A Mirror Theorem for Toric Stacks

We prove a Givental-style mirror theorem for toric Deligne--Mumford stacks X. This determines the genus-zero Gromov--Witten invariants of X in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen--Ruan orbifold cohomology of X.

When X is equipped with an action of an algebraic torus T, there is an induced T-action on the moduli space M g,n,d (X , d). (ev ⋆ 1 a 1 )ψ k 1 1 . . . (ev ⋆ n a n )ψ kn n for a 1 , . . . , a n ∈ H • CR,T (X ). In this paper we consider the case where the moduli space M g,n (X , d) itself may not be proper, but the T-fixed locus is proper. This happens for toric stacks. In this case, we define Tequivariant descendant Gromov-Witten invariants by using the virtual localization formula (see [Liu13]); the invariants then take values in S T = Frac(R T ).
Since our target space X is not necessarily proper, we work over the field S T ∼ = C(χ 1 , . . . , χ d ) of fractions of H • T (pt), where {χ 1 , . . . , χ d } is a basis of characters of the torus T ∼ = (C × ) d . Recall that the T-equivariant Novikov ring is Givental's symplectic vector space is the Λ T nov -module H := H CR,T (X ) ⊗ R T S T ((z −1 ))[[NE(X) ∩ H 2 (X, Z)]] equipped with the symplectic form: The coefficient of Q d in an element of H is a formal Laurent series in z −1 , i.e. a power series of the form ∞ n=n 0 a n z −n for some n 0 ∈ Z. The symplectic form Ω is given by the coefficient of z −1 of the orbifold Poincaré pairing (f (−z), g(z)) CR ; the minus sign reflects the fact that we take the residue at z = ∞ rather than z = 0. Consider the polarization The subspaces H ± are maximally isotropic with respect to Ω, and the symplectic form Ω induces a nondegenerate pairing between H + and H − . Thus we can regard H = H + ⊕ H − as the total space of the cotangent bundle T * H + of H + .
The genus-zero descendant potential F 0 X ,T defines a formal germ of a Lagrangian submanifold L X := (p, q) ∈ H + ⊕ H − : p = d q F 0 X ,T ⊂ T * H + ∼ = H given by the graph of the differential of F 0 X ,T . The submanifold L X may be viewed as a formal subscheme of the formal neighbourhood of −1z in H cut out by the equations Here the expression φ α /(−z −ψ) should be expanded as a power series ∞ n=0 (−z) −n−1 φ αψ n in z −1 . The condition (5) ensures that the expression (4) converges in the (Q, x)-adic topology.
REMARK 1. As we shall see in Section 6, using localization in T-equivariant cohomology, the expression (4) lies in a rational version of Givental's symplectic space: and z is identified with the C × -equivariant parameter. The space H rat is embedded into H by the Laurent expansion at z = ∞. This fact plays an important role in the characterization of points on L X in Section 6.
The Lagrangian submanifold L X has very special geometric properties.
THEOREM 2 [Giv04], [CCIT09], [Tse10]. L X is the formal germ of a Lagrangian cone with vertex at the origin such that each tangent space T to the cone is tangent to the cone exactly along zT .
In other words, if N is a formal neighborhood in H of −1z ∈ L X , then we have the following statements: (a) T ∩ L X = zT ∩ N ; Givental has proven that these statements are equivalent to the string equation, dilaton equation, and topological recursion relations [Giv04, Theorem 1]. The statements (6) imply that: sufficient for our purposes, as in Section 6 we consider twisted Gromov-Witten theory for a T-fixed point of a toric stack. Givental's symplectic formalism for the twisted theory has a subtle but important difference from that in the previous section: we need to work with formal Laurent series in z rather than z −1 .
Let E → X be a vector bundle equipped with a T-linearization; as mentioned above, T here acts trivially on the base X . Consider the virtual vector bundle E g,n,d = Rπ ⋆ ev ⋆ E ∈ K 0 T (M g,n (X , d)) where π : C g,n,d → M g,n (X , d) and ev : C g,n,d → X give the universal family of stable maps: The twisted genus-g descendant potential is: In the twisted theory, we work with the twisted orbifold Poincaré pairing where IE is the inertia stack of the total space of E; IE is a vector bundle over IX such that the fibre over (x, g) ∈ IX is the g-fixed subspace of E x . Givental's symplectic vector space for twisted theory is the Λ T nov -module ] equipped with the symplectic form: Let {φ µ }, {φ µ } be dual bases of H • CR (X ) ⊗ S T with respect to the twisted orbifold Poincaré pairing. They define Darboux co-ordinates {p a,µ , q µ a } on H tw as in (3). The Lagrangian submanifold L tw of the twisted theory is then defined similarly: for some t(z) ∈ H tw + [[x]] satisfying t| x=Q=0 = 0. Note that the expression (7) makes sense as an element of H tw [[x]]. We use here the fact that, as T acts trivially on X , the descendant classesψ i are nilpotent on each moduli space M 0,n (X , d); therefore t(ψ) = ∞ k=0 t kψ k and φ α /(−z−ψ) = ∞ n=0 φ αψ n (−z) −n−1 truncate to finite series on each moduli space M 0,n (X , d).
REMARK 4. The analogue of Theorem 2 holds for L tw .

Toric Deligne-Mumford Stacks
In this Section we discuss some background material on toric stacks. More details can be found in [BCS05,Iwa09a,Iwa09b,FMN10].

