A note on disk counting in toric orbifolds

We compute orbi-disk invariants of compact Gorenstein semi-Fano toric orbifolds by extending the method used for toric Calabi-Yau orbifolds. As a consequence the orbi-disc potential is analytic over complex numbers.


INTRODUCTION
The mirror map plays a central role in the study of mirror symmetry. It provides a canonical local isomorphism between the Kähler moduli and the complex moduli of the mirror near a large complex structure limit. Such an isomorphism is crucial to counting of rational curves using mirror symmetry.
In [5] and [6], we derived an enumerative meaning of the inverse mirror maps for toric Calabi-Yau orbifolds and compact semi-Fano toric manifolds in terms of genus 0 open (orbifold) Gromov-Witten invariants (or (orbi-)disk invariants). Namely, we showed that coefficients of the inverse mirror map are equal to generating functions of virtual counts of stable (orbi-)disks bounded by a regular Lagrangian moment map fiber. In particular it gives a way to effectively compute all such invariants.
In this short note we extend our method in [5] to derive an explicit formula for the orbi-disk invariants in the case of compact Gorenstein semi-Fano toric orbifolds; see Theorem 12 for the explicit formulas. This proves [4,Conjecture] for such orbifolds, generalizing [6, Theorem 1.2]: Theorem 1 (Open Mirror Theorem). For a compact Gorenstein semi-Fano toric orbifold, the orbidisk potential is equal to the (exteneded) Hori-Vafa superpotential via the mirror map. Choose b m , . . . , b m −1 ∈ N so that they are contained in the support of the fan Σ and they generate N over Z. An extended stacky fan in the sense of [24] is the data φ is surjective and yields an exact sequence of groups called the fan sequence: Clearly L Z m −n . Tensoring (1.2) with C × yields the following sequence: which is exact. Note that G is an algebraic torus.
By definition, the set of anti-cones is This terminology is justified because for I ∈ A, the complement of I in {0, 1, ..., m − 1} indexes generators of a cone in Σ. For I ∈ A, the collection {Z i | i ∈ I} generates an ideal in C[Z 0 , . . . , Z m −1 ], which in turn determines a subvariety C I ⊂ C m . Set 3) defines a G-action on C m and hence a G-action on U A . This action is effective and has finite stabilizer groups, because N is torsion-free. The toric orbifold associated to (Σ, {b i } m−1 i=0 ∪ {b j } m −1 j=m ) is defined to be the following quotient stack: The standard (C × ) m -action on U A induces a T-action on X Σ via (1.3).
The coarse moduli space of the toric orbifold X Σ is the toric variety X Σ associated to the fan Σ. In this paper we assume that X Σ is semi-projective, i.e. X Σ admits a T-fixed point, and the natural map X Σ → Spec H 0 (X Σ , O X Σ ) is projective. This assumption is required for the toric mirror theorem of [12] to hold. More detailed discussions on semi-projective toric varieties can be found in [13, Section 7.2].
One can check that there is a bijection between Box bσ and the finite group where Σ (n) is the set of n-dimensional cones in Σ.
Following the description of the inertia orbifold of X Σ in [3], for ν ∈ Box(Σ), we denote by X ν the corresponding component of the inertia orbifold of X := X Σ . Note that X 0 = X Σ as orbifolds. Elements ν ∈ Box (Σ) corresponds to twisted sectors of X , namely non-trivial connected components of the inertia orbifold of X .
Following [8], the direct sum of singular cohomology groups of components of the inertia orbifold of X , subject to a dgree shift, is called the Chen-Ruan orbifold cohomology H * CR (X ; Q) of X . More precisely, where age(ν) is called 1 degree shifting number in [8] of the twisted sector X ν . In case of toric orbifolds, age has a combinatorial description [3]: Using T-actions on twisted sectors induced from that on X , we can define T-equivariant Chen-Ruan orbifold cohomology H * CR,T (X ; Q) by replacing singular cohomology with T-equivariant co- By general properties of equivariant cohomology, H * CR,T (X ; Q) is a module over H * T (pt, Q) and admits a map H * CR,T (X ; Q) → H * CR (X ; Q) called non-equivariant limit. 1 Following Miles Reid, it is now more commonly called age.
which is called the divisor sequence. Line bundles on X = [U A /G] correspond to G-equivariant line bundles on U A . In view of (1.3), T-equivariant line bundles on X correspond to (C × ) m -equivariant line bundles on U A . Because the codimension of ∪ I / ∈A C I ⊂ C m is at least 2, the Picard groups satisfy: The natural map P ic T (X ) → P ic(X ) is identified with the map ψ ∨ : M → L ∨ appearing in the divisor sequence.

