 − D̄4 + 2D̄5 − D̄6 = 0
2D̄7 − D̄8 − D̄9 + 2D̄10 − D̄11 − D̄12 = 0
−D̄2 + D̄3 − D̄5 + D̄6 − D̄8 + D̄9 − D̄11 + D̄12 = 0
−D̄4 − D̄5 − D̄6 + D̄10 + D̄11 + D̄12 − D̄13 + D̄14 = 0
(106)
and the definition eq. (105), one can compute the induced group action on H2(X
We find



1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 3 −1 1 0 0 0
0 3 −1 0 1 0 0
0 0 0 0 1 0 0
3 0 0 0 1 0 −1
0 0 0 0 1 1 −1



. (107)
Second, recall that the G2 action on the divisors of X



1 0 0 0 0 0 0
0 1 0 0 0 0 0
1 3 0 −1 0 0 0
0 3 1 −1 0 0 0
0 0 0 0 1 0 0
1 0 0 0 3 0 −1
0 0 0 0 3 1 −1



, (108)
see Part A eq. (??).
The essentially unique12 identification of divisors on X
and X then turns out to
J̄∗1 = τ1, J̄
2 = τ2, J̄
3 = ψ2, J̄
4 = υ2,
J̄∗5 = φ, J̄
6 = 3τ1 + υ1 − ψ1 = g2(ψ1), J̄
7 = υ1.
(109)
Note that we are identifying divisors on X
with divisors on X in eq.(109), something
that one would usually not do. However, in view of the anticipated self-mirror prop-
erty, X
∗ ∼= X, this is a sensible thing to try to attempt. And, indeed, the identification
above is an isomorphism of the intersection rings.
Regardless of this identification, we now continue to apply mirror symmetry. First,
the second Chern class is
· J̄∗1 = 12, c2
· J̄∗5 = 0, c2
· J̄∗2 = 12,
· J̄∗3 = c2
· J̄∗6 = 24, c2
· J̄∗4 = c2
· J̄∗7 = 12.
(110)
Using this information, we now compute the B-model prepotential
(q1, . . . , q7) = 3q5 + 3q4q5 +
q25 + 3q5q7 + 3q7q5q6 + 3q4q5q7 +
+ 3q3q4q5 +
7 + 3q3q4q5q7 +
+ 3q4q3q2q5 + 3q7q5q4q6 + 3q1q5q6q7 +
q45 +O(q
(111)
Finally, we insert the mirror map and obtain the A-model prepotential on X
. Since
we already identified the bases J̄∗1 , . . . , J̄
7 with the divisors on X , we will use the
same names (but with an added ∗ superscript) for the Fourier-transformed variables
to expand the prepotential. With this notation, we obtain
(P ∗, Q∗1, Q
3, 1) = 3P
∗ + 3
P ∗2 + 1
P ∗3 + 3
P ∗4 + 3
+ 3P ∗Q∗2 +
P ∗2Q∗22 + 3P
∗Q∗2Q
3 + 3P
∗R∗2 +
P ∗2R∗22 + 3P
∗R∗2Q
+ 3P ∗R∗2Q
3 + 3P
∗R∗2R
3 + 3P
∗R∗2R
2 + 3P
∗R∗2R
+ 3P ∗Q∗1R
3 + 3P
∗Q∗1R
2 + 9P
∗Q∗1R
3 + 3P
∗Q∗2Q
+ 3P ∗Q∗2Q
2 + 9P
∗Q∗2Q
total degree ≥ 6
(112)
12Up to the ĝ2 and g
2 ĝ2 symmetry.
see also Part A eq. (??). The instanton numbers on X
are the expansion coefficients
(P ∗, Q∗1, Q
3, 1)
n1,...,n7
(n1,n2,n3,n4,n5,n6,n7)
P ∗n1Q∗n21 Q
. (113)
We see that we almost get the complete instanton expansion eq. (15), we only miss
the expansion in the b∗1 variable which is not computed by the toric mirror symmetry
algorithm. Up to total degree 5, the instanton numbers are
(1,0,0,0,0,0,0) =3 n
(1,0,0,0,0,1,0) =3 n
(1,0,0,0,0,1,1) =3 n
(1,0,1,0,0,0,0) =3
(1,0,1,0,0,1,0) =3 n
(1,0,1,0,0,1,1) =3 n
(1,0,1,1,0,0,0) =3 n
(1,0,1,1,0,1,0) =3
(1,0,1,1,0,1,1) =3 n
(1,1,0,0,0,1,2) =9 n
(1,0,1,2,1,0,0) =9 n
(1,0,1,1,1,1,0) =3
(1,1,1,0,0,1,1) =3 n
(1,1,0,0,0,1,1) =3 n
(1,0,1,1,1,0,0) =3.
(114)
Finally, let us take a look at the G∗2 action, see eq. (85). Of the 7 generators of the
toric Mori cone, eq. (104), only the 3 generators l̄
+ , l̄
+ and l̄
+ are invariant. Not
surprisingly, the dual G∗2-invariant divisors
J̄∗5 = φ, J̄
1 = τ1, J̄
2 = τ2 (115)
were identified with the G2-invariant divisors on X in eq. (109). Therefore, only 3
Kähler parameters survive to the quotient X∗ = X
/G∗2, and we have
= h1,2
= h1,1
= h1,2
= 3. (116)
4.6 Instanton Numbers of X∗
Now that we have the expression eq. (113) for the prepotential on X
, we can again
apply a suitable variable substitution
P ∗, Q∗1, Q
p∗, q∗, r∗, b∗1, b
(117)
and obtain the prepotential on the quotient X∗ = X
/G∗2. The correct way to replace
the variables is determined by the group action on the homology and cohomology as
we explained in Part A. Having computed the G∗2-action in eq. (107), we determine
the descent equation for the prepotential to be13
p∗, q∗, r∗, b∗1, b
|G∗2|
p∗, q∗, b∗2, b
∗, b∗22 , b
2 , b
. (118)
13Interestingly, eq. (118) turns out to be exactly analogous to eq. (16), even though the identifica-
tion of divisors on X
and X is not just a relabeling of divisors.
Using the series expansion of the prepotential for b∗1 = 1 on X
from Subsection 4.5,
we now find that
X∗,0(p
∗, q∗, r∗, 1, b∗2)
Li3(p
2 ) + 4 Li3(p
2 ) + 4 Li3(p
+ 14 Li3(p
∗q∗2b
2 ) + 16 Li3(p
∗q∗r∗b
2 ) + 14 Li3(p
∗r∗2b
+ 40 Li3(p
∗q∗3b
2 ) + 56 Li3(p
∗q∗2rb
2 ) + 56 Li3(p
∗q∗r2b
+ 40 Li3(p
∗r∗3b
2 ) + 105 Li3(p
∗q∗4b
2 ) + 160 Li3(p
∗q∗3r∗b
− 2 Li3(p
∗2q∗b
2 )− 2 Li3(p
∗2r∗b
− 28 Li3(p
∗2q∗2b
2 ) + 32 Li3(p
∗2q∗r∗b
2 )− 28 Li3(p
∗2r∗2b
+ 3Li3(p
∗3q∗) + 3 Li3(p
∗3r∗)
total p∗, q∗, r∗-degree ≥ 5
(119)
The corresponding instanton numbers
X∗,0(p
∗, q∗, r∗, 1, b∗2) =
n1,n2,n3,m2
(n1,n2,n3,m2)
p∗n1q∗n2r∗n3b∗m22
(120)
are listed in Table 3. For comparison purposes, we list the summed instanton numbers
(n1, n2, n3) n
(n1,n2,n3,0)
(n1,n2,n3,1)
(n1,n2,n3,2)
(n1,n2,n3)
(1, 0, 0) 3 3 3 9
(1, 0, 1) 12 12 12 36
(1, 0, 2) 42 42 42 126
(1, 0, 3) 120 120 120 360
(1, 1, 1) 48 48 48 144
(1, 1, 2) 168 168 168 504
(2, 0, 1) −6 −6 −6 −18
(2, 0, 2) −84 −84 −84 −252
(2, 1, 1) 96 96 96 288
(3, 0, 1) 3 0 0 3
Table 3: Instanton numbers nX
(n1,n2,n3,m2)
computed by toric mirror symmetry.
They are invariant under the exchange n2 ↔ n3, so we only display
them for n2 ≤ n3.
on X as well, see eq. (99). One observes that the sum over the more refined instanton
numbers on X∗ equals the summed instanton number on X , another clue towards X
being self-mirror.
4.7 Instanton Numbers Assuming The Self-Mirror Property
So far, we have alluded to X being possibly self-mirror, but not actually made use
of this property. Now we are going to assume the self-mirror property and, hence,
obtain the prepotential on X as
X,0(p, q, r, b1, b2) = F
X∗,0(p, q, r, b1, b2). (121)
Note that at linear and quadratic order in p we can actually recover the b1, b2 expansion
from the summed instanton numbers in Subsection 4.4 and the factorization which
we will prove in Section 6.
In contrast, for the prepotential terms at order p3 we have to use the X∗ pre-
potential to obtain the b2 expansion from eq. (119). Since this is based on a toric
computation on X
, we do not directly obtain the b1 expansion. However, note that
the fact that g1 acted torically, eq. (8a), and g2 non-torically, eq. (8b), is just a conse-
quence of the choice of coordinate system on P2×P1×P2. By a suitable coordinate
choice, we could have made any one of the four Z3 subgroups of G = Z3 × Z3 act
torically. Therefore, any combination of b1, b2 other than 1 = b
2 has to occur in the
same way in the complete series expansion of the prepotential. We conclude that the
prepotential can only depend on b1 and b2 through the combinations
i,j=0
2. (122)
This observation lets us recover the full b1, b2 expansion of the prepotential. To
summarize, we obtain
X∗,0(p, q, r, b1, b2)
i,j=0
Li3(pb
2) + 4 Li3(pqb
2) + 4 Li3(prb
+ 14 Li3(pq
2bi1b
2) + 16 Li3(pqrb
2) + 14 Li3(pr
2bi1b
+ 40 Li3(pq
3bi1b
2) + 56 Li3(pq
2rbi1b
2) + 56 Li3(pqr
2bi1b
+ 40 Li3(pr
3bi1b
2) + 105 Li3(pq
4bi1b
2) + 160 Li3(pq
3rbi1b
+ 196 Li3(pq
2r2bi1b
2) + 160 Li3(pqr
3bi1b
2) + 105 Li3(pr
4bi1b
− 2 Li3(p
2qbi1b
2)− 2 Li3(p
2rbi1b
2)− 28 Li3(p
2q2bi1b
+ 32 Li3(p
2qrbi1b
2)− 28 Li3(p
2r2bi1b
2)− 192 Li3(p
2q3bi1b
+ 440 Li3(p
2q2rbi1b
2) + 440 Li3(p
2qr2bi1b
2)− 192 Li3(p
2r3bi1b
+ 3Li3(p
3q) + 3 Li3(p
+ 9 Li3(p
3q2) + 27
i,j=0
Li3(p
3q2bi1b
+ 9 Li3(p
3q2) + 27
i,j=0
Li3(p
3q2bi1b
+ 27 Li3(p
3qr) + 81
i,j=0
Li3(p
3qrbi1b
total p, q, r-degree ≥ 6
(123)
Obtaining all of these terms required a computation of FB
in eq. (111) up to total
degree 23 in the 7 variables, which is close to the limit of what can be done with
current desktop computers. We list the instanton numbers in Table 4. Observe that
the instanton numbers sometimes do depend on the torsion part of their homology
class.
5 The Self-Mirror Property
When one speaks of a Calabi-Yau manifold Y being self-mirror, one has to indicate
which level of invariants one is referring to. In particular, one might think of four
types of invariants that are natural from the point of view of string theory. The
weakest level is just the Euler number. In general, exchanging complex structure and
Kähler moduli changes the sign of χ(Y ) = 2h11(Y )− 2h21(Y ). Therefore, a necessary
condition for Y and its mirror Y ∗ to be equal is obviously that
χ(Y ) = −χ(Y ∗) = 0. (124)
This level of invariants, however, is much too crude and therefore insufficient. A much
stronger level is based on the fact that the cohomology groups of even degree come with
(1,n2,n3,0,0)
(1,n2,n3,m1,m2)
, (m1, m2) 6= (0, 0)
0 1 2 3 4
0 1 4 14 40 105
1 4 16 56 160
2 14 56 196
3 40 160
4 105
0 1 2 3 4
0 1 4 14 40 105
1 4 16 56 160
2 14 56 196
3 40 160
4 105
(2,n2,n3,0,0)
(2,n2,n3,m1,m2)
, (m1, m2) 6= (0, 0)
0 1 2 3
0 0 −2 −28 −192
1 −2 32 440
2 −28 440
3 −192
0 1 2 3
0 0 −2 −28 −192
1 −2 32 440
2 −28 440
3 −192
(3,n2,n3,0,0)
(3,n2,n3,m1,m2)
, (m1, m2) 6= (0, 0)
0 1 2
0 0 3 36
1 3 108
0 1 2
0 0 0 27
1 0 81
Table 4: Instanton numbers nX
(n1,n2,n3,m1,m2)
computed by mirror symmetry.
The table contains all non-vanishing instanton numbers for n1+n2+
n3 ≤ 5. The entries marked in bold depend non-trivially on the tor-
sion part of their respective homology class.
an integral lattice structure and form a ring, and therefore have a product. Because
of Poincaré duality, that is, H2(Y ) = H4(Y )∨, it is sufficient to look at H2(Y ). There
is a product H2(Y )×H2(Y ) → H2(Y ) whose structure constants κijk are the triple
intersection numbers. These intersection numbers are finer invariants than just the
dimensions of the cohomology groups, and a self-mirror Calabi-Yau threefold should
satisfy
κijk(Y ) = κijk(Y
∗). (125)
For simply connected threefolds with torsion-free homology a theorem of Wall [41]
states that the cohomology groups with the intersection product κijk(Y ) together
with the second Chern class c2(Y ) determine the diffeomorphism type of Y .
If, however, Y and Y ∗ have non-trivial fundamental groups then we cannot con-
clude that easily that they are diffeomorphic. But the non-trivial fundamental group
is often reflected in torsion in homology (for example if π1(Y ) is Abelian). In that
case, the conjecture of [6] says that for any Calabi-Yau threefold Z
. (126)
Therefore, a self-mirror manifold Y = Y ∗ is expected to satisfy
. (127)
Of the many spaces Y satisfying eq. (124) there are only a few which also satisfy
eq. (125).
So far we only considered classical topology, but we know that the ring H2(Y )
experiences quantum corrections when going far away from the large volume limit.
At small volume the intersection numbers are replaced by the three-point functions
Cijk(q) of (topological) conformal field theory in eq. (69). In the large volume limit
q goes to zero and the Cijk(q) go to κijk, as expected. The Cijk(q) are characterized
by the genus zero instanton numbers n
= nd. In mathematical terms, these are
resummations of the Gromow-Witten invariants of Y and characterize the symplectic
structure of Y . This level of invariants is even stronger than the cohomology ring,
since there are examples of diffeomorphic manifolds which have different Calabi-Yau
structures, i.e. different n
[50, 51, 31]. Therefore, a self-mirror Calabi-Yau threefold
Y must satisfy
(Y ) = n
(Y ∗). (128)
One can go even further and couple the topological conformal field theory to topolog-
ical gravity and define higher genus instanton numbers n
, where now
d (Y ) = n
∗), g > 0 (129)
has to hold. These invariants are very difficult to compute, however see [52, 53]
for recent progress. We do not know whether they contain more information about
the symplectic structure than the genus zero invariants. In other words, there are
presently no examples known whose n
agree for g = 0 but differ for g > 0.
Now, one can start with any Y and use some method to construct the mirror
Y ∗. Among these are the Greene-Plesser construction in conformal field theory, or its
geometric generalizations by Batyrev and Borisov for complete intersections in toric
varieties. Then, to show that Y is self-mirror one proceeds to compute the various
invariants. The simplest condition, eq. (124), can directly be checked in terms of the
toric data. This concretely means that one starts with a mirror pair Y and Y ∗ satis-
fying eq. (124) and checks whether eqns. (125), (127), (128), and (129) are satisfied.
In fact, in Section 4 we collected a large amount of evidence in favor of the claim that
X and its Batyrev-Borisov mirror threefold X∗ are the same. Indeed, eqns. (40), (83)
and (116) show that X̃ , X , and X satisfy by construction the constraint eq. (124)
on the Euler number. More interestingly, by the identifications found in eqns. (109)
and (115) we observed that the condition on the intersection ring, eq. (125), is sat-
isfied for X and X , respectively. Next, eq. (97) and Table 3 show that X also fulfils
the requirement eq. (128) on the genus zero instanton numbers. It would be very
interesting to see whether also the condition eq. (129) for higher genus curves can be
Finally, we consider the torsion in cohomology. In Part A ?? we have shown that
≃ Z3 ⊕ Z3, (130)
as we expect from a self-mirror threefold. Moreover, we can actually compute the
fundamental group of the Batyrev-Borisov mirror independently. For that, first notice
that the quotient X
= X̃∗/G∗1 is fixed-point free, see Subsection 4.2. The mirror
permutation G∗2 onX
acts freely as well. Therefore, bothX andX∗ are free quotients
by a group isomorphic to Z3 ⊕ Z3, thus their fundamental groups are
≃ Z3 ⊕ Z3. (131)
Moreover, on can easily show that on a proper14 Calabi-Yau threefold Z one has
H2(Z,Z)tors = π1(Z)ab, the Abelianization of the fundamental group. Hence, we see
H3(X,Z)tors ≃ Z3 ⊕ Z3 ≃ H
2(X∗,Z)tors (132)
and the first of eq. (126) is true. This provides the first evidence for the conjecture
of [6] in a context other than toric hypersurfaces.
Another point of view is that there is a geometrical or rather combinatorial reason
for the self-mirror property in this case. From eqns. (36) and (39) one can easily
see that the lattice points νi, ν6+i, ν13, ν14, i = 1, . . . , 3, span a sub-polytope of ∇
satisfying the same linear relations as all the lattice points ρi of ∆
∗ in eq. (23). Hence,
14A proper Calabi-Yau threefold has holonomy group the full SU(3). In particular, this implies
that the fundamental group is finite.
this sub-polytope is isomorphic to ∆∗. The same is true for the polytopes ∇̄∗ and
∆̄∗. The toric variety P∇̄∗ which is the ambient space of X
can therefore be regarded
as a blow-up of a quotient of P∆̄∗ , the ambient space of X . Actually, this blow-up
makes all 7 divisors of X
toric. Similarly, P∇∗ can be regarded as a blow-up of a
quotient of P∆∗ . As shown in Subsection 3.3 this entails that all 19 Kähler moduli of
X̃∗ are realized torically. Note that it is possible that the mirror polytopes ∆∗ and
∇∗ are actually isomorphic. In fact, for toric hypersurfaces there are 41, 710 self-dual
polytopes [54]. The novel feature in our case is that non-isomorphic polytopes lead
to self-mirror complete intersections, consistent with the nef partitions.
6 Factorization vs. The (3,1,0,0,0) Curve
One interesting observation is that the prepotential F
X,0 at order p, see eq. (123)
in this paper and eq. (??) in Part A [1], factors into
i,j=0 b
2 times a function of
p, q, r only. This means that the instanton number for any pseudo-section (curve
contributing at order p) does not depend on the torsion part of its homology class. In
other words, for any pseudo-section there are 8 other pseudo-sections with the same
class in H2(X,Z)free and together filling up all of H2(X,Z)tors = Z3 ⊕Z3. In contrast,
this factorization does not hold at order p3. For example,
X,0(p, q, r, b1, b2) = · · ·+ 3p
b1 + b
1 + b2 + b1b2 + b
1b2 + b
2 + b1b
2 + b
+ · · · .
(133)
The purpose of this subsection is to understand this behavior.
First, the factorization of the prepotential at any order of p not divisible by 3
follows from an extra symmetry that we have not utilized so far. The covering space
X̃ is, in addition to eqns. (8a) and (8b), also invariant under another Ĝ = Z3 × Z3
action generated by (ζ
ĝ1 :
[x0 : x1 : x2] 7→ [x0 : ζx1 : ζ
[t0 : t1] 7→ [t0 : t1] (no action)
[y0 : y1 : y2] 7→ [y0 : y1 : y2] (no action)
(134a)
ĝ2 :
[x0 : x1 : x2] 7→ [x1 : x2 : x0]
[t0 : t1] 7→ [t0 : t1] (no action)
[y0 : y1 : y2] 7→ [y0 : y1 : y2] (no action)
(134b)
This symmetry has fixed points and, therefore, cannot be used if one is looking for
a smooth quotient of X̃ . However, it commutes with G and hence descends to a
Ĝ = Z3 × Z3 symmetry of X (with fixed points). Clearly, the instanton sum must
observe this additional geometric symmetry. To make use of this symmetry, we have
to express its action on the variables in F
X,0(p, q, r, b1, b2). We can do so by first noting
that the basic 81 curves
s1×s2 ⊂ X̃, s1 ∈MW (B1), s2 ∈MW (B2) (135)
are really one orbit under G × Ĝ. Recall that, after dividing out G, these curves
became the 9 sections in MW (X) = Z3 ⊕ Z3, see Part A ??. We now observe
that MW (X) = {sij} is one Ĝ-orbit; since each of these sections contributes pb
i, j = 0, . . . , 2 the induced Ĝ action on the prepotential must be
ĝ1 : F
X,0(p, q, r, b1, b2) 7→ F
X,0(b1p, q, r, b1, b2),
ĝ2 : F
X,0(p, q, r, b1, b2) 7→ F
X,0(b2p, q, r, b1, b2).
(136)
Clearly, the prepotential must be invariant under the ĝ1, ĝ2 action. While imposing
no constraint on the p3n terms in the prepotential, all other powers of p must appear
in the combination
i,j=0
, n 6≡ 0 mod 3. (137)
This proves the factorization observed at the beginning of this subsection.
Second, we would like to understand the p3q terms in eq. (133). These are the
curves in the homology classes15
(3, 1, 0, ∗, ∗) ∈ Z3 ⊕ Z3 ⊕ Z3 = H2
. (138)
We will show that the rational curves in this class come in a single family, that is, the
moduli space of genus 0 curves on X in these homology classes
X, (3, 1, 0, ∗, ∗)
(139)
is connected. In particular, all such curves have the same homology class (3, 1, 0, 0, 0)
and only contribute to p3q in the prepotential eq. (133). As discussed in Part A ??,
any such map CX : P
1 → X factors
CX //
C eX ��
��������
. (140)
15Recall that the exponent of p is the degree along the base P1. This is why we pick a basis in
H2(X,Z)free such that a curve in (n1, n2, n3,m1,m2) contributes at order p
n1qn2rn3bm11 b
2 in the
prepotential.
The map C
can be written in terms of homogeneous coordinates as a function
: P1[z0:z1] 7→ P
[x0:x1:x2]
×P1[t0:t1] ×P
[y0:y1:y2]
(141)
satisfying the equations (7a) and (7b) defining X̃ ,
F1 ◦ C eX
[z0 : z1]
= 0 = F2 ◦ C eX
[z0 : z1]
∀[z0 : z1] ∈ P
1 . (142)
The curve CX ends up in the homology class (3, 1, 0, ∗, ∗) if and only if the defining
equation (141) is of degree (3, 1, 0) in P2×P1×P2. Hence, eq. (141) is defined by
complex constants αij, βij , γi (up to rescaling) such that
xi = αi0 z0 + αi1 z1 i = 0, 1, 2
ti = βi0 z
0 + βi1 z
0z1 + βi2 z0z
1 + βi3 z
1 i = 0, 1
yi = γi i = 0, 1, 2.
(143)
These constants have to be picked such that the resulting curve lies on the complete
intersection X̃, that is, they have to satisfy eq. (142). Inserting eq. (143), we find
that F1 ◦ C eX
[z0 : z1]
is a homogeneous degree 6 polynomial in [z0 : z1]. Since
the coefficients of zk0z
1 must vanish individually, this yields 7 constraints for the
parameters αij , βij . What makes this system of constraint equations tractable is the
fact that they are all linear in βij ,
F1 ◦ C eX = 0 ⇔

