The Crepant Transformation Conjecture for Toric Complete Intersections

Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier-Mukai transformation between the K-groups of X and Y, via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on X and Y: they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne-Mumford stacks and toric complete intersections, and the Mellin-Barnes method for analytic continuation of hypergeometric functions.


Introduction
A birational map ϕ : X + X − between smooth varieties, orbifolds, or Deligne-Mumford stacks is called a K-equivalence if there exists a smooth variety, orbifold, or Deligne-Mumford stack X and projective birational morphisms f ± : X → X ± such that f − = ϕ • f + and f + K X + = f − K X − : In this case, the celebrated Crepant Transformation Conjecture of Y. Ruan predicts that the quantum (orbifold) cohomology algebras of X + and X − should be related by analytic continuation in the quantum parameters. This conjecture has stimulated a great deal of interest in the connections between quantum cohomology (or Gromov-Witten theory) and birational geometry: see, for example, [9,10,17,18,20,22,23,27,42,46,54,[57][58][59][60]63,69,72,76,77]. Ruan's original conjecture was subsequently refined, revised, and extended to higher genus Gromov-Witten invariants, first by Bryan-Graber [19] under some additional hypotheses, and then by Coates-Iritani-Tseng, Iritani, and Ruan in general [35,36,51]. Recall that a toric Deligne-Mumford stack X can be constructed as a GIT quotient C m / / ω K of C m by an action of a complex torus K, where ω is an appropriate stability condition, and that wall-crossing in the space of stability conditions induces birational transformations between GIT quotients [38,74]. Our main result implies the CIT/Ruan version of the Crepant Transformation Conjecture in genus zero, in the case where X + and X − are complete intersections in toric Deligne-Mumford stacks and ϕ : X + X − arises from a toric wall-crossing. We concentrate initially on the case where X + and X − are toric, deferring the discussion of toric complete intersections to §1.3.
1.1. The Toric Case. We consider toric Deligne-Mumford stacks X ± of the form C m / / ω K , where K is a complex torus, and consider a K-equivalence ϕ : X + X − determined by a wallcrossing in the space of stability conditions ω. The action of T = (C × ) m on C m descends to give (ineffective) actions of T on X ± , and we consider the T -equivariant Chen-Ruan cohomology groups appropriate identification of the spaces of GKZ solutions with the K-groups of the corresponding toric Deligne-Mumford stacks, the analytic continuation of solutions to a GKZ system is induced by a Fourier-Mukai transformation between the K-groups. Our computation may be viewed as a straightforward generalization of theirs. The differences from their situation are: (a) we work with a fully equivariant version, that is, the parameters β j in the GKZ system are arbitrary and we use the equivariant K-groups (here β j corresponds to the equivariant parameter); (b) we compute analytic continuation of the I-function corresponding to the big quantum cohomology; in terms of the GKZ system, we do not assume that lattice vectors in the set 1 A lie on a hyperplane of height one. Since we work equivariantly, we can use the fixed point basis in localized equivariant cohomology to calculate the analytic continuation of the I-functions. It turns out that analytic continuation via the Mellin-Barnes method becomes much easier to handle in the fully equivariant setting, because we only need to evaluate residues at simple poles 2 . It is also straightforward to compute the Fourier-Mukai transformation in terms of the fixed point basis in the localized equivariant K-group, and hence to see that analytic continuation coincides with Fourier-Mukai.
Regarding part (b) above, we choose A to be the set {b 1 , . . . , b m } ⊂ N of ray vectors of an extended stacky fan [11,55]. Since we do not restrict ourselves to the weak Fano case, and since we work with Jiang's extended stacky fans, the generic rank of the GKZ system can be bigger than the rank of H • CR,T (X ± ). To remedy this, we treat one special variable analytically and work formally in the other variables. In fact, the big I-functions are not necessarily convergent in all of the variables, and we analytically continue the I-function with respect to one specific variable y r . This amounts to considering an adic completion of the Borisov-Horja better-behaved GKZ system [14] with respect to the other variables. The analytic continuation in Theorem 1.1 occurs across a "global Kähler moduli space" M • which is treated as an analytic space in one direction and as a formal scheme in the other directions.
1.3. The Toric Complete Intersection Case. Let ϕ : X + X − be a K-equivalence between toric Deligne-Mumford stacks that arises from a toric wall-crossing, as in §1. 1. Let X be the common toric blow-up of X ± and let X 0 denote the common blow-down; X 0 here is a (singular) toric variety, not a stack.
Consider a direct sum of semiample line bundles E 0 → X 0 , and pull this back to give vector bundles E + → X + , E → X, and E − → X − . Let s + ,s, and s − be sections of, respectively, E + , E, and E − that are compatible via f + and f − (so f + s + = s = f − s − ) such that the zero loci of s ± intersect the flopping locus of ϕ transversely. Let Y + , Y , and Y − denote the substacks defined by 1 Recall that Gelfand-Kapranov-Zelevinsky defined the GKZ system in terms of a finite set A ⊂ Z d . They called it the A-hypergeometric system. 2 For an example of the complexities caused by non-simple poles, see the orbifold flop calculation in [27, §7].
the zero loci of, respectively, s + ,s, and s − . In this situation there is a commutative diagram: where the vertical maps are inclusions, the bottom triangle is (1.1), and the squares are Cartesian. The K-equivalence ϕ : X + X − induces a K-equivalence ϕ : Y + Y − . We now consider the Crepant Transformation Conjecture for ϕ : Y + Y − . Since the complete intersections Y ± will not in general be T -invariant we consider non-equivariant Gromov-Witten invariants and the non-equivariant quantum product. (Our assumptions on X ± ensure that the non-equivariant theory makes sense.) Denote by H • amb (Y ± ) the image im ι ± ⊂ H • CR (Y ± ), where ι ± : Y ± → X ± is the inclusion map. If τ ∈ H • amb (Y ± ) then the big quantum product τ preserves the ambient part H • amb (Y ± ) ⊂ H • CR (Y ± ). We can therefore define a quantum connection on the ambient part: This is a pencil of flat connections on the trivial H • amb (Y ± )-bundle over an open set in H • amb (Y ± ) where, as in (1.2), z ∈ C × is the pencil variable, τ ∈ H • amb (Y ± ) is the co-ordinate on the base of the bundle, φ 0 , . . . , φ N are a basis for H • amb (Y ± ), and τ 0 , . . . , τ N are the corresponding co-ordinates of τ .
In §7.1 below we construct an ambient version of the Gamma-integral structure, which is an assignment to each class E in the ambient part of K-theory K 0 amb (Y ± ) = im ι ± ⊂ K 0 (Y ± ) of a flat section s(E) for the quantum connection on the ambient part H • amb (Y ± ). This gives a lattice in the space of flat sections which is isomorphic to the ambient part of (integral) K-theory K 0 amb (Y ± ). Theorem 1.4. Let ϕ : Y + Y − be a K-equivalence between toric complete intersections as above. Then: (1) the quantum connections on the ambient parts H • amb (Y ± ) ⊂ H • CR (Y ± ) become gaugeequivalent after analytic continuation in τ , via a gauge transformation which is homogeneous of degree zero and regular at z = 0. If Y is compact then Θ Y preserves the orbifold Poincaré pairing.
(2) when expressed in terms of the ambient integral structure, the gauge transformation Θ Y coincides with the Fourier-Mukai transformation given by the top triangle in (1.4).
As before, Theorem 1.4 is slightly imprecisely stated: precise statements can be found as Theorems 7.2, 7.9, and 7.11 below. Arguing as in §1.1 shows that Theorem 1.4 implies the CIT/Ruan version of the Crepant Transformation Conjecture for ϕ : Y + Y − whenever it makes sense, with the corresponding map U Y : H amb (Y − ) → H amb (Y + ) between the ambient parts of the Givental spaces for Y ± being given by: where L amb ± are the fundamental solutions for the quantum connections on the ambient parts H • amb (Y ± ). The proof of Theorem 1.4 relies on the Mirror Theorem for toric complete intersections [31], and on non-linear Serre duality [33,43,44,75], which relates the quantum cohomology of Y ± to the quantum cohomology of the total space of the dual bundles E ∨ ± . Since E ∨ ± is toric, it can be analyzed using Theorem 1.1.
Remark 1.5. The idea of using non-linear Serre duality to analyze wall-crossing has been developed independently by Lee-Priddis-Shoemaker [61], in the context of the Landau-Ginzburg/ Calabi-Yau correspondence. Example 1.6. A mirror Y to the quintic 3-fold arises [7,21,48] as a crepant resolution of an anticanonical hypersurface in X = P 4 /(Z/5Z) 3 . A mirror theorem for Y has been proved by Lee-Shoemaker [62]. The variety Y is a Calabi-Yau 3-fold with h 1,1 (Y ) = 101. There are many birational models of Y as toric hypersurfaces, corresponding to the many different lattice triangulations of the boundary of the fan polytope for X. Theorem 1.4 implies that the quantum connections (and quantum cohomology algebras) of all of these birational models become isomorphic after analytic continuation over the Kähler moduli space (which is 101-dimensional), and that the isomorphisms involved arise from Fourier-Mukai transformations.

A Note on Hypotheses.
Since we work with T -equivariant Gromov-Witten invariants of the toric Deligne-Mumford stacks X ± , we do not need to assume that the coarse moduli spaces |X ± | of X ± are projective. We insist instead that |X ± | is semi-projective, i.e. that |X ± | is projective over the affinization Spec(H 0 (|X ± |, O)), and also that X ± contains at least one torus fixed point. These conditions are equivalent to demanding that X ± is obtained as the GIT quotient C m / / ω K of a vector space by the linear action of a complex torus K; they ensure that the equivariant quantum cohomology of X ± admits a non-equivariant limit. In particular, therefore, the non-equivariant version of the Crepant Transformation Conjecture follows automatically from Theorem 1.1.
We do not assume, either, that the stacks X ± or Y ± satisfy any sort of positivity or weak Fano condition; put differently, we do not impose any additional convergence hypotheses on the I-functions for X ± and Y ± . This extra generality is possible because of our hybrid formal/analytic approach, where we single out one variable y r and analytically continue in that variable alone. The same technique allows us to describe the analytic continuation of big quantum cohomology (or its ambient part), as opposed to small quantum cohomology. In general, obtaining convergence results for big quantum cohomology is hard.
1.5. The Hemisphere Partition Function. Recently there was some progress in physics in the exact computation of hemisphere partition functions for gauged linear sigma models. Hori-Romo [50] explained why the Mellin-Barnes analytic continuation of hemisphere partition functions should be compatible with brane transportation [49] in the B-brane category. In the language of this paper, the hemisphere partition function corresponds to a component of the K-theoretic flat section s(E), and brane transportation corresponds to the Fourier-Mukai transformation. Theorem 1.1 thus confirms the result of Hori-Romo. Note that the relevant equivalence between B-brane categories should depend on a choice of a path of analytic continuation, and that the Fourier-Mukai transformation in Theorem 1.1 corresponds to a specific choice of path (see Figure 1). 1.6. Plan of the Paper. We fix notation for equivariant Gromov-Witten invariants and equivariant quantum cohomology in §2, and introduce the equivariant Gamma-integral structure in §3. We establish notation for toric Deligne-Mumford stacks in §4. In §5 we study K-equivalences ϕ : X + X − of toric Deligne-Mumford stacks arising from wall-crossing, constructing global versions of the equivariant quantum connections for X ± . We prove the Crepant Transformation Conjecture for toric Deligne-Mumford stacks (Theorem 1.1) in §6, and the Crepant Transformation Conjecture for toric complete intersections (Theorem 1.4) in §7.

1.7.
Notation. We use the following notation throughout the paper.
• X denotes a general smooth Deligne-Mumford stack in §2 and §3; it denotes a smooth toric Deligne-Mumford stack in §4 and later.
T (pt, C) = Lie(T ) is the character of T = (C × ) m given by projection to the jth factor, so that R T = C[λ 1 , . . . , λ m ]. • S T is the localization of R T with respect to the set of non-zero homogeneous elements.

Equivariant Quantum Cohomology
In this section we establish notation for various objects in equivariant Gromov-Witten theory. We introduce equivariant Chen-Ruan cohomology in §2.2, equivariant Gromov-Witten invariants in §2.3, equivariant quantum cohomology in §2.4, Givental's symplectic formalism in §2.5, and the equivariant quantum connection in §2.6.