Basics
Following Borisov-Chen-Smith [BCS05], a toric Deligne-Mumford stack is defined in terms of a stacky fan where N is a finitely generated abelian group, Σ ⊂ N Q = N ⊗ Z R is a rational simplicial fan, and ρ : Z n → N is a homomorphism. We denote by ρ i the image under ρ of the ith standard basis vector e i of Z n . Let L ⊂ Z n be the kernel of ρ. The exact sequence is called the fan sequence. By assumption, ρ has finite cokernel and the imagesρ i , 1 i n, of the ρ i s under the canonical map N → N Q generate 1-dimensional cones of the simplicial fan Σ.
By abuse of notation, we sometimes identify a cone σ ∈ Σ with the subset {i :ρ i ∈ σ} of {1, . . . , n} and write i ∈ σ instead ofρ i ∈ σ. The set of anti-cones is defined to be A := I ⊂ {1, . . . , n} : where C I ⊂ C n is the subvariety determined by the ideal in C[Z 1 , . . . , Z n ] generated by {Z i : i / ∈ I}. Let ρ ∨ : (Z * ) n → L ∨ be the Gale dual of ρ [BCS05]. Here L ∨ := H 1 (Cone(ρ) * ) is an extension of the ordinary dual L * = Hom(L, Z) by a torsion subgroup. We have Ker(ρ ∨ ) = N * . The exact sequence is called the divisor sequence. Applying where G := Hom Z (L ∨ , C × ). The toric Deligne-Mumford stack X (Σ) associated to Σ is defined to be the quotient stack where G acts on U A via α.
Throughout this paper we assume that the toric Deligne-Mumford stack X (Σ) has semi-projective 8 coarse moduli space, i.e. that the coarse moduli space X(Σ) is a toric variety that has at least one torusfixed point, such that the natural map X(Σ) → Spec H 0 X(Σ), O X(Σ) is projective. In terms of the fan Σ, this is equivalent [CLS11] to demanding that the support |Σ| of the fan Σ is full-dimensional and convex, and that there exists a strictly convex piecewise linear function φ : |Σ| → R.
Let N tor denote the torsion subgroup of N , and set N := N/N tor . For c ∈ N we denote by c ∈ N the image of c under the natural projection N → N . Given a stacky fan Σ = (N, Σ, ρ), one can consider the set Box defined as follows. For a cone σ ∈ Σ, define Box(σ) := b ∈ N :b = i∈σ a iρi for some a i with 0 a i < 1 and set Box(Σ) := σ∈Σ Box(σ). Components of the inertia stack IX (Σ) are indexed by Box; we write IX (Σ) b for the component corresponding to b ∈ Box. The involution inv on IX (Σ) induces an involution b →b on Box(Σ).

Extended Stacky Fans
Following Jiang [Jia08], toric Deligne-Mumford stacks can also be described using extended stacky fans. Let Σ = (N, Σ, ρ) be a stacky fan, and let S be a finite set equipped with a map 1 S → N Σ := {c ∈ N : c ∈ |Σ|}. We label the finite set S by {1, . . . , m}, where m = |S|, and write s j ∈ N for the image of the jth element of S. The S-extended stacky fan is given by the same group N , the same fan Σ, and the fan map ρ S : Z n+m → N defined by: ρ S (e i ) = ρ i 1 i n s i−n n + 1 i n + m Given an S-extended stacky fan (N, Σ, ρ S ), an associated stack may be defined as follows. Define Let L S be the kernel of ρ S : Z n+m → N . Applying Gale duality to the S-extended fan sequence 0 → L S → Z n+m → N gives the S-extended divisor sequence: We consider the quotient stack where G S acts on U A,S via α S . Jiang showed [Jia08] that this stack associated to the S-extended stacky fan (N, Σ, ρ S ) is isomorphic to the stack X (Σ).

Torus Action and Line Bundles
The inclusion (C × ) n ⊂ U A induces an open embedding of the Picard stack T = [(C × ) n /G] into X (Σ).
We have T ∼ = T × BN tor with T := (C × ) n / Im α ∼ = N ⊗ C × and N tor ∼ = Ker α, where α is given in (9). The Picard stack T acts naturally on X (Σ) and the T -action restricts to the T-action on X (Σ).
A line bundle on X (Σ) corresponds to a G-equivariant line bundle on U A , and a T -equivariant line bundle on X (Σ) corresponds to a (C × ) n -equivariant line bundle on U A . Thus we have natural identifications: The natural map Pic T (X (Σ)) → Pic(X (Σ)) is identified with the divisor map ρ ∨ : (Z n ) * → L ∨ in (8). We write u 1 , . . . , u n for the basis of T -equivariant line bundles on X (Σ) corresponding to the standard basis of (Z n ) * and write D 1 , . . . , D n for the corresponding non-equivariant line bundles, i.e. D i = ρ ∨ (u i ). Abusing notation, we also write u i or D i for the corresponding (T-equivariant or nonequivariant) first Chern classes. These are the (T-equivariant or non-equivariant) Poincaré duals of the toric divisors
(ii) N Σ := {c ∈ N : c ∈ σ for some σ ∈ Σ } Similarly, the T-equivariant Chen-Ruan orbifold cohomology of the toric Deligne-Mumford stack X (Σ) is [Liu13]: where: The (T-equivariant or non-equivariant) classes u i , D i in Section 3.3 correspond to y ρ i in the above descriptions. For b ∈ Box(Σ), y b is the identity class supported on the twisted sector IX (Σ) b .