A1 0 0 0 A5 0 0 0
A2 A1 0 0 A6 A5 0 0
A3 A2 A1 0 A7 A6 A5 0
A4 A3 A2 A1 A8 A7 A6 A5
0 A4 A3 A2 0 A8 A7 A6
0 0 A4 A3 0 0 A8 A7
0 0 0 A4 0 0 0 A8



= 0 (144)
where
= α300 + α
10 + α
20 A5
= α00α10α20
= 3α01α
00 + 3α11α
10 + 3α21α
20 + α
20 A6
= (α01α10 + α00α11)α20 + α00α10α21
= 3α201α00 + 3α
11α10 + 3α
21α20 A7
= α01α11α20 + (α01α10 + α00α11)α21
= α301 + α
11 + α
21 A8
= α01α11α21.
(145)
Thinking of this as 7 linear equations for the 8 parameters βij, there is always a non-
zero solution. The solution is generically unique up to an overall factor, and turns
into an Pn for special values of the αij . Moreover, the parameter space of the αij is
connected (essentially, the moduli space of lines in P2). Since we just identified the
parameter space of the (αij, βij) as a blow-up thereof, it is therefore connected as well.
It remains to satisfy F2 ◦ C eX = 0. One can easily see that the only way is to pick
the γi to be simultaneous solutions of
γ30 + γ
1 + γ
2 = 0 = γ1γ2γ3. (146)
Since two cubics intersect in 9 points, there are 9 such solutions, permuted by G.
Therefore, the parameter space of (αij , βij, γi) has 9 connected components, permuted
by the G-action. The moduli space of curves CX on X is the G-quotient of the moduli
space of curves C
on X̃, and therefore has only a single connected component. By
continuity, every curve CX in this connected family has the same homology class,
explaining the piece of the prepotential given in eq. (133).
7 Towards a Closed Formula
Putting all the information together we found out about the prepotential on X , one
can try to divine a closed form for the prepotential. We guess that the order pn terms
have the closed form
X,0(p, q, r, b1, b2)
i,j∈Z3
P (q)4P (r)4
M2n−2(q, r) (147)
if n is not a multiple of 3 and, slightly weaker, that
X,0(p, q, r, 1, 1)
P (q)4P (r)4
M2n−2(q, r) (148)
if n is a multiple of 3. Here,
• P (q) is the usual generating function of partitions eq. (4).
• The M2n−2 are polynomials in the Eisenstein series E2(q), E4(q), E6(q) and
E2(r), E4(r), E6(r), starting with
M−2(q, r) = 0
M0(q, r) = 1
M2(q, r) = E2(q)E2(r)
M4(q, r) =
E4(q)E4(r) +
E4(q)E2(r)
2 + E2(q)
2E4(r)
E2(q)
2E2(r)
M6(q, r) =
E6(q)E6(r) +
E6(q)E4(r)E2(r) + E4(q)E2(q)E6(r)
E6(q)E2(r)
3 + E2(q)
3E6(r)
E4(q)E2(q)E4(r)E2(r)
E2(q)
3E4(r)E2(r) + E4(q)E2(q)E2(r)
E2(q)
3E2(r)
M8(q, r) =
E6(q)E2(q)E6(r)E2(r) +
E4(q)E4(q)E4(r)E4(r)
E6(q)E2(q)E4(r)E4(r) + E4(q)E4(q)E6(r)E2(r)
E6(q)E2(q)E4(r)E2(r)
2 + E4(q)E2(q)
2E6(r)E2(r)
E6(q)E2(q)E2(r)
4 + E2(q)
4E6(r)E2(r)
+ 137
E4(q)E4(q)E4(r)E2(r)
2 + E4(q)E2(q)
2E4(r)E4(r)
E4(q)E4(q)E2(r)
4 + E2(q)
4E4(r)E4(r)
E4(q)E2(q)
2E4(r)E2(r)
2 + 121
E2(q)
4E2(r)
E4(q)E2(q)
2E2(r)
4 + E2(q)
4E4(r)E2(r)
(149)
They are symmetric under the exchange q ↔ r and of weight 2n in q and r
separately. But, for example, M4 above does not factor into a function of q and
a function of r. So theM2n−2 are not the products of the polynomials appearing
in the dP9 prepotential. However, by setting q = 0 or r = 0 one recovers the
corresponding polynomials in the dP9 prepotential [55].
• The E2i are the usual Eisenstein series
E2(q) = 1− 24q − 72q
2 − 96q3 − 168q4 − 144q5 − 288q6 +O(q7)
E4(q) = 1 + 240q + 2160q
2 + 6720q3 + 17520q4 + 30240q5 +O(q6)
E6(q) = 1− 504q − 16632q
2 − 122976q3 − 532728q4 +O(q5).
(150)
Note that the naive Taylor series coefficients of the prepotential are fractional, but
when expanding in terms of Li3’s (which account for the multicover contributions)
one finds integral instanton numbers.
These expressions for the prepotential agree with all instanton numbers computed
in this paper. Unfortunately, we have not been able to guess a closed formula that
includes the b1 and b2 dependence of the prepotential F
X,0(p, q, r, b1, b2)|pn if n is
divisible by 3. We expect that these involve extra functions beyond the Eisenstein
series.
8 Conclusion
In the initial paper Part A [1], we analyzed the topology of the Calabi-Yau manifold
of interest and found that
= Z3 ⊕ Z3 ⊕ Z3. (151)
Although the presence of torsion curve classes complicates the counting of rational
curves, we managed to derive the A-model prepotential to linear order in p.
The goal of this paper is to go beyond the results of Part A using mirror symmetry.
By carefully adapting methods designed for complete intersections in toric varieties, we
can apply mirror symmetry to compute the instanton numbers onX , even thoughX is
not toric. Using thatX is self-mirror, we completely solve this problem and are able to
calculate the complete A-model prepotential to any desired precision (and for arbitrary
degrees in p), limited only by computer power. Carrying out this computation, we
find the first examples of instanton numbers that do depend on the torsion part of
their integral homology class, see Table 4 on Page 35.
Since the self-mirror property of X is important, we investigate it in detail. In
doing so, we go far beyond just checking that the Hodge numbers are self-mirror.
In particular, we find that the intersection rings are identical and that torsion in
homology obeys the conjectured mirror relation [6]. Finally, going beyond classical
geometry, we independently calculate certain instanton numbers onX and its Batyrev-
Borisov mirror X∗. Again, we find that X and X∗ are indistinguishable, providing
strong evidence for X being self-mirror. Both of these results extend those found in
Part A [1].
Using these results, we are able to guess certain closed expressions for the pre-
potential of X in terms of modular forms. In certain limits it specializes to the dP9
prepotential of [55]. There it is known that the coefficients in p of the dP9 prepoten-
tial satisfy a recursion relation. Moreover, there is a gap condition, that is, a certain
number of subsequent terms in a series expansion is absent. This condition provides
sufficient data to determine the integration constants for the recursion and allows to
determine the prepotential completely, even at higher genus. We expect a similar
story to be valid for the prepotential of X .
Acknowledgments
The authors would like to thank Albrecht Klemm, Tony Pantev, and Masa-Hiko Saito
for valuable discussions. We also thank Johanna Knapp for providing a Singular [56]
code to compute the intersection ring of Calabi-Yau manifolds in toric varieties. This
research was supported in part by the Department of Physics and the Math/Physics
Research Group at the University of Pennsylvania under cooperative research agree-
ment DE-FG02-95ER40893 with the U. S. Department of Energy and an NSF Focused
Research Grant DMS0139799 for “The Geometry of Superstrings”, in part by the
Austrian Research Funds FWF grant number P18679-N16, in part by the European
Union RTN contract MRTN-CT-2004-005104, in part by the Italian Ministry of Uni-
versity (MIUR) under the contract PRIN 2005-023102 “Superstringhe, brane e inter-
azioni fondamentali”, and in part by the Marie Curie Grant MERG-2004-006374.E. S.
thanks the Math/Physics Research group at the University of Pennsylvania for kind
hospitality.
A Triangulation of ∇̄∗ and ∇∗
In principle the coherent triangulations of the fan over ∇̄∗ can be computed with
TOPCOM by finding the 720 star triangulations in the total of 230, 832 coherent tri-
angulations of ∇̄∗. The discussion of the symmetry properties is greatly facilitated,
however, by an explicit understanding of their structure. We will work out the trian-
gulations by first triangulating the facets and then checking the compatibility of their
maximal intersections and the coherence of the resulting star triangulations.
We start with a couple of useful definitions. A circuit is a minimal collection of n
affinely dependent points p1, . . . , pn,
λ1p1 + . . . λnpn = 0 with λ1 + . . .+ λn = 0, λi 6= 0, (152)
any proper subset of which is affinely independent. The coefficient vector λn hence has
nonzero entries and is unique up to a prefactor. We indicate the unique separation into
points with positive and negative coefficients with the notation 〈pi1 . . . pis|pis+1 . . . pin〉.
Each circuit admits two different triangulations, which are obtained by dropping one
of the points with positive coefficients and one of the remaining points, respectively.
We indicate this with a hat over the relevant subset. The two resulting triangulations
̂pi1 . . . pis |pis+1 . . . pin
pi1 . . . pis| ̂pis+1 . . . pin
(153)
hence consist of s and n− s simplices, respectively. If the first point is in the convex
hull of the others, that is, s = 1, then only one of the triangulations is maximal (all
points are vertices of at least one simplex).
Furthermore, we introduce the notation:
ai = ν̄i, bi = ν̄3+i, ci = ν̄6+i, di = ν̄12+i, i = 1, 2, 3,
e = ν̄13, f = ν̄14.
(154)
Among these 14 vectors in eq. (154) there are 9 independent linear relations, see
eq. (37),
a1 + a2 + a3 = 0, c1 + c2 + c3 = 0,
e+ f = 0, bi = ai + e, dl = cl + f,
(155)
which imply others like ai + bj = aj + bi and ai + cl = bi + dl or e =
(b1 + b2 + b3)
and f = 1
(d1 + d2 + d3).
Lemma 1. ∇̄∗ has 15 facets, 6 of which are simplicial:
aiajbibjclcmdldm] i<j
aiajd1d2d3
b1b2b3clcm
. (156)
The nine non-simplicial facets form an orbit under the permutation symmetries
Zab3 × Z
3 generated by gab : (
( ai+1
and gcd :
( cl+1
. According to the
linear relations eq. (155) the eight points on each non-simplicial facet form quadratic
circuits ai+ bj = bi+aj , ai+ cl = bi+dl, and cl+dm = cm+dl, which we call mixed if
they contain vertices of both elements of the nef partition 〈aicl|bidl〉, and pure circuits
〈aibj |biaj〉, 〈cldm|cmdl〉 otherwise.
The coherent triangulations of the facets [aiajbibjclcmdldm] are most easily ob-
tained from their Gale transform
1 −1 −1 1 0 0 0 0
1 0 −1 0 1 0 −1 0
0 0 0 0 1 −1 −1 1
 , (157)
which is the coefficient matrix of the basis ai−aj −bi+bj = 0, ai−bi+cl−dl = 0 and
cl− cm−dl+dm = 0 of linear relations. The coherent triangulations are in one-to-one
correspondence to chambers that are seperated by the facets of the cones generated
by all linear bases µ = {v1, v2, v3} with vi selected among the 8 column vectors of the
Gale transform [57, 58]. In the present case the cones over the faces of the parallel-
epiped in Figure 1 are subdivided into 24 chambers, which are indicated by dashed
lines. The triangulations, which we can label by the facet containing and the edge
adjoining the chamber, are obtained as the sets of complements of those bases µ that
span a cone containing the respective chamber.
Hence, each non-simplicial facet has 24 coherent triangulations, which can be
characterized by the triangulations of its 2 pure and of its 4 mixed circuits: Calling
the triangulation 〈âicl|bidl〉 positive and the triangulation 〈aicl|b̂idl〉 negative, and
arranging the cyclic permutations gab and gcd in the horizontal and vertical direction,
respectively, we can assign one of 16 different types ± ±± ± to each triangulation, where
Figure 1: Secondary fan of the non-simplicial facets. Chambers are indicated
by dashed lines.
the signs indicate the induced triangulations of the mixed circuits. The constraints
that reduce the a priori 32 = 26 combinations to 24 all derive from the following rules:
ai aj
bi bj
aicl|b̂idl
âjcl|bjdl
aibj |âjbi
âicl|bidl
ajcl|b̂jdl
âibj |ajbi
〉 (158)
i.e. a triangular prism can be triangulated in 6 different ways, which correlates the a
priori 8 combinations of the triangulations of the 3 squares (with analogous constraints
for the two “horizontal” prisms [aibiclcmdldm] contained in the facet [aiajbibjclcmdldm]).
Putting the pieces together we obtain
Lemma 2. The 24 triangulations of the non-simplicial facets can be assorted as fol-
lows:
• For + ++ + ,
− − the pure circuits are unconstrained, yielding 2 · 2
2 = 8 triangu-
lations.
• For + +− − ,
+ + the pure ab-circuit is unconstrained; with the transposed types
+ − ,
− + this accounts for another 8 triangulations.
• The final 8 triangulations come from the 8 types with an odd number of positive
signs, for which the triangulation of the pure circuits is unique.
• The two types + −− + and
+ − cannot occur because of contradictory implications
for the triangulations of the pure circuits.
The secondary fan and the induced triangulations for the codimension-two faces at
which the non-simplical facets intersect can be obtained from Figure 1 by projection
along the dropped vertices. The secondary fan of the prism of eq. (158), for example,
which is shown in Figure 2, is obtained from Figure 1 by projection along the diagonal
Taibj
〈aicl|bidl〉s
Taicl
〈ajcl|bjdl〉
Tajcl
〈ajbi|aibj〉
Tajbi
〈bidl|aicl〉 s
Tbidl
��〈ajcl|bjdl〉
bi Tbjdl
〈ajbi|aibj〉
Figure 2: Secondary fan of the codimension two face [aiajbibjcldl].
〈cmdm〉. The wall crossings between the six cones in Figure 2 are labeled by the circuits
whose flops relate the adjoining triangulations [57].
For the construction of the complete star triangulation we now observe that the
non-simplicial intersections of the 9 non-simplicial facets [aiajbibjclcmdldm] are given
by the 18 triangular prisms [aiajbibjcldl] and [aibiclcmdldm]. If we interpret the former
as vertices and the latter as links then the resulting compatibility conditions corre-
spond to a graph with the topology of a torus. The vertices of this graph are decorated
by signs as shown in Table 5 and connected by horizontal and vertical links. The re-
1 · 26 9 · 22 18 · 22 6 · 24 36 · 20 36 · 21 9 · 22
Table 5: The 824 = 2 (64 + 36 + 72 + 96 + 36 + 72 + 36) star triangulations of
∇∗, including the 720 = 2 (36 + 36+ 72+ 72+ 36+ 72+ 36) coherent
triangulations.
striction on the compatible signs is due to the absence of the inconsistent types + −− +
and − ++ − as subgraphs on the torus. The multiplicities µ · 2
n come from the number
n of unconstrained pure circuits and from the order µ of the effective part of the
symmetry group generated by transposition and permutations of lines and columns.
We thus find a total of 824 triangulations. The cyclic permutation symmetry that
we want to keep on the Calabi-Yau manifold X
amounts to a diagonal shift, i.e.
its induced action on the graph is generated by gabgcd. We are hence left with the
types
, and the shift symmetry furthermore aligns the triangulations of
the pure circuits and thus reduced the multiplicities from 26 to 22, yielding a total of
8 triangulations for which P∇̄∗ is G
2 symmetric.
The resulting triangulations of the facet [a2a3b2b3c2c3d2d3] are
triangulation of [a2a3b2b3c2c3d2d3]
〈â2b3|a3b2〉, 〈ĉ2d3|c3d2〉 {[a3b2b3d2d3], [b2b3c3d2d3], [a2a3b2d2d3], [b2b3c2c3d2]}
〈â2b3|a3b2〉, 〈c2d3|ĉ3d2〉 {[a3b2b3d2d3], [b2b3c2d2d3], [a2a3b2d2d3], [b2b3c2c3d3]}
〈a2b3|â3b2〉, 〈ĉ2d3|c3d2〉 {[a2b2b3d2d3], [b2b3c3d2d3], [a2a3b3d2d3], [b2b3c2c3d2]}
〈a2b3|â3b2〉, 〈c2d3|ĉ3d2〉 {[a2b2b3d2d3], [b2b3c2d2d3], [a2a3b3d2d3], [b2b3c2c3d3]}
(159)
triangulation of [a2a3b2b3c2c3d2d3]
〈a2b3|â3b2〉, 〈c2d3|ĉ3d2〉 {[a2a3b3c2c3], [a2a3c2c3d3], [a2b2b3c2c3], [a2a3c2d2d3]}
〈a2b3|â3b2〉, 〈ĉ2d3|c3d2〉 {[a2a3b3c2c3], [a2a3c2c3d2], [a2b3b2c2c3], [a2a3c3d2d3]}
〈â2b3|a3b2〉, 〈c2d3|ĉ3d2〉 {[a2a3b2c2c3], [a2a3c2c3d3], [a3b2b3c2c3], [a2a3c2d2d3]}
〈â2b3|a3b2〉, 〈ĉ2d3|c3d2〉 {[a2a3b2c2c3], [a2a3c2c3d2], [a3b2b3c2c3], [a2a3c3d2d3]}
(160)
It can be checked that the triangulations listed in eqns. (159) and (160) come from the
chambers contained in the cones over [a2a3c2c3] and [b2b3d2d3], respectively. For the
first of these triangulations we consider the chamber adjoining the edge [a2c2], which
is contained in the span of the four bases µ1 = {a2c2c3}, µ2 = {a2a3c2}, µ3 = {b3c2c3}
and µ4 = {a2a3d3}, whose complements are [a3b2b3d2d3], [b2b3c3d2d3], [a2a3b2d2d3] and
[b2b3c2c3d2] in agreement with the first triangulation in eq. (159).
Unfortunately, coherent triangulations of the facets that induce the same trian-
gulations on their common (maximal) intersections do not automatically combine to
coherent star triangulations of the polytope, and indeed only 720 of the 824 triangu-
lations in Table 5 turn out to be coherent. The non-coherent ones are easily isolated
by observing that coherent triangulations (via their height functions) induce coherent
triangulations of the prisms [a1a2a3b1b2b2] and [c1c2c3d1d2d3], which eliminates the
triangulations for which Zab3 or Z
3 is not broken by the triangulation of the pure
circuits. For the triangulation types
this reduces the multiplicity from
82 to 62. The only other affected types are the ones in the middle column of Table 5,
which have unbroken horizontal symmetry and for which the multiplicity is reduced
from 12 · 8 to 12 · 6. This poses a problem for the eight Z3-symmetric triangulations,
which are all non-coherent. Coherence of the remaining 720 triangulations can be
established by checking that their Mori cones are all strictly convex [59].
What comes to our rescue is that, even if all projective ambient spaces break the
diagonal Z3 permutation symmetry, it may be preserved on X
if the obstructing ex-
ceptional sets do not overlap with the complete intersection. In the present case these
are the blow-ups of the singularities coming from the pure circuits, i.e. codimension
two sets of the form ai · bj or cl · dm, where we use, for simplicity, the symbol of the
vertex ν̄j for the corresponding divisor Dj . Recall from eq. (77) that X
is given by
the product D̄∗0,1 · D̄
0,2 of the divisors
D̄∗0,1 = a1 + a2+ a3 + b1 + b2 + b3 + e, D̄
0,2 = c1 + c2 + c3+ c1+ d2+ d3+ f (161)
defined by the nef partition. Taking into account the five linear equivalences, we
observe that
a1 + b1 = a2 + b2 = a3 + b3, c1 + d1 = c2 + d2 = c3 + d3,
b1 + b2 + b3 + e = d1 + d2 + d3 + f,
(162)
for divisor classes in the intersection ring. We first show that e and f do not intersect
: In any maximal triangulation e and f belong only to the simplices
b̂1b2b3cmcle
d̂1d2d3amalf
, (163)
respectively, so that
e·ai = e·dl = e·f = 0 ⇒ e·D̄
0,1 = e·(b1+b2+b3+e) = e·(d1+d2+d3+f) = 0 (164)
and similarly f · D̄∗0,2 = 0. Putting everything together, we conclude that
a1 · b2 · D̄
0,1 = a1 · b2 · 3(a3 + b3) = 0 (165)
because none of the facets, and hence no triangle in any of the triangulations contains
{a1, b2, a3} or {a1, b2, b3} as a subset. Similarly cl · dm · D̄
0,2 = 0 in the intersection
ring for l 6= m. Consequently, all exceptional sets arising from triangulations of
pure circuits do not intersect X
and hence do not obstruct the cyclic permutation
symmetry G∗2. We will denote any of the remaining 36 coherent triangulations of type
by T+ = T+(∇̄
∗) and T− = T−(∇̄
∗), respectively.
The polytope ∇∗ of the mirror X̃∗ of the universal cover has 39 lattice points, with
the same 12 vertices as ∇̄∗ but living on the finer lattice M̄ . The 24 additional lattice
points, see eq. (39), are
aij =
(ai + 2aj), bij =
(bi + 2bj), (166)
cij =
(ci + 2cj), dij =
(di + 2dj), (167)
where i 6= j. These additional points are all located on edges of ∇∗. It is natural
to consider triangulations that are refinements of the ones that we just discussed.
Observing that the additional points turn all simplices in eqns. (159), (160) and (163)
into pyramids over a tetrahedron with interior points on opposite edges it is easy to see
that the maximal triangulations are unique and multiply the number 54 = 9 · 4+6 · 3
of triangles in the original triangulations by a factor of 9. The resulting triangulations
have been used to show that the divisors corresponding to the vertices aij and cij do
not intersect X̃∗.
B The Flop of X∗
In Subsection 4.5 we have taken into account only one of the triangulations T+(∇̄
We can repeat the same calculation with one of the triangulations T−. We denote
the resulting Calabi-Yau manifold by X
−. Skipping the details, we find that the
generators of the Mori cone NE(X
−) can be expressed in terms of those of NE(X
in eq. (104) as
− = l̄
+ + 3
+ + l̄
+ + l̄
+ + l̄
− = l̄
+ + 3
+ + l̄
+ + l̄
− = l̄
+ + l̄
− = − l̄
− = − l̄
+ − l̄
+ − l̄
+ − l̄
+ − l̄
− = l̄
− = l̄
(168)
One can also express the dual basis of divisors J̄ ′i on the flop in terms of the dual basis
J̄∗i on X , see eq. (105). We find
J̄ ′1 = J̄
1 , J̄
2 = J̄
2 , J̄
5 = 3J̄
1 + 3J̄
2 − J̄
J̄ ′3 = 3J̄
1 + J̄
3 − J̄
5 , J̄
4 = 3J̄
1 + J̄
3 − J̄
J̄ ′6 = 3J̄
2 − J̄
5 + J̄
6 , J̄
7 = 3J̄
2 − J̄
5 + J̄
(169)
The intersection ring is
J̄ ′21 J̄
2 = 1, J̄
3 = 2, J̄
4 = 1, J̄
5 = 3, J̄
6 = 3,
J̄ ′21 J̄
7 = 3, J̄
2 = 1, J̄
3 = 3, J̄
4 = 3, J̄
5 = 3,
J̄ ′1J̄
6 = 3, J̄
7 = 3, J̄
3 = 6, J̄
4 = 6, J̄
5 = 9,
J̄ ′1J̄
6 = 9, J̄
7 = 9, J̄
4 = 3, J̄
5 = 9, J̄
6 = 9,
J̄ ′1J̄
7 = 9, J̄
5 = 9, J̄
6 = 9, J̄
7 = 9, J̄
6 = 9,
J̄ ′1J̄
7 = 9, J̄
7 = 9, J̄
3 = 3, J̄
4 = 3, J̄
5 = 3,
J̄ ′22 J̄
6 = 2, J̄
7 = 1, J̄
3 = 9, J̄
4 = 9, J̄
5 = 9,
J̄ ′2J̄
6 = 9, J̄
7 = 9, J̄
4 = 9, J̄
5 = 9, J̄
6 = 9,
J̄ ′2J̄
7 = 9, J̄
5 = 9, J̄
6 = 9, J̄
7 = 9, J̄
6 = 6,
J̄ ′2J̄
7 = 6, J̄
7 = 3, J̄
3 = 18, J̄
4 = 18, J̄
5 = 27,
J̄ ′23 J̄
6 = 27, J̄
7 = 27, J̄
4 = 18, J̄
5 = 27, J̄
6 = 27,
J̄ ′3J̄
7 = 27, J̄
5 = 27, J̄
6 = 27, J̄
7 = 27, J̄
6 = 27,
J̄ ′3J̄
7 = 27, J̄
7 = 27, J̄
4 = 9, J̄
5 = 27, J̄
6 = 27,
J̄ ′24 J̄
7 = 27, J̄
5 = 27, J̄
6 = 27, J̄
7 = 27, J̄
6 = 27,
J̄ ′4J̄
7 = 27, J̄
7 = 27, J̄
5 = 27, J̄
6 = 27, J̄
7 = 27,
J̄5J̄
6 = 27, J̄
7 = 27, J̄5J̄
7 = 27, J̄
6 = 18, J̄
7 = 18,
J̄ ′6J̄
7 = 18, J̄
7 = 9.
(170)
The second Chern class is
· J̄ ′1 = 12, c2
· J̄ ′5 = 18, c2
· J̄ ′2 = 12,
· J̄ ′3 = c2
· J̄ ′6 = 24, c2
· J̄ ′4 = c2
· J̄ ′7 = 30.
(171)
We observe that both the intersection ring and the second Chern class cannot be
brought into (11) and (110) by a linear transformation with integer coefficients, re-
spectively. Hence, the second phase really is topologically distinct.
We denote the Fourier-transformed variables in the B-model prepotential (71) by
q′i, i = 1, . . . , 7. With this notation, we obtain
(q′1, . . . , q
7) = 3 q
1 + 3 q
2 + 3 q
+ 3 q′3q
5 + 3 q
+ 3 q′3q
5 + 3 q
6 + 3 q
12333
12333
+ 3 q′1q
+ 3 q′2q
− 6 q′1q
5 − 6 q
+ 3 q′3q
6 + 3 q
7 +O(q
(172)
The instanton numbers on X
− are the expansion coefficients in
(q′1, . . . , q
7, 1) =
n1,...,n7
(n1,n2,n3,n4,n5,n6,n7)
. (173)
Up to degree 4, they read
1,0,0,0,0,0,0 = 3 n
0,1,0,0,0,0,0 = 3 n
0,0,0,0,1,0,0 = 3 n
2,0,0,0,0,0,0 = −6
0,2,0,0,0,0,0 = −6 n
3,0,0,0,0,0,0 = 27 n
0,3,0,0,0,0,0 = 27 n
4,0,0,0,0,0,0 = −192
0,4,0,0,0,0,0 = −192 n
0,0,1,0,1,0,0 = 3 n
0,0,0,0,1,1,0 = 3 n
0,0,1,1,1,0,0 = 3
0,0,1,0,1,1,0 = 3 n
0,0,0,0,1,1,1 = 3 n
1,0,0,3,0,0,0 = 3 n
0,1,0,0,0,0,3 = 3
1,0,1,1,1,0,0 = −6 n
0,1,0,0,1,1,1 = −6 n
0,0,1,1,1,1,0 = 3 n
0,0,1,0,1,1,1 = 3
(174)
It is easy to check that the symmetry G∗2 acts without fixed points on X
− so that there
are two phases of the quotient X∗, too, with h1,1(X∗) = h1,2(X∗) = 3 and fundamental
group π1(X
∗) = Z3×Z3. They correspond to the two classes of triangulations T+ and
T−. The first phase was studied in detail in Subsection 4.5 and 4.6.
We denote the Calabi-Yau manifold in the second phase by X∗− = X
2. From
the linear equivalence relations eq. (106) and the definition eq. (169) we can compute
the induced group action on H2(X
−,Z) and find