The elements {e
Together with the natural quotient maps, they fit into a commutative diagram.
As explained in [23, Section 3.1.2], there is a canonical splitting of the quotient map L ∨ ⊗ Q → H 2 (X ; Q). For m ≤ j ≤ m − 1, let I j ∈ A be the anticone of the cone containing b j . This allows us to write b j = i / ∈I j c ji b i for c ji ∈ Q ≥0 . Tensoring the fan sequence (1.2) with Q, we may find a unique D ∨ j ∈ L ⊗ Q such that values of the natural pairing −, − between L ∨ and L satisfy Using D ∨ j we get a decomposition is can be identified with H 2 (X ; Q) via the map L ∨ ⊗ Q → H 2 (X ; Q). 2 The map ψ ∨ : M → L ∨ is surjective since N is torsion-free.
Define extended Kähler cone of X to be We choose an integral basis such that p a is in the closure of C X for all a and p r +1 , . . . , p r ∈ m −1 i=m R ≥0 D i . We get a nef basis Their equivariant limitsD T i can be expressed as For i = m, . . . , m − 1, we haveD i = 0 in H 2 (X ; R) andD T i = 0. Localization gives the following description of H ≤2 CR,T : be the basis dual to {p 1 , . . . , p r } ⊂ L ∨ . H eff 2 (X ; Q) admits a basis {γ 1 , . . . , γ r }, and we have Elements of K eff should be interpreted as effective curve classes. Elements of K eff ∩ H 2 (X ; R) should be viewed as classes of stable maps P(1, m) → X for some m ∈ Z ≥0 . See e.g. [23, Section 3.1] for more details.

1.2.
Genus 0 open orbifold GW invariants according to [10]. Let (X , ω) be a toric Kähler orbifold of complex dimension n, equipped with the standard toric complex structure J 0 and a toric Kähler structure ω. Denote by (Σ, b) the stacky fan that defines Let L ⊂ X be a Lagrangian torus fiber of the moment map µ 0 : X → M R := M ⊗ Z R, and let β ∈ π 2 (X , L) = H 2 (X , L; Z) be a relative homotopy class.
1.2.1. Holomorphic orbi-disks and their moduli spaces. A holomorphic orbi-disk in X with boundary in L is a continuous map w : (D, ∂D) → (X , L) satisfying the following conditions: is an orbi-disk with interior marked points z + 1 , . . . , z + l . More precisely D is analytically the disk D 2 ⊂ C so that for j = 1, . . . , l, the orbifold structure at z + j is given by a disk neighborhood of z + j uniformized by the branched covering map br : z → z m j for some m j ∈ Z >0 . (If m j = 1, z + j is not an orbifold point.) (2) For any z 0 ∈ D, there is a disk neighborhood of z 0 with a branched covering map br : z → z m , and there is a local chart (V w(z 0 ) , G w(z 0 ) , π w(z 0 ) ) of X at w(z 0 ) and a local holomorphic lifting w z 0 of w satisfying w • br = π w(z 0 ) • w z 0 . (3) The map w is good (in the sense of Chen-Ruan [7]) and representable. In particular, for each z + j , the associated group homomorphism (1.8) between local groups which makes w z + j equivariant, is injective. 4 For a real number λ ∈ R, let λ , λ and {λ} denote the ceiling, floor and fractional part of λ respectively.
The type of a map w as above is defined to be x := (X ν 1 , . . . , X ν l ). Here ν j ∈ Box(Σ) is the image of the generator 1 ∈ Z m j under h j .
There are two notions of Maslov index for an orbi-disk. The desingularized Maslov index µ de is defined by desingularizing the interior singularities (following Chen-Ruan [7]) of the pull-back bundle w * T X in [10, Section 3]. The Chern-Weil (CW) Maslov index is defined as the integral of the curvature of a unitary connection on w * T X which preserves the Lagrangian boundary condition, see [11]. The following lemma, which appeared as [5, Lemma 3.1], computes the CW Maslov indices of disks. Let Ω be a non-zero meromorphic n-form on X which has at worst simple poles. Let D ⊂ X be the pole divisor of Ω. Suppose also that the generic points of D are smooth. Then for a special Lagrangian submanifold L ⊂ X \ D, the CW Maslov index of a class β ∈ π 2 (X , L) is given by Here, β ·D is defined by writing β as a fractional linear combination of homotopy classes of smooth disks.
The classification of orbi-disks in a symplectic toric orbifold has been worked out in [10, Theorem 6.2]. In the classification, the basic disks corresponding to the stacky vectors (and twisted sectors) play a basic role. Denote the homotopy classes of these disks by β 0 , · · · , β m−1 . (2) The holomorphic orbi-disks with one interior orbifold marked point and desingularized Maslov index 0 (modulo T n -action and reparametrizations of the domain) are in bijective correspondence with the twisted sectors ν ∈ Box (Σ) of the toric orbifold X . Denote the homotopy classes of these orbi-disks by β ν .
As in [10], these generators of π 2 (X , L) are called basic disk classes. They are the analogue of Maslov index 2 disk classes in toric manifolds.

