Enumerative meaning of mirror maps for toric Calabi-Yau manifolds
We prove that the inverse of a mirror map for a toric Calabi-Yau manifold of the form , where is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono [FOOO10]. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert [GS11a, Conjecture 0.2 and Remark 5.1] as part of their program, and was later formulated in terms of Fukaya-Oh-Ohta-Ono’s invariants in the toric Calabi-Yau case in [CLL12, Conjecture 1.1].
Key words and phrases:Open Gromov-Witten invariants, mirror maps, GKZ systems, toric manifolds, Calabi-Yau, mirror symmetry
2010 Mathematics Subject Classification:Primary 14N35, 53D45; Secondary 14J33, 53D37, 53D12
- 1 Introduction
- 2 Computing open GW invariants via -functions
- 3 Toric mirror map and SYZ map
- 4 GKZ systems and period integrals
- 5 Discussions
Let be an -dimensional toric Calabi-Yau manifold, i.e. a smooth toric variety with trivial canonical line bundle . Such a manifold is necessarily noncompact. Let . Then is defined by a fan in . Let be the primitive generators of the 1-dimensional cones of . Without loss of generality, we assume that, for ,
for some and . Also, following Gross [Gro01], we assume that the fan has convex support so that is a crepant resolution of an affine toric variety with Gorenstein canonical singularities.
The Picard number of is equal to . Let be a nef basis of and let be the dual basis. Each -cycle corresponds to an integral relation
where . We equip with a toric symplectic structure and regard as a Kähler manifold. We also complexify the Kähler class by adding a B-field and setting .
An important class of examples of toric Calabi-Yau manifolds is given by the total spaces of the canonical line bundles over compact toric Fano manifolds , e.g. .
In [CLL12], Leung and the first two authors of this paper study local mirror symmetry for a toric Calabi-Yau manifold from the viewpoint of the SYZ conjecture [SYZ96]. Starting with a special Lagrangian torus fibration (the Gross fibration) on , we construct the SYZ mirror of using -duality modified by instanton corrections and wall-crossing, generalizing the constructions of Auroux [Aur07, Aur09]. The result is given by the following family of noncompact Calabi-Yau manifolds [CLL12, Theorem 4.37] (see also [AAK12, Section 7]):
is a generating function of disk open Gromov-Witten invariants. Here, denotes the monomial if ; is the cone of effective classes; denotes and can be expressed in terms of the complexified Kähler parameters
the coefficients () are related to the complexified Kähler parameters ’s by
are classes of the basic disks bounded by a Lagrangian torus fiber , and the coefficients are 1-pointed genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono [FOOO10] (see Subsection 2.1 for more precise definitions).
Notice that the SYZ mirror family (1.1) is entirely written in terms of symplectic-geometric data of . Another striking feature is that (1.1) is expected to be inherently written in canonical flat coordinates. This was first conjectured by Gross and Siebert [GS11a, Conjecture 0.2 and Remark 5.1] where they predicted that period integrals of the mirror can be interpreted as counting of tropical disks (instead of holomorphic disks) in the base of an SYZ fibration for a compact Calabi-Yau manifold; see also [GS11b, Example 5.2] where Gross and Siebert observed a relation between the so-called slab functions which appeared in their program and period computations for in [GZ02]. In [CLL12], a more precise form of this conjecture for toric Calabi-Yau manifolds, which we recall and clarify below, is stated in terms of the genus 0 open Gromov-Witten invariants .
Let be the convex hull of the finite set . Then is an -dimensional integral polytope, which can also be viewed as the convex hull of in . Notice that
since is smooth. Denote by the space of Laurent polynomials of the form (i.e. those with Newton polytope ). Also let be the projective toric variety defined by the normal fan of .
A Laurent polynomial and hence the associated affine hypersurface in is called -regular [Bat93] if the intersection of the closure with every torus orbit is a smooth subvariety of codimension one in . Denote by the space of all -regular Laurent polynomials. The algebraic torus acts on by
Following Batyrev [Bat93] and Konishi-Minabe [KM10], we define the complex moduli space of the mirror Calabi-Yau manifold to be the GIT quotient of by this action. Since is inside the interior of , the moduli space is nonempty and has (complex) dimension [Bat93]. Also, as the integral relations among the lattice points are generated by where , in fact we can write down the local coordinates on explicitly as
The moduli space parametrizes a family of open Calabi-Yau manifolds defined by
Classically, a mirror map
from the complex moduli space of the mirror to the complexified Kähler moduli
(where denotes the Kähler cone) is defined by period integrals
over integral cycles which constitute part of an integral basis of the middle homology . Here, is the holomorphic volume form
on . A mirror map gives a local isomorphism from to near and , and hence provides canonical flat (local) coordinates on .