J̄ ′1
J̄ ′2
J̄ ′3
J̄ ′4
J̄ ′5
J̄ ′6
J̄ ′7


1 0 0 0 0 0 0
0 1 0 0 0 0 0
3 0 0 −1 1 0 0
3 0 1 −1 0 0 0
0 0 0 0 1 0 0
0 3 0 0 1 0 −1
0 3 0 0 0 1 −1


J̄ ′1
J̄ ′2
J̄ ′3
J̄ ′4
J̄ ′5
J̄ ′6
J̄ ′7

. (175)
In terms of the three invariant divisors J ′1 = J̄
2 = J̄
3 = J̄
2 the intersection ring
and the second Chern class of X∗− then are
J ′22 J
3 = 1, J
2 = 3, J
3 = 1, J
3 = 3,
J ′21 J
2 = 9, J
3 = 3, J
3 = 9, J
1 = 27,
1 = 18, c2 J
2 = 12, c2 J
3 = 12.
(176)
Again, we observe that there is no linear basis transformation with integer coefficients
that brings both the intersection ring and the second Chern class into (90). Hence,
also the phase X∗− is topologically distinct from X
(n1, n2, n3) n
(n1,n2,n3,0)
(n1,n2,n3,1)
(n1,n2,n3,2)
(n1,n2,n3)
(1, 0, 0) 3 0 0 3
(2, 0, 0) −6 0 0 −6
(3, 0, 0) 18 0 0 18
(0, 1, 0) 3 3 3 9
(1, 1, 0) −6 −6 −6 −18
(1, 1, 1) 12 12 12 36
(2, 1, 0) 15 15 15 45
(1, 2, 0) 12 12 12 36
Table 6: Instanton numbers n
(n1,n2,n3,m2)
computed by toric mirror symmetry.
They are invariant under the exchange n2 ↔ n3, so we only display
them for n2 ≤ n3.
To give a geometrical interpretation of what happens, we look at the induced action
of G∗2 on the toric Mori cone NE(X
−). Only the generators l̄
− , l̄
− and l̄
− in eq. (168)
are invariant. This is exactly as in the first phase. Denoting the invariant generators
± , l̄
± , l̄
± by l
± , l
± , l
± , respectively, we observe that phase T− is obtained from
the phase T+ as a flop by the curve corresponding to the generator l
− = −l
+ , l
− = l
+ + 3l
+ , l
− = l
+ + 3l
+ . (177)
If we use the realization of X in terms of the fiber product of two dP9 surfaces, the
above result means that the base P1 of X has been flopped.
Furthermore, having computed the G∗2-action in eq. (175), we determine the de-
scent equation for the prepotential to be
p′, q′, r′, b′1, b
|G∗2|
p′, q′, b′2, b
′, b′2
2, b′2
2, b′1
. (178)
The corresponding instanton numbers
′, q′, r′, 1, b′2) =
n1,n2,n3,m2
(n1,n2,n3,m2)
p′n1q′n2r′n3b′2
(179)
are listed in Table 6.
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	Introduction
	Calabi-Yau Threefolds
	The Calabi-Yau Threefold X
	The Intermediate Quotient Xbar
	Variables
	Toric Geometry and Mirror Symmetry
	Toric Varieties
	The Batyrev-Borisov Construction
	Toric Intersection Ring
	Mori Cone
	B-Model Prepotential
	Quotienting the B-Model
	The Quotient by G1
	The Quotient by G2
	B-Model on Xbar
	Instanton Numbers of X
	B-Model on the Mirror of Xbar
	Instanton Numbers of the Mirror of X
	Instanton Numbers Assuming The Self-Mirror Property
	The Self-Mirror Property
	Factorization vs. The (3,1,0,0,0) Curve
	Towards a Closed Formula
	Conclusion
	Triangulations
	The Flop of X*
	Bibliography

  We apply mirror symmetry to the problem of counting holomorphic rational
curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. As we found in
Part A [hep-th/0703182], the integral homology group H_2(X,Z)=Z^3 + Z_3 + Z_3
contains torsion curves. Using the B-model on the mirror of X as well as its
covering spaces, we compute the instanton numbers. We observe that X is
self-mirror even at the quantum level. Using the self-mirror property, we
derive the complete prepotential on X, going beyond the results of Part A. In
particular, this yields the first example where the instanton number depends on
the torsion part of its homology class. Another consequence is that the
threefold X provides a non-toric example for the conjectured exchange of
torsion subgroups in mirror manifolds.