Maps to 1-Dimensional Torus Orbits
We next describe toric maps from certain very simple toric orbifolds P r 1 ,r 2 to the toric Deligne-Mumford stack X (Σ). This establishes notation that we will need to state and prove our mirror theorem.
DEFINITION 5. Let r 1 and r 2 be positive integers. There is a unique Deligne-Mumford stack with coarse moduli space equal to P 1 , isotropy group µ r 1 at 0 ∈ P 1 , isotropy group µ r 2 at ∞ ∈ P 1 , and no other non-trivial isotropy groups. We call this stack P r 1 ,r 2 .
These data are related as follows: r i is the order be the fan sequence for X (Σ). A representable toric morphism f : P r 1 ,r 2 → X (Σ) is given by a commutative diagram: for some integers m 1 , m 2 , m and some map η. Given a morphism as in (a), and hence a commutative diagram (12), let b 1 be the unique element of Box(σ 1 ) such that b 1 ≡ η(−1) mod ρ 1 , and let b 2 be the unique element of Box(σ 2 ) such that b 2 ≡ η(1) mod ρ 2 . Then there exist unique non-negative integers q 1 , q 2 such that: Thus a morphism as in (a) determines data as in (b).
Conversely, suppose that we are given v 1 , v 2 , q 1 , q 2 as in (b). Define η : Z → N by setting: and-for instance by looking at images in N -we see that 0 k i < r i . Now set The diagram: , and similarly m 2 ρ 2 = η(r 2 ). Thus: The fan sequence (11) defining X (Σ) is exact at Z 2 , and we deduce that there exists an integer m > 0 such that m 1 r ′ 2 m 2 r ′ 1 = w 2 m w 1 m and hence that the diagram: It is almost immediate that the constructions (a) ⇒ (b) and (b) ⇒ (a) are inverses of each other: the key point is that, if f : P r 1 ,r 2 → X (Σ) is representable, then r i is the order of b i in N/ρ i .
The equivalence (b) ⇔ (c) is immediate: we set q 1 = w 2 l − f 1 , write w 1 l = q 2 + f 2 with f 2 = w 1 l the fractional part and q 2 = ⌊w 1 l⌋ the integer part, and set b 2 = −q 1 ρ 1 − q 2 ρ 2 − b 1 . REMARK 7. The box elements b 1 , b 2 in the above proposition are given by the restrictions of f to 0, ∞ ∈ P r 1 ,r 2 respectively. The rational number l > 0 in (c) measures the "degree" of the map f in the sense that torsion. The degree of the map between the coarse moduli spacesf : NOTATION 8. Let Σ = (N, Σ, ρ) be a stacky fan. We write σ|σ ′ if σ, σ ′ ∈ Σ are top-dimensional cones that meet along a codimension-1 face. Whenever σ|σ ′ , we write j for the unique index such thatρ j is in σ but not in σ ′ , and j ′ for the unique index such thatρ j ′ is in σ ′ but not in σ.

Proof. Let
The result now follows from Proposition 6, with w 1 there equal to w j ′ here, w 2 there equal to w j here, and c equal to lw j .
REMARK 11. Note that the choice of σ, σ ′ , b and c in Proposition 10 determines the map f : P r 1 ,r 2 → X (Σ) uniquely, and hence determines both r 2 and the box element b ′ ∈ Box(σ ′ ) given by the restriction for some q ′ ∈ Z 0 . Note the asymmetry between b and b ′ : the restriction f | 0 givesb = inv(b) and the restriction f | ∞ gives b ′ . This convention is useful in our recursion analysis.
DEFINITION 12. Let σ, σ ′ ∈ Σ be top-dimensional cones satisfying σ|σ ′ . Let j, j ′ be as in Notation 8. Define l(c, σ, j) to be the element of L ⊗ Q ∼ = H 2 (X, Q) given by the unique relation of the form: REMARK 13. When we have a box element b ∈ Box(σ) satisfying c =b j , l(c, σ, j) is the degree of the representable toric morphism f : P r 1 ,r 2 → X (Σ) specified by a rational number c > 0 in Proposition 10(2). We have and f | ∞ give respectively the box elementsb and b ′ . In other words: c > 0 such that c =b j and that (13) holds for some q ′ ∈ Z 0

The Extended Picard Group
In this Section we introduce notions of extended Picard group for a Deligne-Mumford stack X and extended degree for an orbifold stable map f : C → X . There is less here than meets the eye: the extended degree of f amounts in the end to a convenient way of packaging the extra discrete data attached to f , given by the elements of Box(X ) associated to the marked points. In what follows we will use this material only when X is a toric Deligne-Mumford stack, but the definitions make sense for general Deligne-Mumford stacks and we give them in this context.
DEFINITION 15. The box of a Deligne-Mumford stack X , denoted Box X , is the set of generic representable morphisms b : Bµ r → X . In other words, it is the set of connected components of the inertia stack IX . We write the order r of the box element b as r b .
REMARK 16. If X is a toric Deligne-Mumford stack then this reduces to the notion of Box(Σ) given in Section 3.1.
DEFINITION 17. Let X be a Deligne-Mumford stack and let S be a finite set equipped with a map S → Box X . Abusing notation, we denote an element of S and its image in Box X by the same symbol b. The S-extended Picard group of X , denoted by Pic S X , is defined by the exact sequence: In other words, an element of Pic S X is a pair (L, ϕ) where L ∈ Pic X is a line bundle on X , and ϕ : S → Q has the property that ϕ DEFINITION 18. Let f : (C, x 1 , . . . , x k ) → X be an orbifold stable map. An S-decoration of f is an assignment of s j ∈ S to each marking x j such that the element of Box X given by f | x j coincides with the image of s j in Box X . The S-extended degree of an S-decorated orbifold stable map f is an element of (Pic S X ) * defined by The Riemann-Roch theorem for orbifold curves [AGV08] shows that the right-hand side is an integer. The S-extended Mori cone is the cone NE S (X ) ⊂ (Pic S X ) * ⊗ R generated by the S-extended degrees of S-decorated orbifold stable maps. One can easily see that: induced from (14), where NE(X ) denotes the usual Mori cone.
REMARK 19. We can think of elements of S as "states" to be inserted at markings of a stable map. If an S-decorated orbifold stable map has a degree d ∈ H 2 (X, Z) and each "state" b ∈ S is inserted n b times to it, the S-extended degree with respect to (L, ϕ) is given by: The value ϕ(b) can be viewed as the degree of the variable dual to b ∈ S.
REMARK 20. When X is Gorenstein and the subset S consists of those box elements of age 1, the Sextended degree of a stable map is essentially the same thing as the orbifold Neron-Severi degree defined by Bryan-Graber [BG09, §2].