2.1.
Smooth Deligne-Mumford stacks with Torus Action. Let X be a smooth Deligne-Mumford stack of finite type over C equipped with an action of an algebraic torus T ∼ = (C × ) m . Let |X| denote the coarse moduli space of X and let IX denote the inertia stack X × |X| X of X: a point on IX is given by a pair (x, g) with x ∈ X and g ∈ Aut(x). We write for the decomposition of IX into connected components. We assume the following conditions: (1) the coarse moduli space |X| is semi-projective, i.e. is projective over the affinization These conditions allow us to define Gromov-Witten invariants of X and also the equivariant (Dolbeault) index of coherent sheaves on X. The first and second conditions together imply that the fixed set X T is compact. The third condition seems to be closely related to the first two, but it implies for example the localization of equivariant cohomology: the restriction H • T (IX; C) → H • T (IX T ; C) to the T -fixed locus is injective and becomes an isomorphism after localization (see [47]). Later we shall restrict to the case where X is a toric Deligne-Mumford stack, where conditions (1-3) automatically hold, but the definitions in this section make sense for general X satisfying these conditions.

Equivariant Chen-Ruan Cohomology. Let H •
CR,T (X) denote the even part of the Tequivariant orbifold cohomology group of Chen and Ruan. It is defined as the even degree part of the T -equivariant cohomology of the inertia stack IX. The grading of H • CR,T (X) is shifted from that of H • T (IX) by the so-called age or degree shifting number ι v ∈ Q [24]; note that we consider only the even degree classes in H • T (IX). (For toric stacks, all cohomology classes on IX are of even degree.) Equivariant formality of IX gives that H • CR,T (X) is a free module over R T . We write for the equivariant orbifold Poincaré pairing: here inv : IX → IX denotes the involution on the inertia stack IX that sends a point (x, g) with x ∈ X, g ∈ Aut(x) to (x, g −1 ). Since X is not necessarily proper, the equivariant integral on the right-hand side here is defined via the Atiyah-Bott localization formula [3] and takes values in the localization S T of R T with respect to the multiplicative set of non-zero homogeneous elements 3 in R T .

Equivariant Gromov-Witten
Invariants. Let X g,n,d denote the moduli space of degree-d stable maps to X from genus g orbifold curves with n marked points [1,2]; here d ∈ H 2 (|X|; Z). The moduli space carries a T -action and a virtual fundamental cycle [X g,n,d ] vir ∈ A •,T (X g,n,d ; Q). There are T -equivariant evaluation maps ev i : X g,n,d → IX, 1 ≤ i ≤ n, to the rigidified inertia stack IX (see [2]). Let ψ i ∈ H 2 T (X g,n,d ) denote the psi-class at the ith marked point, i.e. the equivariant first Chern class of the ith universal cotangent line bundle L i → X g,n,d . For α 1 , . . . , α n ∈ H • CR,T (X) and non-negative integers k 1 , . . . , k n , the T -equivariant Gromov-Witten invariant is defined to be: The moduli space here is not necessarily proper: the right-hand side is again defined via the Atiyah-Bott localization formula and so belongs to S T . Conditions (1) and (2) in §2.1 ensure that the T -fixed locus X T g,n,d in the moduli space is compact, and thus that the right-hand side of (2.1) is well-defined.
CR,T (X) over R T and let τ 0 , τ 1 , . . . , τ N be the corresponding linear co-ordinates. We assume that φ 0 = 1 and φ 1 , . . . , φ r ∈ H 2 T (X) are degree-two untwisted classes that induce a C-basis of H 2 (X; C) ∼ = H 2 T (X)/H 2 T (pt). We write τ = N i=0 τ i φ i for a general element of H • CR,T (X). The equivariant quantum product τ at τ ∈ H • CR,T (X) is defined by the formula Conditions (1) and (2) in §2.1 ensure that ev 3 : X 0,n+3,d → IX is proper, and thus that the push-forward along ev 3 is well-defined without inverting equivariant parameters. It follows that:  where τ = σ + τ with σ = r i=1 τ i φ i and τ = τ 0 φ 0 + N i=r+1 τ i φ i . The String Equation (ibid.) implies that the right-hand side here is in fact independent of τ 0 .

2.5.
Givental's Lagrangian Cone. Let S T ((z −1 )) denote the ring of formal Laurent series in z −1 with coefficients in S T . Givental's symplectic vector space is the space ]-bilinear alternating form: The space is equipped with a standard polarization and It consists of points of H of the form: in the correlator should be expanded as the power series ∞ k=0 ψ k (−z) −k−1 in z −1 . In a more formal language, we define the notion of a 'point on L X ' as follows. Let It should be thought of as a formal family of elements on L X parametrized by x.
The submanifold L X encodes all genus-zero Gromov-Witten invariants (2.1). It has the following special geometric properties [45]: it is a cone, and a tangent space T of L X is tangent to L X exactly along zT . Knowing Givental's Lagrangian cone L X is equivalent to knowing the data of the quantum product τ , i.e. L X can be reconstructed from τ and vice versa. See Remark 2.5.

The Equivariant Quantum Connection and its Fundamental
The associativity of τ implies that the connection ∇ is flat, that is, [∇ i , ∇ j ] = 0 for all i, j. Let ρ denote the equivariant first Chern class (in the untwisted sector): For φ ∈ H • CR,T (X), we write deg φ for the age-shifted (real) degree of φ, so that φ ∈ H deg φ CR,T (X). The equivariant Euler vector field E and the grading operator µ ∈ End C (H • CR,T (X)) are defined by The quantum connection is compatible with the grading operator in the sense that [Gr, This follows from the virtual dimension formula for the moduli space of stable maps.
] determined by the following conditions: Here φ ∈ H • CR,T (X) and vQ ∂ ∂Q with v ∈ H 2 T (X) acts on Novikov variables as Q d → v, d Q d (it acts by zero when v ∈ H 2 T (pt) ⊂ H 2 T (X)). The flatness equation fixes L(τ, z) up to right multiplication by an endomorphism-valued function g(z; Q) in z and Q; the divisor equation implies that the ambiguity g(z; Q) is independent of Q and commutes with v∪, v ∈ H 2 T (X); finally the initial condition fixes L(τ, z) uniquely. The fundamental solution satisfying these conditions can be written explicitly in terms of (descendant) Gromov-Witten invariants: This is defined over R T (without inverting equivariant parameters) because it can be rewritten in terms of the push-forward along the last evaluation map ev n+2 as in (2.2 where φ, α, β ∈ H • CR,T (X). Remark 2.5 ( [45]). The fundamental solution L(τ, z) is determined by the quantum product τ via differential equations (2.5)-(2.7). Then τ → T τ = L(τ, −z) −1 H + gives a versal family of tangent spaces to Givental's cone L X . The cone L X is reconstructed as L X = τ zT τ .
We now study ∇-flat sections s(τ, z) that are homogeneous of degree zero: Gr(s(τ, z)) = 0. By Proposition 2.4, if a flat section L(τ, z)f (z) is homogeneous of degree zero, then: This differential equation has the fundamental solution: here k is chosen so that all the eigenvalues of kµ are integers. Note that homogeneous flat sections can be multi-valued in z (as they contain log z). We have: Corollary 2.6. The sections s i (τ, z) = L(τ, z)z −µ z ρ φ i , i = 0, . . . , N satisfy ∇s i (τ, z) = Gr s i (τ, z) = 0 and give a basis of homogeneous flat sections. They belong to for a sufficiently large k ∈ N.

Equivariant Gamma-Integral Structure
In this section we introduce one of the main ingredients of our result: an integral structure for equivariant quantum cohomology. This is a K 0 T (pt)-lattice in the space of flat sections for the equivariant quantum connection on X which is isomorphic to the integral equivariant K-group K 0 T (X): it generalizes the integral structure for non-equivariant quantum cohomology constructed by Iritani [51] and Katzarkov-Kontsevich-Pantev [56]. Similar structures have been studied by Okounkov-Pandharipande [67] in the case where X is a Hilbert scheme of points in C 2 , and by Brini-Cavalieri-Ross [16] in the case where X is a 3-dimensional toric Calabi-Yau stack. We define the integral structure in §3.1. In §3.2 we observe that the quantum product, flat sections for the quantum connection, and integral structure continue to make sense when the Novikov variable Q (see §2.4) is specialized to Q = 1.
The integral structure is defined in terms of a T -equivariant characteristic class of X called the Γ-class. One of the key points in this section is that the Γ-class behaves like a square root of the Todd class: see equation 3.4. When combined with the Hirzebruch-Riemann-Roch formula, this leads to one of the fundamental properties of the integral structure: that the so-called framing map is pairing-preserving (Proposition 3.2 below).
3.1. The Equivariant Gamma Class and the Equivariant Gamma-Integral Structure. Let K 0 T (X) denote the Grothendieck group of T -equivariant vector bundles on X. We write H •• T (IX) := p H 2p T (IX). We introduce an orbifold Chern character map ch : K 0 T (X) → H •• T (IX) as follows. Let IX = v∈B X v be the decomposition of the inertia stack IX into connected components, let q v : X v → X be the natural map, and let E be a T -equivariant vector bundle on X. The stabilizer g v along X v acts on the vector bundle q * v E → X v , giving an eigenbundle decomposition where g v acts on E v,f by exp(2πif ). The equivariant Chern character is defined to be . These Chern roots are not actual cohomology classes, but symmetric polynomials in the Chern roots make sense as equivariant cohomology classes on X v . The T -equivariant orbifold Todd class Td(E) ∈ H •• T (IX) is defined to be: We write Td X = Td(T X) for the orbifold Todd class of the tangent bundle.
Recall that, because we are assuming condition (2) from §2.1, all of the T -weights of H 0 (X, O) lie in a strictly convex cone in Lie(T ) * . After changing the identification of T with (C × ) m if necessary, we may assume that this cone is contained within the cone spanned by the standard characters λ 1 , . . . , λ m of H 2 T (pt) = Lie(T ) * defined in §1.7. As is explained in [34], under conditions (1-2) in §2.1 there is a well-defined equivariant Euler characteristic f is the Laurent expansion of a rational function in e λ 1 , . . . , e λm at e λ 1 = · · · = e λm = 0 and we expect that the following equivariant Hirzebruch-Riemann-Roch (HRR) formula should hold: (This holds for toric Deligne-Mumford stacks [34].) Formula (3.3) should be interpreted with care. The right-hand side is defined via the localization formula, and lies in a completion S T of S T : S T := n∈Z a n : a n ∈ S T , deg a n = n, there exists n 0 ∈ Z such that a n = 0 for all n < n 0 There is an inclusion of rings Z[[e λ ]][e −λ ] rat → S T given by Laurent expansion at λ 1 = · · · = λ m = 0 (see [34]), and (3.3) asserts that χ(E) coincides with the right-hand side after this inclusion. We now introduce a lattice in the space of homogeneous flat sections for the quantum connection which is identified with the equivariant K-group of X. The key ingredient in the definition is the characteristic class, called the Gamma class, defined as follows. Let E be a vector bundle on X and consider the bundles E v,f → X v and their equivariant Chern roots δ v,f,i , i = 1, . . . , rank(E v,f ) as above (see (3.1)). The equivariant Gamma class Γ(E) ∈ H •• T (IX) is defined to be: Here the Γ-function on the right-hand side should be expanded as a Taylor series at 1 − f , and then evaluated at where ∪ is the cup product on IX, IX → X is the natural projection, age(q * E) : IX → Q is the locally constant function given by age( T (IX), and inv(v) ∈ B corresponds to the component X inv(v) of IX defined by inv(X v ) = X inv(v) . Note that deg 0 means the degree as a class on IX, not the age-shifted degree as an element of H •• CR,T (X). Definition 3.1. Define the K-group framing by the formula: where k ∈ N is as in Corollary 2.6 and Γ X ∪ is the cup product in H •• T (IX). Corollary 2.6 shows that the image of s is contained in the space of Gr-degree zero flat sections. Note that z −µ maps For T -equivariant vector bundles E, F on X, let χ(E, F ) ∈ Z[[e λ ]][e −λ ] rat denote the equivariant Euler pairing defined by: We use a z-modified version χ z (E, F ) that is given by replacing equivariant parameters λ j in χ(E, F ) with 2πiλ j /z: . Using the unitarity in Proposition 2.4, we have where we set Γ * X = Γ(T * X) and used equation (3.4) in the last line. The last expression equals χ z (E, F ) by the HRR formula (3.3). Remark 3.3. Okounkov-Pandharipande [67] and Braverman-Maulik-Okounkov [15] introduced shift operators S i on quantum cohomology, which induce the shift λ i → λ i + z of equivariant parameters (see [65,Chapter 8] for a detailed description). Our K-theoretic flat sections s(E) are invariant under the shift operators, and our main result suggests that shift operators for toric stacks should be defined globally on the secondary toric variety.
3.2. Specialization of Novikov Variables. In this section we show that the quantum product, the flat sections for the quantum connection, and the K-group framing remain well-defined after the specialization Q = 1 of the Novikov variable Q. Recall that τ 0 , . . . , τ N are co-ordinates on H • CR,T (X) dual to a homogeneous R T -basis {φ 0 , . . . , φ N } of H • CR,T (X), and that: T (X); • φ 1 , . . . , φ r descend to a basis of H 2 (X) = H 2 T (X)/H 2 T (pt).
Without loss of generality we may assume that the images of φ 1 , . . . , φ r in H 2 (X) are nef and integral. It is clear from Remark 2.1 that the specialization Q = 1 of the quantum product is well-defined, and we have: As discussed in Remark 2.1, the product φ i τ φ j is independent of τ 0 . It is explained in [53, §2.5] that the specialization Q = 1 makes sense for L(τ, z), and: The specialization Q = 1 for homogeneous flat sections s(E) in Definition 3.1 (as well as the homogeneous flat sections s i in Corollary 2.6) also makes sense and we have where k ∈ N is such that all the eigenvalues of kµ are integral.