Introduction 1
2 Calabi-Yau Threefolds 5
2.1 The Calabi-Yau Threefold X . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Intermediate Quotient X . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Toric Geometry and Mirror Symmetry 7
3.1 Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 The Batyrev-Borisov Construction . . . . . . . . . . . . . . . . . . . . . 10
3.3 Toric Intersection Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Mori Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 B-Model Prepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Quotienting the B-Model 20
4.1 The Quotient by G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 The Quotient by G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 B-Model on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4 Instanton Numbers of X . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 B-Model on X
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.6 Instanton Numbers of X∗ . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.7 Instanton Numbers Assuming The Self-Mirror Property . . . . . . . . . 33
5 The Self-Mirror Property 34
6 Factorization vs. The (3,1,0,0,0) Curve 38
7 Towards a Closed Formula 41
8 Conclusion 43
A Triangulation of ∇̄∗ and ∇∗ 44
B The Flop of X∗ 50
Bibliography 53
1 Introduction
Counting world sheet instantons (that is, holomorphic curves) on a Calabi-Yau three-
fold has had a large number of applications in mathematics and physics, ever since it
was essentially solved by mirror symmetry several years ago [2]. The purpose of this
paper is to take into account an important subtlety that does not appear in very sim-
ple Calabi-Yau manifolds like hypersurfaces in smooth toric varieties. This subtlety
is the appearance of torsion curve classes. That is, the homology1 group
= Z3 ⊕ Z3 ⊕ Z3 (1)
contains the torsion2 subgroup Z3⊕Z3. Here, the manifold of interest X is a quotient
of one of Schoen’s Calabi-Yau manifolds [3, 4] by a freely acting symmetry group.
There are already a few known examples of such Calabi-Yau manifolds with torsion
curves [5, 6, 7, 8, 9], but the proper instanton counting has never been done before.
The prime motivation for studying these curves is that one would like to compute
the superpotential for the vector bundle moduli [10, 11, 12, 13, 14, 15, 16] in a heterotic
MSSM [17, 18, 19, 20, 21, 22, 23, 24, 25]. Our main result will be that there exist
smooth rigid rational curves in X that are alone in their homology class. This proves
that, in general, no cancellation between contributions to the superpotential W from
instantons in the same homology class can occur.
Therefore we would like to count rational curves on X . In physical terms, we need
to find the instanton correction F
X,0 to the genus zero prepotential of the (A-model)
topological string on X . This is usually written as a (convergent) power series in
h11 variables qa = e
2πita . Each summand is the contribution of an instanton, and
the (integer) coefficients are the multiplicities of instantons in each homology class.
According to [26, 27, 1] the novel feature of the 3-torsion curves on X is that for each
3-torsion generator we need an additional variable bj such that b
j = 1. The Fourier
series of the prepotential on X becomes
X,0(p, q, r, b1, b2) =
n1,n2,n3∈Z
m1,m2∈Z3
n(n1,n2,n3,m1,m2) Li3
pn1qn2rn3bm11 b
, (2)
where n(n1,n2,n3,m1,m2) is the instanton number in the curve class (n1, n2, n3, m1, m2).
For the purpose of computing the prepotential, we can either use directly the A-
model or start with the B-model and apply mirror symmetry. The A-model calculation
was carried out in the companion paper [1], entitled Part A. The results were:
• A set of powerful techniques to compute the torsion subgroups in the integral
homology and cohomology groups of X . They are spectral sequences starting
with the so-called group (co)homology of the group action on the universal cover
1In the following, Z3
= Z/3Z always denotes the integers mod 3. Similarly, we write (Z3)
⊕nZ3 = Z3 ⊕ · · · ⊕ Z3 for the Abelian group generated by n generators of order 3.
2Not to be confused with the torsion tensor of a connection.
• A closed formula for the genus zero prepotential
X,0(p, q, r, b1, b2) =
i,j=0
pbi1b
P (q)4P (r)4+O(p2) =
i,j=0
Li3(pb
2)+· · · (3)
to linear order in p, extending the one computed in [28] for the universal cover X̃ .
Here, if p(k) is the number of partitions of k ∈ Z≥, then P (q) is the generating
function for partitions,
P (q)
p(i)qi =
ln q)
. (4)
• Expanding eq. (3) as an instanton series we find that the number of rational
curves of degree (1, 0, 0, m1, m2) is:
n(1,0,0,m1,m2) = 1, ∀ m1, m2 ∈ Z3. (5)
Furthermore, these curves have normal bundle OP1(−1)⊕OP1(−1). Hence, there
are indeed 9 smooth rigid rational curves which are alone in their homology class.
Alternatively, one can start with the B-model topological string and apply mirror
symmetry, which is what we will do in this paper, entitled Part B. This will allow
us to obtain the higher order terms in p. The order in p up to which one wants to
compute the instanton numbers is only limited by computer power. We will again
find a closed formula at every order in p, however, this time by guessing it from the
instanton calculation, and hence only up to the order given by this limitation. The
way to arrive at this result is as follows:
• The universal cover X̃ admits a simple realization as a complete intersection in
a toric variety. In this situation, mirror symmetry boils down to an algorithm to
compute instanton numbers. Unfortunately, there are many non-toric divisors
which cannot be treated this way. It turns out that, after descending to X ,
precisely the torsion information is lost. In this approach one can only compute
X,0(q1, q2, q3, 1, 1).
• As a pleasant surprise we find strong evidence that the manifold X is self-mirror.
In particular, we attempt to compute the instanton numbers on the mirror X∗
by descending from the covering space X̃∗. The embedding of X̃∗ into a toric
variety is such that all 19 divisors are toric. In principle, this allows for a
complete analysis including the full Z3 ⊕Z3 torsion information, but this is too
demanding in view of current computer power.
• Although the full quotient X = X̃/(Z3 × Z3) is not toric, it turns out that a
certain partial quotient X = X̃/Z3 can be realized as a complete intersection
in a toric variety. That way, one only has to deal with h11(X) = 7 parameters,
which is manageable on a desktop computer. Assuming the self-mirror property,
we work with the mirror X
, for which again all divisors are toric, and we can
compute the expansion of F
X,0(p, q, r, 1, b2) to any desired degree. A symmetry
argument then allows one to recover the b1 dependence as well. Finally, we can
extract the instanton numbers n(n1,n2,n3,m1,m2) including the torsion information.
• As can be seen from the A-model result eq. (3), we observe that the prepotential
X,0 at order p factors into
i,j=0 b
2 times a function of p, q, r only. This
means that the instanton number does not depend on the torsion part of its
homology class. We will explain the underlying reason for this factorization
and show that it breaks down at order p3. This fits nicely with the B-model
computation at order p3, where the instanton numbers do depend on the torsion
part.
• Another consequence of the self-mirror property is that X is a non-toric example
for the conjecture of [6]. By this conjecture, certain torsion subgroups of the
integral homology groups are exchanged under mirror symmetry.
An easily readable overview and a discussion of the physical consequences of our
findings for superpotentials and moduli stabilization of heterotic models was pre-
sented in [27]. The present Part B is self-contained and can be read independently of
Part A [1]. All necessary results from Part A are reproduced in this part.
As a guide through this paper, we start in Section 2 with a brief overview of the
topology of the various spaces involved as determined in Part A [1]. This is followed by
a review of the Batyrev-Borisov construction of mirror pairs of complete intersections
in toric varieties in Section 3. We illustrate this construction by means of the covering
spaces X̃ and X̃∗ of our example. The review includes the techniques to compute the
B-model prepotential and the mirror map. These are applied in Section 4 to the partial
quotients X and X
yielding the main results stated above. This assumes that X as
well as X are self-mirror, and evidence for this property is recapitulated in Section 5.
Moreover, we show how the torsion subgroups are exchanged. Section 6 contains an
explanation for the breakdown of the factorization alluded to above. Putting all the
information together we try to guess a closed form for the prepotential in Section 7.
Finally, we present our conclusions in Section 8. In the course of this work we will
notice that a certain flop of X is very natural from the toric point of view, and we
will present it in Appendix B.
2 Calabi-Yau Threefolds
2.1 The Calabi-Yau Threefold X
The Calabi-Yau manifold X of interest is constructed as a free G
= Z3 × Z3 quo-
tient of its universal covering space X̃ . The latter is one of Schoen’s Calabi-Yau
threefolds [3]. It is simply connected and hence easier to study. Among its various
descriptions are the fiber product of two dP9 surfaces, a resolution of a certain T
orbifold [29], or a complete intersection in a toric variety. In the present Part B,
we will mostly use the latter viewpoint. The simplest way is to introduce the toric
ambient variety P2×P1×P2 with homogeneous coordinates
[x0 : x1 : x2], [t0 : t1], [y0 : y1 : y2]
∈ P2×P1×P2 . (6)
The embedded Calabi-Yau threefold X̃ is then obtained as the complete intersection
of a degree (0, 1, 3) and a degree (3, 1, 0) hypersurface in P2×P1×P2. We restrict
the coefficients of their defining equations Fi = 0 to a particular set of three complex
parameters λ1, λ2, λ3, such that the polynomials Fi read
x30 + x
1 + x
x0x1x2
= F1 (7a)
λ1t0 + t1
y30 + y
1 + y
λ2t0 + λ3t1
y0y1y2
= F2. (7b)
For the special complex structure parametrized by λ1, λ2, λ3 the complete intersection
is invariant under the G = Z3 × Z3 action generated by (ζ
[x0 : x1 : x2] 7→ [x0 : ζx1 : ζ
[t0 : t1] 7→ [t0 : t1] (no action)
[y0 : y1 : y2] 7→ [y0 : ζy1 : ζ
[x0 : x1 : x2] 7→ [x1 : x2 : x0]
[t0 : t1] 7→ [t0 : t1] (no action)
[y0 : y1 : y2] 7→ [y1 : y2 : y0]
One can show that the fixed points of this group action in P2 ×P1 ×P2 do not satisfy
eqns. (7a) and (7b), hence the action on X̃ is free.
2.2 The Intermediate Quotient X
The partial quotient
G1 (9)
will be of particular interest in this paper because this quotient is generated by phase
symmetries, see eq. (8a), and hence is toric. In particular, we will need a basis of
Kähler classes. As usual, we will not distinguish degree-2 cohomology and degree-4
homology classes but identify them via Poincaré duality. Part A [1] ?? shows that3
= H2(X̃,Z)G1 ⊕ Z3 = spanZ
φ, τ1, υ1, ψ1, τ2, υ2, ψ2
⊕ Z3. (10)
Hence, by abuse of notation, we can identify the free generators on X with the G1-
invariant generators on X̃ , see Part A eq. (??), via the pull back by the quotient map.
The triple intersection numbers on X = X̃/Z3 are one-third of the corresponding
intersection numbers on X̃ listed in Part A eq. (??). Hence, the intersection numbers
on X are
φτ1τ2 = 3 φτ1υ2 = 3 φτ1ψ2 = 6 φυ1τ2 = 3 φυ1υ2 = 3
φυ1ψ2 = 6 φψ1τ2 = 6 φψ1υ2 = 6 φψ1ψ2 = 12 τ
1 τ2 = 1
τ 21υ2 = 1 τ
1ψ2 = 2 τ1υ1τ2 = 3 τ1υ1υ2 = 3 τ1υ1ψ2 = 6
τ1ψ1τ2 = 3 τ1ψ1υ2 = 3 τ1ψ1ψ2 = 6 τ1τ
2 = 1 τ1τ2υ2 = 3
τ1τ2ψ2 = 3 τ1υ
2 = 3 τ1υ2ψ2 = 6 τ1ψ
2 = 6 υ
1τ2 = 3
υ21υ2 = 3 υ
1ψ2 = 6 υ1ψ1τ2 = 6 υ1ψ1υ2 = 6 υ1ψ1ψ2 = 12
2 = 1 υ1τ2υ2 = 3 υ1τ2ψ2 = 3 υ1υ
2 = 3 υ1υ2ψ2 = 6
2 = 6 ψ
1τ2 = 6 ψ
1υ2 = 6 ψ
1ψ2 = 12 ψ1τ
2 = 2
ψ1τ2υ2 = 6 ψ1τ2ψ2 = 6 ψ1υ
2 = 6 ψ1υ2ψ2 = 12 ψ1ψ
2 = 12.
Clearly, G2 acts on the partial quotient X . From Part A eq. (??) it follows that,
of the 7 non-torsion divisors above, only 3 are G2-invariant. This invariant part is
particularly manageable and will be important in the following. We find
= span
φ, τ1, τ2
with products 3τ 2i = τiφ. In particular, the triple intersection numbers on X are
τ 21 τ2 = 1, τ1φτ2 = 3, τ1τ
2 = 1, (13)
and 0 otherwise. Finally, the second Chern class of X is c2(X) = 12(τ
1 + τ
2 ). There-
fore,
· τ1 = 12, c2
· φ = 0, c2
· τ2 = 12. (14)
2.3 Variables
As we discussed in Part A ??, the instanton-generated superpotential should be
thought of as a series with one variable for each generator in H2. In particular, we
3The torsion in H2 are just the Wilson lines, that is, first Chern classes of flat line bundles. They
will play no role in the following. The torsion curves in H2, on the other hand, are the focus of this
paper.
Calabi-Yau
threefold
) Free
generators
Torsion
generators
X̃ Z19
p0, q0, . . . , q8, r0, . . . , r8
X = X̃/G1 Z
7 ⊕ Z3
P,Q1, Q2, Q3, R1, R2, R3
X = X̃/G Z3 ⊕ Z3 ⊕ Z3
p, q, r
b1, b2
Table 1: The different Calabi-Yau threefolds, curve classes, and variables used
to expand the prepotential.
will be interested in the Calabi-Yau threefolds X̃ , X , and X . For these, the degree-2
integral homology and the variables used (see Part A [1] for precise definitions) are
in summarized Table 1. Pushing down the curves by the respective quotients lets us
express the prepotential on the quotient in terms of the prepotential on the covering
space. We found in Part A that
P,Q1, Q2, Q3, R1, R2, R3, b1)
PQ51Q
3 b1, Q
2 , Q
3b1, Q3, 1, b1, Q1Q
2 , R
1, R3, 1, b
1, R1R
p, q, r, b1, b2) =
p, q, b2, b2, r, b
2, b1
. (16)
3 Toric Geometry and Mirror Symmetry
In this section we review mirror symmetry and the construction of the B-model for
the mirror of the covering space X̃ . Since X̃ is a complete intersection in a toric
variety, we can use the standard constructions. Because we expect the model to
be self-mirror, we will analyze the B-model for X̃ and its mirror X̃∗. The toric
geometry for X̃ is much simpler4 than for X̃∗, but contains less information. In this
section we will start with the simpler model in order to review the Batyrev-Borisov
construction for the mirror of a complete intersection in a toric variety. Then we will
apply this construction to the more complicated model, now without going into too
many details. We will see that, on the simpler side, not all parameters are toric and
no torsion is visible. However, on the more complicated mirror side, all parameters
are toric which will allow us, in principle, to perform the B-model computation of
4Meaning that X̃ is a complete intersection in the very simple toric variety P2 ×P1 ×P2, whereas
X̃∗ is embedded in a complicated toric ambient variety.
the complete prepotential. As X̃ ∼= X̃∗ is expected to be self-mirror, this determines
the complete prepotential F
eX∗,0
as well. In practice, however, the analysis is
computationally too involved.
Fortunately, the space X = X̃/G1 and its mirror will turn out to be both tractable
with toric methods and sufficiently informative. This quotient will be the subject
of Section 4. Finally, this is also the starting point for arguing in Section 5 that the
self-mirror property persists at the level of instanton corrections.
Recall that, in Subsection 2.1 we defined our Calabi-Yau manifold as the complete
intersection
F1 = 0, F2 = 0
⊂ P2×P1×P2 (17)
with the two polynomials F1, F2 as in eqns. (7a) and (7b), respectively. In order to
construct the mirror manifold following Batyrev and Borisov, we need to reformulate
this definition in terms of toric geometry. We review here some essential ingredients
of toric geometry, for details we refer to [30, 31] and references therein. We will give
the abstract definitions and concepts step by step, and at each step illustrate them
with the example X̃ and its mirror manifold X̃∗.
3.1 Toric Varieties
Given a lattice N of dimension d, a toric variety VΣ is defined in terms of a fan Σ
which is a collection of rational polyhedral5 cones σ ⊂ N such that it contains all
faces and intersections of its elements. VΣ is compact if the support of Σ covers all
of the real extension NR of the lattice N . The resulting d-dimensional variety VΣ is
smooth if all cones are simplicial and if all maximal cones are generated by a lattice
basis.
Let Σ(1) denote the set of one-dimensional cones (rays) with primitive generators ρi,
i = 1, . . . , n. The simplest description of VΣ introduces n homogeneous coordinates zi
corresponding to the generators ρi of the rays in Σ
(1). These homogeneous coordinates
are then subjected to weighted projective identifications
z1 : · · · : zn
1 z1 : · · · : λ
a = 1, . . . , h (18)
for any nonzero complex number λ ∈ C×, where the integer n-vectors q
i are genera-
tors of the linear relations
i ρi = 0 among the primitive lattice vectors
6 ρi. In or-
der to obtain a well-behaved quotient, we must exclude an exceptional set Z(Σ) ⊂ Cn
5Here, the tip of the cone is always the origin of N . A cone is rational if it is spanned by rays
which pass through lattice points (other than the origin), that is, have rational slopes. A cone is
polyhedral if it is the cone over an (d − 1)-dimensional polytope. In other words, curved faces are
not allowed.
6We will use the same symbol ρ∗ to denote the generators in Σ
(1) and the corresponding primitive
lattice vectors in N .
that is defined in terms of the fan, as will be explained below. Hence, the quotient is
Cn − Z(Σ)
, (19)
where Γ ≃ N/ span{ρi} is a finite abelian group. There are h = n−d independent C
identifications, therefore the complex dimension of VΣ equals the rank d of the lattice
N . The identifications by Γ are only non-trivial if the ρi do not span the lattice N .
Refinements of the lattice N with fixed ρi can hence be used to construct quotients
of toric varieties VΣ by discrete phase symmetries such as Z3. Such quotients will be
discussed in Section 4. Note that the rays ρi are in 1–to–1 correspondence with the
(C×)-invariant divisors Di on VΣ, which are defined as
zi = 0
⊂ VΣ. (20)
Conversely, the homogeneous coordinate zi is a section of the line bundle O(Di).
For example, consider the simplest compact toric variety, the projective space Pd.
Its fan Σ = Σ(∆) is generated by the n = d+ 1 vectors
ρ1 = e1, ρ2 = e2, . . . , ρn−1 = ed, ρn = −
ei (21)
of a d-dimensional simplex ∆. They satisfy a single linear relation,
i=1 ρi = 0.
Therefore qi = 1 for all i, and the homogeneous coordinates in eq. (18) are the usual
homogeneous coordinates on Pd.
For products of toric varieties we simply extend the relations for any single factor
by zeros and take the union of them. Hence, the fan of the polyhedron ∆∗ describing
the 5-dimensional toric variety P2×P1×P2 in eq. (17) is generated by the n = 5+3 = 8
vectors
ρ1 = e1, ρ2 = e2, ρ3 = −e1 − e2, ρ4 = e3,
ρ5 = −e3, ρ6 = e4, ρ7 = e5, ρ8 = −e4 − e5
satisfying the linear relations
ρi = 0. (23)
Except for the origin, there are no other lattice points in the interior of ∆∗. The
corresponding homogeneous coordinates will be denoted by
z1 = x0, z2 = x1, z3 = x2,
z4 = t0, z5 = t1,
z6 = y0, z7 = y1, z8 = y2.
In more general situations, given a polytope ∆∗ ⊂ N we will denote the resulting toric
variety by P∆∗ = VΣ(∆∗).
3.2 The Batyrev-Borisov Construction
Batyrev showed that a generic section ofK−1
, the anticanonical bundle of P∆∗, defines
a Calabi-Yau hypersurface if ∆∗ is reflexive, which means, by definition, that ∆∗ and
its dual
x ∈MR
∣∣∣ (x, y) ≥ −1 ∀y ∈ ∆∗
are both lattice polytopes. Here, M = Hom(N,Z) is the lattice dual to N and MR is
its real extension. Mirror symmetry corresponds to the exchange of ∆ and ∆∗ [32].
The generalization of this construction to complete intersections of codimension r > 1
is due to Batyrev and Borisov [33, 34]. For that purpose, they introduced the notion
of a nef partition. Consider a dual pair of d-dimensional reflexive polytopes ∆ ⊂
∗ ⊂ NR. In that context, a partition E = E1 ∪ · · · ∪ Er of the set of vertices
of ∆∗ into disjoint subsets E1, . . . , Er is called a nef-partition if there exist r integral
upper convex Σ(∆∗)-piecewise linear support functions φl : NR → R, l = 1, . . . , r such
φl(ρ) =
1 if ρ ∈ El,
0 otherwise.
Each φl corresponds to a divisor
D0,l =
Dρ (27)
on P∆∗ , and their intersection
Y = D0,1 ∩ · · · ∩D0,r (28)
defines a family Y of Calabi-Yau complete intersections of codimension r. Moreover,
each φl corresponds to a lattice polyhedron ∆l defined as
x ∈MR