Extended Degrees for Toric Stacks
Suppose now that X = X (Σ) is the toric Deligne-Mumford stack associated to a stacky fan Σ = (N, Σ, ρ), and that S is a finite set equipped with a map S → N Σ = {c ∈ N :c ∈ |Σ|}. By composing it with a natural projection N Σ → Box(Σ) we obtain a map S → Box(Σ). We now identify L S∨ with Pic S X (Σ).
Let m = |S| and let s 1 , . . . , s m ∈ N Σ be the images of elements of S in N Σ . The fan sequence and the S-extended fan sequence fit into the following commutative diagram: with exact rows and columns. We give a splitting of the first row over the rational numbers. Define µ : Q m → L S ⊗ Q by sending the j-th standard basis vector to where σ(j) is the minimal cone containings j and the positive numbers s ji are determined by i∈σ(j) s jiρi = s j . The map µ defines a splitting of the first row of (16) over Q: Let r j be the order of the image of s j ∈ N Σ in N/ i∈σ(j) Zρ i . Then we have r j s ji ∈ Z. Therefore the dual of µ gives One can check that the map µ * together with the canonical map ι * : L ∨ → L S∨ fits into the exact sequence: where res maps an element of L ∨ ∼ = Pic(X ) to the ages of the corresponding line bundle at the box elements given by s 1 , . . . , s m and can is the canonical projection. Thus we obtain: The standard decomposition (15) matches with the splitting (17). The Mori cone and the S-extended Mori cone are described, as subsets of L ⊗ R and L S ⊗ R, as follows: (17).
Our semi-projectivity assumption implies that the Mori cone NE(X (Σ)) is strictly convex.
λ i e i such that λ n+j ∈ Z, 1 j m, and λ i ∈ Z if i / ∈ σ and i n. Set Λ S := σ∈Σ Λ S σ . DEFINITION 23. The reduction function is and define: . REMARK 24. Elements of ΛE S b can be interpreted as the S-extended degrees of certain orbifold stable maps, as follows. Let f : (C, x 1 , . . . , x k , x ∞ ) → X be an orbifold stable map such that f | x∞ gives the box elementb ∈ Box(Σ) and the rest of the markings x 1 , . . . , x k are S-decorated, i.e. each x i is assigned an element of S that maps to the box element f | x i . Then f is naturally an (S ⊔ {b})-decorated stable map and has the (S ⊔ {b})-extended degree of the form (κ, 1) ∈ L S⊔{b} ⊂ Z n+m × Z for some κ ∈ Z n+m . On the other hand, we have a "wrong-way map" ǫ : L S⊔{b} ⊗ Q → L S ⊗ Q defined by the commutative diagram where the horizontal arrows are the splitting given in (17). The map ǫ induces a bijection Here we regard f as being S-decorated by forgetting the last marking x ∞ ; we need to generalize Definition 18 by allowing orbifold stable maps with domain curves having unmarked stacky points (and in this case deg S (f )(L, ϕ) is not necessarily an integer).
Here j is the index defined in Notation 8 and c is a positive number such that c =b j and that (13) holds for some q ′ ∈ Z 0 . We need to show that λ := λ ′ + l(c, σ, j) ∈ Λ S b . It suffices to show that v S (λ) = b. First we show λ ∈ Λ S σ . As described in Definition 12, l(c, σ, j) is given by the relation of the form Thus the ith component of λ ∈ L S ⊗ Q ⊂ Q n+m is given by On the other hand, (13) together with the relation (18) shows that This proves λ ′ j ′ + c ′ ∈ Z and hence λ ∈ Λ S σ . Now we show v S (λ) = b. We already know that v S (λ) lies in Box(σ). On the other hand, The last expression is congruent to b modulo i∈σ Zρ i by (13). Therefore v S (λ) = b as claimed. For the converse, if λ ∈ Λ S b , one can argue similarly to show that λ − l(c, σ, j) lies in Λ S b ′ .

Toric Mirror Theorem
In this Section we state the main result of this paper, Theorem 31.
NOTATION 26. Let σ ∈ Σ be a top-dimensional cone. We write u k (σ) for the character of T given by the restriction of the line bundle u k to the T-fixed point X (Σ) σ .
DEFINITION 28. Let Σ = (N, Σ, ρ) be a stacky fan, and let S be a finite set equipped with a map S → N Σ . Set m = |S| and regard L S ⊗ Q as a subspace of Q n+m . The S-extended T-equivariant I-function of X (Σ) is: Some explanations are in order: (i) the summation range ΛE S b ⊂ L S ⊗ Q was introduced in Definition 23. (ii) for each λ ∈ ΛE S b , we write λ i for the ith component of λ as an element of Q n+m . We have λ i =b i for 1 i n and λ i = 0 for n + 1 i n + m.
(iii) u i := 0 if n + 1 i n + m. For i = 1, . . . , n, u i is the T-equivariant first Chern class of the line bundle discussed in Section 3.3. (iv) y b is the identity class supported on the twisted sector IX (Σ) b associated to b ∈ Box(Σ); see Section 3.4.