Toric Deligne-Mumford Stacks as GIT Quotients
In the rest of this paper we consider toric Deligne-Mumford stacks X with semi-projective coarse moduli space such that the torus-fixed set X T is non-empty. This is the class of stacks that arise as GIT quotients of a complex vector space by the action of a complex torus. In this section we establish notation and describe basic properties of these quotients. Good introductions to this material include [4,§VII], [37] and [11] We set ∠ ∅ := {0}. Definition 4.2. Consider now a stability condition ω ∈ L ∨ ⊗ R, and set: The square brackets here indicate that X ω is the stack quotient of U ω (which is K-invariant) by K. We call X ω the toric stack associated to the GIT data (K; L; D 1 , . . . , D m ; ω). We refer to elements of A ω as anticones, for reasons which will become clear in §4.2 below. Assumption 4.3. We assume henceforth that: (1) {1, 2, . . . , m} ∈ A ω ; (2) for each I ∈ A ω , the set {D i : i ∈ I} spans L ∨ ⊗ R over R. These are assumptions on the stability condition ω. The first ensures that X ω is non-empty; the second ensures that X ω is a Deligne-Mumford stack. Under these assumptions, A ω is closed under enlargement of sets, i.e. if I ∈ A ω and I ⊂ J then J ∈ A ω . Let S ⊂ {1, 2, . . . , m} denote the set of indices i such that {1, . . . , m} \ {i} / ∈ A ω . It is easy to see that the characters {D i : i ∈ S} are linearly independent and that every element of A ω contains S as a subset. Therefore we can write for some A ω ⊂ 2 {1,...,m}\S and an open subset U ω of C m−|S| . The toric stack X ω can be also written as the quotient [U ω /G] of U ω for G = Ker(K → (C × ) |S| ): this corresponds to the original construction of toric Deligne-Mumford stacks by Borisov-Chen-Smith [11].
The space of stability conditions ω ∈ L ∨ ⊗ R satisfying Assumption 4.3 has a wall and chamber structure. The chamber C ω to which ω belongs is given by and X ω ∼ = X ω as long as ω ∈ C ω . The GIT quotient X ω changes when ω crosses a codimensionone boundary of C ω . We call C ω the extended ample cone; as we will see in §4.5 below, it is the product of the ample cone for X ω with a simplicial cone.

GIT Data and Stacky Fans.
In the foundational work of Borisov-Chen-Smith [11], toric DM stacks are defined in terms of stacky fans. Jiang [55] introduced the notion of an extended stacky fan, which is a stacky fan with extra data. Our GIT data above are in one-to-one correspondence with extended stacky fans satisfying certain conditions, as we now explain. An S-extended stacky fan is a quadruple Σ = (N, Σ, β, S), where: • N is a finitely generated abelian group 4 ; • Σ is a rational simplicial fan in N ⊗ R; To obtain an extended stacky fan from our GIT data, consider the exact sequence: The extended stacky fan Σ ω = (N, Σ ω , β, S) corresponding to our data consists of the group N and the map β defined above, together with a fan Σ ω in N ⊗ R and S given by 5 : The quotient construction in [55, §2] coincides with that in Definition 4.2, and therefore X ω is the toric Deligne-Mumford stack corresponding to Σ ω . Extended stacky fans (N, Σ ω , β, S) corresponding to GIT data satisfy the following conditions: (1) the support |Σ ω | of the fan is convex and full-dimensional; (2) there is a strictly convex piecewise-linear function f : |Σ ω | → R that is linear on each cone of Σ ω ; (3) the map β : Z m → N is surjective. The first two conditions are geometric constraints on X ω : they are equivalent to saying that the corresponding toric stack X ω is semi-projective and has a torus fixed point. The third condition can be always achieved by adding enough extended vectors.
Conversely, given an extended stacky fan Σ = (N, Σ, β, S) satisfying the conditions (1)-(3) just stated, we can obtain GIT data as follows. Define a free Z-module L by the exact sequence (4.3) and define K := L ⊗ C × . The dual of (4.3) is an exact sequence: A ω = I ⊂ {1, 2, · · · , m} : S ⊂ I, σ I is a cone of Σ and take the stability condition ω ∈ L ∨ ⊗ R to lie in I∈Aω ∠ I ; the condition (2) ensures that this intersection is non-empty. This specifies the data in Definition 4.2.

Torus-Equivariant
Cohomology. The action of T = (C × ) m on U ω descends to a Q := T /K-action on X ω . We also consider an ineffective T -action on X ω induced by the projection T → Q. The Q-equivariant and T -equivariant cohomology of X ω are modules over It is well-known that: and I and J are the ideals of additive and multiplicative relations: Note that u i = 0 for i ∈ S because the corresponding divisor (4.6) is empty (see equation 4.1). Indeed, this relation is contained in the ideal J. The T -equivariant cohomology is given by the extension of scalars: We note that the assumptions at the beginning of §2 are satisfied for toric Deligne-Mumford stacks obtained from GIT data. First, all the Q-weights appearing in the Second, X ω is equivariantly formal since the cohomology group of X ω is generated by Q-invariant cycles [47]. Because each component of IX ω is again a toric stack given by certain GIT data (see §4.8), we have that IX ω is also equivariantly formal. The same conclusions hold for the T -action.

Second Cohomology and Homology.
There is a commutative diagram: with exact rows and columns. Note that we have The leftmost column is identified with the exact sequence (4.4). In particular we have The square at the upper left of (4.7) is a pushout and we have: It follows that the middle row of (4.7) splits canonically: we have a well-defined homomorphism 6 . The class θ(p) can be written as the T -equivariant first Chern class of a certain line bundle L(p) associated to p (see §6.3.2). One advantage of working with T -equivariant cohomology instead of Q-equivariant cohomology is the existence of this canonical splitting.
We also introduce a canonical splitting of the projection Take j ∈ S. The corresponding extended vector b j ∈ N ⊗ R lies in the support of the fan. Let σ I j ∈ Σ, I j ⊂ {1, . . . , m} \ S be the minimal cone 7 containing b j and write b j = i∈I j c ij b i for some c ij ∈ R >0 . By the exact sequence (4.3), there exists an element ξ j ∈ L ⊗ Q such that 6 More precisely (−θ) gives a splitting of the middle row of (4.7). 7 Minimality is not essential here. Let σI ∈ Σ, I ⊂ {1, . . . , m}, be any cone containing bj and write bj = i∈I cijbi for some cij ∈ R ≥0 . In our setting, the vectors {bi : i ∈ I} are linearly independent for any choice of cone σI , and so the coefficients cij here are unique. In particular, therefore, we have that cij = 0 for i ∈ I \ Ij.
Note that one has D i · ξ j = δ ij for i, j ∈ S. Hence {ξ i } i∈S spans a complementary subspace of H 2 (X ω ; R) = j∈S Ker(D j ) ⊂ L ⊗ R and defines a splitting: or, for the dual space, The equivariant first Chern class of T X ω is given by: . This is the non-equivariant Poincaré dual of the toric divisor (4.6), that is, the nonequivariant limit of u i . The cone of ample divisors of X ω is given by The Mori cone is the dual cone of C ω : Fixed Points and Isotropy Groups. Fixed points of the T -action on X ω are in one-to-one correspondence with minimal anticones, that is, with δ ∈ A ω such that |δ| = r. A minimal anticone δ corresponds to the T -fixed point: We now describe the isotropy of the Deligne-Mumford stack X ω , i.e. those elements g ∈ K such that the action of g on U ω has fixed points. Recall that there are canonical isomorphisms K ∼ = L ⊗ C × and Lie(K) ∼ = L⊗C, via which the exponential map Lie(K) → K becomes id ⊗ exp(2πi−) : L⊗C → L ⊗ C × . The kernel of the exponential map is L ⊂ L ⊗ C. Define K ⊂ L ⊗ Q to be the set of f ∈ L ⊗ Q such that: The lattice L acts on K by translation, and elements g ∈ K such that the action of g on U ω has fixed points correspond, via the exponential map, to elements of K/L.

4.7.
Floors, Ceilings, and Fractional Parts. For a rational number q, we write: q for the largest integer n such that n ≤ q; q for the smallest integer n such that q ≤ n; and q for the fractional part q − q of q.
4.8. The Inertia Stack and Chen-Ruan Cohomology. Recall the definition of the inertia stack IX ω from §2.1. Components of IX ω are indexed by elements of K/L: the component X f ω of IX ω corresponding to f ∈ K/L consists of the points (x, g) in IX ω such that g = exp(2πif ).
Recall the set I f defined in (4.13).
The component X f ω in the inertia stack IX ω is the toric Deligne-Mumford stack with GIT data given by K, L, and ω exactly as for X ω , and characters D i ∈ L ∨ for i ∈ I f . We have: ω as a closed substack of the toric stack X ω . According to Borisov-Chen-Smith [11], components of the inertia stack of X ω are indexed by elements of the set Box(X ω ): In fact, we have an isomorphism [51, §3.1.3]: When j ∈ S and b j ∈ Box(X ω ), the element −ξ j ∈ L ⊗ Q defined in (4.9) belongs to K and corresponds to b j . The The T -equivariant Chen-Ruan cohomology of X ω is, as we saw in §2.2, the T -equivariant cohomology of the inertia stack IX ω with age-shifted grading: This contains the T -equivariant cohomology of X ω as a summand, corresponding to the element Recall that the component X f ω of the inertia stack is the toric Deligne-Mumford stack with GIT data (K; L; ω; D i : i ∈ I f ). In particular, therefore, the anticones for X f ω are given by {I ∈ A ω : I ⊂ I f }. T -fixed points on the inertia stack IX ω are indexed by pairs (δ, f ) where δ is a minimal anticone in A ω , f ∈ K/L, and D i · f ∈ Z for all i ∈ δ. The pair (δ, f ) determines a T -fixed point on the component X f ω of the inertia stack: the T -fixed point that corresponds to the minimal anticone δ ⊂ I f .

