GS slab and open GW

Gross-Siebert’s slab functions and open GW invariants for toric Calabi-Yau manifolds

Abstract.

This paper derives an equality between the slab functions in Gross-Siebert program and generating functions of open Gromov-Witten invariants for toric Calabi-Yau manifolds, and thereby confirms a conjecture of Gross-Siebert on symplectic enumerative meaning of slab functions. The proof is based on the open mirror theorem of Chan-Cho-Lau-Tseng [CCLT13]. It shows an instance of correspondence between tropical and symplectic geometry in the open sector.

1. Introduction

The celebrated Gross-Siebert program [GS11a, GHK11] reconstructs mirrors of algebraic varieties by using toric degenerations and tropical geometry. It can be regarded as an algebraic version of the SYZ program [SYZ96], which gives a brilliant way of handling quantum corrections coming from singular strata. In [GS14], Gross and Siebert applied their construction to toric Calabi-Yau manifolds and construct their mirrors. They also discussed enumerative meaning of their slab functions in terms of counting tropical discs.

On the other hand, SYZ construction for toric Calabi-Yau manifolds was carried out by a joint work of the author with Chan and Leung [CLL12] using open Gromov-Witten invariants in symplectic geometry. Moreover, an equality with the Hori-Iqbel-Vafa mirror was proved in [CLT13] for the total space of the canonical line bundle of a toric Fano manifold, and in [CCLT13] for general toric Calabi-Yau orbifolds. As a result mirror maps can be expressed in terms of open Gromov-Witten invariants, which was called the open mirror theorem.

It is natural to ask whether these two different approaches produce the same mirror for toric Calabi-Yau manifolds. The aim of this paper is to give an affirmative answer to this question. In particular, we establish an equality between Gross-Siebert normalized slab functions for toric Calabi-Yau manifolds and the wall-crossing generating function for open Gromov-Witten invariants. This confirms the conjecture by Gross-Siebert [GS11a, Conjecture 0.2] that their slab functions have an enumerative meaning in terms of counting holomorphic discs bounded by a fiber of a Lagrangian torus fibration.

Theorem 1.1.

For a toric Calabi-Yau manifold, let be the Gross-Siebert slab function for a slab and a primitive generator of a ray in the fan. Let denotes the open Gromov-Witten invariants associated to a disc class bounded by a moment-map fiber . We have the equality

where are some explicit curve classes (see Equation (2.1)), is the mirror map, and is an explicit hypergeometric series (see Definition 5.2). The monomials for a vector is explained in Section 2.

The second equality in the above theorem is the main result of the joint work [CCLT13] of the author with Chan, Cho and Tseng. In this paper we will derive the first equality based on the work of [CCLT13] and the combinatorial meaning of hypergeometric series, see the proof of Proposition 6.3 and Section 1.1. As a consequence, this implies that the mirror constructed using tropical geometry by Gross-Siebert equals to that constructed using symplectic geometry by Chan-Lau-Leung. This gives a correspondence between tropical and symplectic geometry in the open sector (while the correspondence in the closed sector has been well-studied, see for instance [Mik05, Gro11]). The geometry underlying such a correspondence is very rich and deserves a further study.

As an immediate consequence,

Corollary 1.2.

For toric Calabi-Yau manifolds, the mirror constructed using tropical geometry by Gross-Siebert equals to the mirror constructed using symplectic geometry by Chan-Lau-Leung.

Remark 1.3.

In [GS14], Gross-Siebert counted tropical discs and trees and explained a conjectural relation with their slab functions. This paper employs symplectic geometry instead of tropical geometry and expressse slab functions in terms of counting holomorphic discs. The statement and proof is independent of their tropical interpretation.

Acknowledgement

I am grateful to Nai-Chung Conan Leung for encouragement and useful advice during preparation of this paper. I express my gratitude to Mark Gross and Bernd Siebert for useful explanations of their program, and to my collaborators Kwokwai Chan, Cheol-Hyun Cho, Hansol Hong, Sang-Hyun Kim, Hsian-Hua Tseng and Baosen Wu for their continuous support. This project is supported by Harvard University.

2. Setup and notations for toric Calabi-Yau manifolds

In this section we briefly recall the essential matarials for toric Calabi-Yau manifolds used in this paper.

Let be a lattice of rank and be the dual lattice. Fix a primitive vector . Let be a (closed) lattice polytope of dimension contained in the affine hyperplane . By choosing a lattice point in the lattice polytope , we have a lattice polytope in the hyperplane . Let be a triangulation of such that each maximal cell is a standard simplex. Then by taking a cone over this triangulation, we obtain a fan supported in . Then is a toric Calabi-Yau manifold, which means that the anti-canonical divisor is linearly equivalent to zero.