I S X (Σ) (Q, z) is a formal power series in Q, x, t with coefficients in the localized equivariant Chen-Ruan cohomology H
) and z is identified with the C × -equivariant parameter; see Remark 1. DEFINITION 29. If Σ = (N, Σ, ρ) and S are as above and the coarse moduli space of X (Σ) is projective, then we define the S-extended (non-equivariant) I-function of X (Σ) by the same equation (19), but with u i , 1 i n, replaced by the non-equivariant first Chern class D i . REMARK 30. One can replace the summation range ΛE S b in the formula (19) with Λ S b without changing the I-function. This is because the summand for λ ∈ Λ S b contains a factor ( i:λ i ∈Z <0 u i )y b which vanishes unless {ρ i : λ i ∈ Z <0 or λ i / ∈ Z} spans a cone; in particular the summand for λ automatically vanishes unless λ lies in NE S (X (Σ)).
The main result of this paper is: THEOREM 31 (Toric Mirror Theorem). Let Σ = (N, Σ, ρ) be a stacky fan giving rise to a smooth toric Deligne-Mumford stack X (Σ) with semi-projective coarse moduli space, and let S be a finite set equipped with a map S → N Σ . The S-extended T-equivariant I-function I Proof. Since the coarse moduli space of X (Σ) is projective, the non-equivariant Chen-Ruan cohomology, S-extended non-equivariant I-function of X (Σ), and non-equivariant Gromov-Witten theory of X (Σ) are well-defined. Pass to the non-equivariant limit in Theorem 31.
REMARK 33. Theorem 31 and Corollary 32 take a particularly simple form when the pair (X (Σ), S) is weak Fano. Roughly speaking, in this case the S-extended I-function I S X (Σ) coincides with (a suitable restriction of) the J-function of X (Σ). See [Iri09, Section 4.1] for more details.
REMARK 34. The non-extended I-function (i.e. the S-extended I-function with S = ∅) typically only determines the restriction of the J-function to the "very small parameter space" H 2 (X ; C) ⊂ H 2 CR (X ; C). Taking S to be non-trivial in Theorem 31 and Corollary 32, however, in practice often allows one to determine the J-function along twisted sectors too. But it is convenient to take S not to be too large-not equal to the whole of Box(X ), for example-as otherwise we may lose control over the asymptotics of the I-function. We will elaborate on these points elsewhere.
The remainder of this paper contains a proof of Theorem 31. We first give a criterion, in Theorem 41, that characterizes points on the Lagrangian cone L X (Σ) . We then show, in Section 7, that the S-extended I-function I S X (Σ) satisfies the criterion in Theorem 41.

Lagrangian Cones in the Toric Case
Let X = X (Σ) be the toric Deligne-Mumford stack associated to a stacky fan Σ = (N, Σ, ρ), as in Section 3.1. In this Section we characterize those points of H which lie on Givental's Lagrangian cone L X associated to T-equivariant Gromov-Witten theory of X (Σ) (see Section 2.3). Recall that the T-fixed points of X (Σ) are in bijection with top-dimensional cones of Σ: given a top-dimensional cone σ ∈ Σ, we have a fixed point Note that X (Σ/σ) ≃ BN (σ), where N (σ) := N/N σ and N σ ⊂ N is the subgroup generated by ρ i , i ∈ σ.
NOTATION 37. Let σ ∈ Σ be a top-dimensional cone. We write H σ for Givental's symplectic vector space associated to the T-fixed point X (Σ) σ . We also write H tw σ and L tw σ for the symplectic vector space and Lagrangian cone corresponding to the Gromov-Witten theory of X (Σ) σ , twisted by the vector bundle T σ X (Σ) and the T-equivariant inverse Euler class e −1 T . More precisely: given by restricting to T-fixed points, and thus an isomorphism of vector spaces: Under this isomorphism, the symplectic form on H corresponds to the direct sum of e −1 T -twisted symplectic forms on σ H σ . For f ∈ H and σ ∈ Σ a top-dimensional cone, we write f σ ∈ H σ for the component of f along H σ ⊂ H. Thus f σ is the restriction of f to the inertia stack IX (Σ) σ of the T-fixed point DEFINITION 39 (Recursion Coefficient). Let Σ = (N, Σ, ρ) be a stacky fan, and let σ, σ ′ ∈ Σ satisfy σ|σ ′ . Let j be as in Notation 8. Fix b ∈ Box(σ), and let c be a positive rational number such that c =b j withb = inv(b). The recursion coefficient associated to (σ, σ ′ , b, c) is the element of S T = Frac(H • T (pt)) ∼ = C(χ 1 , . . . , χ d ) given by: where c ′ , c i are as in Definition 12, i.e. c ′ = D j ′ · l(c, σ, j) and c i = D i · l(c, σ, j) for i ∈ σ ∩ σ ′ .
REMARK 40. The recursion coefficient RC(c) (σ,b) depends only on σ, σ ′ , b, and c. The box element b ′ ∈ Box(σ ′ ) is determined by these data, via Remark 11. THEOREM 41. Let X = X (Σ) be a smooth toric Deligne-Mumford stack associated to a stacky fan Σ = (N, Σ, ρ). Let x = (x 1 , . . . , x m ) be formal variables. Let f be an element of H [[x]] such that ]-valued point of L X if and only if the following three conditions hold: (C1) For each top-dimensional cone σ ∈ Σ and each b ∈ Box(σ), the restriction f (σ,b) is a power series in Q and x such that each coefficient of this power series is an element of S T×C × ∼ = C(χ 1 , . . . , χ d , z) and, as a function in z, it is regular except possibly for a pole at z = 0, a pole at z = ∞, and simple poles at: Here we use Notation 8. (C2) The residues of f σ,b at the simple poles satisfy the following recursion relations: given any σ, σ ′ ∈ Σ such that σ|σ ′ , b ∈ Box(σ) and c > 0 with c =b j , we have: Here we use Notation 8, Definition 12 and Definition 39.
On the other hand, if we take the representable morphism f : P r 1 ,r 2 → X (Σ) associated to σ, c, j, b in Proposition 10, then u j (σ)/c and u j ′ (σ ′ )/c ′ are the induced T-weights of the tangent spaces at 0 and ∞ of the coarse domain curve |P r 1 ,r 2 | ∼ = P 1 (see Definition 12 for c ′ ). Therefore we have u j (σ)/c = −u j ′ (σ ′ )/c ′ and it follows that REMARK 43. Note that (C3) involves analytic continuation: (C1) implies that each coefficient of Q d x k in f σ (z) is a rational function in z, χ 1 , . . . , χ d , and so it makes sense to take the Laurent expansion at z = 0.
Proof of Theorem 41. In outline: Brown's proof for toric bundles [Bro09, Theorem 2] works for toric Deligne-Mumford stacks too. In detail: we argue as follows.
Suppose first that f is a Λ T nov [[x]]-valued point on L X . Then where 1 σ is the identity element in H • CR,T (X (Σ) σ ) and t σ (z) ∈ H σ,+ , and we have where We compute the sum in (22) using localization in T-equivariant cohomology. Chiu-Chu Melissa Liu has produced a detailed and beautifully-written introduction to localization in T-equivariant Gromov-Witten theory of toric stacks [Liu13]; we follow her notation closely. T-fixed strata in the moduli space M 0,n+1 (X , d) are indexed by decorated trees Γ, where: -Each edge e of Γ is labelled by a codimension-1 cone τ e ∈ Σ and a positive integer d e .
-Each flag 2 (e, v) of Γ is labelled with an element k (e,v) of the isotropy group G v of the T-fixed point X (Σ) σv .
-A number of compatibility conditions hold.
The compatibility conditions that we require are spelled out in detail in [Liu13, Definition 9.6]; they include, for example, the requirement if (e, v) is a flag of Γ then the T-fixed point determined by σ v is contained in the closure of the 1-dimensional T-orbit determined by τ e . We denote the set of all decorated trees satisfying the compatibility conditions by G 0,n+1 (X , d), so that T-fixed strata in M 0,n+1 (X , d) are indexed by decorated trees Γ ∈ G 0,n+1 (X , d).
We rewrite equation (22) as where Contr σ,b (Γ) denotes the contribution to the T-equivariant Gromov-Witten invariant d) corresponding to Γ. We will need some notation for graphs. For a decorated graph Γ ∈ G 0,n+1 (X , d), we write: S v = j ∈ {1, 2, . . . , n + 1} : s(j) = v 2 A flag (e, v) of Γ is an edge-vertex pair such that e is incident to v.

