GLOBAL MIRRORS AND DISCREPANT TRANSFORMATIONS FOR TORIC DELIGNE-MUMFORD STACKS

We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules (and of the all-genus Gromov-Witten potentials) under a discrepant toric wallcrossing. In the case of weighted blowups of weak-Fano compact toric stacks along toric centres, we show that an analytic lift of the formal decomposition corresponds, via the Γ̂integral structure, to an Orlov-type semiorthogonal decomposition of topological K-groups. We state a conjectural functoriality of Gromov-Witten theories under discrepant transformations in terms of a Riemann-Hilbert problem.


Introduction
It is a very interesting problem to study how Gromov-Witten invariants (or quantum cohomology) change under birational transformations. When the birational transformation is crepant (or a K-equivalence), a conjecture of Yongbin Ruan [91] says that the quantum cohomology of K-equivalent spaces should be related to each other by analytic continuation in quantum parameters (see e.g. [79,20,77,32,30]). In this paper, we are concerned with discrepant transformations, or more precisely, birational maps ϕ : X + X − between smooth Deligne-Mumford stacks such that there exist projective birational morphisms f ± : X → X ± satisfying f − = ϕ • f + and that f * + K X + − f * − K X − is a non-zero effective divisor. In this case, we write "K X + > K X − " by a slight abuse of notation.
This includes the case where ϕ is a blowup along a smooth subvariey. In this case, we do not expect that the quantum cohomology of X + and X − are related by analytic continuation because their ranks are different. Instead, we expect that the quantum cohomology of X + would contain the quantum cohomology of X − as a direct summand after analytic continuation. This is analogous to the conjecture that D b (X + ) contains D b (X − ) as a semiorthogonal summand [13,71,72,9], where D b (X ± ) denotes the derived category of coherent sheaves on X ± . In this paper, we describe a decomposition of quantum cohomology D-modules (and of all genus Gromov-Witten potentials) for toric Deligne-Mumford stacks under discrepant transformations. Moreover, we show in special cases that the decomposition of quantum cohomology D-modules is induced by a semiorthogonal decomposition of derived categories (or more precisely of topological K-groups) via the Γ-integral structure [63,70]. We also formulate a general conjecture in view of our results in the toric case.
1.1. Quantum cohomology D-modules. Our central objects of study are quantum (cohomology) D-modules. Let X be a smooth Deligne-Mumford stack. The genus-zero Gromov-Witten invariants define a family of (super)commutative product structures τ on the orbifold cohomology group H * CR (X) parametrized by τ ∈ H * CR (X); this is called quantum cohomology. The product τ then defines a meromorphic flat connection ∇ called the quantum connection (or Dubrovin connection) on the trivial H * CR (X)-bundle over H * CR (X) × C. It is given by the formulae where (τ, z) represents a point on the base H * CR (X) × C, {τ i } are linear co-ordinates on H * CR (X) dual to a basis {φ i }, E is the so-called Euler vector field, and µ is the (constant) grading operator. This connection is self-dual with respect to the pairing P between the fibres at (τ, −z) and (τ, z) induced by the orbifold Poincaré pairing. The quantum D-module 1 QDM an (X) of X is, roughly speaking, the module of sections of this vector bundle equipped with the meromorphic flat connection ∇ and the pairing P (see §2.3, §6.1 for the details). We also obtain the formal quantum D-module (or more precisely, the quantum D-module completed in z) by restricting QDM an (X) to the formal neighbourhood of z = 0: When a torus T acts on X, we can also define the T-equivariant quantum D-module QDM an T (X) and its formal version QDM an T (X); they are deformation of the non-equivariant quantum Dmodules.
1.2. Global Landau-Ginzburg models and toric wall-crossings. A smooth semiprojective toric Deligne-Mumford (DM) stack X Σ (in the sense of Borisov-Chen-Smith [14]) can be defined as the Geometric Invariant Theory (GIT) quotient of a vector space C S by a torus L C × = L⊗C × , where L is a free Z-module of finite rank. The torus L C × acts on C S via a group homomorphism L C × → (C × ) S , and by dualizing it, we get the family pr : (C × ) S → L ⊗ C × of tori equipped with the function F = b∈S u b on (C × ) S , where u b is the bth co-ordinate on 1 Here we assume the convergence and the analyticity of quantum cohomology (which are true for toric DM stacks); the superscript "an" means analytic.
This is the Landau-Ginzburg (LG) model mirror to X Σ introduced by Givental [48]. Using the secondary fan of Gelfand-Kapranov-Zelevinsky [45], we can partially compactify this LG model to an LG model of the form (see §3.2) where Y, M are possibly singular toric DM stacks (in the sense of Tyomkin [105]). The base M of this LG model corresponds to the (extended) Kähler moduli space of X Σ (i.e. the base of the quantum D-module) and contains a distinguished point 0 Σ called the large radius limit point of X Σ . It also contains 2 , as torus-fixed points, the large radius limit points 0 Σ of several other toric DM stacks X Σ which can be obtained from X Σ by varying the stability condition for the GIT quotient [C S //L C × ]. Hodge-theoretic mirror symmetry for toric DM stacks established by Coates-Corti-Iritani-Tseng [26,27] implies that, for each smooth toric DM stack X Σ whose large radius limit point appears in M, we have a mirror map defined on the formal neighbourhood of 0 Σ mir : (M, 0 Σ ) −→ a partial compactification of H * CR (X Σ )/2πiH 2 (X Σ , Z) and a mirror isomorphism GM(F ) Σ ∼ = mir * QDM an (X).
Here GM(F ) denotes the Gauss-Manin system associated with the LG potential F and the sub/superscripts Σ means the completion at the large-radius limit point 0 Σ (see § §4. 1-4.2).
Using the convergence result from [61,27], we show that this mirror isomorphism can be extended to a small analytic neighbourhood of 0 Σ as an isomorphism between a certain analytified Gauss-Manin system GM an (F ) Σ and the formal quantum D-module QDM an (X Σ ) (see Theorem 4.32). This enables us to compare the quantum D-modules of various birational models of X Σ over the mirror moduli space M. Then we arrive at the following result (in this theorem, we do not assume compactness of X ± or (semi-)positivity of c 1 (X ± )).
Theorem 1.1 (Theorem 5.16). Let ϕ : X + X − be a discrepant transformation between semiprojective toric DM stacks induced by a single wall-crossing in the space of GIT stability conditions. Suppose that K X + > K X − . Then we have a formal decomposition of the Tequivariant quantum D-modules (1.1) mir * + QDM an T (X + ) ∼ = mir * − QDM an T (X − ) ⊕ R over a non-empty open subset U 0 of M × Lie T, where mir ± denotes the mirror map for X ± and R is a locally free O an 2 In fact, by choosing a suitable presentation of X Σ as the GIT quotient [C S //L C × ], we can arrange that the large radius limit point of any given smooth toric DM stack X Σ having the same affinization and the same generic stabilizer as X Σ appears in the base space M. This theorem is a generalization of the result of González-Woodward [51] who showed a decomposition of the quantum cohomology algebras under a running of the toric minimal model programme. In this theorem, we consider analytic continuation over a neighbourhood of a toric curve C ⊂ M connecting the large radius limit points 0 ± for X ± . When ϕ is an isomorphism in codimension one ("flip"), the curve C is asymptotically close, near the large radius limit points 0 ± , to the curve in the boundary of the Kähler moduli space given by the extremal curve class. When ϕ (or ϕ −1 ) contracts a divisor, the curve C is asymptotically close to the curve corresponding to the extremal class near 0 + (resp. 0 − ) and to the line spanned by a cohomology class of degree greater than 2 near 0 − (resp. of degree less than 2 near 0 + ), see Remark 5.9. In either case, at least one of the mirror maps mir ± involves negative degree variables with respect to the Euler vector field (an instance of the generalized mirror transformation [69,28,62]) and the formal decomposition occurs over the base of the big quantum cohomology in general. We also note that the decomposition (1.1) is defined only over the formal power series ring C [[z]] and the completion in z is unavoidable. In fact, as Theorem 1.3 below shows, the Stokes structure does not admit an orthogonal decomposition.