Wall-Crossing in Toric Gromov-Witten Theory
In this section we consider crepant birational transformations X + X − between toric Deligne-Mumford stacks which arise from variation of GIT. We use the Mirror Theorem for toric Deligne-Mumford stacks [26,30] to construct a global equivariant quantum connection over (a certain part of) the secondary toric variety for X ± ; this gives an analytic continuation of the equivariant quantum connections for X + and X − . 5.1. Birational Transformations from Wall-Crossing. Recall that our GIT data in §4.1 consist of a torus K ∼ = (C × ) r , the lattice L = Hom(C × , K) of C × -subgroups of K, and characters D 1 , . . . , D m ∈ L ∨ . Recall further that a choice of stability condition ω ∈ L ∨ ⊗ R satisfying Assumption 4.3 determines a toric Deligne-Mumford stack X ω = U ω /K . The space L ∨ ⊗ R of stability conditions is divided into chambers by the closures of the sets ∠ I , |I| = r − 1, and the Deligne-Mumford stack X ω depends on ω only via the chamber containing ω. For any stability condition ω satisfying Assumption 4.3, the set U ω contains the big torus T = (C × ) m , and thus for any two such stability conditions ω 1 , ω 2 there is a canonical birational map X ω 1 X ω 2 , induced by the identity transformation between T /K ⊂ X ω 1 and T /K ⊂ X ω 2 . Our setup is as follows. Let C + , C − be chambers in L ∨ ⊗ R that are separated by a hyperplane wall W , so that W ∩ C + is a facet of C + , W ∩ C − is a facet of C − , and W ∩ C + = W ∩ C − . Choose stability conditions ω + ∈ C + , ω − ∈ C − satisfying Assumption 4.3 and set X + := X ω + , X − := X ω − , and Then C ± = I∈A ± ∠ I . Let ϕ : X + X − be the birational transformation induced by the toric wall-crossing and suppose that m i=1 D i ∈ W As we will see below this amounts to requiring that ϕ is crepant. Let e ∈ L denote the primitive lattice vector in W ⊥ such that e is positive on C + and negative on C − .
Remark 5.1. The situation considered here is quite general. We do not require X + , X − to have projective coarse moduli space (they are required to be semi-projective). We do not require that X + , X − are weak Fano, or that they satisfy the extended weak Fano condition in [51, §3.1.4]. In other words, we do not require m i=1 D i ∈ W to lie in the boundary W ∩ C + = W ∩ C − of the extended ample cones.
Choose ω 0 from the relative interior of W ∩ C + = W ∩ C − . The stability condition ω 0 does not satisfy our Assumption 4.3, but we can still consider: comes from a Q-Cartier divisor on the underlying singular toric variety X 0 = C m / / ω 0 K associated to the fan Σ ω 0 . On the other hand, in §6.3, we shall construct a toric Deligne-Mumford stack X equipped with proper birational morphisms f ± : X → X ± such that the diagram (1.3) commutes. Then f + (K X + ) and f − (K X − ) coincide since they are the pull-backs of a Q-Cartier divisor on X 0 . This is what is meant by the birational map ϕ being crepant 8 . Set: Our assumptions imply that both M + and M − are non-empty. The following lemma is easy to check: Lemma 5.2. Set: Then one has M 0 ∈ A thin 0 and Remark 5.3. Let Σ ± be the fans of X ± . In terms of fans, a toric wall-crossing can be described as a modification along a circuit [12,40], where 'circuit' means a minimal linearly dependent set of vectors. In our wall-crossing, the relevant circuit is There are three types of possible crepant toric wall-crossings: (I) X + and X − are isomorphic in codimension one ("flop"), (II) ϕ induces a morphism X + → |X − | or X − → |X + | contracting a divisor to a toric subvariety ("crepant resolution") and (III) the rigidifications 9 X rig + , X rig − are isomorphic (only the gerbe structures change; we call it a "gerbe flop"). Define: Proposition 5.4. The intersection S 0 := S + ∩ S − is contained in M 0 . Moreover, one and only one of the following holds: Next we claim that: (III) ϕ induces an isomorphism X rig The geometric picture in each case can be seen from the stacky fans: (I) the sets of one-dimensional cones are the same; (II-i) the fan Σ − is obtained by deleting the ray R ≥0 b i from Σ + ; σ M + ∈ Σ − is a minimal cone containing b i ; ϕ contracts the toric divisor {z i = 0} to the closed subvariety associated with σ M + ; (II-ii) similar; (III) the stacky fan Σ − is obtained from Σ + by Example 5.6.

Decompositions of Extended Ample Cones.
Recall the decomposition (4.11) of the vector space L ∨ ⊗ R and the decomposition (4.12) of the extended ample cone. In the case at hand, we have two (possibly different) decompositions of L ∨ ⊗ R associated to the GIT quotients X + and X − : be a common facet of C + and C − , and write C W for the relative interior of C W . We now show that these decompositions of the cones C + , C − are compatible along the wall.
and so the decompositions (5.1) restrict to the same decomposition of W : Under this decomposition of W , the cone C W decomposes as for some cone C W in W . With cases as in Proposition 5.4, we have: The rest of the statements follow easily. Suppose that i ∈ S 0 . Recall the definition of ξ ± i in (4.9). Let σ I ∈ Σ + be the minimal cone containing b i . Then I ∈ A + . If I ∈ A − , we have ξ + i = ξ − i by the definition of ξ ± i . Suppose that I / ∈ A − . By Lemma 5.2, I contains M − but not M + . We have a relation of the form: with c j > 0. By adding to the right-hand side of (5.3) a suitable positive multiple of the relation given by e ∈ L via (4.3), we obtain a relation of the form is isomorphic to the equivariant quantum connection for X + (respectively for X − ). Thus the equivariant quantum connections for X + and X − can be analytically continued to each other. Roughly speaking, the space M will be a covering of a neighbourhood of a certain curve in the secondary toric variety for X ± ; in this section we introduce notation for and local co-ordinates on this secondary toric variety.
The wall and chamber structure of L ∨ ⊗ R described in §5.1 defines a fan in L ∨ ⊗ R, called the secondary fan or Gelfand-Kapranov-Zelevinsky (GKZ) fan. The toric variety associated to the GKZ fan is called the secondary toric variety. We consider the subfan of the GKZ fan consisting of the cones C + , C − and their faces, and consider the toric variety M associated to this fan. (Thus M is an open subset of the secondary toric variety.) In the context of mirror symmetry, M arises as the moduli space of Landau-Ginzburg models mirror to X ± . It contains the torus fixed points P + and P − associated to the cones C + and C − , which are called the large radius limit points for X + and X − . More precisely, because we want to impose only very weak convergence hypotheses on the equivariant quantum products for X ± , we restrict our attention to the formal neighbourhood of the torus-invariant curve C ⊂ M connecting P + and P − : C is the closed toric subvariety associated to the cone C W = W ∩ C + = W ∩ C − .
Our secondary toric variety M is covered by two open charts Since the cones C ± are not necessarily simplicial, M is in general singular. For our purpose, it is convenient to use a lattice structure different from L and to work with a smooth cover M reg of M. We will define the cover M reg by choosing suitable co-ordinates. As in §4.6, consider the subsets K ± ⊂ L ⊗ Q: and define L + (respectively L − ) to be the Z-submodule of L ⊗ Q generated by K + (respectively by K − ). Note that L + and L − are free (because they are submodules of L ⊗ Q, which is torsion free) of rank equal to the rank of L; they are overlattices of L.
The decomposition (5.1) of L ∨ ⊗ R is compatible with the integral lattice L ∨ ± : one has where we regard H 2 (X ± ; R) as a subspace of L ∨ ⊗ R via the isomorphism H 2 (X ± ; R) ∼ = j∈S ± Ker(ξ ± j ). The lattices L ∨ + and L ∨ − are compatible along the wall; one has (see equation 5.2): Proof. Equation (5.5) holds for both X + and X − and we omit the subscript ± in what follows. Since every element in A contains S, we have D j · f ∈ Z for all j ∈ S and f ∈ K. This shows that Next we prove (5.6). First we claim that for every element f ∈ K + \ K − , there exists α ∈ Q such that f + αe ∈ K − . This follows easily from the definition of K ± and Lemma 5.2. It follows from the claim that for any f ∈ L + , there exists α ∈ Q such that f + αe ∈ L − . Suppose that v ∈ W ∩ L ∨ − . For any f ∈ L + , taking α ∈ Q as above, one has The reverse inclusion follows similarly. The second equality in (5.6) follows from (5.5) and Proposition 5.7.
Remark 5.9. We have H 2 (X ± ; R) ∩ L ± = H 2 (|X ± |; Z). (III) + = − = . Using Lemma 5.8, we can choose 10 integral bases of L ∨ ± such that • p + 1 , . . . , p + + lie in the nef cone C + ⊂ H 2 (X + ; R); . . , . These bases give co-ordinates on the toric charts (5.4). For d ∈ L, we write y d for the corresponding element in the group ring C[L]. The homomorphisms which are resolutions of (respectively) Spec C[C ∨ + ∩ L] and Spec C[C ∨ − ∩ L]. We reorder the bases (5.7) 10 This can be seen from the fact that given a lattice L and a full-dimensional cone C in L ⊗ R, we can choose a basis of L that consists of elements of C.
in such a way that p + i = p − i ∈ W for i = 1, . . . , r − 1 and p ± r is the unique vector (in each basis) that does not lie on the wall W . Let be the corresponding reordering of the co-ordinates. Then the change of co-ordinates is of the form:ỹ for some c ∈ Q >0 and c i ∈ Q. The numbers c i , c here arise from the transition matrix of the two bases (5.7). We find a common denominator for c, c i and write c = A/B and The smooth manifold M reg is defined by gluing the two charts and U − = Spec C[ỹ 1 , . . . ,ỹ r−1 ,ỹ 1/A r ] via the change of variables (5.8). The large radius limit points P + ∈ U + and P − ∈ U − are given respectively by y 1 = · · · = y r = 0 andỹ 1 = · · · =ỹ r = 0. Note that the last variables y r ,ỹ r correspond to the direction of e ∈ L: one has y e = y p + r ·e r The torus-invariant rational curve C reg ⊂ M reg associated to C W is given by y 1 = · · · = y r−1 = 0 on U + and byỹ 1 = · · · =ỹ r−1 = 0 on U − . Let M reg be the formal neighbourhood of C reg in M reg . Since the global quantum connection is an analytic object, we need to work with a suitable analytification of M reg : we include analytic functions in the last variable y r in the structure sheaf and use the analytic topology on C reg ∼ = P 1 . The underlying topological space of M reg is therefore P 1,an ; M reg is covered by two charts U + and U − with structure sheaves: where C + and C − denote the complex plane with co-ordinates y 1/B r andỹ 1/A r respectively and the superscript "an" means analytic (space or structure sheaf). In other words, we regard U + , U − , and M reg as ringed spaces respectively on C + , C − and P 1,an .
The same construction works over an arbitrary C-algebra R. We define M reg (R) by replacing the structure sheaves in (5. Remark 5.10. Taking an overlattice L ± of L corresponds to taking a finite cover of M. This is necessary because the power series defining the I-function (see §5.4) is indexed by elements in L ± . If one takes into consideration Galois symmetry [51] of the quantum connection, one can see that the quantum connection (near P ± ) descends to the secondary toric variety with respect to the original lattice L.

5.4.
The I-Function. Recall Givental's Lagrangian cone introduced in Definition 2.2. We consider the Givental cone L Xω associated to the toric Deligne-Mumford stack X ω . Under the decomposition (4.10) of L ⊗ R, we decompose d ∈ L ⊗ R as: a: a = D j ·d ,a≤D j ·d (u j + az) where K is introduced in §4.6, x = (x j : j ∈ S) and σ ∈ H 2 T (X ω ) are variables, and [−d] is the equivalence class of −d in K/L (recall from §4.8 that K/L parametrizes inertia components). The subscript 'temp' reflects the fact that we are just about to change notation, by specializing certain parameters, and so this notation for the I-function is only temporary. One can see that the summand of I temp ω corresponding to d ∈ K vanishes unless d ∈ C ∨ ω . Therefore the summation ranges over all d ∈ K such that d lies in the Mori cone NE(X ω ) = C ∨ ω and D j · d ≥ 0 for all j ∈ S. The Mirror Theorem for toric Deligne-Mumford stacks can be stated as follows: We adapt the above theorem to the situation of toric wall-crossing. Let I temp ± denote the Ifunction of X ± . We introduce a variant I ± of the I-function which gives a cohomology-valued function on a neighbourhood of P ± in M reg . The I-function I ± is obtained from I temp ± by the following specialization: T (X ± ; C) are the maps introduced in (4.8) and c 0 (λ) = λ 1 + · · · + λ m . Note that we have More explicitly, one can write I + as: where recall that (y 1 , . . . , y r ) = (y i , x j : 1 ≤ i ≤ + , j ∈ S + ) are co-ordinates on U + ⊂ M reg and that The I-function I + belongs to the space: The series e −σ + /z I + (y, z) is homogeneous of degree two with respect to the (age-shifted) grading on H • CR,T (X + ) and the degrees for variables given by: The extra factor e c 0 (λ)/z in the I-function makes the mirror map compatible with Euler vector fields.
We now show that I + (y, z) is analytic in the last variable y r , so that it defines an analytic function on M reg . Lemma 5.13. Expand the I-function as and let U + be the universal covering of the space {y r ∈ C : y p + r ·e r = c}. Each coefficient converges to give an analytic function on the region (5.14) (λ 1 , . . . , λ m , z, y r ) ∈ C m × C × U + : |λ i | < |z| for some > 0 that is independent of i, k 1 , . . . , k r−1 .
Proof. In §6.2 below we will compute the analytic continuation of the I-function after restriction to a fixed point. There we introduce a function called the H-function, which is related to the I-function by a constant linear transformation, and give a Mellin-Barnes integral representation for the H-function. This integral representation makes clear that e −σ + /(2πi) H can be analytically continued to U + . The linear transformation between the I-function and the H-function involves the factor z −µ Γ X (see equation 6.8) and this factor has poles at non-zero (λ 1 /z, . . . , λ m /z). Therefore we obtain the analyticity of the I-function on a region of the form (5.14).
An entirely parallel statement holds for I − (y, z).