Based on our mirror construction, we define a map in the reverse direction:
We define the SYZ map
Then we have the following conjecture:
Conjecture 2 (Conjecture 1.1 in [Cll12]).
There exist integral cycles forming part of an integral basis of the middle homology such that
where is the SYZ map defined in Definition 1 in terms of generating functions of the genus 0 open Gromov-Witten invariants . In other words, the SYZ map coincides with the inverse of a mirror map.
In [CLL12, Conjecture 1.1], it was wrongly asserted that the integral cycles gave an integral basis of . The correct conjecture should be as stated above. We are grateful to the referees for pointing this out.
We expect that Conjecture 2 can be generalized to include the bulk-deformed genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono in [FOOO11]. Namely, we claim that any period integral over an integral cycle can be written in terms of certain generating functions of bulk-deformed genus 0 open Gromov-Witten invariants. Analogous results for was obtained by Gross in his work [Gro10] on tropical geometry and mirror symmetry.
Conjecture 2 not only provides an enumerative meaning to the inverse mirror map, but also explains the integrality of the coefficients of its Taylor series expansions which has been observed earlier (see e.g. Zhou [Zho10]). This also shows that one can write down, using generating functions of disk open Gromov-Witten invariants, Gross-Siebert’s slab functions which satisfy a normalization condition that is necessary to run the Gross-Siebert program (see [GS11a, Remark 5.1] and [GS11b, Example 5.2]). In [CLL12, Section 5.3], evidences of Conjecture 2 were given for the toric Calabi-Yau surface and the toric Calabi-Yau 3-folds , , . In the joint work [LLW12] of the second author with Leung and Wu, Conjecture 2 was verified for all toric Calabi-Yau surfaces.
In this paper, we prove the above conjecture for toric Calabi-Yau manifolds of the form in a weaker sense that the -cycles , , are allowed to have complex coefficients instead of being integral. The precise statement is as follows:
For a toric Calabi-Yau manifold of the form where is a compact toric Fano manifold, there exist linearly independent cycles such that
where is the SYZ map in Definition 1 defined in terms of generating functions of the genus 0 open Gromov-Witten invariants .
Our proof, which mainly relies on the formula proved in [Cha11] and the toric mirror theorem [Giv98, LLY99] for compact semi-Fano toric manifolds, can be outlined as follows. First, by the main result of [Cha11], genus 0 open Gromov-Witten invariants of involved in Conjecture 2 can be equated with certain genus 0 closed Gromov-Witten invariants of the -bundle
over . We observe that the closed Gromov-Witten invariants needed here occur in a certain coefficient of the -function of . Since is semi-Fano (i.e. the anticanonical bundle is numerically effective), the toric mirror theorem of Givental [Giv98] and Lian-Liu-Yau [LLY99] can be applied and it says that is equal to the combinatorially and explicitly defined -function , via a toric mirror map.
Here comes another key observation: the toric mirror map, which is defined as certain coefficients of the -function, can be written entirely in terms of a single function which is precisely the reciprocal of the generating function of genus 0 open Gromov-Witten invariants of . It is known that components of the toric mirror map of satisfy certain GKZ-type differential equations. Combining with the previous observations, it is then easy to deduce that inverse of the SYZ map for gives solutions to the GKZ hypergeometric system associated to . Our main result Theorem 5 then follows by noting that period integrals give a basis of solutions of the GKZ system.
To prove the stronger version, Conjecture 2, we need to show that, after a suitable normalization of , there exist integral cycles such that the period integrals have logarithmic terms of the form . This is closely related to integral structures coming from the central charge formula [Hos06]. We plan to address this problem in the future; see the last subsection for more discussions.