Using the Givental-Teleman formula [49, 103,19,108], we obtain a decomposition for the (all-genus) ancestor Gromov-Witten potentials. The result is stated in terms of Givental's quantization formalism; we refer to §5.5 for the notation. Theorem 1.2 (Theorem 5. 19). Let X ± be toric DM stacks as in Theorem 1.1. Let A ±,τ denote the ancestor potentials of X ± at τ ∈ H * CR,T (X ± ). For (q, χ) ∈ M × Lie T in a non-empty open set, we have T s U q,χ A +,mir + (q,χ) = A −,mir − (q,χ) ⊗ T ⊗ rank R where χ is the T-equivariant parameter, U q,χ is the quantization of a symplectic transformation associated with the decomposition (1.1), T is the Witten-Kontsevich tau-function (the ancestor poential of a point) and T s is a certain shift operator.

Analytic lift and
Orlov's decomposition. The non-equivariant quantum D-module has (in general) irregular singularities at z = 0 and the formal quantum D-module misses analytic information such as the Stokes structure at z = 0. For a compact toric DM stack, at least in the non-equivariant limit and over the semisimple locus, the formal structure of the quantum D-module is very poor 3 , since it is determined only by eigenvalues of the Euler multiplication. We will restore the missing information by describing the analytic lift of the formal decomposition (1.1). By the Hukuhara-Turrittin theorem (see §6.1), the decomposition (1.1) in the non-equivariant limit can be locally lifted to an analytic isomorphism 4 : (1.2) mir * + QDM an (X + ) where B is a small open subset of M and I is an angular sector {z : | arg(z) − φ| < π 2 + } with > 0. We call it the analytic lift or a sectorial decomposition; its uniqueness is ensured by the fact that the angle of the sector is bigger than π. The analytic lift induces a decomposition (depending on B and I) of the local system underlying QDM an (X + ). On the other hand, the Γ-integral structure [63,70] identifies the complexified topological K-group K(X) ⊗ C with the space of multi-valued flat sections of the quantum D-module; for toric stacks, it corresponds to the integral structure on the GKZ system identified by Borisov-Horja [15]. We show in some special cases that the decomposition of the local system given by the analytic lift corresponds to a semiorthogonal decomposition of the topological K-group K(X + ) via the Γ-integral structure. An important ingredient here is the fact [63] that the Γ-integral structure coincides with the natural integral structure of the Gauss-Manin system under mirror symmetry. By describing the analytic lift in terms of mirror oscillatory integrals and studying the relationship between the local systems of Lefschetz thimbles (see Theorems 7.22,7.31), we obtain the following result. Theorem 1.3 (Theorems 7. 25, 7.31, 7.33). Let X − be a weak-Fano compact toric stack satisfying a mild technical assumption as described in §7.1 and let ϕ : X + → X − be a weighted blowup along a toric substack Z ⊂ X − . We assume that X + is also weak-Fano. Then there exist a submersion f from an open set W of H * CR (X + ) to H * CR (X − ) and an angular sector I (of angle greater than π) such that we have an analytic decomposition over the sector which induces, via the Γ-integral structure, a semiorthogonal decomposition of the K-group Here E is the exceptional divisor of ϕ, ϕ E = ϕ| E : E → Z, i E : E → X + is the inclusion and J = k 0 + · · · + k c − 1 when a fibre of ϕ E : E → Z is given by the weighted projective space P(k 0 , . . . , k c ). Remark 1.4. (1) In this theorem, we allow Z to be of codimension one (i.e. a toric divisor); in this case X + is obtained from X − by a root construction (see [21]) along Z.
(2) When X − is a smooth projective variety and ϕ : X + → X − is the blowup along a smooth subvariety Z, the decomposition (1.3) is induced by Orlov's semiorthogonal decomposition [86] for D b (X + ). In general, the Orlov-type decomposition (1.3) arises from a sectorial decomposition of the quantum D-module at a point which is far from the large radius limit point. On the other hand, we can idenfity explicitly the locus in the mirror moduli space M where the analytic lift (1.2) induces the pull-back ϕ * : K(X − ) → K(X + ) in K-theory (Theorem 7.25).
(3) The result suggests that each residual piece R an i of the sectorial decomposition should be related to the quantum D-module of the blowup centre Z; they certainly have the same formal structure, but we do not know if the Stokes structures are related (although we expect their relationship from homological mirror symmetry).
(4) We need the weak-Fano assumption when we apply results from [63]. We hope that the same result holds without such assumptions, but it may require some technical advances. (5) In §8, we formulate a general conjecture relating the decomposition of topological Kgroups and that of quantum D-modules under discrepant transformations. Under the conjecture, the decomposition of K-groups in principle determines the relationship between the quantum D-modules of X + and X − (including the map f ). This involves solving a Riemann-Hilbert problem; see Proposition 8.5.
1.4. Related works. We mention to some of the earlier works that are closely related to the present paper.
The relationship between quantum cohomology and derived category has been suggested by Dubrovin [39]. Our Theorem 1.3 can be viewed as a variation on this theme (see also Gamma conjecture [44] or Dubrovin-type conjecture [100]). Bayer [12] showed that the semisimplicity of quantum cohomology is preserved under blowup at a point in connection with Dubrovin's conjecture [39]. His computation [12,Lemma 3.4.2] for the spectral cover is compatible with the picture in this paper.
Acknowledgements. I would like to thank Pedro Acosta, Arend Bayer, Andrea Brini, Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, Eduardo González, Claus Hertling, Yuki Hirano, Paul Horja, Yunfeng Jiang, Yuan-Pin Lee, Chiu-Chu Melissa Liu, Wanmin Liu, Thomas Reichelt, Yongbin Ruan, Kyoji Saito, Fumihiko Sanda, Christian Sevenheck, Yota Shamoto, Mark Shoemaker, Takuro Mochizuki, Mauricio Romo, Hsian-Hua Tseng, Chris Woodward for many insightful discussions and explanations. This work is supported by EP-SRC grant EP/E022162/1, and JSPS Kakenhi Grants Number 22740042, 23224002, 24224001, 25400069, 26610008, 16K05127, 16H06335, 16H06337 and 17H06127. Part of this work was done while I was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring semester of 2018 and the stay was supported by the National Science Foundation under Grant No.DMS-1440140. I thank the organizers and participants of the programme for many stimulating discussions.