Global Equivariant Quantum Connection.
In this section we use the I-function I + to construct a global quantum connection on the universal cover where U + is the open chart (5.9) of M reg and y e = y p + r ·e r is a function on U + . The action of µ B on U + is by deck transformations of y 1/B r → y r . As in Lemma 5.13, we denote by U + the universal cover of {y r ∈ C : y p + r ·e r = c}. The space U + is the underlying topological space of M + , and M + is a formal thickening of U + ; more precisely, M + is the ringed space In a neighbourhood of P + , the global quantum connection that we will construct can be identified with the equivariant quantum connection of X + . The main result in this section is: Theorem 5.14. There exist the following data: • an open subset U • + ⊂ U + such that P + ∈ U • + and that the complement U + \ U • + is a discrete set; we write M • . . , y r−1 ]]; • a flat connection ∇ + = d + z −1 A + (y) on F + of the form: . . , y r−1 ]]; • a vector field E + on M + (R T ), called the Euler vector field, defined by: • a mirror map τ + : M + (R T ) → H • CR,T (X + ) of the form: . . , y r−1 ]] τ + | y 1 =···=yr=0 = 0 such that ∇ + equals the pull-back τ * + ∇ + of the equivariant quantum connection ∇ + of X + by τ + , that is: and that the push-forward of E + by τ + is the Euler vector field E + for X + defined in equation 2.4. Moreover, there exists a global section Υ + 0 (y, z) of F + such that I + (y, z) = zL + (τ + (y), z) −1 Υ + 0 (y, z) where L + (τ, z) is the fundamental solution for the quantum connection of X + in Proposition 2.4.   . By construction, the mirror map τ + here depends on how much we have extended vectors b j , j ∈ S + in the extended stacky fan. If we add sufficiently many extended vectors, we can make it submersive near P + and Theorem 5.14 gives an analytic continuation of the big quantum cohomology. In fact we have α j x j + higher order terms.
Here α j = i∈I j u n ij i 1 [−ξ j ] , where ξ j ∈ K + is given in (4.9), I j ⊂ {1, . . . , m} \ S + is such that σ I j contains b j , and b j = i∈I j (n ij + ij )b i with ij ∈ [0, 1) and n ij ∈ Z ≥0 . Note that 1 [−ξ j ] corresponds to the Box element b j − i∈I j n ij b i ∈ Box(X + ).
Remark 5.19. The logarithmic singularity of ∇ + along j∈S + x j = 0 is not very important: this can be eliminated by shifting the mirror map τ by j∈S + λ j log x j ; see (5.10).
The rest of this section is devoted to the proof of Theorem 5.14. First we recall how to compute the quantum connection of X + using the I-function (cf. [31]). By the Mirror Theorem 5.11, I temp + (σ, x, −z) is a point on the Givental cone L + := L X + for X + . Recall from Remark 2.5 that the cone L + is ruled by its tangent spaces (multiplied by z): This implies that one has: The map (σ, x) → τ (σ, x) is called the mirror map: this will be determined below. In Lemma 5.22 we will construct differential operators P i = P i (z∂), i = 0, . . . , N which depend polynomially on z and on the vector fields z∂ v , v ∈ H 2 T (X + ), and z∂ x j , j ∈ S + , and which satisfy: Then: Here τ * ∇ is the pull-back of the quantum connection of X + via the mirror map τ , and we used the fact that one has ∂ v • L + (τ, z) −1 = L + (τ, z) −1 • (τ * ∇) v for any vector field v on (σ, x)-space. Note that: x]] does not contain negative powers of z; • L + (τ, z) does not contain positive powers of z; and • L + (τ, z) = id +O(z −1 ). Thus the right-hand side of (5.15) can be regarded as the Birkhoff factorization of the left-hand side (see [70]), when we view both sides as elements in the loop group LGL N +1 with z the loop parameter. The properties of P i listed above ensure that the left-hand side of (5.15) is invertible at Q = σ = x = 0, and that its Birkhoff factorization can be determined recursively in powers of Q, σ and x (see Lemma 5.23). Thus the I-function determines L + (τ, z) −1 as a function of (σ, x), via Birkhoff factorization. The mirror map τ = τ (σ, x) is determined by the asymptotics and L + (τ, z) −1 determines the pulled-back quantum connection τ * ∇.
We perform the above procedure globally on M reg , using the I-function I + obtained from I temp + by the specialization Q = 1, σ = σ + . It will be convenient to assume the following condition.
Remark 5.21. Assumption 5.20 is harmless: it can be always achieved by adding enough extended vectors to the extended stacky fan and in fact Theorem 5.14 holds without this assumption (see Remark 5.26).
Recall from §4.8 that H • CR,T (X + ) is the direct sum of sectors H • T (X f + ), f ∈ K + /L and recall from §4.3 that each sector H • T (X f + ) is generated by divisor classes. Thus we can take an R T -basis of H • CR,T (X + ) of the form: where F f,i (a 1 , . . . , a + ) ∈ C[a 1 , . . . , a + ] is a homogeneous polynomial. Recall from §4.8 that elements in K + /L are in one-to-one correspondence with elements in Box(X + ). Let v f ∈ Box(X + ) be the element corresponding to f ∈ K + /L. By Assumption 5.20, there exist non-negative integers n f,j , j = 1, . . . , m, such that On the other hand, taking a minimal cone σ f in Σ + containing v f , we can write for some c f,j ∈ [0, 1). We set c f,j = 0 if j ∈ S + or b j / ∈ σ f . Then m j=1 (n f,j − c f,j )b j = 0 and by (4.3), there exists an element d f ∈ L ⊗ Q such that D j · d f = n f,j − c f,j . By definition of K + , d f ∈ K + and [−d f ] = f in K + /L by (4.14) and (5.16). Set D j = r a=1 µ ja p + a for some µ ja ∈ Z.
Define differential operators D j , ∆ f as The following Lemma was proved in [51,Lemma 4.7], in the non-equivariant and compact case. The proof works verbatim here.
Lemma 5. 22. Let F f,i , φ f,i , ∆ f be as above. Define the differential operator P + f,i by Then we have: ).

Applying the differential operators
, to I + , we obtain a matrix of the form: where I + is regarded as a column vector written in the basis {φ f,i } of H • CR,T (X + ) and I + (y, z) = id +O(y) is a square matrix. We may also view I + (y, z) as an End(H • CR (X + ))-valued function via the basis {φ f,i }. By the homogeneity of e −σ + /z I + and P + f,i , we find that the endomorphism I + (y, z) is homogeneous of degree-zero with respect to the degree (5.12) of variables and the grading on H • CR (X + ), i.e. that: (5.18) z ∂ ∂z + E + + ad(µ + ) I + (y, z) = 0 As in (5.15), we consider the Birkhoff factorization of (5.17). Since e σ + /z = id +O(z −1 ), it suffices to consider the Birkhoff factorization of I + (y, z). Set: γ(y r , z) := I + (y, z) . By Lemma 5.13, z → γ(y r , z) is a loop in End(H • CR (X + )) that depends analytically on y r ∈ U + . We first consider the Birkhoff factorization of γ(y r , z). Since γ(y r , z) is homogeneous, it is a Laurent polynomial in z and both factors of the Birkhoff factorization γ(y r , z) = γ − (z)γ + (z) are also homogeneous if the factorization exists. Therefore the Birkhoff factorization is equivalent to the block LU decomposition of γ(y r , 1): where each block corresponds to a homogeneous component of H • CR (X + ) and I r denotes the identity matrix of size r. The block LU decomposition of γ(y r , 1) exists if and only if H = (γ(y r , 1)H ≤p ) ⊕ H >p holds for all p ∈ Q, where H = H • CR (X + ) and H ≤p (resp. H >p ) denotes the subspace of degree less than or equal to p (resp. greater than p). This is a Zariski open condition for γ(y r , 1). Since γ(y r = 0, 1) = id, it follows that γ(y r , z) admits a Birkhoff factorization on the complement U • + of a discrete set in U + . Clearly one has P + ∈ U • + . Proof. It suffices to show that Γ = γ −1 − Γγ −1 + admits a Birkhoff factorization Γ = Γ − Γ + . Expanding Γ and Γ ± in power series in s 1 , . . . , s l , one can determine the coefficients recursively from the equation Γ = Γ − Γ + .
By equation (5.20), L + (y, z) determines the quantum connection pulled-back by the mirror map τ + (y). Set τ * where the term dσ + gives a logarithmic singularity (see equation 5.10): Thus the connection form A + (y) is a global 1-form on M + satisfying the properties in Theorem 5.14.
Remark 5.25. Note that A + (y) is independent of z: in the formal neighbourhood of P + = {y 1 = · · · = y r = 0} this follows from the fact that d + z −1 A + (y) is the pulled-back quantum connection, and this is true everywhere by analytic continuation.
Finally we see that E + corresponds to E + . Choose a homogeneous R T -basis {φ i } of H • CR,T (X + ) such that φ 0 = 1 and φ i = θ(p + i ) for 1 ≤ i ≤ + and write τ i + (y) for the ith component of τ + (y) with respect to this basis. One needs to check that E + τ i The homogeneity of L −1 + shows thatτ + (y) is homogeneous of (real) degree two: this implies that E +τ i . The rest is a straightforward computation. This completes the proof of Theorem 5.14.
Remark 5.26. For the existence of a global quantum connection in Theorem 5.14 and other main results in this paper, we do not need Assumption 5.20. Let us write M S + for M + to emphasize the dependence on the extension data S. Then one has: Suppose that an S-extended stacky fan does not satisfy Assumption 5.20. By taking a bigger S ⊃ S, we can achieve Assumption 5.20 and construct a global quantum connection on M S + . Then we obtain a global quantum connection on M S + by restriction. In this way, the global quantum connections form a projective system over all extension data S. Assumption 5.20 ensures that F + is generated by a section Υ + 0 and its covariant derivatives. For the convenience of discussion, we will sometimes use Assumption 5.20 in the rest of the paper, but this does not affect the final conclusion.