The rest of this paper is organized as follows. In Section 2, we recall the formula equating open and closed Gromov-Witten invariants in [Cha11] and explicitly compute the generating functions for using -functions and the toric mirror theorem. In Section 3, we compute the toric mirror map for in terms of the functions . In Section 4, we first deduce that components of the inverse of the SYZ map for are solutions to the GKZ hypergeometric system associated to . Then we prove our main result Theorem 5 by showing that the period integrals give a basis of solutions of the GKZ systems attached to . We end with some discussions about definitions of mirror maps and ways to enhance Theorem 5 to Conjecture 2 in Section 5.
We are grateful to Mark Gross and Bernd Siebert for sharing their insights and ideas, and to Lev Borisov, Herb Clemens, Shinobu Hosono, Yukiko Konishi and Satoshi Minabe for enlightening discussions and valuable comments on GKZ systems and period integrals. We thank Conan Leung for encouragement and related collaborations. We would also like to express our deep gratitude to the referees for reading our manuscript very carefully and for many useful comments and suggestions which led to a significant improvement in the exposition of this paper.
Part of this work was done when K. C. was working as a project researcher at the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), University of Tokyo and visiting IHÉS. He would like to thank both institutes for hospitality and providing an excellent research environment. S.-C. L. also expresses his deep gratitude to Kavli IPMU for hospitality and providing a very nice research and living environment. The work of K. C. described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK404412). The work of H.-H. T. was supported in part by NSF grant DMS-1047777.
2. Computing open GW invariants via -functions
In this section, we will establish a formula relating the generating functions of disk open Gromov-Witten invariants of and the toric mirror map for the -bundle over . The main result is Theorem 9.
2.1. Open Gromov-Witten invariants of
In this subsection we recall the formula computing open Gromov-Witten invariants of in terms of closed Gromov-Witten invariants, proved in [Cha11]. To begin with, let us briefly recall the definition of the genus 0 open Gromov-Witten invariants following Fukaya-Oh-Ohta-Ono [FOOO10].
Let be a toric manifold of complex dimension , equipped with a toric Kähler structure . Let
be a Lagrangian torus fiber of the moment map associated to the Hamiltonian -action on . Let be a relative homotopy class with Maslov index . Consider the moduli space of holomorphic disks in with boundaries lying in and one boundary marked point representing the class . A compactification of ,
is given by the moduli space of stable maps from genus 0 bordered Riemann surfaces to with one boundary marked point and class .
It is shown in [FOOO09, FOOO10] that is a Kuranishi space of virtual (real) dimension . Let be its virtual fundamental cycle. This is an -cycle instead of a chain because is of minimal Maslox index and consequently . The pushforward of this cycle by the evaluation map at the boundary marked point defines a genus 0 open Gromov-Witten invariant
In [FOOO10], it is shown that the number is independent of the choice of perturbations by multi-sections and hence is indeed an invariant of .
Since the moment map of does not contain singular fibers, there is no wall-crossing; in other words, the invariants remain unchanged when we move the Lagrangian torus fiber .
Let be the primitive generators of the fan defining and let be the associated toric prime divisors (i.e. irreducible torus-invariant hypersurfaces) respectively. Holomorphic disks in with boundaries in give an additive basis
of such that . In [CO06], Cho and Oh proved that, for , there exists a unique (up to automorphisms of the domain) holomorphic disk passing through a generic point in and representing the class . We call such holomorphic disks the basic disks and hence their classes the basic disk classes.
Suppose that is semi-Fano, i.e. the anticanonical divisor is nef. Then it follows from the results of Cho-Oh [CO06] and Fukaya-Oh-Ohta-Ono [FOOO10] that only when for some and some with (see e.g. [Cha11, Lemma 5.1]).
In what follows we will focus on the case when is the total space of the canonical line bundle over a compact toric Fano manifold . Such an is a (noncompact) Calabi-Yau manifold, i.e. . By our convention, the primitive generators of the fan are chosen to be of the form for and so that . Without loss of generality, we also require that form the primitive generators of the fan in defining . The toric prime divisor is then nothing but the zero section .
Since is the only compact toric prime divisor in , the invariant is non-zero only when either for some or for some . By the results of Cho-Oh [CO06] mentioned above, we already know that for any . The other invariants are computed by the following formula:
Theorem 7 (Theorem 1.1 in [Cha11]).