Let denote the number of lattice points lying in the polytope , and let be generators of rays of the fan (which are one-to-one corresponding to the lattice points in ). By reordering ’s if necessary, we assume belongs to a cone of the fan. In particular it forms a basis of . Then we can write for , and if .

Denote by the toric prime divisor corresponding to . Each toric prime divisor corresponds to a basic disc class bounded by a Lagrangian torus fiber (see [CO06] for detailed discussions on basic disc classes). We have and the exact sequence

Since is Calabi-Yau, we have for any . For a disc class , its Maslov index is (see [CO06]). In particular the basic disc classes have Maslov index two.

Define the curve classes

(2.1)

for . Then forms a basis of . The dual basis of is . The corresponding Kähler parameters and mirror complex parameters are denoted as and respectively. We set , and so for .

Remark 2.1.

In the language of Gross-Siebert program, the choice of the basis corresponds to the choice of a set of piecewise linear functions supported on . corresponds to the piecewise linear function which takes value on , on for , and zero on for .

Now we define the mirror complex variables which are used to define the mirror of the toric Calabi-Yau manifold . Denote by for the monomial in variables . Then for . This gives a correspondence between lattice points in and monomials in variables. On the other hand, we need a correspondence between lattice points in and monomials in variables.

Fix and set for all . Then (since by for all ). Fix a basis of . Then for some , and we define for . Since can be expressed as an integer combination of ’s, can be regarded as monomials in the coordinates ’s. for will be used as complex coordinates for the mirror of .

In the next two sections we review the construction of mirror varieties by tropical geometry and symplectic geometry respectively.

3. Gross-Siebert slab functions

In [GS11a] Gross-Siebert developed a construction of mirror varieties by extracting tropical geometric data from a toric degeneration of a smooth algebraic variety. In a recent paper [GS14] they applied their construction to toric Calabi-Yau manifolds. They constructed a toric degeneration of a toric Calabi-Yau manifold, and use it to construct its mirror Calabi-Yau. The survey paper [GS11b] provides an excellent exposition of their program.

In the following we will just sketch their construction in a very brief way, and focus on the slab functions and the resulting mirror varieties which are the main subjects of this paper. Some of the notations here are different from that in [GS14] for convenience of comparing with the SYZ mirror constructed from symplectic geometry discussed in the next section.

The Gross-Siebert program starts with a toric degeneration of an algebraic variety , which roughly speaking is a family of varieties where for and is a union of toric varieties (of the same dimension) glued along toric strata. From the intersection complex of the central fiber , an affine manifold with singularities is constructed, together with some initial ‘walls’ which are certain codimension-one strata attached with ‘slab functions’. From this initial data, a structure of walls on can be constructed order-by-order by working on scattering diagrams. Intuitively by gluing all the connected components of the complement of the walls in by using the slab functions, one obtain a family of varieties which is defined as the mirror.

In the case of a toric Calabi-Yau manifold, which can be regarded as a local piece of a compact Calabi-Yau variety, the construction simplifies drastically. can be identified as topologically, and there is exactly one wall which can be identified as . The discriminant locus is contained in this wall, which can be topologically identified as the dual of the triangulation on the polytope . consists of several connected components, which have one-to-one correspondence with the lattice points in .

Slabs and slab functions play a key role in constructing the mirror family. In this case the slabs are identified as the maximal cells in the triangulation of . Slab functions are elements in attached to slabs; in this case we have a slab function attached to each pair , where is a slab and is a vertex of . The slab functions are determined by the following properties.

Definition 3.1 (Slab functions).

Slab functions are elements in the collection , where , determined by the following properties:

  1. The constant term of each (as a series in and ) is .

  2. If and are adjacent vertices of , then

  3. has no term of the form where and .

  4. If , then . Hence actually does not depend on .

(3) is the most essential condition. It is called to be the normalization condition. Roughly speaking it means is counting tropical discs, and hence terms corresponding to tropical curves cannot appear. From these four conditions one can compute the slab functions order-by-order.

Having the slab functions , the outcome of the construction of Gross-Siebert program in this case is the following:

Theorem 3.2 ([Gs14]).

The Gross-Siebert mirror of a toric Calabi-Yau manifold is

(3.1)

for a chosen . The superscript ‘tg’ stands for tropical geometry.

By Property (2) and (4) of Definition 3.1, the varieties for different ’s are isomorphic to each other, and so we do not need to worry about the choice of .

4. Open GW invariants and SYZ mirrors of toric Calabi-Yau manifolds

In [CLL12], an SYZ construction of the mirror of toric Calabi-Yau manifolds was developed based on symplectic geometry. This section reviews the construction briefly and states the end result.