The Crepant Resolution Conjecture
We now come to the main result in this paper. In Theorem 5.14, we constructed a global quantum is an open subset of the universal cover M + of ( U + \ {y e = c})/µ B . By applying Theorem 5.14 to X − rather than X + , we obtain a global quantum connection ( = c −1 }. The underlying topological space of M is the universal cover U of C reg \ {y e = 0, c, ∞}. We have natural maps π ± : U → U ± and set is the open dense subset from Theorem 5.14. Note that U \ U • is a discrete set. Since we use P ± ∈ C reg as base points of the universal covers U ± , we need to specify a path from P + to P − in C reg \ {y e = c} in order to identify the maps U → U ± between universal covers. We consider a path in the log(y e )-plane starting from log(y e ) = −∞ and ending at log(y e ) = ∞ such that it avoids log(c) + 2πiZ. We use a path γ as in Figure 1 passing through the interval log |c| + πi(w − 1), log |c| + πi(w + 1) in the log(y e )-plane, where w := −1 − j:D j ·e<0 (D j · e) = −1 + j:D j ·e>0 (D j · e). (1) I + (y, z) = UI − (y, z) after analytic continuation in y e along the path γ in Figure 1; (2) U•(g − v∪) = (g + v∪)•U for all v ∈ H 2 T (X 0 ), where g ± : X ± → X 0 is the common blow-down appearing in the diagram (1.3); (3) there exists a Fourier-Mukai transformation FM : K 0 T (X − ) → K 0 T (X + ) such that the following diagram commutes: where the vertical map Ψ ± : taking values 13 in the "multi-valued Givental space": Here k ∈ N is an integer such that all the eigenvalues of kµ + , kµ − are integers. Theorem 6.1 will be proved in §6.2 and §6.3. The Fourier-Mukai kernel will be described in §6.3: it is given by a toric common blow-up X of X ± . 12 We use the usual grading on H • CR,T (X±), RT = H • T (pt) and set deg z = 2. 13 Cf. Corollary 2.6. Notation 6.2. In Theorem 6.1, ρ ± = c T 1 (T X ± ) ∈ H 2 T (X ± ), µ ± is the grading operator (2.4) on H • CR,T (X ± ) and deg 0 : is the degree operator as in §3.1.
Remark 6.4. The symplectic transformation U in Theorem 6.1 and the gauge transformation Θ in Theorem 6.3 are related by where L ± is the fundamental solution for the quantum connection of X ± in Proposition 2.4. The gauge transformation Θ sends the section Υ − 0 ∈ F − to the section Υ + 0 ∈ F + , where Υ ± 0 are as in Theorem 5.14.
Remark 6.5. Theorems 6.1 and 6.3 can be interpreted as the statement that the symplectic transformation U matches up the Givental cones L ± associated to X ± after analytic continuation of L ± : U(−z)L − = L + . In fact, Remark 2.5 suggests that we may analytically continue the Lagrangian cones by the formula: and then equation (6.2) would imply (6.3). As discussed in the Introduction, to avoid subtleties in defining the analytic continuation of Givental cones in the equivariant setting, in this paper we state our results in terms of analytic continuation of the I-function (Theorem 6.1) or in terms of the equivariant quantum connection and gauge transformations (Theorem 6.3).
Remark 6.6. Theorem 6.3 implies that the global quantum connections of X + and X − can be glued together to give a flat connection over M • . This flat connection descends to the formal neighbourhood M of C in the secondary toric variety M via Galois symmetry as in Remark 5.10. This global connection, or D-module, on M can be described by explicit GKZ-type differential equations: it is a completed version of Borisov-Horja's better-behaved GKZ system 14 [14]. In the papers [51,71], the toric quantum connection is described in terms of GKZ-type differential equations through mirror symmetry. The I-functions I ± (q, z) are local solutions to these differential equations around the large radius limit points.
Proof that Theorem 6.1 implies Theorem 6.3. One can easily check that the change of variables (5.8) preserves degree, and that E + = E − . By Theorem 6.1, we have Similarly, the discussion in §5.5 applied to X − yields a global section Υ − 0 of F − and a (global) fundamental solution L − (y, z)e −σ − /z for ∇ − = d + z −1 A − (y) such that: Comparing (6.5) with (6.6) and using (6.4), since U is independent of the base variables y. In particular, it follows that Υ − is invertible. Setting Θ = Υ + ( Υ − ) −1 , we obtain: Since e σ ± /z L −1 ± are fundamental solutions for ∇ ± , Θ gives a gauge transformation between ∇ − and ∇ + , i.e. Θ • ∇ − = ∇ + • Θ. One may assume that the first columns of Υ + and Υ − are given respectively by Υ + 0 and Υ − 0 , and therefore Θ(Υ − 0 ) = Υ + 0 . Next we see that Θ preserves the grading and the pairing. Part (2) in Theorem 6.1 implies that This together with (6.7) implies that: Since deg y r = 0, we know that all of the factors in this equation except for Θ are homogeneous of degree zero; thus Θ is also homogeneous of degree zero. The fundamental solutions e σ ± /z L −1 ± preserve the pairing by Proposition 2.4 (we saw in §5.5 that they coincide with the fundamental solutions from Proposition 2.4 via the mirror maps τ ± ) and U also preserves the pairing. Thus Θ preserves the pairing. Finally we consider the analytic continuation of K-theoretic flat sections. Note that the flat section s(E)(τ − (y), z) is analytically continued along M • by the right-hand side of the formula where Ψ − is the map in Theorem 6.1. Using (6.7), we obtain: Part (3) of Theorem 6.1 shows that this is equal to s(FM(E))(τ + (y), z).

Mellin-Barnes Analytic Continuation.
In this section, we compute the analytic continuation of the I-function and determine the linear transformation U in Theorem 6.1. 6.2.1. The H-Function. It will be convenient to introduce another cohomology-valued hypergeometric function called the H-function. Noting that the I-function can be written in terms of ratios of Γ-functions: we set: and similarly for H − . Formally speaking, H + belongs to the space: Noting that the T -equivariant Gamma class of X + is given by we obtain the relationship between the H-function and the I-function: where ρ + , µ + , deg 0 are as in Notation 6.2 and The relationship between H − and I − is similar.
Remark 6.7. The H-function H + has an analytic properties analogous to those of the I-function stated in Lemma 5.13. Namely e −σ + /(2πi) H + (y) is a formal power series in y 1 , . . . , y r−1 with coefficients of the form N i=0 f i (λ, y r )φ i where {φ i } is an R T -basis of H • T (IX + ) and f i (λ, y r ) is analytic in (λ 1 , . . . , λ r , y r ) ∈ C m × U + . Note that the H-function has better analytic behaviour with respect to λ. The analytic continuation of H-functions performed below should be understood as analytic continuation of the coefficient functions f i (λ, y r ).