Preliminaries
In this section, in order to fix notation, we review quantum cohomology, quantum Dmodules and the Gamma-integral structure. Our main interest in this paper lies in the case where X is a toric DM stack, but all the materials in this section make sense for a general smooth DM stack X satisfying mild assumptions.
Let X be a smooth DM (Deligne-Mumford) stack over C. We write X for the coarse moduli space of X. Recall that the inertia stack IX is the fibre product X × X×X X of the two diagonal morphisms X → X × X. A point on IX is given by a pair (x, g) of a point x ∈ X and a stabilizer g ∈ Aut(x). We write IX = v∈Box X v for the decomposition of IX into connected components, where Box is the index set. We have a distinguished element 0 ∈ Box that corresponds to the untwisted sector X 0 ∼ = X consisting of points (x, g = 1) with the trivial stabilizer. The orbifold cohomology H * CR (X) of Chen and Ruan [24] is defined to be (2.1) H * CR (X) := H * −2 age (IX, C) = v∈Box H * −2 age(v) (X v , C) where age : IX → Q ≥0 is a locally constant function giving a shift of degrees and we write age(v) = age | Xv (see §3.1.3 for the age in the case of toric DM stacks). The right-hand side means the cohomology group of the underlying complex analytic space of IX and we use complex coefficients unless otherwise specified. We also restrict ourselves to cohomology classes of "even parity", i.e. we only consider cohomology classes of even degrees 5 on IX. For toric DM stacks, every orbifold cohomology class has even parity. When X is proper, the orbifold Poincaré pairing on H * CR (X) is defined to be where inv : IX → IX is the involution sending (x, g) to (x, g −1 ). For d ∈ H 2 (X, Q) and l ∈ Z ≥0 , let X g,l,d denote the moduli stack of genus-g twisted stable maps to X of degree d (this was denoted by K g,l (X, d) in [2]). It carries a virtual fundamental class [X g,l,d ] vir ∈ A * (X g,l,d , Q) and the evaluation maps ev i : X g,l,d → IX, i = 1, . . . , l to the rigidified cyclotomic inertia stack IX (see [2, §3.4]). When the moduli stack X g,l,d is proper (this happens when X has a projective coarse moduli space), we define genus-zero descendant Gromov-Witten invariants by α 1 ψ k 1 , . . . , α l ψ k l g,l,d where α 1 , . . . , α l ∈ H * CR (X), d ∈ H 2 (X, Q) and ψ i denotes the ψ-class (see [2, §8.3]) at the ith marking. Here note that the rigidified cyclotomic inertia stack IX has the same coarse moduli space as IX, and thus α i can be regarded as a cohomology class of IX.
We assume that there exists a finitely generated monoid Λ + ⊂ H 2 (X, Q) such that R ≥0 Λ + is a strictly convex full-dimensional cone in H 2 (X, R) and that Λ + contains classes of any orbifold stable curves. We also assume that Λ + is saturated, i.e. Λ + = Λ ∩ R ≥0 Λ + for Λ := ZΛ + . The monoid Λ + for toric DM stacks will be described in §3. 3  ]. When we want to emphasize the dependence of on the parameter τ , we shall write α τ β in place of α β.
The Gromov-Witten invariants and the quantum product can be generalized to the equivariant setting or to a non-projective space X. Suppose that an algebraic torus T acts on X. The equivariant orbifold cohomology H * CR,T (X) is defined to be the T-equivariant cohomology of IX with the same degree shift as before: We assume that (a) IX is equivariantly formal, i.e. the Serre spectral sequence for IX × T ET → BT collapses at the E 2 -term (over Q); this implies that H * CR,T (X) is a free R T := H * T (pt, C)module of rank dim H * CR (X); (b) the T-fixed set of X is projective; (c) the evaluation maps ev i : X 0,l,d → IX are proper.
By the assumption (a), the C-basis {φ i } s i=0 ⊂ H * CR (X) can be lifted to an R T -basis of H * CR,T (X), which we denote by the same symbol. This induces R T -linear co-ordinates (τ 0 , . . . , τ s ) → τ = s i=0 τ i φ i on H * CR,T (X) as before. Under the assumption (b), we can define the equivariant orbifold Poincaré pairing and the equivariant Gromov-Witten invariants via the virtual localization formula [53]. They take values in the fraction field S T := Frac(R T ) of R T . Under the assumption (c), we can define the quantum product using the push-forward by the proper map ev 3 as follows: where PD stands for the Poincaré duality on IX. Therefore we get the quantum product defined on the space H * CR,T (X) ] without inverting equivariant parameters, whereas the Gromov-Witten invariants themselves lie in S T in general.
Remark 2.1. As remarked in [20], the assumption (c) is satisfied if the coarse moduli space X is semi-projective, i.e. projective over an affine variety. We will impose the semi-projectivity assumption on toric DM stacks.
2.2. Kähler moduli space. In this section, we specialize the Novikov variable Q to one with the aid of the divisor equation, and introduce the Kähler moduli space M A (X) parameterizing the quantum product.
As before, let φ 0 , φ 1 , . . . , φ s be a homogeneous basis of H * CR (X). We assume that φ 0 = 1 is the identity class and that {φ 1 , . . . , φ r }, r ≤ s is a basis of the degree-two untwisted sector H 2 (X) ⊂ H 2 CR (X). We write τ = σ + τ with σ = r i=1 τ i φ i ∈ H 2 (X) and τ = This shows that the quantum product τ depends only on the equivalence class where the element of C[Λ + ] corresponding to d ∈ Λ + represents the function q d = e τ ·d . The space M A (X) gives a partial compactification of H * CR (X)/2πiΛ depending on the choice of the monoid Λ + . By setting Q = 1, we may view τ as a family of products parameterized by the formal neighbourhood of the "origin" (that is, q d = 0 for all non-zero d ∈ Λ + and τ 0 = τ r+1 = · · · = τ s = 0) in M A (X). This origin is called the large radius limit point.
Quantum cohomology for smooth DM stacks has the additional symmetry called the Galois symmetry [63, §2.2]. Let H 2 (X, Z) denote the sheaf cohomology of the constant sheaf Z on the topological stack (orbifold) underlying X; an element ξ ∈ H 2 (X, Z) corresponds to a topological orbi-line bundle L ξ over X. For a connected component X v of IX, the stabilizer along X v acts on fibres of L ξ by a constant scalar exp(2πif v (ξ)) for some (2.3) dg(ξ)(α 1 )ψ k 1 , . . . , dg(ξ)(α l )ψ k l g,l,d = e 2πiξ·d α 1 ψ k 1 , . . . , α l ψ k l g,l,d and thus the quantum product satisfies The map g(ξ) induces the action of H 2 (X, Z)/Λ on H * CR (X)/2πiΛ ; this action naturally extends to the partial compactification M A (X). In view of this symmetry, we can regard the quantum products τ as a family of products parametrized by the formal neighbourhood of the origin (large radius limit) of the stack: We refer to M A (X) or to the quotient stack above as the Kähler moduli space, where the subscript A stands for the A-model.
The above construction can be adapted to the equivariant quantum cohomology. We choose a homogeneous R T -basis The equivariant Kähler moduli spaces are given by replacing the ground ring C with R T .
It is fibred over Spec The T-equivariant quantum product τ can be viewed as a family of product structures parameterized by the formal neighbourhood of the origin in [M A,T (X)/(H 2 (X, Z)/Λ )].