The construction starts with a Lagrangian torus fibration of a toric Calabi-Yau manifold, which was constructed by Goldstein [Gol01] and Gross [Gro01] independently. The base of the fibration can be identified topologically as the upper half space for a constant . The codimension-two discriminant locus , like in the construction in the last section, is contained in the hyperplane and can be topologically identified as the dual of the triangulation on the polytope . The connected components of have one-to-one correspondence with the lattice points in .

We fix a choice of to define the following contractible open subset of :

The torus fibration trivializes over , and hence can be identified with . Then for any fiber , can be identified with .

One essential ingredient of the construction is open Gromov-Witten invariant of a Lagrangian fiber, which is roughly speaking counting holomorphic discs bounded by a Lagrangian fiber. It turns out that open Gromov-Witten invariants of fibers at points above and below the wall differ drastically (and the invariants are not well-defined for fibers at the wall), and this is known as wall-crossing phenomenon which was studied by [Aur07] in this context. The invariants for fibers below the wall are only non-trivial for exactly one disc class, while the invariants for fibers above the wall can be identified with that of moment-map fibers.

Definition 4.1 (Open Gromov-Witten invariants).

Let be a toric Calabi-Yau manifold, a regular moment-map fiber, and a disc class bounded by the fiber . Let be the moduli space of stable discs with one boundary marked point representing . The open Gromov-Witten invariant associated to is

where is the evaluation map at the boundary marked point.

By dimension counting, the open Gromov-Witten invariants are non-zero only when has Maslov index two. Moreover stable disc classes of a moment-map fiber must take the form , where is a basic disc class and is an effective curve class (see [FOOO10]).

Having the open Gromov-Witten invariants , the outcome of the construction in [CLL12] is the following:

Theorem 4.2.

The SYZ mirror of a toric Calabi-Yau manifold is

(4.1)

for chosen in the construction, where

(4.2)

The superscript ‘sg’ stands for symplectic geometry.

given in Equation (4.2) is called to be the wall-crossing generating function of open Gromov-Witten invariants because it describes the change in generating function of open invariants when crossing the wall .

5. Open mirror theorem

In [CCLT13] an equality between open Gromov-Witten invariants and mirror map was derived, which is a main ingredient in our proof of Theorem 1.1. In this section we recall the result, which was called the open mirror theorem.

The main object of study of the open mirror theorem is the generating function of open Gromov-Witten invariants (see Definition 4.1).

Definition 5.1 (Generating function of open Gromov-Witten invariants).

Let be a toric Calabi-Yau manifold. The generating function of open Gromov-Witten invariants associated to a toric divisor is defined to be which takes the form . The summation is over all curve classes .

Mirror map plays a central role in mirror symmetry. It gives a canonical local isomorphism between the Kähler moduli and the mirror complex moduli. In our context, the generating functions defined above are expressed in terms of the mirror map.

Definition 5.2 (Mirror map).

For , define the hypergeometric functions

(5.1)

where the summation is over all effective curve classes satisfying

Then the mirror map is defined as

Now we are prepared to state the open mirror theorem:

Theorem 5.3 (Open mirror theorem [Cclt13]).

For a toric Calabi-Yau manifold,

where is the inverse of the mirror map in Definition 5.2.

The proof involves identification of the generating functions of open invariants with certain generating functions of closed invariants of a toric compactification of , which depends on the basic disc class . Then the closed invariants can be expressed in terms of hypergeometric functions and mirror maps by using the closed toric mirror theorem [Giv98, LLY97, LLY99a, LLY99b]. Note that the compactification may contain orbifold strata, and so a natural class to study for this purpose is toric Calabi-Yau orbifolds. Nevertheless, the theorem itself can be stated within the manifold setting.

Remark 5.4.

Similar result holds for toric semi-Fano manifolds [CLLT12], in which the proof involves Seidel representation and degeneration techniques. Open mirror theorem is useful for studying global mirror symmetry, as illustrated in [CCLT14, Lau13].

6. Proof of Theorem 1.1

We are now ready to prove the main Theorem 1.1. The strategy is the following. First wall-crossing generating function of open Gromov-Witten invariants can be expressed in terms of hypergeometric functions and mirror maps by using Theorem 5.3 (the main result of [CCLT13]). Then we prove that it satisfies the defining properties of slab functions by observing a combinatorial meaning of hypergeometric functions (as counting tropical curves, roughly speaking), and hence the equality between (see Equation (4.2)) and (see Definition 3.1) follows. This implies Gross-Siebert mirror constructed from tropical geometry equals to the SYZ mirror constructed from symplectic geometry for toric Calabi-Yau manifolds.