Restriction of the H-Function to T -Fixed Points.
Recall that the T -fixed points on X + are indexed by minimal anticones δ ∈ A + , and that the T -fixed points on the inertia stack IX + are indexed by pairs (δ, f ) with δ ∈ A + a minimal anticone and f ∈ K + /L satisfying D i · f ∈ Z for all i ∈ δ. The minimal anticone δ determines a T -fixed point x δ on X + and the pair (δ, f ) determines a T -fixed point x (δ,f ) on the component X f + of the inertia stack IX + . Let i δ and i (δ,f ) denote the inclusion maps x δ → X + and x (δ,f ) → IX + respectively. Set u j (δ) = i δ u j ∈ H 2 T (pt), noting that u j (δ) = 0 if and only if j ∈ δ. We have that: where σ + (δ) := i δ σ + . Consider the factor j∈δ Γ 1 + D j · d −1 in the summand: since d ≡ f mod L and since D j · f ∈ Z for all j ∈ δ, the term D j · d here is an integer. Thus the factor The H-function is a sum over the subset K eff + of K + , K eff + = f ∈ L ⊗ Q : i ∈ {1, 2, . . . , m} : D i · f ∈ Z ≥0 ∈ A + which is in general quite complicated, but the restriction i (δ,f ) H + of H + to a T -fixed point in IX + is a sum over the much simpler set δ ∨ . 6.2.3. Analytic Continuation of the H-Function. The Localization Theorem in T -equivariant cohomology [3,8,47] implies that one can compute the analytic continuation of H + by computing the analytic continuation of the restriction i (δ,f ) H + to each T -fixed point x (δ,f ) ∈ IX + . The restriction i (δ,f ) H + is a H •• T (pt)-valued function. During the course of analytic continuation, we regard the equivariant parameters λ 1 , . . . , λ m as generic complex numbers. There are two cases: • δ ∈ A + ∩ A − ; • δ ∈ A + but δ ∈ A − . The anticone δ determines a T -fixed point x δ in X + , and in the first case it also determines a fixed point in X − . In the first case the birational transformation ϕ : X + X − is an isomorphism in a neighbourhood of x δ , and it is clear from (6.9) that i (δ,f ) H + = i (δ,f ) H − , noting that u j (δ) is the same for X + and X − . In the second case x δ lies in the flopping locus of ϕ, and we will see that the analytic continuation of i (δ,f ) H + is a linear combination of restrictions i (δ − ,f − ) H − for appropriate δ − ∈ A − and f − ∈ K − . Note that in the second case, δ has the form {j 1 , . . . , j r−1 , j + } with D j 1 , . . . , D j r−1 ∈ W and 15 D j+ · e > 0 (see Lemma 5.2). Definition 6.8. Let δ + ∈ A + and δ − ∈ A − be minimal anticones. We say that δ + is next to δ − , written δ + |δ − , if δ + = {j 1 , . . . , j r−1 , j + } and δ − = {j 1 , . . . , j r−1 , j − } with D j 1 , . . . , D j r−1 ∈ W , D j + · e > 0, and D j− · e < 0. In this case δ + / ∈ A − and δ − / ∈ A + .
Proof of Theorem 6.13. The first statement follows immediately from (6.9) and Corollary 6.12. In this case, i (δ + ,f + ) H + (respectively i (δ + ,f + ) H − ) is a formal power series in y 1 , . . . , y r−1 (respectively inỹ 1 , . . . ,ỹ r−1 ) with coefficients that are polynomials in y r (respectively inỹ r ), and the series i (δ + ,f + ) H + , i (δ + ,f + ) H − match under the change (5.8) of co-ordinates. Consider now where δ + ∈ A + but δ + ∈ A − . We can write d ∈ δ ∨ + uniquely as d = d + + ke with k a non-negative integer, d + ∈ δ ∨ + , and d + − e ∈ δ ∨ + . Then: Consider the second sum here. This is: where we used Γ(y)Γ(1 − y) = π/(sin πy). Thus (6.13) is: Res s=k Γ(s)Γ(1 − s)e πis (y e ) s j:D j ·e<0 Consider now the contour integral 2πi + D j · d + + sD j · e e −πiw y e s ds where the contour C, shown in Figure 2, is chosen such that the poles at s = n are on the right of C and the poles at s = −1 − n and at are on the left of C; here n is a non-negative integer. Note that all poles of the integrand are simple. By assumption we have that m j=1 D j ∈ W , and hence that m j=1 D j · e = 0. Let c ∈ C be the conifold point (5.13). Lemma A.6 in [12] implies that: • the contour integral (6.15) is convergent and analytic as a function of y e in the domain {y e : | arg(y e ) − wπ| < π}; • for |y e | < |c|, the integral is equal to the sum of residues on the right of C; and • for |y e | > |c|, the integral is equal to minus the sum of residues on the left of C. The residues at s = −1 − n vanish, where n is a non-negative integer: each such residue contains a factor j∈δ + Γ 1 + D j · d + − (n + 1)e −1 and d + − (n + 1)e ∈ δ ∨ + , so at least one of the Γ-functions is evaluated at a negative integer. After analytic continuation in y e , therefore, (6.14) becomes minus the sum of residues at the poles (6.16). The residue at the pole This simplifies dramatically. Set n = k(−D j − · e) + l with 0 ≤ l < (−D j − · e), and thus the residue (6.17) is: where we used p = − k and Corollary 6.12.
Let f − denote the equivalence class of d − in K − /L, noting that (δ + , f + )|(δ − , f − ) and that for some integer N . (Here we used Notation 6.10.) The dependence of (6.18) on N cancels, giving: and minus the sum of these residues gives the analytic continuation of (6.14). After analytic continuation in y e = y p + r ·e r , therefore, we have that: where (δ, f ) and (δ + , f + ) index T -fixed points in IX + , 1 δ,f = i (δ,f ) 1 and N δ,f := T x (δ,f ) X f + . Then Theorem 6.13 can be restated as: H + = U H H − Define the linear transformation U so that the following diagram commutes: where the vertical maps are defined by Ψ ± (α) = z −µ ± z ρ ± ( Γ X ± ∪ (2πi) deg 0 2 inv * α) and k ∈ N is as in Theorem 6.1. The relationship (6.8) between the H-function and the I-function implies part (1) of Theorem 6.1: Since the I-function contains neither log z nor non-integral powers of z, it follows that U is in fact a linear transformation: (6.20) gives that U is automatically degree-preserving. We show that U satisfies part (2) of Theorem 6.1. Noting that p + i = p − i , i = 1, . . . , r − 1 are on the wall W , it suffices to show that This follows from equation (6.21) and the monodromy properties of the I-functions: Note that y j → e 2πi y j corresponds toỹ j → e 2πiỹ j under the change (5.8) of variables. It remains to show that: • U is symplectic; • U is defined over R T ((z −1 )), i.e. that U admits a non-equivariant limit. These properties follow from the identification of U H with the Fourier-Mukai transformation defined in the next section. We will discuss these points in §6.5 below.
6.3. The Fourier-Mukai Transform. We now construct a diagram (1.1) canonically associated to the toric birational transformation ϕ : X + X − , where X is a toric Deligne-Mumford stack and f + , f − are toric blow-ups, and compute the Fourier-Mukai transformation: In §6.4 below we will see that this transformation coincides, via the equivariant integral structure in Definition 3.1, with the transformation U from §6.2.4 given by analytic continuation. 6.3.1. The Common Blow-Up of X + and X − . Recall from §4.2 that X + and X − are defined in terms of an exact sequence: where the map L → Z m is given by (D 1 , . . . , D m ). This sequence defines an action of K = (C × ) r on C m , and X ± = U ω ± K for appropriate stability conditions ω + , ω − ∈ L ∨ ⊗ R. Let b 1 , . . . , b m denote the images of the standard basis elements for Z m under the map β. Consider now the action of K × C × on C m+1 defined by the exact sequence: The mapβ is the direct sum of β with the map Z → N defined by the element so the images of the standard basis elements for Z m ⊕Z under the mapβ are b 1 , . . . , b m+1 . Consider the chambers C + , C − , and C in (L ⊕ Z) ∨ ⊗ R that contain, respectively, the stability conditions where ω 0 is a point in the relative interior of W ∩ C + = W ∩ C − as in §5.1, and ε is a very small positive real number. Let X denote the toric Deligne-Mumford stack defined by the stability conditionω. Proof. Straightforward. Lemma 6.16. We have the following statements.
(1) The toric Deligne-Mumford stack corresponding to the chamber C + is X + .
(2) The toric Deligne-Mumford stack corresponding to the chamber C − is X − .
(3) There is a commutative diagram as in (1.1) , where: • f + : X → X + is a toric blow-up, arising from wall-crossing from the chamber C to C + ; and • f − : X → X − is a toric blow-up, arising from wall-crossing from the chamber C to C − .
Proof. In view of §4.1, the description of Aω ± in Lemma 6.15 proves (1) and (2). The birational transformations f + : X X + and f − : X X − determined by the toric wall-crossings are each morphisms which contract the toric divisor defined by the (m + 1)-st homogeneous co-ordinate. Indeed, f + is induced by the identity birational map Uω Uω + , and a point (z 1 , . . . , z m , z m+1 ) ∈ Uω + is equivalent to the point (z 1 z l 1 m+1 , . . . , z m z lm m+1 , 1) ∈ U ω + × {1} under the action of the C ×subgroup of K × C × corresponding to e ⊕ 1 ∈ L ⊕ Z, where we set l i := max(−D i · e, 0) for 1 ≤ i ≤ m. Therefore f + is induced by a morphism (6.22) Uω → U ω + (z 1 , . . . , z m , z m+1 ) → (z 1 z l 1 m+1 , . . . , z m z lm m+1 ) which is equivariant with respect to the group homomorphism (quotient by the C × -subgroup given by e ⊕ 1) Using Lemma 6.15, one can easily check that the map (6.22) indeed sends Uω to U ω + . We obtain a similar description for f − by considering the C × -subgroup given by 0 ⊕ 1 ∈ L ⊕ Z instead of e ⊕ 1.
Remark 6.17. Torus fixed points on X lying on the exceptional divisor {z m+1 = 0} of f ± correspond to minimal anticonesδ ∈ Aω such thatδ ∈ A thick 0 andδ ∩ M 0 ∈ A thin 0 . These minimal anticones take the formδ = {j 1 , . . . , j r−1 , j + , j − } where D j 1 , . . . , D j r−1 ∈ W , D j + · e > 0 and D j − · e < 0. The birational morphism f ± maps the corresponding torus fixed point xδ ∈ X to the torus fixed point x δ ± ∈ X ± with Torus fixed points on X lying away from the exceptional divisor {z m+1 = 0} corresponds to minimal anticonesδ ∈ Aω of the formδ = δ ∪ {m + 1}, δ ∈ A thick 0 = A + ∩ A − . The morphisms f ± are isomorphisms in neighbourhoods of these fixed points, and the torus fixed point xδ maps to the fixed point x δ in X + or in X − . Remark 6.18. The stacky fan Σ for X is obtained from the stacky fans Σ ± for X ± by adding the extra ray b m+1 = j: is a minimal linear relation (or circuit) in Σ ± , see Remark 5.3. So our discussion here is a rephrasing in terms of GIT data of the material in [12, §5]. The T -invariant divisor {z i = 0} on X ω defined in (4.6) determines a T -equivariant line bundle O({z i = 0}) on X ω , and we denote the class of this line bundle in T -equivariant K-theory by R i . For the spaces X + , X − , and X we write these classes as Let us write: An irreducible K-representation p ∈ Hom(K, C × ) = L ∨ defines a line bundle L(p) → X ω : , v], t ∈ T and thus defines a class in K 0 T (X ω ). We write L + (p) for the corresponding line bundle on X + and L − (p) for the corresponding line bundle on X − . We have stands for the irreducible T -representation given by the ith projection T → C × . In particular we have c T 1 (L ± (p)) = θ ± (p) for the map θ in (4.8). Similarly a character (p, n) ∈ Hom(K × C × , C × ) = L ∨ ⊕ Z defines a T -equivariant line bundle L(p, n) → X and we have: The classes L ± (p) (respectively the classes L(p, n)) generate the equivariant K-group K 0 T (X ± ) (respectively K 0 T ( X)) over Z[T ]. Let δ − ∈ A − be a minimal anticone, x δ − be the corresponding T -fixed point on X − , i δ − : x δ − → X − be the inclusion of the fixed point, and G δ − be the isotropy group of x δ − . We have that x δ − ∼ = BG δ − , and that i δ − R i = 1 for i ∈ δ − . A basis for K 0 T (X − ), after inverting non-zero elements of Z[T ], is given by: We need to specify a T -linearization on (i δ − ) . Choosing a liftˆ ∈ Hom(K, C × ) = L ∨ of each G δ − -representation : G δ − → C × , we write any element in (6.24) in the form: Then {e δ − , } is a basis for the localized T -equivariant K-theory of X − . There is an entirely analogous basis {e δ + , } for the localized T -equivariant K-theory of X + . We will describe the action of the Fourier-Mukai transform in terms of these bases.

6.3.3.
Computing the Fourier-Mukai Transform. Consider the diagram (1.1) and the associated Fourier-Mukai transform FM : K 0 T (X − ) → K 0 T (X + ). In this section we prove: Theorem 6.19. If δ − ∈ A − is a minimal anticone such that δ − ∈ A + then FM(e δ − , ) = e δ − , where on the left-hand side of the equality δ − is regarded as a minimal anticone for X − and on the right-hand side δ − is regarded as a minimal anticone for X + . If δ − is a minimal anticone in A − such that δ − ∈ A + then FM(e δ − , ) is equal to where j − is the unique element of δ − such that D j − · e < 0, l = −D j − · e and T = ζ · (R + j − ) 1/l : ζ ∈ µ l . Remark 6.20. We have 1 l t∈T t n = (R + j − ) n/l if l divides n; 0 otherwise.
Thus 1 l t∈T f (t) makes sense as an element K 0 T (X + ) for a Laurent polynomial f (t) in t. Note that each summand appearing in the formula for FM(e δ − , ) is in fact a Laurent polynomial in t, since the factor  Let k i := max(D i · e, 0) and l i := max(−D i · e, 0). Then: Proof. These statements follows from the description of f ± : X → X ± in the proof of Lemma 6.16; see (6.22) and (6.23).
We now analyze the push-forward of classes supported on torus fixed points of X.
Proof of Theorem 6.19. Suppose first that δ − ∈ A + ∩ A − . Then, as discussed, ϕ gives an isomorphism between neighbourhoods of the fixed points corresponding to δ − . Thus FM(e δ − , ) = e δ − , . Suppose now that δ − ∈ A − but δ − ∈ A + , so that δ − = {j 1 , . . . , j r−1 , j − } with D j 1 · e = · · · = D j r−1 · e = 0 and D j − · e < 0. Proposition 6.21 gives: where the index i in the product satisfies i ≤ m. This restricts to zero at a fixed point xδ ∈ X unless xδ is in f −1 + (x δ − ), that is, unlessδ has the form δ − ∪ {j + } with D j + · e > 0. The Localization Theorem in T -equivariant K-theory [34] gives: where i, j ≤ m and the sum runs overδ = δ − ∪ {j + } such that D j + · e > 0. Restricted to such a T -fixed point, S j + becomes trivial, so the numerator in (6.26) contains a factor (iδ) (1 − S k j + m+1 ) that is divisible by (iδ) (1 − S m+1 ). Thus (6.26) depends polynomially on S m+1 . Now: where we used part (3) of Proposition 6.22. This is: Applying the Localization Theorem again gives the result. Here we need to check that the restriction of to the fixed point corresponding to δ ∈ A + ∩ A − vanishes. If there exists j ∈ δ with j / ∈ δ − and D j · e ≤ 0 then the restriction vanishes since i δ S + j = 1. Otherwise one has δ \ δ − ⊂ M + . In this case j − ∈ δ and there exists j 0 ∈ δ ∩ M + . Thus the restriction contains the factor We start by computing the Chern characters of certain line bundles. It is easy to see that: In view of this, we define ch(t) := f ∈K + /L ζe 2πiD j − ·f /l e u j − /l 1 f for t = ζ(R + j − ) 1/l ∈ T appearing in Theorem 6.19. Here we fix lifts K + /L → K + , K − /L → K − as in Notation 6.10 and identify f ∈ K + /L with its lift in K + . Lemma 6.24. Suppose that (δ + , f + ) indexes a T -fixed point on X + , that (δ − , f − ) indexes a T -fixed point on X − , and that (δ + , f + )|(δ − , f − ). Let j − ∈ δ − be the unique index such that D j − · e < 0 and write l = −D j − · e. Setting t = e −2πiD j − ·f − /l (R + j − ) 1/l , we have: We also have: and: (δ + ,f + ) are the coefficients appearing in Theorem 6.13. Proof. This is just a calculation. Recall from Notation 6.10 that f − = f + + αe for some α ∈ Q.
The formulae (6.28) easily follow from Lemma 6.11 and (6.11). The formula (6.29) is an easy consequence of (6.28). To see (6.30), we calculate, using (6.28), where we used the fact that D j − · f − ∈ Z. Using Lemma 6.11 again to calculate the exponential factor, we arrive at the expression for C Proof of Theorem 6.23. We first show that the commutative diagram holds over S T . Then it follows that U H has a non-equivariant limit, as FM does. Consider the element e δ, ∈ K 0 T (X − ) with δ ∈ A + ∩ A − . Theorem 6.19 and the definition (6.19) of U H show that ch(FM(e δ, )) = ch(e δ, ) = U H ( ch(e δ, )).
Consider now e δ − , ∈ K 0 T (X − ) for δ − ∈ A − \ A + . It is clear that ch(FM(e δ − , )) is supported only on fixed points x (δ + ,f + ) ∈ IX + such that δ + |δ − . By the definition (6.19) of U H , it suffices to show that: We may rewrite the result in Theorem 6.19 as We have a one-to-one correspondence between the index of summation f − in (6.31) and the index of summation t ∈ T in (6.32) given by is the unique element satisfying D j − · e < 0 and l = −D j − · e. Therefore (6.31) follows from (6.32), (6.29) and (6.30). The Theorem is proved. 6.5. Completing the Proof of Theorem 6.1. Combining the commutative diagrams (6.20) and (6.27), we obtain the commutative diagram (6.1) in Theorem 6.1. Since the Fourier-Mukai transformation FM can be defined non-equivariantly, U also admits a non-equivariant limit. Finally we show that U is symplectic, i.e. that (U(−z)α, U(z)β) = (α, β) for all α, β. Since FM is induced by an equivalence of derived categories [34], it preserves the Euler pairing χ(E, F ) given in (3.5). The proof of Proposition 3.2 shows that the vertical maps Ψ ± in (6.1) preserve the pairing in the sense that: The commutative diagram (6.1) now shows that U is symplectic. This completes the proof of Theorem 6.1.