Remark 2.2. Unlike the non-equivariant case, the equivariant Kähler moduli space M A,T (X) is not a partial compactification of H * CR,T (X)/2πiΛ ; we do not even have a natural map H * CR,T (X) → M A,T (X). If we regard H * CR,T (X) as a locally free coherent sheaf over Spec R T and write H CR,T (X) = Spec(Sym • R T (H * CR,T (X) ∨ )) for the total space of the associated vector bundle (where (−) ∨ stands for the dual as an R T -module), then we have an open dense embedding H CR,T (X)/2πiΛ → M A,T (X). In this paper, we take the view that the equivariant quantum product τ is parametrized by points in H CR,T (X) rather than by equivariant cohomology classes. Note that equivariant cohomology classes correspond to sections of Remark 2.3. The equivariant Kähler moduli space M A,T (X) depends on the choice of a basis φ 1 , . . . , φ r . In fact, in the equivariant case, we can replace the basis for some c i ∈ H 2 T (pt) without violoating the homogeneity. Then the corresponding 0-th co- In other words, the construction of the equivariant Kähler moduli space requires the choice of a splitting of the sequence 0 → H 2 Remark 2.4. In the above discussion, we considered the specialization Q = 1 and the equivalence class of τ in H * CR (X)/2πiΛ . This is equivalent to considering the restriction to τ 1 = · · · = τ r = 0 and the substitution of q d for Q d .
Remark 2.5. Henceforth we specialize the Novikov variable Q to one in the quantum product τ , unless otherwise stated.
2.3. Quantum D-module. The quantum product defines a meromorphic flat connection on a vector bundle (with fibre orbifold cohomology) over the Kähler moduli space, called the quantum connection. The quantum connection, the grading and the orbifold Poincaré pairing constitute the quantum D-module of X.
We start by explaining the equivariant version; we get the non-equivariant version by taking non-equivariant limit. As before, we fix a homogeneous R T -free basis {φ 0 , . . . , φ s } of H * CR,T (X) that satisfies (2.4). We use this basis to construct the equivariant Kähler moduli spaces M A,T (X). Consider the vector bundle is the complex plane with co-ordinate z and H CR,T (X) denotes a vector bundle over Spec R T corresponding to H * CR,T (X) (see Remark 2.2). The Galois symmetry in the previous section induces the (H 2 (X, Z)/Λ )-action on this vector bundle defined by the map dg(ξ) × g(ξ) × id Cz . By the Galois symmetry, this vector bundle descends to a vector bundle on the quotient stack The quantum connection is a meromorphic flat partial connection on this vector bundle over the formal neighbourhood of the origin in M A,T (X) (times C z ); it is given by This connection is partial in the sense that it is defined only in the τ -direction and not in the z-direction or in the direction of equivariant parameters (the first term d means the relative differential over Spec R T [z]). The connection has simple poles along z = 0 and has logarithmic singularities along the toric boundary of Spec C[Λ + ]. Note that dτ i with 1 ≤ i ≤ r defines a logarithmic 1-form on Spec C[Λ + ]. In a more formal language, the module (which we regard as the module of sections of the bundle (2.6) over the formal neighbourhood of the origin in ]. This is compatible with the quantum connection in the sense that Let P denote the pairing between the fibres of the bundle (2.6) at (q, τ , χ, −z) and at (q, τ , χ, z) induced by the orbifold Poincaré pairing.
where (·, ·) denotes the orbifold Poincaré pairing (2.2). The pairing P satisfies ] and the compatibility equations with ∇ and E: . This is often called a z-connection [93].
The non-equivariant quantum D-module is the restriction of the equivariant one to the origin 0 ∈ Spec R T = Lie T. It is a quadruple ], ∇, Gr, P where the pairing P is defined only when X is proper. In the non-equivariant case, we can define the connection in the z-direction by and this preserves the pairing P . Here E denotes the non-equivariant limit of (2.7) (and thus does not contain the term n i=1 χ i ∂ ∂χ i ) and is the section of QDM(X) corresponding to E and µ ∈ End(H * CR (X)) is the endomorphism given by µ( [70,Proposition 3.1] is an integral lattice in the space of flat sections of the quantum D-module. The integral lattice is identified with the topological K-group of X. We review its definition only in the nonequivariant case (see [31, §3] for the equivariant Γ-integral structure). For simplicity, we assume that the quantum product τ is convergent in a neighbourhood of the large radius limit point. Then the quantum connection is also analytic in the same neighbourhood; it has no singularities on the intersection of the open subset and the convergence domain of τ . Introduce the following End(H * CR (X))-valued function: Then by [63, Proposition 2.4], L(τ, z)z −µ z c 1 (X) φ i , i = 0, . . . , s form a basis of (multi-valued) ∇flat sections (which are flat also in the z-direction with respect to (2.11)), i.e. L(τ, z)z −µ z c 1 (X) is a fundamental solution of the quantum connection.
We introduce the Chern character and the Γ-class for a smooth DM stack X. Let π : IX → X denote the natural projection. Recall the decomposition IX = v∈Box X v into twisted sectors. For an orbi-vector bundle V on X, the stabilizer along X v acts on (π * V )| Xv and decomposes it into the sum of eigenbundles where the stabilizer acts on V v,f by exp(2πif ). The Chern character of V is defined to be: Consider now the tangent bundle V = T X and let δ v,f,j , j = 1, . . . , rank(V v,f ) denote the Chern roots of T X v,f . The Γ-class of X is defined to be: where in the right-hand side we expand the Euler Γ-function Γ(z) in series at z = 1 − f . This is an algebraic cohomology class defined over transcendental numbers.
Definition 2.7 ([63, Definition 2.9]). Let K(X) denote the Grothendieck group of topological orbi-vector bundles on X. For V ∈ K(X), we define a (multi-valued) flat section s V (τ, z) of the quantum D-module by where deg 0 denotes the grading operator on H * (IX) without age shift, i.e. deg 0 (α) = pα for α ∈ H p (IX). The map V → s V defines an integral lattice in the space of flat sections, which we call the Γ-integral structure.
Important properties of the Γ-integral structure are as follows [63, Proposition 2.10]: • it is monodromy-invariant around the large radius limit point: we have where L ξ is the line bundle corresponding to ξ ∈ H 2 (X, Z); therefore it defines a Z-local system underlying the quantum D-module; • it intertwines the Euler pairing with the orbifold Poincaré pairing: if V 1 , V 2 are holomorphic orbi-vector bundles on X, we have Remark 2.8. The fundamental solution L(τ, z)z −µ z c 1 (X) is multi-valued in (τ, z), but it has a standard determination when τ is a real class in H * CR (X; R) = H * (IX; R) (which is sufficiently close to the large radius limit point) and z is positive real. We will sometimes use such a point as a base point.

Global Landau-Ginzburg mirrors of toric DM stacks
In this section, we construct a global Landau-Ginzburg model (LG model) which is simultaneously mirror to several smooth toric DM stacks. For background materials on toric (Deligne-Mumford) stacks, we refer the reader to [14,42,66,67,105,34].
3.1. Toric data. Throughout the paper, we fix the data (N, Π), where • N is a finitely generated abelian group of rank n (possibly having torsion) and • Π is a full-dimensional, convex, rational polyhedral cone in N R := N ⊗ R.