Consider the -bundle over . Let be the fiber class and the Poincaré dual of a point. Denote by the 1-point genus zero closed Gromov-Witten invariant of with insertion the point class . Then we have the equality
between open and closed Gromov-Witten invariants.
2.2. Computation via -function
The -bundle is a toric manifold defined by the fan generated by together with the additional ray spanned by . We have and . Choose a nef basis (recall that ) of so that gives a positive basis of with dual basis and such that gives the dual basis in . As usual, we denote by the toric prime divisors associated to the generators respectively.
By definition, the small J-function of is given by [Giv98]
where , (here ’s are regarded as formal variables), is a homogeneous additive basis, and is its dual basis (with respect to the Poincaré pairing). Expanding (2.2) into a power series in yields
where in the second equality we use the string equation.
We observe that the closed Gromov-Witten invariants (2.1) we need occur in the coefficient of the -term of that takes values in . Indeed, since (2.1) has no descendant insertions, we look at the terms in the above expansion with . Furthermore, to get (2.1) we need and thus .
In order to extract (2.1) from , we use the explicit formula for given by the toric mirror theorem. Recall that the -function of is given by
where , (again we regard the ’s as formal variables) and we identify with its cohomology class in . Here, the product should be expanded into a -series by writing . Note that both and are -valued formal functions. (In fact, by [Iri09, Lemma 4.2], the -function and hence, via the mirror theorem stated below, the -function are convergent power series near and respectively.)
The -function has the asymptotics
where is a (multi-valued) function with values in . We define the toric mirror map for to be the map . The toric mirror theorem applied to the semi-Fano toric manifold then says the following
The -function and the -function coincides via the toric mirror map , i.e.
In view of this, in order to extract the invariants (2.1), we should look for the coefficient of the -term of that takes values in .
Consider the expansion of the factor
into a -series, achieved by writing . Since we want terms with values in , we cannot have involved. There are three possibilities:
: the only term in the expansion of (2.4) that does not contain is the leading term
: the quotient (2.4) is just 1 in this case.
: in this case the quotient (2.4) is proportional to , because of the factor corresponding to .
Thus for the terms we need, only cases (1) and (2) can occur.
Note that . Therefore, in the sum
the part of the -term that takes values in is
where the sum is over all such that and for all . Note that the term in with and the factor do not contribute. So the coefficient of the -term of that takes values in is given exactly by (2.5).
For such that and for all , there are two possibilities:
either for some and for all , or
and for all .
Case (i) is impossible as is compact, so we are left with case (ii). Let be as in case (ii). Note that , and and for any . Hence, must be of the form for some . Then we have since . So must be corresponding to an integral relation of the form . The only such class is the fiber class , and therefore we conclude that the sum (2.5) is simply given by .
For the -bundle over a compact toric Fano manifold , we have the following formula:
where is the inverse of the toric mirror map for and
is the generating function for open Gromov-Witten invariants of ; here .
By comparing the coefficients of the -terms that takes values in on both sides, we have
By dimension reasons, the invariant is nonzero only when ; and we know that for , if and only if for some . Hence, we have
where is a monomial in the variables . Note that . The proposition now follows from the formula in Theorem 7. ∎
3. Toric mirror map and SYZ map
In this section, we show that the inverse of the toric mirror map for the -bundle contains the SYZ map (Definition 1) for the toric Calabi-Yau manifold . The key observation is that both maps are completely determined by one and the same function .
Recall that the toric mirror map for is given by the coefficient of the -term of the -function that takes values in . By analyzing the quotient (2.4) as we did in the previous subsection, it is not hard to see that the term (depending on )
in will contribute to only when there exists at most one such that . If for all , then (3.1) is of the form
which does not contribute to the toric mirror map at all since cannot be equal to 1. On the other hand, if there exists such that and for all , then (3.1) is of the form
which contributes to the toric mirror map whenever .
This shows that the -function expands as
Writing (), the toric mirror map
can then be expressed as
For the -bundle over a toric Fano manifold , the function is nonzero only when . This can be seen by applying [GI12, Proposition 4.3] which says that the function is nonzero if and only if is not a vertex of the fan polytope (recall that the fan polytope is the convex hull of the primitive generators of the fan ). Moreover, depends only on the variables since for every with . Therefore we have
The toric mirror map for the -bundle over a compact toric Fano manifold is given by
where is a function of the variables .