Toric Complete Intersections
We now turn to the Crepant Transformation Conjecture for toric complete intersections. Consider toric Deligne-Mumford stacks X ± of the form C m / / ω K , where K = (C × ) r is a complex torus, and consider a K-equivalence ϕ : X + X − determined by a wall-crossing in the space of stability conditions ω as in §5. We use notation as there, so that L = Hom(C × , K) is the lattice of cocharacters of K; the space of stability conditions is L ∨ ⊗ R; and the birational map ϕ is induced by the wall-crossing from a chamber C + ⊂ L ∨ ⊗ R to a chamber C − ⊂ L ∨ ⊗ R, where C + and C − are separated by a wall W . Consider characters E 1 , . . . , E k of K such that: (7.1) • each E i lies in W ∩ C + = W ∩ C − ; • for each i, the line bundle L X + (E i ) → X + corresponding to E i is a pull-back from the coarse moduli space |X + |; • for each i, the line bundle L X − (E i ) → X − corresponding to E i is a pull-back from the coarse moduli space |X − |; where L X ± (E i ) are the line bundles on X ± associated to the character E i in §6.3.2. Let: Let s + , s − be regular sections of, respectively, the vector bundles E + → X + and E − → X − such that: • s + and s − are compatible via ϕ : X + X − ; • the zero loci of s ± intersect the flopping locus of ϕ transversely; and let Y + ⊂ X + , Y − ⊂ X − be the complete intersection substacks defined by s + , s − . The birational transformation ϕ then induces a K-equivalence ϕ : Y + Y − . In this section we establish the Crepant Transformation Conjecture for ϕ : Y + Y − . 7.1. The Ambient Part of Quantum Cohomology. Under our standing hypotheses on the ambient toric stacks X ± , the complete intersections Y ± automatically have semi-projective coarse moduli spaces, and so the (non-equivariant) quantum products on H • CR (Y ± ) are well-defined. Thus we have a well-defined quantum connection where τ is the non-equivariant big quantum product, defined exactly as in (2.2). This is a pencil ∇ of flat connections on the trivial H • CR (Y ± )-bundle over an open set in H • CR (Y ± ); here, as in the equivariant case, z ∈ C × is the pencil variable, τ ∈ H • CR (Y ± ) is the co-ordinate on the base of the bundle, φ 0 , . . . , φ N are a basis for H • CR (Y ± ), and τ 0 , . . . , τ N are the corresponding co-ordinates of τ ∈ H • CR (Y ± ), so that τ = N i=0 τ i φ i . We consider now a similar structure on the ambient part of H • CR (Y ± ), that is, on: H • amb (Y ± ) := im ι ± ⊂ H • CR (Y ± ) where ι ± : Y ± → X ± are the inclusion maps. If τ ∈ H • amb (Y ± ) then the big quantum product τ preserves H • amb (Y ± ) [53, Corollary 2.5], and so (7.2) restricts to give a well-defined quantum connection on the ambient part of H • CR (Y ± ). The restriction of the fundamental solution L ± (τ, z) for (7.2), defined exactly as in (2.8), gives a fundamental solution L amb ± (τ, z) for the quantum connection on the ambient part.
There is also an ambient part of K 0 (Y ± ), given by K 0 amb (Y ± ) := im ι ± , and an ambient K-group framing (cf. Definition 3.1) where µ and ρ are the grading operator and first Chern class for Y ± , k ∈ N is such that the eigenvalues of kµ are integers, and Γ Y ± is the non-equivariant Γ-class of Y ± . As in §3, the image of s is contained in the space of flat sections for the quantum connection on the ambient part of H • CR (Y ± ) which are homogeneous of degree zero. 7.2. I-Functions for Toric Complete Intersections. Recall from §5.4 that the GIT data for X + determine a cohomology-valued hypergeometric function I + . The I-function I X + := I + is a multi-valued function of y 1 , . . . , y r , depending analytically on y r and formally on y 1 , . . . y r−1 , defined near the large-radius limit point (y 1 , . . . , y r ) = (0, . . . , 0) in M reg . The GIT data for the total space of E ∨ + (regarded as a non-compact toric stack) is obtained from the GIT data for X + by adding extra toric divisors −E 1 , . . . , −E k . It is easy to see that the corresponding I-function I E ∨ + is also a multi-valued function of y 1 , . . . , y r , depending analytically on y r and formally on y 1 , . . . y r−1 , which is defined near the same large-radius limit point (y 1 , . . . , y r ) = (0, . . . , 0) in M reg . The global quantum connections for X + and E ∨ + were constructed, in §5.5, using the I-functions I X + and I E ∨ + . We now introduce a closely-related I-function, defined in terms of GIT data for X + and the characters E 1 , . . . , E k , that will allow us to globalize the quantum connection on the ambient part of H • CR (Y ± ). With notation as in §5.4, except with u i now denoting the non-equivariant class Poincaré-dual to the ith toric divisor (4.6) and with v j ∈ H 2 (X + ), 1 ≤ j ≤ k, given by the non-equivariant first Chern class of the line bundle corresponding to the character E j , define a H • CR (X + )-valued hypergeometric series I temp X + ,Y + (σ, x, z) ∈ H • CR (X + ) ⊗ C((z −1 ))[[Q, σ, x]] by: Note that for each d ∈ K and each j ∈ {1, 2, . . . , k}, E j · d is a non-negative integer. (The subscript 'temp' here again reflects the fact that this notation for the I-function is only temporary: we are just about to change notation, by specializing certain parameters.) Under our hypotheses (7.1) on the line bundles L X + (E j ), we have a Mirror Theorem for the toric complete intersection Y + : We define the I-function I X + ,Y + to be the function obtained from I temp X + ,Y + by the specialization Q = 1, σ = σ + := θ + ( r i=1 p + i log y i ) where θ + is as in (4.8). Thus: a: a = D j ·d ,a≤0 (u j + az) a: a = D j ·d ,a≤D j ·d (u j + az) where (y 1 , . . . , y r ) are as in §5. 4. Repeating the analysis in Lemma 5.13 shows that I X + ,Y + , just like I X + and I E ∨ + , is a multi-valued function of y 1 , . . . , y r that depends analytically on y r and formally on y 1 , . . . y r−1 , defined near the large-radius limit point (y 1 , . . . , y r ) = (0, . . . , 0) in M reg . The arguments in §5.5 can now be applied verbatim to I Y + := ι + I X + ,Y + , and thus we construct a global version of the quantum connection on the ambient part H • amb (Y + ), defined over the base M • + . The analog of Theorem 5.14 holds, with the same proof: Theorem 7.2. There exist the following data: • an open subset U • + ⊂ U + such that P + ∈ U • + and that the complement U + \ U • + is a discrete set; we write M • • a mirror map τ + : M + → H • amb (Y + ) of the form: τ + = ι + σ + +τ +τ+ ∈ H • amb (Y + ) ⊗ O U • + [[y 1 , . . . , y r−1 ]] τ + | y 1 =···=yr=0 = 0 such that ∇ + equals the pull-back τ * + ∇ + of the (non-equivariant) quantum connection ∇ + on the ambient part of H • CR (Y + ) by τ + , that is: ∂τ k + (y) ∂ log y i (φ k τ + (y) ) 1 ≤ i ≤ + C j (y) = N k=0 ∂τ k + (y) ∂x j (φ k τ + (y) ) j ∈ S + and that the push-forward of E + by τ + is the (non-equivariant) Euler vector field E + on the ambient part H • amb (Y + ). Moreover, there exists a global section Υ + 0 (y, z) of F + such that I Y + (y, z) = zL amb + (τ + (y), z) −1 Υ + 0 (y, z) where L amb + (τ, z) is the ambient fundamental solution from §7.1 Remark 7.3. Here, as in Theorem 5.14, the Novikov variable Q has been specialized to 1.
Remark 7.4. Entirely parallel results hold for Y − .

Analytic Continuation of I-Functions.
To prove the Crepant Transformation Conjecture in this context, we need to establish the analog of Theorem 6.1. To do this, we will compare the analytic continuation of the I-functions I X ± ,Y ± with the analytic continuation of I E ∨ ± . Let T = (C × ) m denote the torus acting on X ± , and T = (C × ) m+k denote the torus acting on E ∨ ± . The splitting T = T × (C × ) k gives R T = R T [κ 1 , . . . , κ k ] where κ j , 1 ≤ j ≤ k, is the character of (C × ) k given by projection to the jth factor of the product (C × ) k . We regard T as acting on X ± via the given action of T ⊂ T and the trivial action of (C × ) k ⊂ T , so that: FM : K 0 (E ∨ − ) → K 0 (E ∨ + ) coincide under the natural identification of K 0 (X ± ) with K 0 (E ∨ ± ). The same statement holds equivariantly.
Proof. Consider the fiber diagram: where the bottom triangle is (1.1) and the top triangle is the analog of (1.1) for E ∨ ± , and apply the flat base change theorem.
Let U E ∨ be the symplectic transformation from Theorem 6.1 applied to E ∨ ± . Combining Lemma 7.5 with Theorem 6.1 gives a commutative diagram: where ρ ± ∈ H 2 T (E ∨ ± ) is the T -equivariant first Chern class of E ∨ ± and µ ± are the T -equivariant grading operators. Recall that and that the Chern roots of E ∨ ± are pulled back from the common blow-down X 0 of X ± . Part (2) of Theorem 6.1 thus implies that we can factor out the contributions of Γ(E ∨ ± ) and c T 1 (E ∨ ± ) from the vertical maps in (7.3), replacing the vertical arrows by: This proves: Lemma 7.6. The transformations U X : H(X − ) → H(X + ) and U E ∨ : H(E ∨ − ) → H(E ∨ + ) coincide under the natural identifications of H(X ± ) with H(E ∨ ± ). In particular, U E ∨ is independent of κ 1 , . . . , κ k .
The I-functions I X + ,Y + and I E ∨ + are related 16 by: λ=0,κ=−z = e πic 1 (E ∨ + )/z I X + ,Y + (±y) where the subscript on the left-hand side denotes the specialization: (7.4) λ i = 0 1 ≤ i ≤ m κ j = −z 1 ≤ j ≤ k and the ± on the right-hand side denotes the change of variables: The specialization (7.4) is given by a shift S : κ j → κ j − z in the equivariant parameters followed by passing to the non-equivariant limit. Note that the change of variables (7.5) maps y d to (−1) −c 1 (E ∨ + )·d y d . Recall from Theorem 6.1 that, after analytic continuation, we have I E ∨ + = U E ∨ I E ∨ − . Since U E ∨ is independent of κ j , 1 ≤ j ≤ k, it follows that U E ∨ commutes with the shift S. Since the Chern roots of E ∨ are pulled back from the common blow-down X 0 of X ± , it follows that U E ∨ e πic 1 (E ∨ − )/z = e πic 1 (E ∨ + )/z U E ∨ Setting λ = 0 and κ j = −z in the equality I E ∨ + = U E ∨ I E ∨ − , and replacing H(E ∨ ± ) and U E ∨ with their non-equivariant limits H(E ∨ ± ) := H • CR (E ∨ ± ) ⊗ C((z −1 )) and U E ∨ : H(E ∨ − ) → H(E ∨ + ) we find that I X + ,Y + = U E ∨ I X − ,Y − after analytic continuation. Thus: I X + ,Y + = U X I X − ,Y − after analytic continuation. 7.4. Compatibility of Fourier-Mukai Transformations. For the analogue of part (3) of Theorem 6.1, we need to compare the Fourier-Mukai transformation associated to X + X − with the Fourier-Mukai transformation associated to Y + Y − . This is a base change question (cf. Lemma 7.5), but this time we do not have flatness. By assumption, we have: where the vertical maps are inclusions, the bottom triangle is (1.1) and the top triangle is the analog of (1.1) for Y ± . The substacksỸ is defined by the vanishing of a sections : X → E, where E → X is the direct sum of line bundles 16 An analogous relationship holds between IX − ,Y − and I E ∨ − .