∣∣∣ (x, y) ≥ −φl(y) ∀y ∈ NR
. (29)
The lattice points m ∈ ∆l correspond to monomials
〈m,ρi〉
i ∈ Γ (P∆∗ ,O(D0,l)) . (30)
One can show that the sum of the functions φl is equal to the support function of K
and, therefore, the corresponding Minkowski sum is ∆1 + · · · + ∆r = ∆. Moreover,
the knowledge of the decomposition E = E1 ∪ · · · ∪Er is equivalent to that of the set
of supporting polyhedra Π(∆) = {∆1, . . . ,∆r}, and therefore this data is often also
called a nef partition.
It can be shown that given a nef partition Π(∆) the polytopes7
{0} ∪ El
⊂ NR (31)
define again a nef partition Π∗(∇) = {∇1, . . . ,∇r} such that the Minkowski sum
∇ = ∇1 + · · · + ∇r is a reflexive polytope. This is the combinatorial manifestation
of mirror symmetry in terms of dual pairs of nef partitions of ∆∗ and ∇∗, which we
summarize in the diagram
∆ = ∆1 + . . .+∆r ∆
∇1, . . . ,∇r
Mirror
Symmetry
uujjj
MR NR
∆1, . . . ,∆r
(∆l,∇l′) ≥ −δl l′ ∇ = ∇1 + . . .+∇r
. (32)
In the horizontal direction, we have the duality between the lattices M and N and
mirror symmetry goes from the upper right to the lower left. The other diagonal has
also a meaning in terms of mirror symmetry as we will explain below. The complete
intersections Y ⊂ P∆∗ and Y
∗ ⊂ P∇∗ associated to the dual nef partitions are then
mirror Calabi-Yau varieties.
Let us now apply the Batyrev-Borisov construction to the complete intersection
eq. (17), hence r = 2. There exist several nef-partitions of ∆∗. The one which has
the correct degrees (3, 1, 0) and (0, 1, 3) is, up to exchange of t0 and t1, E1 = {ρi |i =
1, . . . , 4} and E2 = {ρi |i = 5, . . . , 8}. Adding the origin and taking the convex hull
yields the polytopes
ρ1, . . . , ρ4, 0
, ∇2 =
ρ5, . . . , ρ8, 0
, (33)
where the ρi are defined in eq. (22). The two divisors cutting out the Calabi-Yau
threefold are, according to eq. (27),
D0,1 =
Di, D0,2 =
Di ⇒ X̃ = D0,1 ∩D0,2 ⊂ P∆∗ (34)
Note that, while ∆∗ has no further lattice points, its dual ∆ has 18 vertices and 300
lattice points. Using the computer package PALP [35], we determine the associated
polytopes ∆1 and ∆2 of the global sections of O(D0,1) and O(D0,2), respectively. In
an appropriate lattice basis there is, up to symmetry, a unique nef partition consisting
ν1, . . . , ν6, 0
, ∆2 =
ν7, . . . , ν12, 0
, (35)
7The brackets
· · ·
denote the convex hull.
where
ν1 = 2e1 − e2, ν2 = −e1 + 2e2, ν3 = −e1 − e2,
ν4 = 2e1 − e2 − e3, ν5 = −e1 + 2e2 − e3, ν6 = −e1 − e2 − e3,
ν7 = 2e4 − e5, ν8 = −e4 + 2e5, ν9 = −e4 − e5,
ν10 = e3 + 2e4 − e5, ν11 = e3 − e4 + 2e5, ν12 = e3 − e4 − e5.
Among these 12 vectors there are the 7 independent linear relations
3ν3 + ν4 + ν5 − 2ν6 = 0, 3ν9 + ν10 + ν11 − 2ν12 = 0,
ν1 − ν3 − ν4 + ν6 = 0, −ν1 + ν2 + ν4 − ν5 = 0,
ν7 − ν9 − ν10 + ν12 = 0, −ν7 + ν8 + ν10 − ν11 = 0,
−ν2 + ν5 − ν8 + ν11 = 0.
The convex hull ∇∗ = 〈∆1,∆2〉 yields the fan Σ(∇
∗) and, consequently, the toric
variety P∇∗ . Let D
i , i = 1, . . . , 12 be the divisors associated to the vertices νi. Then,
by eq. (27), the nef partition eq. (35) defines the divisors
D∗0,1 =
D∗i , D
0,2 =
D∗i , ⇒ X̃
∗ = D∗0,1 ∩D
0,2 ⊂ P∇∗ (38)
cutting out the mirror complete intersection X̃∗. In contrast to ∆∗, the polytope ∇∗
contains extra integral points. We find that it contains, in addition to the origin and
the vertices in eq. (36), the 26 points
ν13 =
(ν4 + ν5 + ν6) = −e3, ν12+6k+i+j =
(ν3k+i + 2ν3k+j),
ν14 =
(ν10 + ν11 + ν12) = e3, ν15+6k+i+j =
(ν3k+j + 2ν3k+i)
∀ k ∈ {0, . . . , 3}, (i, j) ∈
(1, 2), (1, 3), (2, 3)
For completeness, note that the dual polytope ∇ has 15 vertices and 24 lattice points.
Running PALP to compute the Hodge numbers using the formula of [36], we obtain
= h1,2
= h1,1
= h1,2
= 19, (40)
in agreement with Part A [1], eq. (??).
So far, we have mainly focused on the information contained in the reflexive poly-
topes ∆∗ and ∇∗ and ignored their duals. We have already mentioned that in the
reflexive case a generic section of K−1
defines a Calabi-Yau manifold, and that such
sections are provided by the lattice points of ∆. In other words, ∆ and ∇ are the
Newton polytopes of Y and Y ∗, respectively. That is, the complete intersection Y
(Y ∗) is defined by r polynomial equations, and the exponents of the monomials in
each equation are the lattice points in ∆ (∇). More precisely, the Minkowski sum for,
say, ∆ = ∆1 + · · ·+∆r defines r homogeneous polynomials
Fl(z) =
∇l′∩N
〈m,ρi〉+δl l′
i , l = 1, . . . , r (41)
with coefficients al,m ∈ C. The simultaneous vanishing of F1, . . . , Fr then defines
the complete intersection Calabi-Yau manifold Y ⊂ P∆∗ . Exchanging ∆l and ∇l′ in
eq. (41) yields the equations F ∗l defining the mirror manifold Y
∗. It is in this sense
that the map from the upper left to the lower right in eq. (32) is also a manifestation
of mirror symmetry. Since we will not need the actual polynomials for X̃ and X̃∗, we
refrain from writing them explicitly. Instead, we refer the reader to Section 4, where
we determine the equations in a simpler situation.
3.3 Toric Intersection Ring
Up to now we have only considered one of the ingredients in the fan Σ, namely, the
generators ρ ∈ Σ(1) which defined the C× action in eq. (19). The second ingredient
is the exceptional set Z(Σ). It corresponds to fixed loci of a continuous subgroup of
for which the quotient eq. (19) is not well defined. Therefore, these loci have
to be removed. In terms of the homogeneous coordinates zi, this happens precisely
when a subset {zi |i ∈ I}, I ⊆ {1, . . . , n}, of the coordinates vanishes simultaneously
such that there is no cone σ ∈ Σ containing all of the ρi ⊆ σ, i ∈ I. Hence, the set
Z(Σ) is the union of the sets ZI = {[z1 : · · · : zn] |zi = 0 ∀i ∈ I}. Minimal index
sets I with this property are called primitive collections [37]. In order to determine
the index sets I we need a coherent8 triangulation T = T (∆∗) of the polytope ∆∗
for which all simplices contain the origin. Different triangulations will yield different
exceptional sets and, hence, different toric varieties. However, for simplicity, we will
mostly suppress the choice of a triangulation in the notation. In the case of complete
intersections, only those triangulations of ∆∗ are compatible with a given nef partition
that can be lifted to a triangulation of the corresponding Gorenstein cone, see [38].
The polytope defining projective space Pd admits a unique triangulation with
the required properties, and this triangulation consists of n = d + 1 simplices. The
only primitive collection is I = {1, . . . , n}. This is well-known from the definition
of projective space, where we have to remove the origin z1 = · · · = zd+1 = 0 from
Cd+1. Similarly, the polyhedron ∆∗ for the ambient space P∆∗ of X̃ admits a unique
triangulation, and the primitive collections are those of its factors, that is,
I1 = {1, 2, 3}, I2 = {4, 5}, I3 = {6, 7, 8}. (42)
8Coherent triangulations, sometimes also called regular triangulations, satisfy some technical
property that is equivalent to the associated toric quotient being Kähler.
The mirror polyhedron∇∗, on the other hand, admits a huge number of triangulations.
We will discuss particularly interesting triangulations of the mirror polyhedron at the
end of Appendix A.
The primitive collections determine the cohomology ring of toric varieties and,
together with the nef partition, complete intersections. Recall that if the collection
ρi1 , . . . , ρik of rays is not contained in at least one cone, then the corresponding ho-
mogeneous coordinates zil are not allowed to vanish simultaneously. Therefore, the
corresponding divisors Dil have no common intersection. Hence, we obtain non-linear
relations RI = Di1 · . . . ·Dik = 0 in the intersection ring. It can be shown that all such
relations are generated by the primitive collections I = {i1, . . . , ik} defined above.
The ideal generated by these RI is called Stanley-Reisner ideal
ISR =
RI , I primitive collection
⊂ Z[D1, . . . , Dn], (43)
and Z[D1, . . . , Dn]/ISR is the Stanley-Reisner ring. The intersection ring of a non-
singular compact toric variety PΣ is [39]
= Z [D1, . . . , Dn]
(m, ρi)Di
. (44)
In other words, the intersection ring can be obtained from the Stanley-Reisner ring by
adding the linear relations
(m, ρi)Di = 0, where it is sufficient to take a set of basis
vectors for m ∈ M . In particular, the intersection number of the divisors spanning a
maximal-dimensional simplicial cone σ = span
R≥{ρi1 , . . . , ρid} is
Di1 · . . . ·Did =
Vol(σ)
, (45)
where Vol(σ) is the lattice-volume, that is, the geometric volume divided by the
volume 1
of a basic simplex. For practical purposes it is sufficient to compute one
of these volumes, the remaining intersections can be obtained using the linear and
non-linear relations.
Having found the intersection ring of the ambient toric variety, we now turn to
the complete intersection Y ⊂ P∆∗ . The toric part of its even-degree intersection ring
is [40]
Hevtoric
= Q [D1, . . . , Dn]
IY , (46)
where IY is the ideal quotient
(m, ρi)Di
D0,l. (47)
Note that it can happen that some of the Di appear as generators of IY . This means
that they can be set to zero in the intersection ring. Geometrically, this means that
these divisors do not intersect a generic complete intersection Y . While the inter-
section ring depends on the triangulation T (∆∗) through the primitive collections
defining the Stanley-Reisner ideal, we conjecture that the divisors Di not intersecting
Y are independent of the choice of triangulation. This conjecture is proven for r = 1
and supported by a large amount of empirical evidence for r > 1. We conclude that
the dimension dimH2toric(Y ) is in general smaller than h
1,1(Y ) for the following two
reasons: Only h = n − d = dimH2(P∆∗,Z) divisors are realized in the ambient toric
variety P∆∗ , and some of them may not descend to the complete intersection Y . Using
the adjunction formula we can compute the the Chern classes of Y by expanding
c(Y ) =
(1 +Di)
(1 +D0,l)
. (48)
The intersection ring together with the second Chern class determine the diffeomor-
phism type of a simply-connected Calabi-Yau manifold [41]. If we consider the coho-
mology with integral coefficients there can be torsion and, in fact, this is what this
paper is all about. Unfortunately, a combinatorial formula in terms of the fan Σ(∆)
for the torsion in the integral cohomology of a toric Calabi-Yau manifold is only known
in the hypersurface case [6].
We now illustrate these concepts in the example of the complete intersection X̃ ⊂
P∆∗ = P
2 × P1 × P2 and its mirror manifold X̃∗. In eq. (42) we already determined
the primitive collections, hence the corresponding Stanley-Reisner ideal is
ISR =
D1D2D3, D4D5, D6D7D8
. (49)
The linear equivalences are D1 = D2, D1 = D3, D4 = D5, D6 = D7, D6 = D8 and,
hence, we can choose K1 = D4, K2 = D1, K3 = D6 as a basis for H
2(P∆∗). In terms
of this basis, we obtain D0,1 = K1+3K2 and D0,2 = K1+3K3, see eq. (27). Therefore,
the ideal I
in eq. (34) is
2K2 −K2
2K3, K1K2 − 3K2
2, K1K3 − 3K3
2, K1
2, K2
3, K3
. (50)
Next, we define the restriction of the Ki to X̃ to be the divisors
J̃i = Ki · X̃ = Ki(K1 + 3K2)(K1 + 3K3). (51)
We need to compute one of the intersection numbers directly from the volume of a
cone, say, J̃1J̃2J̃3 = K1K2K3(K1 + 3K2)(K1 + 3K3) = 9K1K
3 , where we made use
of the relations in I
. Using eq. (45), this intersection can be evaluated to be
3 = 9D1D2D4D6D7 = 9/Vol
〈ρ1, ρ2, ρ4, ρ6, ρ7〉
= 9/Vol
〈e1, e2, e3, e4, e5〉
Then, again using eq. (50), we see that the only non-vanishing intersection numbers
and the second Chern class are
J̃22 J̃3 = 3, J̃1J̃2J̃3 = 9, J̃2J̃
3 = 3,
· J̃1 = 0, c2
· J̃2 = 36, c2
· J̃3 = 36.
Note that only h
toric(X̃) = 3 of the h
1,1(X̃) = 19 parameters are realized torically.
Comparing the triple intersection numbers with eq. (13), it is clear that these 3 toric
divisors are precisely the G-invariant divisors on X̃ .
A similar, though much more complicated, calculation can be done for X̃∗ ⊂ P∇∗ .
Using the results of Appendix A one can show that, among the points in eq. (39), the
14 divisors D∗13, D
14, D
12+6k+i+j , D
15+6k+i+j, k = 0, 2 appear as generators of eq. (47)
and, therefore, do not intersect X̃∗. Subtracting from the remaining 24 divisors in
eqns. (36) and (39) the remaining 5 linear relations in eq. (37), we find that all
toric(X̃
∗) = h1,1(X̃∗) = 19 moduli are realized torically.
3.4 Mori Cone
As we have just seen, the cohomology classes Di span H
2(PΣ,R) = H
1,1(PΣ). The
Kähler classes of a smooth projective toric variety PΣ form an open cone in H
1,1(PΣ)
called the Kähler cone K(PΣ). This cone has a combinatorial description in terms of
the fan Σ, which we now review.
First, define a support function to be a continuous function ψ : NR → R given on
each cone σ ∈ Σ by an mσ ∈MR via
ψ(ρ) = (mσ, ρ) ∀ρ ∈ σ ⊂ NR. (54)
A support function determines a divisor D =
ψ(ρi)Di. We say that D is convex if
ψ is a convex function on NR. The convex classes form a non-empty strongly convex
polyhedral cone in H1,1(PΣ) whose interior is the Kähler cone K(PΣ). Such a support
function is strictly convex if and only if
ψ(ρi1 + · · ·+ ρik) > ψ(ρi1) + · · ·+ ψ(ρik) (55)
for every primitive collection I = {i1, . . . , ik} [40]. The dual of the Kähler cone
K(PΣ) is called the Mori cone or the cone of numerically effective curves NE(PΣ).
Its generators can be described by vectors l(a) of the corresponding linear relations∑
i ρi = 0. Each face of the Kähler cone K(PΣ) is dual to an edge of NE(PΣ).
These edges are generated by curves c(a), and the entries of the vector l(a) are
= c(a) ·Di. (56)
A practical algorithm to find the generators for l(a) in terms of the triangulation T (∆∗)
is described in [42]. Of course, we are not interested in the ambient space but in a
complete intersection Y ⊂ P∆∗ . The restriction of a Kähler class on the ambient space
yields a Kähler class on Y , but not every Kähler class on Y arises that way. We define
the toric part of the Kähler cone on Y as the restriction [43]
K(Y )toric = K(PΣ)
⊂ K(Y ). (57)
In the simplicial case, we can always take the basis Ji of H
toric(Y,Q) to be edges of the
Kähler cone. The dual of the toric Kähler cone of Y is the (toric) Mori cone NE(Y )toric.
This is sufficient for mirror symmetry purposes, however, it can be larger than the
actual cone of effective curves. Once the generators l(a) of NE(P∆∗) are determined,
we need to add the information about the nef partition. For this purpose, we define
= −D0,m · c
(a) m = 1, . . . , r. (58)
Finally, it is customary to write the generators of the Mori cone NE(Y )toric as
l(a) =
0,1 , . . . , l
0,r ; l
1 , . . . , l
, (59)
which are, by abuse of notation, again denoted by l(a). The knowledge of the (toric)
Mori cone is important for several reasons. It defines the local coordinates on the
complex structure moduli space of the mirror Y ∗ near the point of maximal unipotent
monodromy. Moreover, the generators enter the coefficients of the fundamental period
which is a solution of the Picard-Fuchs equations as we will review in Subsection 3.5.
For example, using the unique primitive collections in eq. (42), the Mori cone for
P∆∗ is generated
l(1) =(0, 0, 0, 1, 1, 0, 0, 0)
l(2) =(1, 1, 1, 0, 0, 0, 0, 0)
l(3) =(0, 0, 0, 0, 0, 1, 1, 1).
Recalling the nef partition D0,1 = D1 + · · ·+D4, D0,2 = D5 + · · ·+D8, we prepend
(−D0,1 · c
(a),−D0,2 · c
(a)) = (−3, 0), (−1,−1), (0,−3), a = 1, 2, 3, to obtain the gener-
ators
l(1) =(−1,−1; 0, 0, 0, 1, 1, 0, 0, 0)
l(2) =(−3, 0; 1, 1, 1, 0, 0, 0, 0, 0)
l(3) =( 0,−3; 0, 0, 0, 0, 0, 1, 1, 1)
of the Mori cone NE(X̃)toric. Due to the large number of toric moduli, the calculation
for the Mori cone NE(P∇∗) of the ambient toric variety of the mirror X̃
∗ is much more
complex.
9We sort the Mori cone generators such that the first one corresponds to the P1 of the ambient
space, and the second and third generator are the hyperplane sections of the two P2. In other words,
we have J̃a · c
(b) = δb
. This is the basis of curves that we used for the A-model computation.
3.5 B-Model Prepotential
Mirror symmetry identifies the quantum corrected Kähler moduli space of Y with
the classical complex structure moduli space of Y ∗, see the excellent treatise in [43]
for details. The deformations of the complex structure of Y ∗ are encoded in the
periods ̟ =
Ω and the latter can be computed from the equations F ∗l that cut
out Y ∗ ⊂ P∇∗ . Given the Mori cone eq. (59) and the classical intersections numbers
κabc = Ja · Jb · Jc we follow [44, 45, 38, 43] to write down a local expansion of the
periods, convergent near the large complex structure point, which is characterized
by its maximal unipotent monodromy. In the following, we will review just the bare
essentials.
The coefficients ai in the polynomial constraints F
l of the complete intersection Y
see eq. (41), define the complex structure of Y ∗. A particular set of local coordinates
ua on the complex structure moduli space on Y
∗ is defined by
i b = 1, . . . , h (62)
where h
toric(Y ) and am,0 is the coefficient in (41) corresponding to the origin in
∇l. In these coordinates, the point of maximal unipotent monodromy is at ub = 0.
We define the cohomology-valued period
̟(u, J) =
{na≥=0}
0,mJa
a=1 l
0,mna
a=1 l
una+Jaa . (63)
where (x)n = Γ(x + n)/Γ(x) is the Pochhammer symbol. Note that the choice of
triangulation is implicit in the generators l(a) of the Mori cone. Expanding ̟(u, J)
by cohomology degree yields
̟(u, J) = ̟(0)(u) +
̟(1)a (u)Ja +
̟(2)a (u)κabcJbJc −̟
(3)(u) dVol, (64)
where dVol is the volume form. The coefficients in eq. (64) are the fundamental period
̟(0)(u), that is, the unique solution to the Picard-Fuchs equations holomorphic at
ua = 0, and
̟(1)a (u) = ∂Ja̟(u, J)|J=0, ̟
a (u) =
κabc∂Jb∂Jc̟(u, J)|J=0,
̟(3)(u) = −
κabc∂Ja∂Jb∂Jc̟(u, J)|J=0.
These coefficients coincide with the basis of solutions of the Picard- Fuchs equations
obtained from the Frobenius method in [46, 31]. The B-model prepotential FBY ∗,0 is
Y ∗,0(u) =
2̟(0)(u)2
̟(0)(u)̟(3)(u) +
̟(1)a (u)̟
a (u)
. (66)
At the large complex structure point the mirror map defines natural flat coordinates
on the Kähler moduli space of the original manifold Y , which are
i (u)
̟0(u)
, i = 1, . . . , h. (67)
We also define qj = e
2πitj = uj + O(u
2). One way to obtain the prepotential is to
compute its third derivatives
C∗abc = DaDbDcF
Y ∗,0 =
Ω ∧ ∂a∂b∂cΩ, (68)
and apply the Picard-Fuchs operators. This leads to linear differential equations,
which determine C∗abc up to a common constant, see again [46, 43] for details. The
quantum corrected three point function Cijk(q) on Y follows from C
abc(u) using the
inverse mirror map eq. (67) u = u(t), and one obtains
Cijk(q) =
̟(0)(u(q))2
C∗abc(u(q)). (69)
In practice, we use the formula
Cijk(q) = ∂ti∂tj
k (u(q))
̟(0)(u(q))
. (70)
Integrating three times with respect to ti yields the prepotential F
Y ∗,0(t) up to a
polynomial of degree three in ti which can be determined partially by the topological
data of Y .
Mirror symmetry then ensures that the B-model prepotential, eq. (66), is equal to
the A-model prepotential. That is,
FY,0(q) = F
Y ∗,0(u(q)). (71)
This allows us to compute the instanton numbers nd. For the case of interest,
X̃ ∈ P∆∗ = P
2 × P1 × P2, (72)
we refer to [28] where this program been carried out in detail. The same calculation
can in principle be done on the mirror X∗, but the large number of toric moduli
again makes it highly extensive. Instead, we refer to the next section where a suitable
quotient of X̃∗ will be treated in detail for which the computations are reasonably
simple.
4 Quotienting the B-Model
In this section we consider the quotientX = X̃/G in terms of toric geometry and study
the mirror of X in this context. In order to achieve this, we first analyze the partial
quotient X = X̃/G1. Using the techniques introduced in Section 3, we construct the
mirror X
. Using their toric realization, we perform the B-model computation for the
non-perturbative prepotentials F
and F
, respectively. Finally, we explain how
one can implement the quotient by G2 on both sides in order to obtain X and X
4.1 The Quotient by G1
We start with a review of the general discussion of free quotients of complete intersec-
tions in toric geometry in [31]. Consider a fan Σ ⊂ NR and pick a lattice refinement
N̄ such that Γ = N̄/N is a finite abelian group. Such a lattice refinement consists of
a finite sequence of lattice refinements of the form N → N +wpZ which are described