22
A MIRROR THEOREM FOR TORIC STACKS -E v for the set of edges incident to the vertex v ∈ V (Γ): Liu [Liu13, Theorem 9.32] shows that the contribution from M Γ to the Gromov-Witten invariant γ 1ψ a 1 1 , . . . , γ n+1ψ a n+1 n+1 0,n+1,d is: where: G e is the generic stabilizer of the one-dimensional toric substack X (Σ/τ e ) f e : C e → X is the toric map to the one-dimensional toric substack X (Σ/τ e ) determined by the edge e and the decorations τ e , d e , {k (e,v) : v is a vertex incident to e} H i (C e , f * e T X ) mov denotes the moving part with respect to the T-action w (e,v) = e T (T y(e,v) C e ), where y(e, v) is the marked point on C e determined by (e, v) is determined by the decorations k j , j ∈ S v , and k (e,v) , e ∈ E v (T σv X ) 0,val(v),0 is the twisting bundle associated to the vector bundle T σv X over the T-fixed here in the unstable cases val(v) = 1 and val(v) = 2 are defined as in [Liu13, (9.12)-(9.14)]. The twisting bundles (T σv X ) 0,val(v),0 in the unstable cases val(v) = 1 and val(v) = 2 are defined to be (T σv X ) k (e,v) ; see the end of [Liu13, §9.3.4].
Consider now the graph sum in (23). Each graph Γ in the sum contains a distinguished vertex v that carries the first marked point. We may assume both that σ v = σ and that the label k 1 of the first marking is equal tob, as otherwise the contribution of Γ is zero. There are two possibilities: (B) v has valence at least 3.
In the first case we say that Γ has type A, and in the second case we say that Γ has type B; see Figures 1 and 2. As we will see below, the contributions from type A graphs have simple poles at points of the form u j (σ)/c as described in the statement of the Theorem, and the contributions from type B graphs are polynomials in z −1 . The condition (C1) then follows.
Consider a graph Γ of type A. Let e ∈ E(Γ) be the edge incident to v. Then Γ is obtained from another decorated graph Γ ′ by adding the decorated vertex v and the decorated edge e. See Figure 1. Let v ′ be the other vertex incident to e. The graph Γ ′ is assigned the first marking at v ′ instead of the edge e. The map f e : C e → X determined by the edge e has C e ≃ P r (e,v) ,r (e,v ′ ) , f e (0) = X σv , and f e (∞) = X σ v ′ ; the restriction f e | 0 : Bµ r (e,v) → X σv givesb ∈ Box(σ v ). Let c ∈ Q and b ′ ∈ Box(σ v ′ ) be the rational number and box element determined by applying Proposition 10 and Remark 11 to f e , and write σ ′ = σ v ′ . Sinceψ 1 = −r (e,v) w (e,v) , we obtain: Calculating the ratio c Γ /c Γ ′ and evaluating the integral over M (b,b) 0,2 (BG v ) using [Liu13, (9.14)] yields: Liu has computed the ratio of Euler classes here [Liu13, Lemma 9.25], and in our notation this gives: . (See Remark 44 below for a detailed comparison between Liu's notation and ours.) Note that the degree of the map f e : C e → X is l(c, σ, j); see Definition 12. Note also that if we hold the decorated vertex v and the decorated edge e constant (or in other words, if we hold the map f e : C e → X constant) then the sum of Contr(Γ ′ ) σ ′ ,b ′ over all compatible trees Γ ′ is exactly 3 the graph sum that defines f (σ ′ ,b ′ ) . Thus the contribution to from all graphs Γ of type A is: This proves (C2).
Write: quantity in equation 25) and: We have that: Consider the contribution to (26) given by the sum over decorated graphs Γ of type B such that the distinguished vertex v has valence l and that the label k 1 of the distinguished vertex is equal tob ∈ Box(σ). Each such graph Γ gives contributions of the form (24). We evaluate these contributions by integrating over all the factors M b ′ 0,val(v ′ ) (BG v ′ ) except that associated with the distinguished vertex v, obtaining an expression of the form 4 : for some elements b 2 , . . . , b l ∈ Box(σ) and some polynomials h i (t,ψ i ) in t 0 , t 1 ,. . . , Q, andψ i . Suppose that Γ is obtained from type A subgraphs Γ 2 , . . . , Γ l by joining them at the distinguished vertex v, as in Figure 2. If Γ i consists of one vertex with two markings (such as Γ 4 in Figure 2) then h i (t,ψ i ) = Otherwise h i (t,ψ i ) records a more complicated contribution determined by the subgraph Γ i ; we have: where d i is the total degree of the contribution from Γ i . Now fix v and all other parts of Γ except the subtree Γ i , and sum over all possible subtrees Γ i : the total contribution of the h i (t,ψ i )s is (25) with b = b i and z =ψ i . Thus the contribution to (26) given by the sum over decorated graphs Γ of type B such that the distinguished vertex has valence l and that the label k 1 of the distinguished vertex is equal tob ∈ Box(σ) is: These are twisted Gromov-Witten invariants of the T-fixed point X (Σ) σ . Summarizing, we see that (26) becomes: The superscript 'tw' indicates that these are Gromov-Witten invariants of X (Σ) σ twisted by the vector bundle T σ X (Σ) and the T-equivariant inverse Euler class e −1 T . Using (7), we see that the Laurent expansion at z = 0 of f σ lies in L tw σ . Thus we have proved (C3). Conversely, suppose that f ∈ H [[x]] satisfies f | Q=x=0 = −1z and conditions (C1), (C2), and (C3) in the statement of the Theorem. Conditions (C1) and (C2) together imply that: for some t σ ∈ H σ,+ [[x]] with t σ | Q=t=0 = 0. The remainder O(z −1 ) is a formal power series in Q and x with coefficients in z −1 S T [z −1 ]. Let t GW ∈ H + be the unique element such that its restriction to IX (Σ) σ is t σ , and let f GW be the element of L X defined by (21) with t = t GW . Then, in view of the first part of the proof, we have that f GW and f both satisfy conditions (C1-C3), and both give rise to the same values t σ in (27). It therefore suffices to show that f can be reconstructed uniquely from the collection t σ : σ ∈ Σ is a top-dimensional cone (28) using condition (C3).
We argue by induction on the degree with respect to Q and x. Pick a Kähler class ω of X (Σ) and assign the degree d ω + m i=1 k i to the monomial Q d x k 1 1 · · · x km m . Let κ 0 = d 0 ω > 0 be the minimal possible degree of a non-constant stable map. Suppose that f is uniquely determined from the collection (28) up to order κ. We shall show that f is determined up to order κ + κ 0 . We know by (27) that f σ is determined up to order κ + κ 0 except for the term O(z −1 ). On the other hand, under the Laurent expansion at z = 0, all the quantities in the first line of (27) lie in H tw σ,+ . Therefore, in view of (7), the term O(z −1 ) is uniquely determined up to order κ + κ 0 from the quantities in the first line by condition (C3), i.e. that the Laurent expansion at z = 0 of f σ lies in L tw σ . This completes the induction and the proof of Theorem 41.
REMARK 44. For the convenience of the reader, we compare Liu's notation [Liu13, Lemma 9.25] with ours. Consider a decorated graph Γ occurring in the proof above, and an edge e ∈ E(Γ) with incident vertices v, v ′ ∈ V (Γ). The edge e corresponds to a toric representable morphism f = f e : P r 1 ,r 2 → X (Σ) given by σ, σ ′ , c, b in Proposition 10, where σ = σ v and σ ′ = σ v ′ . Let j and j ′ be the indices in Notation 8. Recall (from Definition 12) that the degree l(c, σ, j) ∈ L ⊗ Q of the map f is given by the relation: r (e,v) , r (e,v ′ ) , w (e,v) , w (e,v ′ ) are given in our notation as: d := the degree of the map (f : |P r 1 ,r 2 | → |X (Σ) τ |) between the coarse curves ( ∼ = P 1 ) where we set N (τ ) = N/ i∈τ Zρ i , N (τ ) = N (τ )/N (τ ) tor and N (τ ) tor is the torsion part of N (τ ).