We do not require that Π is strictly convex; Π will be the support of the fan of a smooth toric DM stack. To construct a mirror family of LG models, we choose a finite subset S ⊂ N such that The set S specifies the set of monomials appearing in the LG model. We make the following technical assumption to ensure that the base of the mirror family has no generic stabilizers, that is, generic LG models have no automorphisms of diagonal symmetry. This is not an essential restriction. In fact, if S does not satisfy this assumption, we can first construct the mirror family by taking a bigger set S ⊃ S satisfying the assumption, and then restrict the family to the subspace of the base corresponding to S (then, the subspace has generic stabilizers). Notation 3.2. We write N tor for the torsion part of N and write N = N/N tor for the torsion-free quotient. For v ∈ N, we write v for the image of v in N. The subscripts Q, R, etc means the tensor product with Q, R over Z, e.g. N R = N ⊗ Z R. For subsets A ⊂ N and B ⊂ N R , we write A ∩ B := {a ∈ A : a ∈ B}. Definition 3.3. A stacky fan adapted to S is a triple Σ = (N, Σ, R) such that (i) Σ is a rational simplicial fan defined on the vector space N R ; (ii) the support |Σ| = σ∈Σ σ of the fan Σ equals Π = b∈S R ≥0 b; (iii) there exists a strictly convex piecewise linear function η : Π → R which is linear on each cone of Σ; (iv) R ⊂ S is a subset such that the map R b → R ≥0 b gives a bijection between R and the set Σ(1) of one-dimensional cones of Σ.
The data (N, Σ, R) gives a stacky fan in the sense of Borisov, Chen and Smith [14]. We write X Σ for the smooth toric DM stack (toric stack for short) defined by Σ. We also set R(Σ) := R ("rays") and G(Σ) := S \ R ("ghost rays"). We denote by Fan(S) the set of stacky fans adapted to S.
Remark 3.4. The above conditions (ii), (iii) imply that the corresponding toric DM stack X Σ is semiprojective [34, §7.2], that is, the coarse moduli space X Σ of X Σ is projective over the affine variety Spec H 0 (X Σ , O) and has a torus fixed point. Conversely, any semiprojective toric DM stack arises from some toric data (N, Π, S, Σ, R) in this section.
3.1.1. Toric DM stacks. Let Σ = (N, Σ, R) ∈ Fan(S) be a stacky fan adapted to S. When we start from the stacky fan Σ, the set G(Σ) = S \ R can be viewed as the data of an extended stacky fan in the sense of Jiang [68]. The extended stacky fan is given by the pair (Σ, G(Σ)) of the stacky fan Σ and the finite subset G(Σ) ⊂ N ∩ |Σ|. We recall a definition of the smooth toric DM stack X Σ [14,68] in terms of the extended stacky fan. The stacky fan Σ defines the fan sequence where β(Σ) : Z R(Σ) → N sends the basis e b ∈ Z R(Σ) corresponding to b ∈ R(Σ) to b ∈ N and L Σ := Ker(β(Σ)). The dual sequence is called the divisor sequence, where (L Σ ) ∨ := H 1 (Cone(β(Σ)) ) is the Gale dual of β(Σ) (see [14]). The extended fan sequence is the sequence: where the map β : Z S → N sends the basis e b ∈ Z S corresponding to b ∈ S to b ∈ N and L := Ker(β). Note that β is surjective by Assumption 3.1. The dual sequence is called the extended divisor sequence, where M := Hom(N, Z) and D : (Z S ) → L is dual to L → Z S and Cok(D) ∼ = Ext 1 (N, Z). The torus L C × := L ⊗ C × acts on C S via the natural map L C × → (C × ) S induced by L → Z S . The toric stack X Σ is defined as a GIT quotient of C S by the L C × -action. Set where we regard C I with I ⊂ S as a co-ordinate subspace of C S (we set C ∅ = {0}). Since every element of S Σ is contained in R(Σ), we may also write: We define: Since L C × acts on U Σ with at most finite stabilizers, X Σ is a smooth DM stack. The toric stack X Σ depends only on Σ = (N, Σ, R) and does not depend on the choice of the extension G(Σ) = S \ R (see [68]). The coarse moduli space X Σ of X Σ is the toric variety associated with the fan Σ. The (C × ) S -action on U Σ induces the T-action on X Σ , where T is the torus where L C × acts on the second factor C by the character ξ. We denote by D b := D(e b ) the image of the standard basis e b ∈ (Z S ) under D. We can see that D b with b ∈ G(Σ) yields the trivial line bundle on X Σ , and the correspondence ξ → L ξ gives the identification: This group is isomorphic to the Gale dual (L Σ ) ∨ of β(Σ) appearing in (3.2). The torsion free quotient of Pic(X Σ ) can be identified with the ordinary dual (L Σ ) . Moreover we have the identifications so that the divisor sequence (3.2) over Q is identified with . This is the class of a toric divisor. We also write for the T-equivariant cohomology of a point.

3.1.3.
Orbifold cohomology. For a cone σ of Σ, we introduce Box(σ) ⊂ N as and set Box(Σ) = σ∈Σ Box(σ). The set Box(Σ) parametrizes connected components of the inertia stack IX Σ [14]. We write X Σ,v for the component of IX Σ corresponding to v ∈ Box(Σ). The age of a box element v ∈ Box(σ) is given by age Chen and Ruan [24] is given by (as a graded vector space): As a ring, it is generated by the fundamental classes 1 v on IX Σ,v with v ∈ Box(Σ) and the toric divisor classes Here we regard D b as a class supported on the untwisted sector X Σ,0 = X Σ . The T-equivariant orbifold cohomology H * CR,T (X Σ ) is defined by replacing each factor in the right-hand side of (3.7) with the T-equivariant cohomology 3.2. Landau-Ginzburg model. We construct a global family of mirror LG models which are simultaneously mirror to all the toric stacks X Σ with Σ ∈ Fan(S). The uncompactified LG model [48,58,63] is the family of tori obtained from the extended fan sequence (3.3) by applying Hom(−, C × ), together with the function F : where u b denotes the C × -valued co-ordinate on (C × ) S given by the projection to the bth factor. We shall partially compactify this family to include all the large radius limit points of X Σ with Σ ∈ Fan(S). We construct partial compactifications of (C × ) S and L ⊗ C × as possibly singular toric DM stacks in the sense of Tyomkin [105]. According to Tyomkin [105, §4.1], a singular toric DM stack can be described by toric stacky data (L, Ξ, ) such that: • L is a finitely generated free abelian group; • Ξ is a (not necessarily simplicial) rational fan on L R = L ⊗ R; • ⊂ |Ξ| is a subset in the support |Ξ| of Ξ such that for each cone σ ∈ Ξ, there exists a finite index sublattice L(σ) ⊂ L such that ∩ σ = L(σ) ∩ σ. We call the integral structure of the toric stacky data. Note that this is a generalization of a stacky fan (N, Σ, R) of Borisov, Chen and Smith [14] when the group N has no torsion. For a given stacky fan (N, Σ, R) with free N, we can assign a toric stacky data (N, Σ, ) by taking to be the union of the monoids σ Z = Z ≥0 (R ∩ σ) for all σ ∈ Σ. Tyomkin constructed a singular toric DM stack from (L, Ξ, ) by gluing affine charts; its coarse moduli space is the toric variety X Ξ associated with the fan Ξ. We refer the reader to [105, §4.1] for the details (the construction of the affine charts in our case will be reviewed in §3.4).