by a vector wp =
αpiρi with αpi ∈ Z. The group Γ is then isomorphic to
Zkp .
Let Σ̄ be the fan obtained from Σ by relating everything to the lattice N̄ . In this
context, we make some additional identifications in the toric quotient eq. (19) [47].
One finds that VΣ̄ = VΣ/Γ is the quotient of VΣ by the finite abelian group Γ. Its
action on the homogeneous coordinates is by multiplication by phases
z1 : · · · : zn
ξα1z1 : · · · : ξ
, ξ = e
k , (73)
for every cyclic subgroup of order k. We will denote such group actions by Zk :
(α1, . . . , αn). If VΣ is a compact toric variety, then the quotient VΣ̄ is never free [39].
However, a hypersurface or complete intersection in VΣ need not intersect the set
of fixed points, and in that case we get a smooth quotient manifold with nontrivial
fundamental group.
We now apply this to P∆∗ = P
2 × P1 × P2 defined in eq. (22). The first step in
performing the quotient of P∆∗ by G1 thus amounts to a refinement N̄ = wZ+N of
the lattice N with index |G1| = 3. From the definition eq. (8a) of the action of G1 on
P∆∗ and eq. (24) we read off that the refinement is by a vector
ρ2 + 2ρ3 + ρ7 + 2ρ8
+ Z5. (74)
The resulting polytope ∆̄∗ admits the same nef partition as ∆∗ in eq. (33),
∇̄1 = 〈ρ̄1, . . . , ρ̄4, 0〉, ∇̄2 = 〈ρ̄5, . . . , ρ̄8, 0〉. (75)
where we express the generators ρ̄ in terms of ρ as
ρ̄i = ρi, i = 1, . . . , 6 ,
ρ̄7 = ρ7 + e1 + 2e2 + e4 + 2e5, ρ̄8 = ρ8 − e1 − 2e2 − e4 − 2e5.
It is easy to check that the ρ̄i satisfy the same linear relations eq. (23) as the ρi,
and that w = 1
(ρ̄1 − ρ̄2 + ρ̄6 − ρ̄7) = −e2 − e5. The ρ̄i together with w therefore
indeed generate the lattice N̄ . Note that, while all 8 non-zero lattice points of ∆̄∗
are vertices, the dual polytope ∆̄ has 18 vertices and 102 points. Using PALP [35]
again, we compute the lattice points of the polytope ∇̄∗ = 〈∆̄1, ∆̄2〉 ⊂MR, which will
describe the ambient space of the mirror X
of X . We find
∆̄1 = 〈ν̄1, . . . , ν̄6, 0〉, ∆̄2 = 〈ν̄7, . . . , ν̄12, 0〉, (77)
where we express the vertices ν̄i in terms of the vertices νi of ∇
ν̄3k+1 = ν3k+1, ν̄3k+2 = ν3k+2 − e5, ν̄3k+3 = ν3k+3 + e5, k = 0, . . . , 3. (78)
Again, it is easy to check that the ν̄i satisfy the same linear relations eq. (37) as the
νi. It turns out that the lattice points of ∇̄
∗ generate a sublattice M̄ of index 3 in M ,
and the lattice refinement is generated by
ν̄1 + 2ν̄2 + 2ν̄7 + ν̄8
= e2 + e4 − e5. (79)
Among the points of ∇∗ listed in eq. (39) only ν13 and ν14 are also lattice points of
the sublattice M̄ . In fact, we have ν̄13 = ν13 and ν̄14 = ν14. Hence, ∇̄
∗ has 12 vertices
and 15 lattice points; its dual ∇̄ = ∇̄1 + ∇̄2 has 42 lattice points among which 15 are
vertices10.
Once we have the polytopes ∆̄∗ and ∇̄∗, we can construct X and X
as complete
intersections entirely analogous to X̃ and X̃∗, see Section 3. That is, using eq. (27),
we define
X = D̄0,1 ∩ D̄0,2, X
= D̄∗0,1 ∩ D̄
0,2 (81)
in terms of the nef partitions eq. (75) and (77), respectively. Here, D̄i and D̄
i denote
the divisors associated to the generators ρ̄i and ν̄i, respectively. The absence of fixed
points of the G1 action on the complete intersection X̃ is guaranteed by the fact
that the resulting polytope ∆̄∗ ⊂ N̄R has no additional lattice points [31]. Hence,
X = X̃/G1 has a non-trivial fundamental group π1(X) = Z3. Surprisingly, it turns
out that the mirror X
is a free quotient as well. To see this recall that, as noticed
above, the lattice points of ∇̄∗ generate a sublattice M̄ of index 3 inM . Furthermore,
∇̄∗ also has no additional lattice points with respect to ∇∗. Therefore, there is a
10Note that all of our polytopes differ from the non-free Z3 × Z3 quotient of ∆
∗ defined in [28],
Proposition 7.1. In the notation of [31] their quotient is
∇∗ 6= P
1 1 1 0 0 0 0 0
0 0 0 1 1 1 0 0
0 0 0 0 0 0 1 1
Z3 : 0 1 2 0 0 0 0 0
Z3 : 0 0 0 0 1 2 0 0
and has 21 points and 8 vertices in the lattice N .
group G∗1 ≃ Z3 acting torically on P∇∗ . On the homogeneous coordinates this action
g∗1 :
z1 : · · · : z12
ζz1 : ζ
2z2 : z3 : · · · : z6 : ζ
2z7 : ζz8 : z9 : · · · : z12
. (82)
Hence, X
= X̃∗/G∗1 also has a non-trivial fundamental group π1(X
) = Z3. Note
that this never happens for hypersurfaces in toric varieties [6]. Having the toric
representation of X and X
, we can now compute their Hodge numbers. It turns out
= h1,2
= h1,1
= h1,2
= 7, (83)
in agreement with Part A [1], eq. (??).
4.2 The Quotient by G2
We now turn to the G2 action, which does not act torically. Hence, we cannot, in
principle, find a toric variety containing X = X/G2 as we did for the G1 quotient
above. However, at least we have to ensure that X and X
are G2-symmetric. This
can be achieved via suitable symmetries in the toric data.
The easy part of the toric data for X is the polytope ∆̄∗. The G2 action on the
ambient space permutes the homogeneous coordinates, see eq. (8b). In terms of toric
geometry, this means that it permutes the corresponding points of the polytope. That
is11,
g2 : ρ̄i 7→ ρ̄1+(i mod 3) ∀i ∈ {1, 2, 3},
g2 : ρ̄4 7→ ρ̄4, ρ̄5 7→ ρ̄5,
g2 : ρ̄5+i 7→ ρ̄6+(i mod 3) ∀i ∈ {1, 2, 3}.
It induces a mirror group actionG∗2 onX
which is geometrical, rather than a quantum
symmetry as discussed in [48]. The action of G∗2 is obviously the dual group action
on the dual lattice M , which again must be a symmetry of the relevant polytope ∇̄∗.
We find that
g∗2 : ν̄3k+i 7→ ν̄3k+1+(i mod 3) ∀k = 0, . . . , 3, i ∈ {1, 2, 3}. (85)
As a check on the mirror group action, note that the matrix of scalar products, see
eq. (87) below, is invariant. That is,
g2(ρ̄l), g
2(ν̄l′)
ρ̄l, ν̄l′
∀ l, l′. (86)
By abuse of notation, we denote the corresponding cyclic permutation of homogeneous
coordinates by g∗2 as well. Using this action, we define the mirror of X to be X
11We define the modulus operation such that (i mod 3) ∈ {0, 1, 2}.
/G∗2. This idea has already been used for the construction of mirrors of orbifolds
of the quintic [49] soon after the discovery of the first mirror construction by Greene
and Plesser.
Following eq. (41), the equations for the Calabi-Yau complete intersections X and
are defined by evaluating the matrix of scalar products 〈ρ̄i, ν̄j〉+ δl l′ , which are
〈 , 〉+ δl l′ ν̄1 ν̄2 ν̄3 ν̄4 ν̄5 ν̄6 ν̄13 ν̄7 ν̄8 ν̄9 ν̄10 ν̄11 ν̄12 ν̄14
ρ̄1 3 0 0 3 0 0 1 0 0 0 0 0 0 0
ρ̄2 0 3 0 0 3 0 1 0 0 0 0 0 0 0
ρ̄3 0 0 3 0 0 3 1 0 0 0 0 0 0 0
ρ̄4 1 1 1 0 0 0 0 0 0 0 1 1 1 1
ρ̄5 0 0 0 1 1 1 1 1 1 1 0 0 0 0
ρ̄6 0 0 0 0 0 0 0 3 0 0 3 0 0 1
ρ̄7 0 0 0 0 0 0 0 0 3 0 0 3 0 1
ρ̄8 0 0 0 0 0 0 0 0 0 3 0 0 3 1
The equations of X can now be read off from the columns of eq. (87), and one finds
F1 = (λ5t0 + λ6t1)(x
0 + x
1 + x
2) + (λ7t0 + λ8t1)x0x1x2, (88a)
F2 = (λ1t0 + λ4t1)(y
0 + y
1 + y
2) + (λ2t0 + λ3t1)y0y1y2, (88b)
where the G2-symmetry has been imposed. Note that the last monomial in each
equation corresponds to the vector 0 ∈ ∆̄l, l = 1, 2. Two of the eight coefficients λm
can be fixed by normalizing the equations, say λ4 = λ5 = 1, and three correspond to
the symmetries of P1, that is, SL(2) transformations of [t0 : t1]. Hence, we can, for
example, set λ6 = λ7 = λ8 = 0. This leaves us with 3 complex structure deformations
λ1, λ2, and λ3, see eqns. (7a) and (7b).
The equations defining X∗ correspond to the rows of eq. (87), that is,
F ∗1 = a1(z
4 + z
5 + z
6)z13 + (a2z10z11z12z14 + a3z4z5z6z13)z1z2z3, (89a)
F ∗2 = a4(z
10 + z
11 + z
12)z14 + (a5z4z5z6z13 + a6z10z11z12z14)z7z8z9, (89b)
where, again, invariance under G∗2 has been imposed and the last monomial of each
equation comes from the lattice point 0 ∈ ∇̄l, , l = 1, 2. Both equations are homo-
geneous with respect to all seven scaling degrees that follow from the linear relations
eq. (37). Among the twelve scalings of the coordinates zi, six are compatible with the
cyclic permutations g∗2, see eq. (85). Subtracting the three G2 symmetric indepen-
dent scalings among the relations eq. (37), there remains one torus action that acts
effectively on the parameters plus two normalizations of the equations. As expected,
the six parameters am of the equations of X
∗ thus become the 3 complex structure
moduli.
So far, we only considered the polytopes ∆̄∗ and ∇̄∗. However, this is only part of
the toric data defining the manifolds X and X
, respectively. In addition, we need the
triangulations and the corresponding exceptional sets. A change in the triangulation
corresponds to a flop of the toric variety. The very real danger is that not all, and
perhaps none, of the flopped Calabi-Yau manifolds are G2-symmetric. For X ⊂ P∆̄∗
this turns out to be unproblematic, but for X
⊂ P∇̄∗ we will find a condition for the
choice of a triangulation.
4.3 B-Model on X
We now return to the discussion of the triangulations and the intersection ring of X .
The analogous, but technically much more involved discussion of X
will be presented
in Subsection 4.5.
For X everything is straightforward since the G1-quotient did not introduce ad-
ditional lattice points in the associated polytope ∆̄∗. Therefore, just like for the
polytope ∆∗ of the covering space X̃ , there exists a unique triangulation. In particu-
lar the primitive collections, the Stanley-Reisner ideal, and the ideal IX are identical
to the ones in eqns. (42), (49), and (50) since they are derived from the same triangu-
lation. Moreover, one can easily see that this triangulation is G2-invariant and, hence,
X is G2 symmetric.
The only change is in the normalization of the intersection ring in eq. (52), since
the total volume has to be divided by 3 = |G1|. This can also be seen in eq. (76),
where the volume of the cone is now 3 instead of 1. Hence, on X the intersection ring
and the second Chern class are
J̄22 J̄3 = 1, J̄1J̄2J̄3 = 3, J̄2J̄
3 = 1,
· J̄1 = 0, c2
· J̄2 = 12, c2
· J̄3 = 12.
Comparing these intersection numbers with eq. (13), it is clear that the toric divisors
should be identified with the G1-invariant divisors on X as
J̄1 = φ, J̄2 = τ1, J̄3 = τ2. (91)
The curves spanning the Mori cone on the cover turn out to be G1-invariant as well.
Therefore, the Mori cones NE(P∆̄∗) and NE(X)toric are identical to those in eqns. (60)
and (61), respectively.
Following the steps given in Section 3 we now want to compute the B-model pre-
potential FB
, plug in the mirror map, and obtain the prepotential on X
(P,Q1, Q2, Q3, R1, R2, R3, b1). (92)
We immediately realize the following two caveats:
• We do not know how to incorporate the torsion curves H2(X,Z)tors = Z3 into
the toric mirror symmetry calculation.
• Of the 7 Kähler classes on X , only 3 are toric.
This means that only 3 out of the 7 + 1 variables in the prepotential are accessible,
and the remaining ones are set to one. Looking at the intersection numbers eq. (90),
it is clear that the 3 divisors are precisely the G2-invariant divisors on X , see eq. (13).
Therefore, these 3 variables must be those that map to the variables p, q, and r on
X . By comparing with eq. (16), we see that the corresponding variables on X are P ,
Q1, and R1. Hence, we actually only compute
(P,Q1, 1, 1, R1, 1, 1, 1) =
n1,n2,n3
nX(n1,n2,n3) Li3
P n1Qn21 R
. (93)
In effect, this means that the resulting instanton numbers are not just the instantons
in a single integral homology class, but the instanton numbers in a whole set of integral
homology classes. The instanton numbers sum over all curve classes that cannot be
distinguished by P,Q1, R1 ∈ Hom
H2(X,Z),C
. Up to total degree 4 and the
symmetry
nX(n1,n2,n3) = n
(n1,n3,n2)
, (94)
the resulting instanton numbers are
nX(1,0,0) = 27 n
(1,0,1) = 108 n
(1,0,2) = 378 n
(1,0,3) = 1080
nX(1,1,1) = 432 n
(1,1,2) = 1512 n
(2,0,1) = −54 n
(2,0,2) = −756
nX(2,1,1) = 864 n
(3,0,1) = 9.
4.4 Instanton Numbers of X
Knowing the prepotential on X , we now want to divide out the free G2 action and
arrive at the prepotential on X . Since we do not know the complete expansion but
only eq. (93), we have to set b1 = b2 = 1 in the descent equation (16). This yields
p, q, r, 1, 1) =
p, q, 1, 1, r, 1, 1, 1
n1,n2,n3
nX(n1,n2,n3) Li3
pn1qn2rn3
Up to the symmetry nX
(n1,n2,n3)
(n1,n3,n2)
, the non-vanishing instanton numbers for
X up to total degree 5 are
nX(1,0,0) = 9 n
(1,0,1) = 36 n
(1,0,2) = 126 n
(1,0,3) = 360
nX(1,0,4) = 945 n
(1,1,1) = 144 n
(1,1,2) = 504 n
(1,1,3) = 1440
nX(1,2,2) = 1764 n
(2,0,1) = −18 n
(2,0,2) = −252 n
(2,0,3) = −1728
nX(2,1,1) = 288 n
(2,1,2) = 3960 n
(3,0,1) = 3 n
(3,0,2) = 252
nX(3,1,1) = 756,
Unfortunately, this direct calculation misses the torsion information and only yields
the expansion F
X,0(p, q, r, 1, 1). The b1 dependence was lost because the toric methods
do not yield this part, and the b2 dependence was lost because the relevant divisor on
X was not toric. Comparing with the full expansion of the prepotential
p, q, r, b1, b2) =
n1,n2,n3
m1,m2
nX(n1,n2,n3,m1,m2) Li3
pn1qn2rn3bm11 b
, (98)
see Part A eq. (??), this means we only obtain the sum of the instanton numbers over
all torsion classes
nX(n1,n2,n3) =
m1,m2=0
nX(n1,n2,n3,m1,m2). (99)
Clearly, this destroys the torsion information, that is, the instanton numbers nX
(n1,n2,n3)
do not depend on the torsion part of the integral homology. For comparison purposes,
we list the instanton numbers nX(n1,n2,n3) for 0 ≤ n1, n2, n3 ≤ 5 in Table 2.
4.5 B-Model on X
We now study the mirror X
, which sits in a more complicated ambient toric variety.
Consequently, the analysis is more involved. The big advantage, however, will turn
out to be that all h11(X
) = 7 Kähler moduli are toric, which will enable us to obtain
the full instanton expansion.
Since the polytope ∇̄∗ in eq. (78) is not simplicial, we have to specify a resolution of
the singularities, that is, a triangulation T (∇̄∗). Moreover, not any triangulation will
do, but we have to make sure that it is compatible with the action of the permutation
group G∗2. While a tedious technicality, the existence of such a resolution has to
be shown in order to establish the existence of a geometrical mirror family of X .
In particular, we show in Appendix A that there is no projective resolution of the
ambient space among the 720 coherent star triangulations of ∇̄∗ that respects the
permutation symmetry eq. (85). In other words, if one demands G∗2 symmetry then
the ambient toric variety cannot be chosen to be Kähler, but only a complex manifold.
Clearly, in that case there is no Kähler cone and the usual toric mirror symmetry
algorithm does not work. What comes to the rescue is that there are two classes of
non-symmetric projective resolutions for which the symmetry-violating exceptional
sets do not intersect X
. Hence the complete intersection is G2-symmetric, even
though the ambient space is not.
We conclude that the extended Kähler moduli space of X
contains two sym-
metric phases. We will denote these two classes of triangulations by T± = T±(∇̄
see Appendix A. In fact, the two phases are topologically distinct, and only the tri-
angulation T+ describes the threefold X
that we are interested in. In Appendix B,
we will investigate the other triangulation T− which describes a flop of X
(0,n2,n3)
(3,n2,n3)
0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 0 0 0 0 0
2 0 0 0 0 0 0
3 0 0 0 0 0 0
4 0 0 0 0 0 0
5 0 0 0 0 0 0
0 1 2 3 4 5
0 0 3 252 4158 40173 287415
1 3 756 15390 164280 1259685 7763364
2 252 15390 426708 5427684 46537092 310465062
3 4158 164280 5427684 73971360 657552966 4487097816
4 40173 1259685 46537092 657552966 5948103483 41016575313
5 287415 7763364 310465062 4487097816 41016575313 284581389204
nX(1,n2,n3) n
(4,n2,n3)
0 1 2 3 4 5
0 9 36 126 360 945 2268
1 36 144 504 1440 3780 9072
2 126 504 1764 5040 13230 31752
3 360 1440 5040 14400 37800 90720
4 945 3780 13230 37800 99225 238140
5 2268 9072 31752 90720 238140 571536
0 1 2 3 4 5
0 0 0 −144 −6048 −107280 −1235520
1 0 −306 −12348 −207000 −2273400 −19066500
2 −144 −12348 348480 14609520 235219680 2505155400
3 −6048 −207000 14609520 520226784 8245864800 87989812560
4 −107280 −2273400 235219680 8245864800 131759049600 1417949658000
5 −1235520 −19066500 2505155400 87989812560 1417949658000 15365394415800
(2,n2,n3)
(5,n2,n3)
0 1 2 3 4 5
0 0 −18 −252 −1728 −9000 −38808
1 −18 288 3960 27648 143748 620928
2 −252 3960 54432 380160 1976472 8537760
3 −1728 27648 380160 2654208 13799808 59609088
4 −9000 143748 1976472 13799808 71748000 309920688
5 −38808 620928 8537760 59609088 309920688 1338720768
0 1 2 3 4 5
0 0 0 45 5670 189990 3508920
1 0 36 13140 474840 8793648 111499020
2 45 13140 1112886 38961252 777759975 10723515300
3 5670 474840 38961252 1952428464 47357606430 732897531720
4 189990 8793648 777759975 47357606430 1237373786439 19911043749420
5 3508920 111499020 10723515300 732897531720 19911043749420 327006066948660
Table 2: Summed instanton numbers nX
(n1,n2,n3)
m1,m2
(n1,n2,n3,m1,m2)
(hence not distinguishing torsion) com-
puted by mirror symmetry. The table contains all non-vanishing instanton numbers for 0 ≤ n1, n2, n3 ≤ 6.
Following Subsection 3.3, given the triangulation T+, we can determine the prim-
itive collections. This immediately yields the Stanley-Reisner ideal
ISR =
D̄1D̄13, D̄2D̄4, D̄2D̄13, D̄3D̄4, D̄3D̄5, D̄3D̄13, D̄4D̄10, D̄4D̄11, D̄4D̄12,
D4D̄14, D̄5D̄10, D̄5D̄11, D̄5D̄12, D̄5D̄14, D̄6D̄10, D̄6D̄11, D̄6D̄12,
D̄6D̄14, D̄13D̄10, D̄13D̄11, D̄13D̄12, D̄13D̄14, D̄7D̄14, D̄8D̄10, D̄8D̄12,
D̄8D̄14, D̄9D10, D̄9D̄14, D̄1D̄2D̄3, D̄1D̄2D̄6, D1D̄5D̄6, D̄4D̄5D̄6,
D̄7D̄8D̄9, D̄7D̄9D̄11, D̄7D̄11D̄12, D̄10D̄11D̄12
(100)
where we dropped the superscript ∗ on D̄ for ease of notation. From this, in turn, we
obtain the generators l̄
+ of the Mori cone NE(P∇̄∗):
+ =( 0, 0, 0, 0, 0, 0, 1, 0, 0,−1, 0, 0, 0, 1)
+ =( 1, 0, 0,−1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)
+ =(−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
+ =( 0,−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0, 0)
+ =( 0, 0,−1, 0, 0, 1, 0,−1, 0, 0, 1, 0, 0, 0)
+ =( 0, 0, 0, 0, 0, 0,−1, 0, 1, 1, 0,−1, 0, 0)
+ =( 0, 0, 0, 0, 0, 0, 0, 1,−1, 0,−1, 1, 0, 0)
+ =( 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0,−3, 0)
+ =( 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0,−3).
(101)
A dual basis for the generators of the Kähler cone K(P∇̄∗) is
K̄1 = D̄13 + 2 D̄1 − D̄2 − D̄3 + D̄9 + D̄7 + D̄8 + 3 D̄4,
K̄2 = 3 D̄1 + D̄13 + 3 D̄4,
K̄3 = D̄13 + 2 D̄1 + 3 D̄4,
K̄4 = D̄13 + 2 D̄1 − D̄2 + 3 D̄4,
K̄5 = D̄13 + 2 D̄1 − D̄2 − D̄3 + 3 D̄4,
K̄6 = D̄13 + 2 D̄1 − D̄2 − D̄3 + D̄9 + D̄8 + 3 D̄4,
K̄7 = D̄8 + D̄13 + 2 D̄1 − D̄2 − D̄3 + 3 D̄4,
K̄8 = D̄4 + D̄1,
K̄9 = D̄10 + D̄7.
(102)
The Calabi-Yau complete intersection X
is then defined by X
= K̄1K̄2. It turns
out that the divisors D̄13, D̄14 do not intersect X
. Therefore, all
toric
= h1,1
= 7 (103)
Kähler moduli are realized torically. Since there are two divisors that do not intersect,
finding the Mori cone is somewhat subtle. First, we have to restrict the lattice of linear
relations to the sublattice orthogonal to these two directions. For the generators of
the toric Mori cone NE(X
)toric, this means that l̄
+ → 3l̄
+ + l̄
+ , l̄
+ → 3l̄
+ + l̄
and that we drop l̄
+ , l̄
+ as well as the entries corresponding to intersections with
D̄13, D̄14. In addition, we prepend the intersection numbers with D̄0,1 and D̄0,2. This
yields
+ =(−3, 0; 0, 0, 0, 0, 0, 0, 3, 0, 0,−2, 1, 1)
+ =( 0,−3; 3, 0, 0,−2, 1, 1, 0, 0, 0, 0, 0, 0)
+ =( 0, 0;−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0)
+ =( 0, 0; 0,−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0)
+ =( 0, 0; 0, 0,−1, 0, 0, 1, 0,−1, 0, 0, 1, 0)
+ =( 0, 0; 0, 0, 0, 0, 0, 0,−1, 0, 1, 1, 0,−1)
+ =( 0, 0; 0, 0, 0, 0, 0, 0, 0, 1,−1, 0,−1, 1).
(104)
The dual basis of divisors is
J̄∗1 =
K̄21K̄2, J̄
K̄1K̄
2 , J̄
5 = K̄1K̄2K̄5,
J̄∗3 = K̄1K̄2K̄3, J̄
4 = K̄1K̄2K̄4,
J̄∗6 = K̄1K̄2K̄6, J̄
7 = K̄1K̄2K̄7.
(105)
We now try to identify this basis J̄∗1 , . . . , J̄
7 of divisors onX
with the basis {φ, τ1, υ1, ψ1, τ2, υ2, ψ2}
of divisors on X in eq. (10). It turns out that there is more than one way to identify
the bases if one only wants to preserve the triple intersection numbers. To obtain a
unique answer, we also need to identify the actions by G∗2 and G2 as well. First, the
G∗2 action on H
2(P∇̄∗ ,Z) is defined by eq. (85). Using the linear equivalence relations
2D̄1 − D̄2 − D̄3 + 2D̄4 − D̄5 − D̄6 = 0
−D̄1 + 2D̄2 − D̄3 − D̄4 + 2D̄5 − D̄6 = 0
2D̄7 − D̄8 − D̄9 + 2D̄10 − D̄11 − D̄12 = 0
−D̄2 + D̄3 − D̄5 + D̄6 − D̄8 + D̄9 − D̄11 + D̄12 = 0
−D̄4 − D̄5 − D̄6 + D̄10 + D̄11 + D̄12 − D̄13 + D̄14 = 0
(106)
and the definition eq. (105), one can compute the induced group action on H2(X
We find