Proof of Theorem 31
In this Section we complete the proof of Theorem 31, by showing that the S-extended I-function I S X (Σ) (Q, −z) satisfies the conditions in Theorem 41. This amounts to proving Propositions 45, 46, and 47 below. Note that the sign of z should be flipped when we consider the I-function.
For a top-dimensional cone σ ∈ Σ and b ∈ Box(σ), we write I S σ (Q, z) and I S (σ,b) (Q, z) for the restrictions of I S X (Σ) (Q, z) to the inertia stack IX (Σ) σ of the T-fixed point X (Σ) σ and the component IX (Σ) σ,b of IX (Σ) σ respectively. PROPOSITION 45. The extended I-function satisfies condition (C1) in Theorem 41. In other words, for each top-dimensional cone σ ∈ Σ and b ∈ Box(σ), I S (σ,b) (Q, z) is a power series in the extended Novikov variablesQ and t such that each coefficient of this power series lies in S T×C × = C(χ 1 , . . . , χ d , z) and, as a function of z, it is regular except possibly for a pole at z = 0, a pole at z = ∞, and simple poles at: : ∃σ ′ ∈ Σ such that σ|σ ′ and j ∈ σ \ σ ′ , c > 0 is such that c =b j Here we use Notation 8.
PROPOSITION 46. The extended I-function satisfies condition (C2) in Theorem 41. In other words, for any σ, σ ′ ∈ Σ such that σ|σ ′ , we have: PROPOSITION 47. The extended I-function satisfies condition (C3) in Theorem 41. In other words, if σ ∈ Σ is a top-dimensional cone, then the Laurent expansion at