The graph of η c is the union of "lower faces" of the convex cone in N R ⊕ R generated by (b, c b ), b ∈ S and (0, 1).
For a stacky fan Σ = (N, Σ, R) adapted to S, we define a full-dimensional strictly-convex cone CPL + (Σ) ⊂ (R S ) by (3.8) CPL where CPL stands for "convex piecewise linear" (notation borrowed from [85]) and the subscript + means non-negative. Note that η c in the definition is determined only by Note that ∩ CPL + (Σ) equals the intersection of the following sublattice of (Z S ) with the cone CPL + (Σ). We also define  (1) Define Y to be the singular toric DM stack corresponding to the toric stacky data ((Z S ) , Ξ, ). We call Y the total space of the LG model. (2) Define M to be the singular toric DM stack corresponding to the toric stacky data (L , Ξ, ). We call M the secondary toric stack.   (1) The fan Ξ on L R defined by the cones cpl(Σ) is called the secondary fan or the GKZ fan after the work of Gelfand, Kapranov and Zelevinsky [45] (see also Oda-Park [85]). The fan Ξ gives a lift of the secondary fan Ξ to (R S ) . The support of Ξ is the positive orthant (R ≥0 ) S , and therefore Y can be viewed as an iterated weighted blowup of C S . The fibre of the map D : | Ξ| = (R ≥0 ) S → L R at an interior point ω of cpl(Σ) ⊂ L R can be identified with the image of the moment map µ : X Σ → M R of the T-action on X Σ with respect to the reduced symplectic form associated with ω. Therefore the fan Ξ can be viewed as the total space of the moment polytope fibration over the secondary fan Ξ.
(2) Diemer, Katzarkov and Kerr [36] introduced a closely related (but slightly different) compactification of the LG model in the case where S lies in a hyperplane of integral distance one from the origin. In this case (i.e. when S lies in a hyperplane of height one), our space Y can be obtained from their total Lafforgue stack [36] by contracting a divisor (the zerosection), at least on the level of coarse moduli spaces. Their secondary stack [36] and our secondary toric stack are the same on the level of coarse moduli spaces (the coarse moduli spaces are the toric variety defined by the fan Ξ), however it is not clear to the author if the stack structures are the same. Remark 3.9. Cones of Ξ, Ξ can be described more explicitly as follows. A possibly degenerate fan [85] on N R is a finite collection Σ of convex (but not necessarily strictly convex) rational polyhedral cones in N R such that (1) if σ ∈ Σ and τ is a face of σ, then τ ∈ Σ; and (2) if σ, τ ∈ Σ then the intersection σ ∩ τ is a common face of σ and τ . Let Σ be a possibly degenerate fan on N R . Each cone σ ∈ Σ contains the linear subspace V = σ ∩ (−σ) as a face, and the linear subspace V does not depend on σ. When V = 0, Σ is a fan in the usual sense. A spanning set [85] of Σ is a finite subset R ⊂ N such that each cone σ ∈ Σ is generated by a subset of R over R ≥0 . Let (Σ, R , σ) be a triple such that Σ is a possibly degenerate fan on N R with support |Σ| = Π which admits a strictly convex piecewise linear function η : Π → R linear on each cone of Σ, R ⊂ S is a spanning set of Σ, and σ ∈ Σ is a cone. For such a triple, we define the cone CPL + (Σ, R , σ) ⊂ (R S ) as: When Σ = (N, Σ, R) is a stacky fan adapted to S, R is a spanning set for Σ and we have CPL + (Σ) = CPL + (Σ, R, {0}). Then the fan Ξ consists of the cones CPL + (Σ, R , σ) and the fan Ξ consists of the cones cpl(Σ, R ) = D(CPL + (Σ, R , σ)) (which are independent of σ).

3.3.
Extended refined fan sequence and extended Mori cone. The refined fan sequence [27] is an extension of the fan sequence (3.1) by a finite group. In this section we describe an extended version of the refined fan sequence for Σ ∈ Fan(S) (extended by ghost vectors G(Σ) = S \ R(Σ)). This will be used to describe a local chart of the global LG model (pr : Y → M, F ). For v ∈ Π, we take a cone σ ∈ Σ containing v, and write v = b∈R(Σ)∩σ c b b. Then Ψ Σ (v) = (Ψ Σ b (v)) b∈S is given by The map Ψ Σ gives a section of the map β : We define O(Σ) ⊂ Q S ⊕ N to be the sugbroup: and define Λ(Σ) ⊂ L Q to be These groups define the extended refined fan sequence: where the map O(Σ) → N is given by the second projection. This is compatible with the extended fan sequence and we have the following decompositions: Q contains classes of all orbifold stable maps to X Σ . We also have an isomorphism (Λ Σ ) ∼ = Pic(X Σ ) [27, Lemma 4.8], where Pic(X Σ ) is the Picard group of the coarse moduli space X Σ . On the other hand, we expect that O Σ is the set of classes of orbi-discs in H 2 (X Σ , L; Q) ∼ = Q R(Σ) with boundaries in a Lagrangian torus fibre L ⊂ X Σ . The notation O indicates 'open'.
We introduce the (extended) Mori cones and their open analogues. Let Σ(n) denote the set of n-dimensional (i.e. maximal) cones of Σ. For σ ∈ Σ(n), we define where λ b with b ∈ S denotes the bth component of λ ∈ R S . We define the extended Mori cone The unextended versions are given by is the usual Mori cone, that is, the cone spanned by effective curves in X Σ . The corresponding monoids are given as follows: Similarly, the unextended versions are given by: (3.16) The following lemma follows immediately from [27, Lemma 2.7]. We introduce a pairing between Pic st (X Σ ) := Pic(X Σ )/ Pic(X Σ ) and the lattices O(Σ), Λ(Σ), where Pic(X Σ ) denotes the Picard group of the coarse moduli space X Σ of X Σ . This pairing corresponds to the Galois symmetry for quantum cohomology in §2.2. Note that we have where the first isomorphism follows from the comparison of the extended fan sequence (3.3) and the extended refined fan sequence (3.11); the second isomorphism follows from the fact that Via (3.17), we obtain the following pairings: Pic st (X Σ ) × Λ(Σ) → C × , (ξ, λ) → e 2πi age(ξ,λ) .   Lemma 4.7]). For a box element b ∈ Box(Σ), age(ξ, (Ψ Σ (b), b)) is the age of the line bundle L ξ corresponding to ξ ∈ Pic(X Σ ) along the sector corresponding to the 3.4. Local charts of the LG model. We describe the local charts of Y and M corresponding to Σ ∈ Fan(S). By definition, the local chart of Y corresponding to Σ is given by (see [105, §4.1]): . Similarly, the local chart of M corresponding to Σ is given by: The coarse moduli space of M Σ is Spec(C[cpl(Σ) ∨ ∩ L]). Comparing the extended divisor sequence (3.4) with the sequence 0 → M → PL Z (Σ) → pl Z (Σ) → 0, we obtain the exact sequence where N tor = Hom(N tor , C × ) denotes the Pontrjagin dual of N tor . We have Pic(X Σ ) ∼ = L / b∈G(Σ) ZD b by (3.5), and that the Picard group Pic(X Σ ) of the coarse moduli space is given by pl Z (Σ)/ b∈G(Σ) ZD b (see [34, §4.2]). Therefore and the above sequence (3.19) identifies G Σ with the subgroup of Pic st (X Σ ) on which the generic stabilizers N tor of X Σ acts trivially. We give another description of the local chart Y Σ → M Σ , which shows that the LG model on this chart is the same as the LG model considered in [27, §4]. Proof. First we prove the statement on the dual cones. Observe that the cone CPL + (Σ) can be written as the intersection of the following simplicial cones K σ for all maximal cones σ ∈ Σ(n): , and the linear function Recall that OE(X Σ ) is the sum of the cones C Σ,σ defined in (3.14). Therefore, in order to prove CPL + (Σ) ∨ = OE(X Σ ), it suffices to show that Then the cone K σ is defined by the linear inequalities: On the other hand, for λ ∈ R S and c ∈ (R S ) , we have Hence the dual cone K ∨ σ is defined by the inequalities The latter inequality is equivalent to β(λ) ∈ σ, and thus K ∨ σ = C Σ,σ . Hence CPL + (Σ) ∨ = OE(X Σ ). The equality cpl(Σ) ∨ = NE(X Σ ) follows from this and D(CPL + (Σ)) = cpl(Σ), NE(X Σ ) = OE(X Σ ) ∩ L R .