First we do a change of coordinates in , such that in the new coordinates has an explicit closed expression in terms of the mirror complex parameters (rather than ):

Lemma 6.1.

Let be the wall-crossing generating function defined by Equation (4.2). There exists a change of coordinates such that

Proof.

Take the change of coordinates for . Then for . In particular this gives a change of coordinates for , .

Then for ,

Hence

where the second equality follows from the definition of the mirror map

and the last equality follows from Theorem 5.3 that . ∎

Now we classify the curve classes appeared in the hypergeometric functions (see Equation (5.1)).

Lemma 6.2.

For , and for all if and only if

for some , satisfying . Here denotes the basic disc class corresponding to the toric divisor of .

Proof.

If and for all , then for for and . Moreover since , . Then , and so . Hence , and . Now for any . Thus and

Conversely, if for some , satisfying , then . Thus and . ∎

Using the above two lemmas, we can then prove that satisfies the normalization condition of slab functions.

Proposition 6.3.

has no term of the form where and .

Proof.

By Lemma 6.1, can be written as up to a change of coordinates in . Since a change of coordinates in does not affect whether has term of the form or not, it suffices to prove that of this expression has no term of the form .

We will prove that the part which has no dependence of

is , and hence

has no term of the form . Taking the inverse mirror map (which does not affect whether the expression has terms with no -dependence or not), we obtain the required statement.

A term with no -dependence in takes the form where . By Lemma 6.2, it corresponds to

with the property that and for all . Moreover .

Here comes the key: appears times in the expansion of . Hence the part with no -dependence of

is

Remark 6.4.

Note that the part with no -dependence of has a natural tropical meaning: it is a counting of tropical curves formed by unions of tropical discs corresponding to some vectors , and the requirement of no -dependence is the balancing condition on ’s. This gives a tropical meaning of the hypergeometric functions ’s. In order to make this intuitive interpretation rigorous, more foundational theory on counting tropical discs (and its relation with counting tropical curves) is needed. Nevertheless our proof does not rely on this tropical interpretation.

The remaining three conditions of being slab functions simply follow from the definition of , and hence is a slab function.

Proposition 6.5.

If we set for all and , then satisfies the defining properties of slab functions in Definition 3.1.

Proof.

For , the constant term is obtained from the summand with and , which is . This proves Condition (1).

From the expression of we see that Thus Condition (2) follows.

Since we set , does not depend on which is Condition (4).

Condition (3) is proved in Proposition 6.3. ∎

This finishes the proof of Theorem 1.1. From Equation (3.1) and (4.1), it follows immediately that the mirror constructed using tropical geometry by Gross-Siebert equals to the mirror constructed using symplectic geometry by Chan-Lau-Leung, which is Corollary 1.2.

7. Examples

Example 7.1.

. Let be the closed interval , and take the triangulation given by cutting the interval into two pieces and . The fan is obtained by coning over the triangulation of , and the generators of rays of the fan are and . All effective curve classes are multiples of the exceptional curve class , which corresponds to the Kähler parameter and mirror complex parameter . See Figure 1.

Figure 1. The fan (in the middle) and moment-map polytope (on the right) of . Non-trivial open Gromov-Witten invariants come from disc classes of the form shown on the right.

In the following we will consider the wall-crossing generating function for , which is

In this case it turns out that (which follows from computing the inverse mirror map).

By open mirror theorem and the mirror map , we have

where

and is the mirror map. We show that it satisfies the normalization condition (3).

where . The summands of the second term have constant part (which has no dependence on ) only when for some . Thus the constant part of the above expression is

Hence the wall-crossing generating function for open Gromov-Witten invariants satisfies Gross-Siebert normalization condition.

Example 7.2.

. Let be the lattice polytope in whose vertices are and , and take the triangulation given by dividing into three pieces using the interior lattice point . The fan is obtained by coning over the triangulation of , and the generators of rays of the fan are , . All effective curve classes are multiples of the exceptional curve class (which is the line class in ), which corresponds to the Kähler parameter and mirror complex parameter . See Figure 2.

Figure 2. The fan (in the middle) and moment-map polytope (on the right) of . Non-trivial open Gromov-Witten invariants come from disc classes of the form shown on the right.

In the following we will consider the wall-crossing generating function for , which is

By open mirror theorem and the mirror map , we have

where

and is the mirror map. In this case is an infinite series

We show that satisfies the normalization condition (3) in Definition 3.1.

where and . The summands of the second term have constant part (which has no dependence on ) only when for some . Thus the constant part of the above expression is

Hence the wall-crossing generating function for open Gromov-Witten invariants satisfies Gross-Siebert normalization condition.

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