Poles of the Extended I-Function
In this subsection we prove Proposition 45. Let σ be a top-dimensional cone and take b ∈ Box(σ). The restriction I S (σ,b) of the I-function to the fixed point IX (Σ) σ,b takes the form: where the index i ranges over {1, . . . , n+m} and we regard σ ⊂ {1, . . . , n} as a subset of {1, . . . , n+m}. We also used u i (σ) = 0 for i / ∈ σ. For λ ∈ Λ S b , we have that λ i ∈ Z for all i / ∈ σ because λ i =b i and b ∈ Box(σ). Note also that one may assume that λ i ∈ Z 0 for i / ∈ σ in the above sum, as otherwise the contribution is zero. We see that I S (σ,b) has poles possibly at z = 0 and z = ∞ and simple poles at −u i (σ)/a with 0 < a λ i , a = λ i =b i , i ∈ σ for λ ∈ ΛE S b contributing to the sum. It suffices to see that, if λ i 0 > 0 for some i 0 ∈ σ, then there exists a top-dimensional cone σ ′ such that σ|σ ′ and i 0 ∈ σ \ σ ′ , i.e. i 0 = j in Notation 8. We have i∈σ (−λ i )ρ i = i:1 i n i / ∈σ where s 1 , . . . , s m are the images of elements of S in N Σ . As we remarked above, we may assume that λ i ∈ Z 0 for i / ∈ σ and hence the right-hand side belongs to the support |Σ| of the fan. Therefore i∈σ (−λ i )ρ i ∈ |Σ|. Because |Σ| is convex, the positivity of λ i 0 implies that there exists a topdimensional cone σ ′ ∈ Σ such that σ|σ ′ and i 0 ∈ σ \ σ ′ . Proposition 45 is proved.

Recursion for the Extended I-Function
In this subsection we prove Proposition 46. Let σ ∈ Σ be a top-dimensional cone and let b ∈ Box(σ). Fix another top-dimensional cone σ ′ with σ|σ ′ and a positive rational number c such that c =b j , where j is the index in Notation 8. We examine the residue of I S (σ,b) at z = −u j (σ)/c. Write ✄ λ,i,σ (z) = a = λ i ,a 0 (u i (σ) + az) a = λ i ,a λ i (u i (σ) + az) for λ ∈ Λ S and 1 i n + m. The residue of (29) at z = −u j (σ)/c is given by: Recall from Remark 30 that the summation range can be taken to be Λ S b instead of ΛE S b . Let l(c, σ, j) ∈ ΛE σ ′ ,b ′ σ,b ⊂ L ⊗ Q be the degree from Definition 12. We now consider the change of variables λ = λ ′ + l(c, σ, j) and replace the sum over λ ∈ Λ S b with the sum over λ ′ ∈ Λ S b ′ using Lemma 25. We write c i for the components of l(c, σ, j) ∈ L ⊗ Q ⊂ L S ⊗ Q as an element of Q n+m . Using the notation in Definition 12, we have c i = D i · l(c, σ, j) for 1 i n, c j = c, c j ′ = c ′ and c i = 0 for n + 1 i n + m.
LEMMA 48. Let λ, λ ′ be as above. We have: X (Σ) σ is: where 1 v S (λ) ∈ H • CR (X (Σ) σ ) is the identity class supported on the twisted sector corresponding to v S (λ) ∈ Box(σ). We want to show that this lies on the Lagrangian cone L tw σ . We claim that it suffices to show that I S σ (Q, −z)| t=0 lies on L tw σ . By the String Equation, L tw σ is invariant under multiplication by e − n i=1 u i (σ)t i /z and thus we can remove the factor e − n i=1 u i (σ)t i /z . Since the T-fixed point X (Σ) σ has no Novikov variables, we can regardQ in I S σ (Q, −z) as variables rather than elements of the ground ring. (In other words, L tw σ is defined over S T .) Therefore we can absorb the factor e λt intoQ by rescaling Q. The claim follows.
Define rational numbers a ij for i / ∈ σ, j ∈ σ byρ i = j∈σ a ijρj for 1 i n ands i−n = j∈σ a ijρj for n + 1 i n + m. Then the equation (30) shows that for λ ∈ Λ S σ and j ∈ σ. Henceforth we regard λ j for j ∈ σ as a linear function of (λ i : i / ∈ σ) via this relation. We introduce variables (q i : i / ∈ σ) dual to (λ i : i / ∈ σ) and consider the change of variables: We also have Now it suffices to show that −z is a S T [[q]]-valued point on L tw σ , where ℓ = n + m − dim X (Σ). Jarvis-Kimura [JK02] calculated the Gromov-Witten theory of BG with G a finite group, and it follows from their result that the J-function of BN (σ) ∼ = X (Σ) σ is: (See [CCIT09, Proposition 6.1].) Comparing this with (37), we find that the expression (37) is the hypergeometric modification of J BN (σ) , in the sense of [CG07,CCIT09]. The J-function (38) lies on the Lagrangian cone of the Gromov-Witten theory of BN (σ) (see Remark 3), and we now use the argument of [CCIT09] to show that the hypergeometric modification of the J-function (37) lies on the cone L tw σ of the twisted theory.