1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 3 −1 1 0 0 0
0 3 −1 0 1 0 0
0 0 0 0 1 0 0
3 0 0 0 1 0 −1
0 0 0 0 1 1 −1



. (107)
Second, recall that the G2 action on the divisors of X



1 0 0 0 0 0 0
0 1 0 0 0 0 0
1 3 0 −1 0 0 0
0 3 1 −1 0 0 0
0 0 0 0 1 0 0
1 0 0 0 3 0 −1
0 0 0 0 3 1 −1



, (108)
see Part A eq. (??).
The essentially unique12 identification of divisors on X
and X then turns out to
J̄∗1 = τ1, J̄
2 = τ2, J̄
3 = ψ2, J̄
4 = υ2,
J̄∗5 = φ, J̄
6 = 3τ1 + υ1 − ψ1 = g2(ψ1), J̄
7 = υ1.
(109)
Note that we are identifying divisors on X
with divisors on X in eq.(109), something
that one would usually not do. However, in view of the anticipated self-mirror prop-
erty, X
∗ ∼= X, this is a sensible thing to try to attempt. And, indeed, the identification
above is an isomorphism of the intersection rings.
Regardless of this identification, we now continue to apply mirror symmetry. First,
the second Chern class is
· J̄∗1 = 12, c2
· J̄∗5 = 0, c2
· J̄∗2 = 12,
· J̄∗3 = c2
· J̄∗6 = 24, c2
· J̄∗4 = c2
· J̄∗7 = 12.
(110)
Using this information, we now compute the B-model prepotential
(q1, . . . , q7) = 3q5 + 3q4q5 +
q25 + 3q5q7 + 3q7q5q6 + 3q4q5q7 +
+ 3q3q4q5 +
7 + 3q3q4q5q7 +
+ 3q4q3q2q5 + 3q7q5q4q6 + 3q1q5q6q7 +
q45 +O(q
(111)
Finally, we insert the mirror map and obtain the A-model prepotential on X
. Since
we already identified the bases J̄∗1 , . . . , J̄
7 with the divisors on X , we will use the
same names (but with an added ∗ superscript) for the Fourier-transformed variables
to expand the prepotential. With this notation, we obtain
(P ∗, Q∗1, Q
3, 1) = 3P
∗ + 3
P ∗2 + 1
P ∗3 + 3
P ∗4 + 3
+ 3P ∗Q∗2 +
P ∗2Q∗22 + 3P
∗Q∗2Q
3 + 3P
∗R∗2 +
P ∗2R∗22 + 3P
∗R∗2Q
+ 3P ∗R∗2Q
3 + 3P
∗R∗2R
3 + 3P
∗R∗2R
2 + 3P
∗R∗2R
+ 3P ∗Q∗1R
3 + 3P
∗Q∗1R
2 + 9P
∗Q∗1R
3 + 3P
∗Q∗2Q
+ 3P ∗Q∗2Q
2 + 9P
∗Q∗2Q
total degree ≥ 6
(112)
12Up to the ĝ2 and g
2 ĝ2 symmetry.
see also Part A eq. (??). The instanton numbers on X
are the expansion coefficients
(P ∗, Q∗1, Q
3, 1)
n1,...,n7
(n1,n2,n3,n4,n5,n6,n7)
P ∗n1Q∗n21 Q
. (113)
We see that we almost get the complete instanton expansion eq. (15), we only miss
the expansion in the b∗1 variable which is not computed by the toric mirror symmetry
algorithm. Up to total degree 5, the instanton numbers are
(1,0,0,0,0,0,0) =3 n
(1,0,0,0,0,1,0) =3 n
(1,0,0,0,0,1,1) =3 n
(1,0,1,0,0,0,0) =3
(1,0,1,0,0,1,0) =3 n
(1,0,1,0,0,1,1) =3 n
(1,0,1,1,0,0,0) =3 n
(1,0,1,1,0,1,0) =3
(1,0,1,1,0,1,1) =3 n
(1,1,0,0,0,1,2) =9 n
(1,0,1,2,1,0,0) =9 n
(1,0,1,1,1,1,0) =3
(1,1,1,0,0,1,1) =3 n
(1,1,0,0,0,1,1) =3 n
(1,0,1,1,1,0,0) =3.
(114)
Finally, let us take a look at the G∗2 action, see eq. (85). Of the 7 generators of the
toric Mori cone, eq. (104), only the 3 generators l̄
+ , l̄
+ and l̄
+ are invariant. Not
surprisingly, the dual G∗2-invariant divisors
J̄∗5 = φ, J̄
1 = τ1, J̄
2 = τ2 (115)
were identified with the G2-invariant divisors on X in eq. (109). Therefore, only 3
Kähler parameters survive to the quotient X∗ = X
/G∗2, and we have
= h1,2
= h1,1
= h1,2
= 3. (116)
4.6 Instanton Numbers of X∗
Now that we have the expression eq. (113) for the prepotential on X
, we can again
apply a suitable variable substitution
P ∗, Q∗1, Q
p∗, q∗, r∗, b∗1, b
(117)
and obtain the prepotential on the quotient X∗ = X
/G∗2. The correct way to replace
the variables is determined by the group action on the homology and cohomology as
we explained in Part A. Having computed the G∗2-action in eq. (107), we determine
the descent equation for the prepotential to be13
p∗, q∗, r∗, b∗1, b
|G∗2|
p∗, q∗, b∗2, b
∗, b∗22 , b
2 , b
. (118)
13Interestingly, eq. (118) turns out to be exactly analogous to eq. (16), even though the identifica-
tion of divisors on X
and X is not just a relabeling of divisors.
Using the series expansion of the prepotential for b∗1 = 1 on X
from Subsection 4.5,
we now find that
X∗,0(p
∗, q∗, r∗, 1, b∗2)
Li3(p
2 ) + 4 Li3(p
2 ) + 4 Li3(p
+ 14 Li3(p
∗q∗2b
2 ) + 16 Li3(p
∗q∗r∗b
2 ) + 14 Li3(p
∗r∗2b
+ 40 Li3(p
∗q∗3b
2 ) + 56 Li3(p
∗q∗2rb
2 ) + 56 Li3(p
∗q∗r2b
+ 40 Li3(p
∗r∗3b
2 ) + 105 Li3(p
∗q∗4b
2 ) + 160 Li3(p
∗q∗3r∗b
− 2 Li3(p
∗2q∗b
2 )− 2 Li3(p
∗2r∗b
− 28 Li3(p
∗2q∗2b
2 ) + 32 Li3(p
∗2q∗r∗b
2 )− 28 Li3(p
∗2r∗2b
+ 3Li3(p
∗3q∗) + 3 Li3(p
∗3r∗)
total p∗, q∗, r∗-degree ≥ 5
(119)
The corresponding instanton numbers
X∗,0(p
∗, q∗, r∗, 1, b∗2) =
n1,n2,n3,m2
(n1,n2,n3,m2)
p∗n1q∗n2r∗n3b∗m22
(120)
are listed in Table 3. For comparison purposes, we list the summed instanton numbers
(n1, n2, n3) n
(n1,n2,n3,0)
(n1,n2,n3,1)
(n1,n2,n3,2)
(n1,n2,n3)
(1, 0, 0) 3 3 3 9
(1, 0, 1) 12 12 12 36
(1, 0, 2) 42 42 42 126
(1, 0, 3) 120 120 120 360
(1, 1, 1) 48 48 48 144
(1, 1, 2) 168 168 168 504
(2, 0, 1) −6 −6 −6 −18
(2, 0, 2) −84 −84 −84 −252
(2, 1, 1) 96 96 96 288
(3, 0, 1) 3 0 0 3
Table 3: Instanton numbers nX
(n1,n2,n3,m2)
computed by toric mirror symmetry.
They are invariant under the exchange n2 ↔ n3, so we only display
them for n2 ≤ n3.
on X as well, see eq. (99). One observes that the sum over the more refined instanton
numbers on X∗ equals the summed instanton number on X , another clue towards X
being self-mirror.
4.7 Instanton Numbers Assuming The Self-Mirror Property
So far, we have alluded to X being possibly self-mirror, but not actually made use
of this property. Now we are going to assume the self-mirror property and, hence,
obtain the prepotential on X as
X,0(p, q, r, b1, b2) = F
X∗,0(p, q, r, b1, b2). (121)
Note that at linear and quadratic order in p we can actually recover the b1, b2 expansion
from the summed instanton numbers in Subsection 4.4 and the factorization which
we will prove in Section 6.
In contrast, for the prepotential terms at order p3 we have to use the X∗ pre-
potential to obtain the b2 expansion from eq. (119). Since this is based on a toric
computation on X
, we do not directly obtain the b1 expansion. However, note that
the fact that g1 acted torically, eq. (8a), and g2 non-torically, eq. (8b), is just a conse-
quence of the choice of coordinate system on P2×P1×P2. By a suitable coordinate
choice, we could have made any one of the four Z3 subgroups of G = Z3 × Z3 act
torically. Therefore, any combination of b1, b2 other than 1 = b
2 has to occur in the
same way in the complete series expansion of the prepotential. We conclude that the
prepotential can only depend on b1 and b2 through the combinations
i,j=0
2. (122)
This observation lets us recover the full b1, b2 expansion of the prepotential. To
summarize, we obtain
X∗,0(p, q, r, b1, b2)
i,j=0
Li3(pb
2) + 4 Li3(pqb
2) + 4 Li3(prb
+ 14 Li3(pq
2bi1b
2) + 16 Li3(pqrb
2) + 14 Li3(pr
2bi1b
+ 40 Li3(pq
3bi1b
2) + 56 Li3(pq
2rbi1b
2) + 56 Li3(pqr
2bi1b
+ 40 Li3(pr
3bi1b
2) + 105 Li3(pq
4bi1b
2) + 160 Li3(pq
3rbi1b
+ 196 Li3(pq
2r2bi1b
2) + 160 Li3(pqr
3bi1b
2) + 105 Li3(pr
4bi1b
− 2 Li3(p
2qbi1b
2)− 2 Li3(p
2rbi1b
2)− 28 Li3(p
2q2bi1b
+ 32 Li3(p
2qrbi1b
2)− 28 Li3(p
2r2bi1b
2)− 192 Li3(p
2q3bi1b
+ 440 Li3(p
2q2rbi1b
2) + 440 Li3(p
2qr2bi1b
2)− 192 Li3(p
2r3bi1b
+ 3Li3(p
3q) + 3 Li3(p
+ 9 Li3(p
3q2) + 27
i,j=0
Li3(p
3q2bi1b
+ 9 Li3(p
3q2) + 27
i,j=0
Li3(p
3q2bi1b
+ 27 Li3(p
3qr) + 81
i,j=0
Li3(p
3qrbi1b
total p, q, r-degree ≥ 6
(123)
Obtaining all of these terms required a computation of FB
in eq. (111) up to total
degree 23 in the 7 variables, which is close to the limit of what can be done with
current desktop computers. We list the instanton numbers in Table 4. Observe that
the instanton numbers sometimes do depend on the torsion part of their homology
class.
5 The Self-Mirror Property
When one speaks of a Calabi-Yau manifold Y being self-mirror, one has to indicate
which level of invariants one is referring to. In particular, one might think of four
types of invariants that are natural from the point of view of string theory. The
weakest level is just the Euler number. In general, exchanging complex structure and
Kähler moduli changes the sign of χ(Y ) = 2h11(Y )− 2h21(Y ). Therefore, a necessary
condition for Y and its mirror Y ∗ to be equal is obviously that
χ(Y ) = −χ(Y ∗) = 0. (124)
This level of invariants, however, is much too crude and therefore insufficient. A much
stronger level is based on the fact that the cohomology groups of even degree come with
(1,n2,n3,0,0)
(1,n2,n3,m1,m2)
, (m1, m2) 6= (0, 0)
0 1 2 3 4
0 1 4 14 40 105
1 4 16 56 160
2 14 56 196
3 40 160
4 105
0 1 2 3 4
0 1 4 14 40 105
1 4 16 56 160
2 14 56 196
3 40 160
4 105
(2,n2,n3,0,0)
(2,n2,n3,m1,m2)
, (m1, m2) 6= (0, 0)
0 1 2 3
0 0 −2 −28 −192
1 −2 32 440
2 −28 440
3 −192
0 1 2 3
0 0 −2 −28 −192
1 −2 32 440
2 −28 440
3 −192
(3,n2,n3,0,0)
(3,n2,n3,m1,m2)
, (m1, m2) 6= (0, 0)
0 1 2
0 0 3 36
1 3 108
0 1 2
0 0 0 27
1 0 81
Table 4: Instanton numbers nX
(n1,n2,n3,m1,m2)
computed by mirror symmetry.
The table contains all non-vanishing instanton numbers for n1+n2+
n3 ≤ 5. The entries marked in bold depend non-trivially on the tor-
sion part of their respective homology class.
an integral lattice structure and form a ring, and therefore have a product. Because
of Poincaré duality, that is, H2(Y ) = H4(Y )∨, it is sufficient to look at H2(Y ). There
is a product H2(Y )×H2(Y ) → H2(Y ) whose structure constants κijk are the triple
intersection numbers. These intersection numbers are finer invariants than just the
dimensions of the cohomology groups, and a self-mirror Calabi-Yau threefold should
satisfy
κijk(Y ) = κijk(Y
∗). (125)
For simply connected threefolds with torsion-free homology a theorem of Wall [41]
states that the cohomology groups with the intersection product κijk(Y ) together
with the second Chern class c2(Y ) determine the diffeomorphism type of Y .
If, however, Y and Y ∗ have non-trivial fundamental groups then we cannot con-
clude that easily that they are diffeomorphic. But the non-trivial fundamental group
is often reflected in torsion in homology (for example if π1(Y ) is Abelian). In that
case, the conjecture of [6] says that for any Calabi-Yau threefold Z
. (126)
Therefore, a self-mirror manifold Y = Y ∗ is expected to satisfy
. (127)
Of the many spaces Y satisfying eq. (124) there are only a few which also satisfy
eq. (125).
So far we only considered classical topology, but we know that the ring H2(Y )
experiences quantum corrections when going far away from the large volume limit.
At small volume the intersection numbers are replaced by the three-point functions
Cijk(q) of (topological) conformal field theory in eq. (69). In the large volume limit
q goes to zero and the Cijk(q) go to κijk, as expected. The Cijk(q) are characterized
by the genus zero instanton numbers n
= nd. In mathematical terms, these are
resummations of the Gromow-Witten invariants of Y and characterize the symplectic
structure of Y . This level of invariants is even stronger than the cohomology ring,
since there are examples of diffeomorphic manifolds which have different Calabi-Yau
structures, i.e. different n
[50, 51, 31]. Therefore, a self-mirror Calabi-Yau threefold
Y must satisfy
(Y ) = n
(Y ∗). (128)
One can go even further and couple the topological conformal field theory to topolog-
ical gravity and define higher genus instanton numbers n
, where now
d (Y ) = n
∗), g > 0 (129)
has to hold. These invariants are very difficult to compute, however see [52, 53]
for recent progress. We do not know whether they contain more information about
the symplectic structure than the genus zero invariants. In other words, there are
presently no examples known whose n
agree for g = 0 but differ for g > 0.
Now, one can start with any Y and use some method to construct the mirror
Y ∗. Among these are the Greene-Plesser construction in conformal field theory, or its
geometric generalizations by Batyrev and Borisov for complete intersections in toric
varieties. Then, to show that Y is self-mirror one proceeds to compute the various
invariants. The simplest condition, eq. (124), can directly be checked in terms of the
toric data. This concretely means that one starts with a mirror pair Y and Y ∗ satis-
fying eq. (124) and checks whether eqns. (125), (127), (128), and (129) are satisfied.
In fact, in Section 4 we collected a large amount of evidence in favor of the claim that
X and its Batyrev-Borisov mirror threefold X∗ are the same. Indeed, eqns. (40), (83)
and (116) show that X̃ , X , and X satisfy by construction the constraint eq. (124)
on the Euler number. More interestingly, by the identifications found in eqns. (109)
and (115) we observed that the condition on the intersection ring, eq. (125), is sat-
isfied for X and X , respectively. Next, eq. (97) and Table 3 show that X also fulfils
the requirement eq. (128) on the genus zero instanton numbers. It would be very
interesting to see whether also the condition eq. (129) for higher genus curves can be
Finally, we consider the torsion in cohomology. In Part A ?? we have shown that
≃ Z3 ⊕ Z3, (130)
as we expect from a self-mirror threefold. Moreover, we can actually compute the
fundamental group of the Batyrev-Borisov mirror independently. For that, first notice
that the quotient X
= X̃∗/G∗1 is fixed-point free, see Subsection 4.2. The mirror
permutation G∗2 onX
acts freely as well. Therefore, bothX andX∗ are free quotients
by a group isomorphic to Z3 ⊕ Z3, thus their fundamental groups are
≃ Z3 ⊕ Z3. (131)
Moreover, on can easily show that on a proper14 Calabi-Yau threefold Z one has
H2(Z,Z)tors = π1(Z)ab, the Abelianization of the fundamental group. Hence, we see
H3(X,Z)tors ≃ Z3 ⊕ Z3 ≃ H
2(X∗,Z)tors (132)
and the first of eq. (126) is true. This provides the first evidence for the conjecture
of [6] in a context other than toric hypersurfaces.
Another point of view is that there is a geometrical or rather combinatorial reason
for the self-mirror property in this case. From eqns. (36) and (39) one can easily
see that the lattice points νi, ν6+i, ν13, ν14, i = 1, . . . , 3, span a sub-polytope of ∇
satisfying the same linear relations as all the lattice points ρi of ∆
∗ in eq. (23). Hence,
14A proper Calabi-Yau threefold has holonomy group the full SU(3). In particular, this implies
that the fundamental group is finite.
this sub-polytope is isomorphic to ∆∗. The same is true for the polytopes ∇̄∗ and
∆̄∗. The toric variety P∇̄∗ which is the ambient space of X
can therefore be regarded
as a blow-up of a quotient of P∆̄∗ , the ambient space of X . Actually, this blow-up
makes all 7 divisors of X
toric. Similarly, P∇∗ can be regarded as a blow-up of a
quotient of P∆∗ . As shown in Subsection 3.3 this entails that all 19 Kähler moduli of
X̃∗ are realized torically. Note that it is possible that the mirror polytopes ∆∗ and
∇∗ are actually isomorphic. In fact, for toric hypersurfaces there are 41, 710 self-dual
polytopes [54]. The novel feature in our case is that non-isomorphic polytopes lead
to self-mirror complete intersections, consistent with the nef partitions.
6 Factorization vs. The (3,1,0,0,0) Curve
One interesting observation is that the prepotential F
X,0 at order p, see eq. (123)
in this paper and eq. (??) in Part A [1], factors into
i,j=0 b
2 times a function of
p, q, r only. This means that the instanton number for any pseudo-section (curve
contributing at order p) does not depend on the torsion part of its homology class. In
other words, for any pseudo-section there are 8 other pseudo-sections with the same
class in H2(X,Z)free and together filling up all of H2(X,Z)tors = Z3 ⊕Z3. In contrast,
this factorization does not hold at order p3. For example,
X,0(p, q, r, b1, b2) = · · ·+ 3p
b1 + b
1 + b2 + b1b2 + b
1b2 + b
2 + b1b
2 + b
+ · · · .
(133)
The purpose of this subsection is to understand this behavior.
First, the factorization of the prepotential at any order of p not divisible by 3
follows from an extra symmetry that we have not utilized so far. The covering space
X̃ is, in addition to eqns. (8a) and (8b), also invariant under another Ĝ = Z3 × Z3
action generated by (ζ
ĝ1 :
[x0 : x1 : x2] 7→ [x0 : ζx1 : ζ
[t0 : t1] 7→ [t0 : t1] (no action)
[y0 : y1 : y2] 7→ [y0 : y1 : y2] (no action)
(134a)
ĝ2 :
[x0 : x1 : x2] 7→ [x1 : x2 : x0]
[t0 : t1] 7→ [t0 : t1] (no action)
[y0 : y1 : y2] 7→ [y0 : y1 : y2] (no action)
(134b)
This symmetry has fixed points and, therefore, cannot be used if one is looking for
a smooth quotient of X̃ . However, it commutes with G and hence descends to a
Ĝ = Z3 × Z3 symmetry of X (with fixed points). Clearly, the instanton sum must
observe this additional geometric symmetry. To make use of this symmetry, we have
to express its action on the variables in F
X,0(p, q, r, b1, b2). We can do so by first noting
that the basic 81 curves
s1×s2 ⊂ X̃, s1 ∈MW (B1), s2 ∈MW (B2) (135)
are really one orbit under G × Ĝ. Recall that, after dividing out G, these curves
became the 9 sections in MW (X) = Z3 ⊕ Z3, see Part A ??. We now observe
that MW (X) = {sij} is one Ĝ-orbit; since each of these sections contributes pb
i, j = 0, . . . , 2 the induced Ĝ action on the prepotential must be
ĝ1 : F
X,0(p, q, r, b1, b2) 7→ F
X,0(b1p, q, r, b1, b2),
ĝ2 : F
X,0(p, q, r, b1, b2) 7→ F
X,0(b2p, q, r, b1, b2).
(136)
Clearly, the prepotential must be invariant under the ĝ1, ĝ2 action. While imposing
no constraint on the p3n terms in the prepotential, all other powers of p must appear
in the combination
i,j=0
, n 6≡ 0 mod 3. (137)
This proves the factorization observed at the beginning of this subsection.
Second, we would like to understand the p3q terms in eq. (133). These are the
curves in the homology classes15
(3, 1, 0, ∗, ∗) ∈ Z3 ⊕ Z3 ⊕ Z3 = H2
. (138)
We will show that the rational curves in this class come in a single family, that is, the
moduli space of genus 0 curves on X in these homology classes
X, (3, 1, 0, ∗, ∗)
(139)
is connected. In particular, all such curves have the same homology class (3, 1, 0, 0, 0)
and only contribute to p3q in the prepotential eq. (133). As discussed in Part A ??,
any such map CX : P
1 → X factors
CX //
C eX ��
��������
. (140)
15Recall that the exponent of p is the degree along the base P1. This is why we pick a basis in
H2(X,Z)free such that a curve in (n1, n2, n3,m1,m2) contributes at order p
n1qn2rn3bm11 b
2 in the
prepotential.
The map C
can be written in terms of homogeneous coordinates as a function
: P1[z0:z1] 7→ P
[x0:x1:x2]
×P1[t0:t1] ×P
[y0:y1:y2]
(141)
satisfying the equations (7a) and (7b) defining X̃ ,
F1 ◦ C eX
[z0 : z1]
= 0 = F2 ◦ C eX
[z0 : z1]
∀[z0 : z1] ∈ P
1 . (142)
The curve CX ends up in the homology class (3, 1, 0, ∗, ∗) if and only if the defining
equation (141) is of degree (3, 1, 0) in P2×P1×P2. Hence, eq. (141) is defined by
complex constants αij, βij , γi (up to rescaling) such that
xi = αi0 z0 + αi1 z1 i = 0, 1, 2
ti = βi0 z
0 + βi1 z
0z1 + βi2 z0z
1 + βi3 z
1 i = 0, 1
yi = γi i = 0, 1, 2.
(143)
These constants have to be picked such that the resulting curve lies on the complete
intersection X̃, that is, they have to satisfy eq. (142). Inserting eq. (143), we find
that F1 ◦ C eX
[z0 : z1]
is a homogeneous degree 6 polynomial in [z0 : z1]. Since
the coefficients of zk0z
1 must vanish individually, this yields 7 constraints for the
parameters αij , βij . What makes this system of constraint equations tractable is the
fact that they are all linear in βij ,
F1 ◦ C eX = 0 ⇔

A1 0 0 0 A5 0 0 0
A2 A1 0 0 A6 A5 0 0
A3 A2 A1 0 A7 A6 A5 0
A4 A3 A2 A1 A8 A7 A6 A5
0 A4 A3 A2 0 A8 A7 A6
0 0 A4 A3 0 0 A8 A7
0 0 0 A4 0 0 0 A8