Next we study the dual lattices of PL Z (Σ), pl Z (Σ). For c ∈ (R S ) and a maximal cone The lattice PL Z (Σ) is the intersection of the following lattices L σ for all σ ∈ Σ(n): Therefore PL Z (Σ) = σ∈Σ(n) L σ . For λ ∈ R S and c ∈ (R S ) , equation (3.20) can be rewritten as: Therefore we have: Proof. By the previous Lemma 3.14, we have O(Σ) + /N tor = CPL + (Σ) ∨ ∩ PL Z (Σ) and Λ(Σ) + = cpl(Σ) ∨ ∩ pl Z (Σ) . Therefore we have the natural maps: It follows from the above proposition that the local chart Y Σ → M Σ is a quotient (by Pic st (X)) of the LG model considered in [27, §4]. Indeed, the decompositions (3.16) induce the isomorphisms 3.5. Co-ordinate system on the local chart. Using the presentation in Proposition 3.15, we introduce a convenient co-ordinate system on the local chart for b ∈ S and λ ∈ Λ(Σ). When λ lies in Λ Σ ⊂ Λ(Σ), we also write q λ for q λ . We choose a splitting ς : N → O Σ of the refined fan sequence (3.12) of the form ς(v) = (ς(v), v), where ς : N → Q R(Σ) defines a splitting of the fan sequence (3.1) over Q. For v ∈ N and b ∈ S, we define Note that x v does not necessarily belong to C[O(Σ) + ] and that t b = 1 for b ∈ R(Σ). Then we have for b ∈ S, We can regard q = (q, t) as co-ordinates on the base M Σ and x as co-ordinates on fibres of Y Σ → M Σ . The LG potential can then be viewed as a family of Laurent polynomials in x with the fixed set S of exponents. Furthermore, we use the following co-ordinate expressions when necessary.
• Choosing an isomorphism N ∼ = Z n × N tor , we write x b = x b 1 1 · · · x bn n x ζ when b ∈ N corresponds to (b 1 , . . . , b n , ζ) ∈ Z n × N tor ; x 1 , . . . , x n can be viewed as co-ordinates along fibres of Y → M.
Remark 3.16. We compare the notation of [65,27] with the present one. In these papers, the LG potential was given in the form: where {y b } b∈S are deformation parameters. In the present paper, we set 6 y b = 1 for all b ∈ R(Σ). The variables Q and the other variables y b , b ∈ G(Σ) correspond to our q 1 , . . . , q r and t b with b ∈ G(Σ), and w b corresponds to our

Examples.
We give examples of partially compactified LG models for surface singularities.
3.6.2. Blowup of C 2 . We take N = Z 2 and S = {(1, 0), (0, 1), (1, 1)}. The possible fan structures Σ 1 , Σ 2 are shown in Figure 3. The fan Σ 1 corresponds to C 2 and Σ 2 corresponds to the blowup Bl 0 (C 2 ) at the origin.  In this case, the LG model has no orbifold singularities. The chart Y Σ 1 → M Σ 1 is given by The chart Y Σ 2 → M Σ 2 is given by The two charts are glued by t = q −1 .
The pictures of the fans Ξ, Ξ are similar to Figure 2.

Mirror symmetry
In this section, we review mirror symmetry for smooth toric DM stacks proved by Coates-Corti-Iritani-Tseng [26,27] and discuss its analytification. We construct various versions (algebraic, completed, analytified) of Gauss-Manin systems associated with the Landau-Ginzburg model around the limit point 0 Σ and compare them with the quantum cohomology D-module of the toric stack X Σ . We fix the data (N, Π, S) from §3.1. Note that the toric stacks Y, M have natural log structures defined by their toric boundaries (see e.g. [54,Ch 3]). With respect to these log structures, the family Y → M is log-smooth. The sheaves of logarithmic one-forms and logarithmic vector fields on Y are globally free and given respectively by Let x 1 , . . . , x n be the co-ordinates on fibres of Y → M given by the choice of an isomorphism N ∼ = Z n × N tor (see §3.5). Then the sheaf of relative logarithmic k-forms 8 are The sheaf Θ Y/M of relative logarithmic vector fields is generated by x 1 ∂ ∂x 1 , . . . , x n ∂ ∂xn . Let {χ 1 , . . . , χ n } denote the basis of M dual to the chosen isomorphism N ∼ = Z n ; then the relative vector field x i ∂ ∂x i acts on functions (on the chart Y Σ ) as For ξ ∈ L C , ξq ∂ ∂q denotes a vector field on M such that We define a generator ω of Ω n Y/M by This is normalized so that the integral over the maximal compact subgroup Hom(N, S 1 ) of Hom(N, C × ) equals (2πi) n . Informally speaking, the non-equivariant Gauss-Manin system below is a D-module on M×C z consisting of certain cohomology classes of relative differential forms f ω ∈ pr * Ω n Y/M such that oscillatory integrals are solutions to the D-module, where Y q := pr −1 (q). In the equivariant case, the phase function F should be replaced with The action of χ i commutes with the O M [z]-module structure, and thus GM T (F ) has the structure of an The relative k-forms are independent of the choice of a splitting ς in §3.5, although the co-ordinates xi themselves depend on ς.
(2) The (non-equivariant) Gauss-Manin system GM(F ) is defined to be the non-equivariant limit GM T (F )/M C · GM T (F ) of the equivariant Gauss-Manin system. This has the structure of an O M [z]-module. The flat connection and the grading operator on GM T (F ) descends to a flat connection ∇ : M and an operator Gr ∈ End C (GM(F )).