1.2.2.
Orbi-disk invariants. Pick twisted sectors X ν 1 , . . . , X ν l of the toric orbifold X . Consider the moduli space of good representable stable maps from bordered orbifold Riemann surfaces of genus zero with one boundary marked point and l interior orbifold marked points of type x = (X ν 1 , . . . , X ν l ) representing the class β ∈ π 2 (X , L). According to [10], M op,main 1,l (X , L, β, x) can be equipped with a virtual fundamental chain, which has an expected dimension n if the following equality holds: We assume that the toric orbifold X is semi-Fano (see Definition 3) and Gorenstein 5 . Then the age of every twisted sector of X is a non-negative integer. Since a basic orbi-disk class β ν has Maslov index 2age(ν), we see that every non-constant stable disk class has at least Maslov index 2.
We further assume 6 that the type x consists of twisted sectors with age ≤ 1. Then the virtual fundamental chain [M op,main 1,l (X , L, β, x)] vir has expected dimension n when µ CW (β) = 2, and in fact we get a virtual fundamental cycle because β attains the minimal Maslov index, thus preventing disk bubbling to occur. Therefore the following definition of genus 0 open orbifold GW invariants (also known as orbi-disk invariants) is independent of the choice of perturbations of the Kuranishi structures 7 : Definition 7 (Orbi-disk invariants). Let β ∈ π 2 (X , L) be a relative homotopy class with Maslov index given by (1.11). Suppose that the moduli space M op,main is the fundamental class of the twisted sector X ν j . 5 This means that K X is Cartier. 6 This assumption does not impose any restriction in the construction of the SYZ mirror over H ≤2 CR (X ). We do not discuss mirror construction in this paper. 7 In the general case one may restrict to torus-equivariant perturbations, as did in [16,17,15] 2. GEOMETRIC CONSTRUCTIONS Let β ∈ π 2 (X , L) be a disk class with µ CW (β) = 2. By the discussion in Section 1.2, we can write β = β d + α with α ∈ H 2 (X , Z), c 1 (X ) · α = 0 and either β d ∈ {β 0 , ..., β m−1 } or β d ∈ Box (X ) is of age 1. Denote by b d ∈ N the element corresponding to β d .
Recall that the fan polytope P ⊂ N R is the convex hull of the vectors b 0 , ..., b m−1 . Note that b d ∈ P. Denote by F (b d ) the minimal face of the fan polytope P that contains the vector b d . Let F be a top-dimensional face of P that contains F (b d ). Let Σ β d ⊂ Σ be the minimal convex subfan containing all {b 0 , ..., b m−1 } ∩ F . The vectors determine a fan map. Let X β d ⊂ X be the associated toric suborbifold (of the same dimension). Since X is Gorenstein, we have Lemma 8. X β d is a toric Calabi-Yau orbifold.
Note that X β d depends on the choice of the face F , not just β d . We use X β d to compute open Gromov-Witten invariants of X in class β = β d + α.
In what follows we show that X β d ⊂ X contains all stable orbi-disks of X of class β. First, we have the following analogue of [6, Proposition 5.6].
Lemma 9. Let f : D ∪ C → X be a stable orbi-disk map in the class β = β d + α, where D is a (possibly orbifold) disk and C is a (possibly orbifold) rational curve such that f * [D] = β d and f * [C] = α with c 1 (α) = 0. Then we have Proof. Since c 1 (α) = 0, C should lie in toric divisors of X . Suppose β d is a smooth disk class. Then the sphere component C 0 meeting the disk component D maps into the divisor D d and it should have non-negative intersection with other toric divisors. By [21,Lemma 4.5] which easily extends to the simplicial setting, we have desired statement for f (C 0 ).
If β d is an orbi-disk class, then we can write the corresponding b Let C 1 ⊂ C be a sphere component meeting C 0 , then we have f (C 1 ) ⊂ F (b j ) for some b j ∈ F (b d ) by the intersection condition. Now, we can follow the proof of [6,Proposition 5.6] shows that Lemma 10. Let f : D ∪ C → X be as in Lemma 9. Then we have f (D ∪ C) ⊂ X β d .
Proof. Certainly f (D) ⊂ X β d . We claim that from which the lemma follows.
To see ( Since −K X is nef and −K X · α = 0, we have −K X · A = 0 = −K X · B. Write B = k c k B k as an effective linear combination of the classes B k of irreducible 1-dimensional torus-invariant orbits in X . Again because −K X is nef, we have −K X · B k = 0 for all k. Each B k corresponds to an (n − 1)-dimensional cone σ k ∈ Σ. In the expression B = k c k B k , there is at least one (non-zero) Since X β d is a toric Calabi-Yau orbifold, the open Gromov-Witten invariants n X β d 1,l,β ([pt] L ; 1 ν 1 , . . . , 1 ν l ) have been computed in [5]. By Proposition 11, this gives open Gromov-Witten invariants of X . Explicitly they are given as follows.
log q a = log y a +

4)
Theorem 12. If β i 0 is a basic smooth disk class corresponding to the ray generated by b i 0 for some i 0 ∈ {0, 1, . . . , m − 1}, then we have via the inverse y = y(q, τ ) of the toric mirror map (2.4).
Theorem 12 provides a formula for the open GW invariants n X 1,l,β 112 ([pt] L ; l i=1 1 ν i ) where ν i is either ν 112 or ν 122 for each i. To write down the invariants more systematically, we consider the open GW potential as follows.