= 0 (144)
where
= α300 + α
10 + α
20 A5
= α00α10α20
= 3α01α
00 + 3α11α
10 + 3α21α
20 + α
20 A6
= (α01α10 + α00α11)α20 + α00α10α21
= 3α201α00 + 3α
11α10 + 3α
21α20 A7
= α01α11α20 + (α01α10 + α00α11)α21
= α301 + α
11 + α
21 A8
= α01α11α21.
(145)
Thinking of this as 7 linear equations for the 8 parameters βij, there is always a non-
zero solution. The solution is generically unique up to an overall factor, and turns
into an Pn for special values of the αij . Moreover, the parameter space of the αij is
connected (essentially, the moduli space of lines in P2). Since we just identified the
parameter space of the (αij, βij) as a blow-up thereof, it is therefore connected as well.
It remains to satisfy F2 ◦ C eX = 0. One can easily see that the only way is to pick
the γi to be simultaneous solutions of
γ30 + γ
1 + γ
2 = 0 = γ1γ2γ3. (146)
Since two cubics intersect in 9 points, there are 9 such solutions, permuted by G.
Therefore, the parameter space of (αij , βij, γi) has 9 connected components, permuted
by the G-action. The moduli space of curves CX on X is the G-quotient of the moduli
space of curves C
on X̃, and therefore has only a single connected component. By
continuity, every curve CX in this connected family has the same homology class,
explaining the piece of the prepotential given in eq. (133).
7 Towards a Closed Formula
Putting all the information together we found out about the prepotential on X , one
can try to divine a closed form for the prepotential. We guess that the order pn terms
have the closed form
X,0(p, q, r, b1, b2)
i,j∈Z3
P (q)4P (r)4
M2n−2(q, r) (147)
if n is not a multiple of 3 and, slightly weaker, that
X,0(p, q, r, 1, 1)
P (q)4P (r)4
M2n−2(q, r) (148)
if n is a multiple of 3. Here,
• P (q) is the usual generating function of partitions eq. (4).
• The M2n−2 are polynomials in the Eisenstein series E2(q), E4(q), E6(q) and
E2(r), E4(r), E6(r), starting with
M−2(q, r) = 0
M0(q, r) = 1
M2(q, r) = E2(q)E2(r)
M4(q, r) =
E4(q)E4(r) +
E4(q)E2(r)
2 + E2(q)
2E4(r)
E2(q)
2E2(r)
M6(q, r) =
E6(q)E6(r) +
E6(q)E4(r)E2(r) + E4(q)E2(q)E6(r)
E6(q)E2(r)
3 + E2(q)
3E6(r)
E4(q)E2(q)E4(r)E2(r)
E2(q)
3E4(r)E2(r) + E4(q)E2(q)E2(r)
E2(q)
3E2(r)
M8(q, r) =
E6(q)E2(q)E6(r)E2(r) +
E4(q)E4(q)E4(r)E4(r)
E6(q)E2(q)E4(r)E4(r) + E4(q)E4(q)E6(r)E2(r)
E6(q)E2(q)E4(r)E2(r)
2 + E4(q)E2(q)
2E6(r)E2(r)
E6(q)E2(q)E2(r)
4 + E2(q)
4E6(r)E2(r)
+ 137
E4(q)E4(q)E4(r)E2(r)
2 + E4(q)E2(q)
2E4(r)E4(r)
E4(q)E4(q)E2(r)
4 + E2(q)
4E4(r)E4(r)
E4(q)E2(q)
2E4(r)E2(r)
2 + 121
E2(q)
4E2(r)
E4(q)E2(q)
2E2(r)
4 + E2(q)
4E4(r)E2(r)
(149)
They are symmetric under the exchange q ↔ r and of weight 2n in q and r
separately. But, for example, M4 above does not factor into a function of q and
a function of r. So theM2n−2 are not the products of the polynomials appearing
in the dP9 prepotential. However, by setting q = 0 or r = 0 one recovers the
corresponding polynomials in the dP9 prepotential [55].
• The E2i are the usual Eisenstein series
E2(q) = 1− 24q − 72q
2 − 96q3 − 168q4 − 144q5 − 288q6 +O(q7)
E4(q) = 1 + 240q + 2160q
2 + 6720q3 + 17520q4 + 30240q5 +O(q6)
E6(q) = 1− 504q − 16632q
2 − 122976q3 − 532728q4 +O(q5).
(150)
Note that the naive Taylor series coefficients of the prepotential are fractional, but
when expanding in terms of Li3’s (which account for the multicover contributions)
one finds integral instanton numbers.
These expressions for the prepotential agree with all instanton numbers computed
in this paper. Unfortunately, we have not been able to guess a closed formula that
includes the b1 and b2 dependence of the prepotential F
X,0(p, q, r, b1, b2)|pn if n is
divisible by 3. We expect that these involve extra functions beyond the Eisenstein
series.
8 Conclusion
In the initial paper Part A [1], we analyzed the topology of the Calabi-Yau manifold
of interest and found that
= Z3 ⊕ Z3 ⊕ Z3. (151)
Although the presence of torsion curve classes complicates the counting of rational
curves, we managed to derive the A-model prepotential to linear order in p.
The goal of this paper is to go beyond the results of Part A using mirror symmetry.
By carefully adapting methods designed for complete intersections in toric varieties, we
can apply mirror symmetry to compute the instanton numbers onX , even thoughX is
not toric. Using thatX is self-mirror, we completely solve this problem and are able to
calculate the complete A-model prepotential to any desired precision (and for arbitrary
degrees in p), limited only by computer power. Carrying out this computation, we
find the first examples of instanton numbers that do depend on the torsion part of
their integral homology class, see Table 4 on Page 35.
Since the self-mirror property of X is important, we investigate it in detail. In
doing so, we go far beyond just checking that the Hodge numbers are self-mirror.
In particular, we find that the intersection rings are identical and that torsion in
homology obeys the conjectured mirror relation [6]. Finally, going beyond classical
geometry, we independently calculate certain instanton numbers onX and its Batyrev-
Borisov mirror X∗. Again, we find that X and X∗ are indistinguishable, providing
strong evidence for X being self-mirror. Both of these results extend those found in
Part A [1].
Using these results, we are able to guess certain closed expressions for the pre-
potential of X in terms of modular forms. In certain limits it specializes to the dP9
prepotential of [55]. There it is known that the coefficients in p of the dP9 prepoten-
tial satisfy a recursion relation. Moreover, there is a gap condition, that is, a certain
number of subsequent terms in a series expansion is absent. This condition provides
sufficient data to determine the integration constants for the recursion and allows to
determine the prepotential completely, even at higher genus. We expect a similar
story to be valid for the prepotential of X .
Acknowledgments
The authors would like to thank Albrecht Klemm, Tony Pantev, and Masa-Hiko Saito
for valuable discussions. We also thank Johanna Knapp for providing a Singular [56]
code to compute the intersection ring of Calabi-Yau manifolds in toric varieties. This
research was supported in part by the Department of Physics and the Math/Physics
Research Group at the University of Pennsylvania under cooperative research agree-
ment DE-FG02-95ER40893 with the U. S. Department of Energy and an NSF Focused
Research Grant DMS0139799 for “The Geometry of Superstrings”, in part by the
Austrian Research Funds FWF grant number P18679-N16, in part by the European
Union RTN contract MRTN-CT-2004-005104, in part by the Italian Ministry of Uni-
versity (MIUR) under the contract PRIN 2005-023102 “Superstringhe, brane e inter-
azioni fondamentali”, and in part by the Marie Curie Grant MERG-2004-006374.E. S.
thanks the Math/Physics Research group at the University of Pennsylvania for kind
hospitality.
A Triangulation of ∇̄∗ and ∇∗
In principle the coherent triangulations of the fan over ∇̄∗ can be computed with
TOPCOM by finding the 720 star triangulations in the total of 230, 832 coherent tri-
angulations of ∇̄∗. The discussion of the symmetry properties is greatly facilitated,
however, by an explicit understanding of their structure. We will work out the trian-
gulations by first triangulating the facets and then checking the compatibility of their
maximal intersections and the coherence of the resulting star triangulations.
We start with a couple of useful definitions. A circuit is a minimal collection of n
affinely dependent points p1, . . . , pn,
λ1p1 + . . . λnpn = 0 with λ1 + . . .+ λn = 0, λi 6= 0, (152)
any proper subset of which is affinely independent. The coefficient vector λn hence has
nonzero entries and is unique up to a prefactor. We indicate the unique separation into
points with positive and negative coefficients with the notation 〈pi1 . . . pis|pis+1 . . . pin〉.
Each circuit admits two different triangulations, which are obtained by dropping one
of the points with positive coefficients and one of the remaining points, respectively.
We indicate this with a hat over the relevant subset. The two resulting triangulations
̂pi1 . . . pis |pis+1 . . . pin
pi1 . . . pis| ̂pis+1 . . . pin
(153)
hence consist of s and n− s simplices, respectively. If the first point is in the convex
hull of the others, that is, s = 1, then only one of the triangulations is maximal (all
points are vertices of at least one simplex).
Furthermore, we introduce the notation:
ai = ν̄i, bi = ν̄3+i, ci = ν̄6+i, di = ν̄12+i, i = 1, 2, 3,
e = ν̄13, f = ν̄14.
(154)
Among these 14 vectors in eq. (154) there are 9 independent linear relations, see
eq. (37),
a1 + a2 + a3 = 0, c1 + c2 + c3 = 0,
e+ f = 0, bi = ai + e, dl = cl + f,
(155)
which imply others like ai + bj = aj + bi and ai + cl = bi + dl or e =
(b1 + b2 + b3)
and f = 1
(d1 + d2 + d3).
Lemma 1. ∇̄∗ has 15 facets, 6 of which are simplicial:
aiajbibjclcmdldm] i<j
aiajd1d2d3
b1b2b3clcm
. (156)
The nine non-simplicial facets form an orbit under the permutation symmetries
Zab3 × Z
3 generated by gab : (
( ai+1
and gcd :
( cl+1
. According to the
linear relations eq. (155) the eight points on each non-simplicial facet form quadratic
circuits ai+ bj = bi+aj , ai+ cl = bi+dl, and cl+dm = cm+dl, which we call mixed if
they contain vertices of both elements of the nef partition 〈aicl|bidl〉, and pure circuits
〈aibj |biaj〉, 〈cldm|cmdl〉 otherwise.
The coherent triangulations of the facets [aiajbibjclcmdldm] are most easily ob-
tained from their Gale transform
1 −1 −1 1 0 0 0 0
1 0 −1 0 1 0 −1 0
0 0 0 0 1 −1 −1 1
 , (157)
which is the coefficient matrix of the basis ai−aj −bi+bj = 0, ai−bi+cl−dl = 0 and
cl− cm−dl+dm = 0 of linear relations. The coherent triangulations are in one-to-one
correspondence to chambers that are seperated by the facets of the cones generated
by all linear bases µ = {v1, v2, v3} with vi selected among the 8 column vectors of the
Gale transform [57, 58]. In the present case the cones over the faces of the parallel-
epiped in Figure 1 are subdivided into 24 chambers, which are indicated by dashed
lines. The triangulations, which we can label by the facet containing and the edge
adjoining the chamber, are obtained as the sets of complements of those bases µ that
span a cone containing the respective chamber.
Hence, each non-simplicial facet has 24 coherent triangulations, which can be
characterized by the triangulations of its 2 pure and of its 4 mixed circuits: Calling
the triangulation 〈âicl|bidl〉 positive and the triangulation 〈aicl|b̂idl〉 negative, and
arranging the cyclic permutations gab and gcd in the horizontal and vertical direction,
respectively, we can assign one of 16 different types ± ±± ± to each triangulation, where
Figure 1: Secondary fan of the non-simplicial facets. Chambers are indicated
by dashed lines.
the signs indicate the induced triangulations of the mixed circuits. The constraints
that reduce the a priori 32 = 26 combinations to 24 all derive from the following rules:
ai aj
bi bj
aicl|b̂idl
âjcl|bjdl
aibj |âjbi
âicl|bidl
ajcl|b̂jdl
âibj |ajbi
〉 (158)
i.e. a triangular prism can be triangulated in 6 different ways, which correlates the a
priori 8 combinations of the triangulations of the 3 squares (with analogous constraints
for the two “horizontal” prisms [aibiclcmdldm] contained in the facet [aiajbibjclcmdldm]).
Putting the pieces together we obtain
Lemma 2. The 24 triangulations of the non-simplicial facets can be assorted as fol-
lows:
• For + ++ + ,
− − the pure circuits are unconstrained, yielding 2 · 2
2 = 8 triangu-
lations.
• For + +− − ,
+ + the pure ab-circuit is unconstrained; with the transposed types
+ − ,
− + this accounts for another 8 triangulations.
• The final 8 triangulations come from the 8 types with an odd number of positive
signs, for which the triangulation of the pure circuits is unique.
• The two types + −− + and
+ − cannot occur because of contradictory implications
for the triangulations of the pure circuits.
The secondary fan and the induced triangulations for the codimension-two faces at
which the non-simplical facets intersect can be obtained from Figure 1 by projection
along the dropped vertices. The secondary fan of the prism of eq. (158), for example,
which is shown in Figure 2, is obtained from Figure 1 by projection along the diagonal
Taibj
〈aicl|bidl〉s
Taicl
〈ajcl|bjdl〉
Tajcl
〈ajbi|aibj〉
Tajbi
〈bidl|aicl〉 s
Tbidl
��〈ajcl|bjdl〉
bi Tbjdl
〈ajbi|aibj〉
Figure 2: Secondary fan of the codimension two face [aiajbibjcldl].
〈cmdm〉. The wall crossings between the six cones in Figure 2 are labeled by the circuits
whose flops relate the adjoining triangulations [57].
For the construction of the complete star triangulation we now observe that the
non-simplicial intersections of the 9 non-simplicial facets [aiajbibjclcmdldm] are given
by the 18 triangular prisms [aiajbibjcldl] and [aibiclcmdldm]. If we interpret the former
as vertices and the latter as links then the resulting compatibility conditions corre-
spond to a graph with the topology of a torus. The vertices of this graph are decorated
by signs as shown in Table 5 and connected by horizontal and vertical links. The re-
1 · 26 9 · 22 18 · 22 6 · 24 36 · 20 36 · 21 9 · 22
Table 5: The 824 = 2 (64 + 36 + 72 + 96 + 36 + 72 + 36) star triangulations of
∇∗, including the 720 = 2 (36 + 36+ 72+ 72+ 36+ 72+ 36) coherent
triangulations.
striction on the compatible signs is due to the absence of the inconsistent types + −− +
and − ++ − as subgraphs on the torus. The multiplicities µ · 2
n come from the number
n of unconstrained pure circuits and from the order µ of the effective part of the
symmetry group generated by transposition and permutations of lines and columns.
We thus find a total of 824 triangulations. The cyclic permutation symmetry that
we want to keep on the Calabi-Yau manifold X
amounts to a diagonal shift, i.e.
its induced action on the graph is generated by gabgcd. We are hence left with the
types
, and the shift symmetry furthermore aligns the triangulations of
the pure circuits and thus reduced the multiplicities from 26 to 22, yielding a total of
8 triangulations for which P∇̄∗ is G
2 symmetric.
The resulting triangulations of the facet [a2a3b2b3c2c3d2d3] are
triangulation of [a2a3b2b3c2c3d2d3]
〈â2b3|a3b2〉, 〈ĉ2d3|c3d2〉 {[a3b2b3d2d3], [b2b3c3d2d3], [a2a3b2d2d3], [b2b3c2c3d2]}
〈â2b3|a3b2〉, 〈c2d3|ĉ3d2〉 {[a3b2b3d2d3], [b2b3c2d2d3], [a2a3b2d2d3], [b2b3c2c3d3]}
〈a2b3|â3b2〉, 〈ĉ2d3|c3d2〉 {[a2b2b3d2d3], [b2b3c3d2d3], [a2a3b3d2d3], [b2b3c2c3d2]}
〈a2b3|â3b2〉, 〈c2d3|ĉ3d2〉 {[a2b2b3d2d3], [b2b3c2d2d3], [a2a3b3d2d3], [b2b3c2c3d3]}
(159)
triangulation of [a2a3b2b3c2c3d2d3]
〈a2b3|â3b2〉, 〈c2d3|ĉ3d2〉 {[a2a3b3c2c3], [a2a3c2c3d3], [a2b2b3c2c3], [a2a3c2d2d3]}
〈a2b3|â3b2〉, 〈ĉ2d3|c3d2〉 {[a2a3b3c2c3], [a2a3c2c3d2], [a2b3b2c2c3], [a2a3c3d2d3]}
〈â2b3|a3b2〉, 〈c2d3|ĉ3d2〉 {[a2a3b2c2c3], [a2a3c2c3d3], [a3b2b3c2c3], [a2a3c2d2d3]}
〈â2b3|a3b2〉, 〈ĉ2d3|c3d2〉 {[a2a3b2c2c3], [a2a3c2c3d2], [a3b2b3c2c3], [a2a3c3d2d3]}
(160)
It can be checked that the triangulations listed in eqns. (159) and (160) come from the
chambers contained in the cones over [a2a3c2c3] and [b2b3d2d3], respectively. For the
first of these triangulations we consider the chamber adjoining the edge [a2c2], which
is contained in the span of the four bases µ1 = {a2c2c3}, µ2 = {a2a3c2}, µ3 = {b3c2c3}
and µ4 = {a2a3d3}, whose complements are [a3b2b3d2d3], [b2b3c3d2d3], [a2a3b2d2d3] and
[b2b3c2c3d2] in agreement with the first triangulation in eq. (159).
Unfortunately, coherent triangulations of the facets that induce the same trian-
gulations on their common (maximal) intersections do not automatically combine to
coherent star triangulations of the polytope, and indeed only 720 of the 824 triangu-
lations in Table 5 turn out to be coherent. The non-coherent ones are easily isolated
by observing that coherent triangulations (via their height functions) induce coherent
triangulations of the prisms [a1a2a3b1b2b2] and [c1c2c3d1d2d3], which eliminates the
triangulations for which Zab3 or Z
3 is not broken by the triangulation of the pure
circuits. For the triangulation types
this reduces the multiplicity from
82 to 62. The only other affected types are the ones in the middle column of Table 5,
which have unbroken horizontal symmetry and for which the multiplicity is reduced
from 12 · 8 to 12 · 6. This poses a problem for the eight Z3-symmetric triangulations,
which are all non-coherent. Coherence of the remaining 720 triangulations can be
established by checking that their Mori cones are all strictly convex [59].
What comes to our rescue is that, even if all projective ambient spaces break the
diagonal Z3 permutation symmetry, it may be preserved on X
if the obstructing ex-
ceptional sets do not overlap with the complete intersection. In the present case these
are the blow-ups of the singularities coming from the pure circuits, i.e. codimension
two sets of the form ai · bj or cl · dm, where we use, for simplicity, the symbol of the
vertex ν̄j for the corresponding divisor Dj . Recall from eq. (77) that X
is given by
the product D̄∗0,1 · D̄
0,2 of the divisors
D̄∗0,1 = a1 + a2+ a3 + b1 + b2 + b3 + e, D̄
0,2 = c1 + c2 + c3+ c1+ d2+ d3+ f (161)
defined by the nef partition. Taking into account the five linear equivalences, we
observe that
a1 + b1 = a2 + b2 = a3 + b3, c1 + d1 = c2 + d2 = c3 + d3,
b1 + b2 + b3 + e = d1 + d2 + d3 + f,
(162)
for divisor classes in the intersection ring. We first show that e and f do not intersect
: In any maximal triangulation e and f belong only to the simplices
b̂1b2b3cmcle
d̂1d2d3amalf
, (163)
respectively, so that
e·ai = e·dl = e·f = 0 ⇒ e·D̄
0,1 = e·(b1+b2+b3+e) = e·(d1+d2+d3+f) = 0 (164)
and similarly f · D̄∗0,2 = 0. Putting everything together, we conclude that
a1 · b2 · D̄
0,1 = a1 · b2 · 3(a3 + b3) = 0 (165)
because none of the facets, and hence no triangle in any of the triangulations contains
{a1, b2, a3} or {a1, b2, b3} as a subset. Similarly cl · dm · D̄
0,2 = 0 in the intersection
ring for l 6= m. Consequently, all exceptional sets arising from triangulations of
pure circuits do not intersect X
and hence do not obstruct the cyclic permutation
symmetry G∗2. We will denote any of the remaining 36 coherent triangulations of type
by T+ = T+(∇̄
∗) and T− = T−(∇̄
∗), respectively.
The polytope ∇∗ of the mirror X̃∗ of the universal cover has 39 lattice points, with
the same 12 vertices as ∇̄∗ but living on the finer lattice M̄ . The 24 additional lattice
points, see eq. (39), are
aij =
(ai + 2aj), bij =
(bi + 2bj), (166)
cij =
(ci + 2cj), dij =
(di + 2dj), (167)
where i 6= j. These additional points are all located on edges of ∇∗. It is natural
to consider triangulations that are refinements of the ones that we just discussed.
Observing that the additional points turn all simplices in eqns. (159), (160) and (163)
into pyramids over a tetrahedron with interior points on opposite edges it is easy to see
that the maximal triangulations are unique and multiply the number 54 = 9 · 4+6 · 3
of triangles in the original triangulations by a factor of 9. The resulting triangulations
have been used to show that the divisors corresponding to the vertices aij and cij do
not intersect X̃∗.
B The Flop of X∗
In Subsection 4.5 we have taken into account only one of the triangulations T+(∇̄
We can repeat the same calculation with one of the triangulations T−. We denote
the resulting Calabi-Yau manifold by X
−. Skipping the details, we find that the
generators of the Mori cone NE(X
−) can be expressed in terms of those of NE(X
in eq. (104) as
− = l̄
+ + 3
+ + l̄
+ + l̄
+ + l̄
− = l̄
+ + 3
+ + l̄
+ + l̄
− = l̄
+ + l̄
− = − l̄
− = − l̄
+ − l̄
+ − l̄
+ − l̄
+ − l̄
− = l̄
− = l̄
(168)
One can also express the dual basis of divisors J̄ ′i on the flop in terms of the dual basis
J̄∗i on X , see eq. (105). We find
J̄ ′1 = J̄
1 , J̄
2 = J̄
2 , J̄
5 = 3J̄
1 + 3J̄
2 − J̄
J̄ ′3 = 3J̄
1 + J̄
3 − J̄
5 , J̄
4 = 3J̄
1 + J̄
3 − J̄
J̄ ′6 = 3J̄
2 − J̄
5 + J̄
6 , J̄
7 = 3J̄
2 − J̄
5 + J̄
(169)
The intersection ring is
J̄ ′21 J̄
2 = 1, J̄
3 = 2, J̄
4 = 1, J̄
5 = 3, J̄
6 = 3,
J̄ ′21 J̄
7 = 3, J̄
2 = 1, J̄
3 = 3, J̄
4 = 3, J̄
5 = 3,
J̄ ′1J̄
6 = 3, J̄
7 = 3, J̄
3 = 6, J̄
4 = 6, J̄
5 = 9,
J̄ ′1J̄
6 = 9, J̄
7 = 9, J̄
4 = 3, J̄
5 = 9, J̄
6 = 9,
J̄ ′1J̄
7 = 9, J̄
5 = 9, J̄
6 = 9, J̄
7 = 9, J̄
6 = 9,
J̄ ′1J̄
7 = 9, J̄
7 = 9, J̄
3 = 3, J̄
4 = 3, J̄
5 = 3,
J̄ ′22 J̄
6 = 2, J̄
7 = 1, J̄
3 = 9, J̄
4 = 9, J̄
5 = 9,
J̄ ′2J̄
6 = 9, J̄
7 = 9, J̄
4 = 9, J̄
5 = 9, J̄
6 = 9,
J̄ ′2J̄
7 = 9, J̄
5 = 9, J̄
6 = 9, J̄
7 = 9, J̄
6 = 6,
J̄ ′2J̄
7 = 6, J̄
7 = 3, J̄
3 = 18, J̄
4 = 18, J̄
5 = 27,
J̄ ′23 J̄
6 = 27, J̄
7 = 27, J̄
4 = 18, J̄
5 = 27, J̄
6 = 27,
J̄ ′3J̄
7 = 27, J̄
5 = 27, J̄
6 = 27, J̄
7 = 27, J̄
6 = 27,
J̄ ′3J̄
7 = 27, J̄
7 = 27, J̄
4 = 9, J̄
5 = 27, J̄
6 = 27,
J̄ ′24 J̄
7 = 27, J̄
5 = 27, J̄
6 = 27, J̄
7 = 27, J̄
6 = 27,
J̄ ′4J̄
7 = 27, J̄
7 = 27, J̄
5 = 27, J̄
6 = 27, J̄
7 = 27,
J̄5J̄
6 = 27, J̄
7 = 27, J̄5J̄
7 = 27, J̄
6 = 18, J̄
7 = 18,
J̄ ′6J̄
7 = 18, J̄
7 = 9.
(170)
The second Chern class is
· J̄ ′1 = 12, c2
· J̄ ′5 = 18, c2
· J̄ ′2 = 12,
· J̄ ′3 = c2
· J̄ ′6 = 24, c2
· J̄ ′4 = c2
· J̄ ′7 = 30.
(171)
We observe that both the intersection ring and the second Chern class cannot be
brought into (11) and (110) by a linear transformation with integer coefficients, re-
spectively. Hence, the second phase really is topologically distinct.
We denote the Fourier-transformed variables in the B-model prepotential (71) by
q′i, i = 1, . . . , 7. With this notation, we obtain
(q′1, . . . , q
7) = 3 q
1 + 3 q
2 + 3 q
+ 3 q′3q
5 + 3 q
+ 3 q′3q
5 + 3 q
6 + 3 q
12333
12333
+ 3 q′1q
+ 3 q′2q
− 6 q′1q
5 − 6 q
+ 3 q′3q
6 + 3 q
7 +O(q
(172)
The instanton numbers on X
− are the expansion coefficients in
(q′1, . . . , q
7, 1) =
n1,...,n7
(n1,n2,n3,n4,n5,n6,n7)
. (173)
Up to degree 4, they read
1,0,0,0,0,0,0 = 3 n
0,1,0,0,0,0,0 = 3 n
0,0,0,0,1,0,0 = 3 n
2,0,0,0,0,0,0 = −6
0,2,0,0,0,0,0 = −6 n
3,0,0,0,0,0,0 = 27 n
0,3,0,0,0,0,0 = 27 n
4,0,0,0,0,0,0 = −192
0,4,0,0,0,0,0 = −192 n
0,0,1,0,1,0,0 = 3 n
0,0,0,0,1,1,0 = 3 n
0,0,1,1,1,0,0 = 3
0,0,1,0,1,1,0 = 3 n
0,0,0,0,1,1,1 = 3 n
1,0,0,3,0,0,0 = 3 n
0,1,0,0,0,0,3 = 3
1,0,1,1,1,0,0 = −6 n
0,1,0,0,1,1,1 = −6 n
0,0,1,1,1,1,0 = 3 n
0,0,1,0,1,1,1 = 3
(174)
It is easy to check that the symmetry G∗2 acts without fixed points on X
− so that there
are two phases of the quotient X∗, too, with h1,1(X∗) = h1,2(X∗) = 3 and fundamental
group π1(X
∗) = Z3×Z3. They correspond to the two classes of triangulations T+ and
T−. The first phase was studied in detail in Subsection 4.5 and 4.6.
We denote the Calabi-Yau manifold in the second phase by X∗− = X
2. From
the linear equivalence relations eq. (106) and the definition eq. (169) we can compute
the induced group action on H2(X
−,Z) and find

J̄ ′1
J̄ ′2
J̄ ′3
J̄ ′4
J̄ ′5
J̄ ′6
J̄ ′7


1 0 0 0 0 0 0
0 1 0 0 0 0 0
3 0 0 −1 1 0 0
3 0 1 −1 0 0 0
0 0 0 0 1 0 0
0 3 0 0 1 0 −1
0 3 0 0 0 1 −1


J̄ ′1
J̄ ′2
J̄ ′3
J̄ ′4
J̄ ′5
J̄ ′6
J̄ ′7

. (175)
In terms of the three invariant divisors J ′1 = J̄
2 = J̄
3 = J̄
2 the intersection ring
and the second Chern class of X∗− then are
J ′22 J
3 = 1, J
2 = 3, J
3 = 1, J
3 = 3,
J ′21 J
2 = 9, J
3 = 3, J
3 = 9, J
1 = 27,
1 = 18, c2 J
2 = 12, c2 J
3 = 12.
(176)
Again, we observe that there is no linear basis transformation with integer coefficients
that brings both the intersection ring and the second Chern class into (90). Hence,
also the phase X∗− is topologically distinct from X
(n1, n2, n3) n
(n1,n2,n3,0)
(n1,n2,n3,1)
(n1,n2,n3,2)
(n1,n2,n3)
(1, 0, 0) 3 0 0 3
(2, 0, 0) −6 0 0 −6
(3, 0, 0) 18 0 0 18
(0, 1, 0) 3 3 3 9
(1, 1, 0) −6 −6 −6 −18
(1, 1, 1) 12 12 12 36
(2, 1, 0) 15 15 15 45
(1, 2, 0) 12 12 12 36
Table 6: Instanton numbers n
(n1,n2,n3,m2)
computed by toric mirror symmetry.
They are invariant under the exchange n2 ↔ n3, so we only display
them for n2 ≤ n3.
To give a geometrical interpretation of what happens, we look at the induced action
of G∗2 on the toric Mori cone NE(X
−). Only the generators l̄
− , l̄
− and l̄
− in eq. (168)
are invariant. This is exactly as in the first phase. Denoting the invariant generators
± , l̄
± , l̄
± by l
± , l
± , l
± , respectively, we observe that phase T− is obtained from
the phase T+ as a flop by the curve corresponding to the generator l
− = −l
+ , l
− = l
+ + 3l
+ , l
− = l
+ + 3l
+ . (177)
If we use the realization of X in terms of the fiber product of two dP9 surfaces, the
above result means that the base P1 of X has been flopped.
Furthermore, having computed the G∗2-action in eq. (175), we determine the de-
scent equation for the prepotential to be
p′, q′, r′, b′1, b
|G∗2|
p′, q′, b′2, b
′, b′2
2, b′2
2, b′1
. (178)
The corresponding instanton numbers
′, q′, r′, 1, b′2) =
n1,n2,n3,m2
(n1,n2,n3,m2)
p′n1q′n2r′n3b′2
(179)
are listed in Table 6.
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	Introduction
	Calabi-Yau Threefolds
	The Calabi-Yau Threefold X
	The Intermediate Quotient Xbar
	Variables
	Toric Geometry and Mirror Symmetry
	Toric Varieties
	The Batyrev-Borisov Construction
	Toric Intersection Ring
	Mori Cone
	B-Model Prepotential
	Quotienting the B-Model
	The Quotient by G1
	The Quotient by G2
	B-Model on Xbar
	Instanton Numbers of X
	B-Model on the Mirror of Xbar
	Instanton Numbers of the Mirror of X
	Instanton Numbers Assuming The Self-Mirror Property
	The Self-Mirror Property
	Factorization vs. The (3,1,0,0,0) Curve
	Towards a Closed Formula
	Conclusion
	Triangulations
	The Flop of X*
	Bibliography