By forgetting the action of the fibre co-ordinates x 1 , . . . , x n , we shall regard GM T (F ) as an O M ⊗ R T [z]-module; then GM T (F ) is not of rank one as such 9 . We shall also regard GM T (F ) as a flat connection (i.e. D-module) over M. First we regard it as a module over the ring of differential operators denotes the lift of ξ under the splitting. By this splitting, we can regard GM T (F ) as a module over A different choice of splittings shifts the action of zξq ∂ ∂q ∈ Θ M by an element of M C . When the choice of a splitting is understood, we write z∇ ξq ∂ ∂q for z∇ξ u ∂ ∂u . The grading operator Gr , where E is the Euler vector field defined by Remark 4.2. The action of x 1 , . . . , x n forgotten in the above process corresponds to the Seidel representation (or shift operator) on quantum cohomology. This defines the structure of a difference module with respect to the equivariant parameters χ i , i.e. the action of x i shifts χ j as χ j → χ j − δ i,j z (note that we have the commutation relation [χ i , x j ] = zδ i,j x j as operators acting on GM T (F ) where H n (−) means the cohomology sheaf of a complex of sheaves (not the hypercohomology R n pr * ). This definition involves the choice of co-ordinates x 1 , . . . , x n , which corresponds to the choice of a splitting as above. To see that the first isomorphism holds, note that the nth cohomology is the cokernel of zd+dF T ∧ : [χ] and the relations given by Im(zd + dF T ∧) define the action of . The second isomorphism follows from the first.
, it follows from that ∇ has no singularities along the divisor t b = 0 with b ∈ G(Σ). Therefore, a "smaller" log structure (than the one given by toric boundaries) suffices to describe the logarithmic singularities of ∇. Note that in the equivariant case, the choice of co-ordinates q i , t b , x i determines the splitting of the extended divisor sequence.
Remark 4.5. In the non-equivariant Gauss-Manin system, the connection ∇ and the grading operator Gr together define the connection ∇ z ∂ ∂z = Gr −∇ E − dim X/2 in the z-direction as in the case of non-equivariant quantum D-modules, where E = b∈S D b q ∂ ∂q denotes the non-equivariant Euler vector field. Cf. (2.11).

Completion and mirror isomorphism.
We introduce a completion of the Gauss-Manin system at the large radius limit point 0 Σ ∈ M Σ of X Σ (see Definition 3.7) and recall a statement on mirror symmetry from [27].
Let Σ ∈ Fan(S) be a stacky fan adapted to S.
Definition 4.6. The completed (equivariant and non-equivariant) Gauss-Manin systems at 0 Σ are defined to be: The completed equivariant Gauss- denote the formal neighbourhood of 0 Σ ×Spec R T in M T . We regard GM T (F ) Σ as a Pic st (X)equivariant module over M T,Σ . We again choose a splitting ς : N → O Σ of the refined fan sequence (3.12); via the decomposition (3.13), ς induces a splitting of the extended refined fan sequence (3.11) and that of the extended divisor sequence (3.4) over C. As explained in the previous section ( §4.1), this splitting enables us to regard GM T (F ) Σ as a partial connection over M T,Σ . (This partial connection was explicitly described in Remark 4.4 by choosing co-ordinates q i , t b , x i given by ς).
The splitting ς also defines a splitting H 2 given by this basis is the same as the splitting induced by ς. This basis defines the equivariant Kähler moduli space M A,T (X Σ ) and the equivariant quantum D-module QDM T (X) (2.10), see § §2.2-2.3. Recall that QDM T (X) is an H 2 (X Σ , Z)/(Λ Σ ) -equivariant module over the formal neighbourhood of the origin in M A,T (X Σ ); we denote this formal neighbourhood by such that (1) Mir intertwines the Gauss-Manin connection with the quantum connection; (2) mir preserves the Euler vector fields mir * (E) = E (see (2.7), (4.2)) and Mir intertwines the grading operators (see (2.8), (4.1));    (3.21). Writing (q, t = {t v } v∈G(Σ) ) for the co-ordinates on M T,Σ as in §3.5, we have that the mirror map (q, t) → (q, τ = τ 0 φ 0 + s i=r+1 τ i φ i ) has the following asymptotic form: Here we assumed that is chosen to be a Z-basis of (Λ Σ ) and that its dual basis defines co-ordinates q i , i = 1, . . . , r as in §3.5; O(q, t 2 ) denotes an element of the ideal generated by .
The above mirror isomorphism shows that GM . This fact shows the following two propositions. The first one proves a non-equivariant version of Theorem 4.7.
Proposition 4.10. We have an isomorphism where mir denotes the non-equivariant limit of the mirror map in Theorem 4.7.
Proof. We apply Lemma 4.12 below to ]-module of finite rank, and in particular m Σ -adically complete. This proves the proposition.
Proposition 4.11. We have Proof. We apply Lemma 4.12 to module of finite rank, and in particular m Σ -adically complete. The conclusion follows.
where means the m Σ -adic completion. These isomorphisms follow from the same argument as in Remark 4.3 Analytification of the completed Gauss-Manin system. In this section, we construct an analytification of the completed Gauss-Manin system. There is a trade-off 10 between the analyticity along M T = M × Lie T and that along the z-plane: the analytified Gauss-Manin system is analytic in the M T -direction but formal in the variable z.

Analytification of algebras. First we study the restriction of GM
form a single Pic st (X Σ )-orbit (recall the sequence (3.19)). We mean by0 Σ either a single point in Y Σ or the corresponding finite subset in Spec C[O(Σ) + ] depending on the context. We have pr(0 Σ ) = (0 Σ , 0). In order to avoid the heavy notation, in this section §4.3, we will sometimes omit the subscript Σ for 0 Σ and0 Σ , writing 0 ∈ M Σ for 0 Σ and0 ∈ Y Σ for0 Σ .

Proof.
Note that (3.22)). Therefore the scheme theoretic fibre at (0, 0) is the spectrum of 10 By the convergence result from [26] reviewed in §4.4, we can make the analytified Gauss-Manin system fully analytic both in the M T -direction and in the z-direction. However, this analytic structure is different from the one induced from the original (algebraic) Gauss-Manin system.
where Ψ Σ is given in Notation 3.10 and the product on the right-hand side is given by It follows that the ring (4.4) is precisely the presentation of the orbifold cohomology ring H * CR (X Σ ) due to Borisov-Chen-Smith [14]. Since elements in H >0 CR (X Σ ) are nilpotent, the set-theoretical fibre at (0, 0) equals Spec(H 0 CR (X Σ )) ∼ = Spec(C[N tor ]) =0. We use the following elementary lemma on general topology. Lemma 4.15. Let X and Y be locally compact Hausdorff spaces and let f :  Proof. We apply Lemma 4.15 for f = pr, X = Y, Y = M T and y 0 = (0, 0). Note that Lemma 4.14 gives f −1 (y 0 ) =0. The former statement follows from part (1) of Lemma 4.15 and the latter statement on stalks follows from part (2). We will use the notation O an to denote the complex-analytic structure sheaf. The following lemma shows that pr * O an B (which is coherent by Grauert's Direct Image Theorem) gives an analytification of GM Here is the completion of R T = C[χ 1 , . . . , χ n ], and for a ring K, Proof. We show that the natural maps are isomorphisms. First we prove that the map (4.6) is an isomorphism. The left-hand side of (4.6) is the m an (0,0) -adic completion of ( pr is the ideal of (0, 0). Recall from Lemma 4.14 that the set-theoretic fibre of pr at (0, 0) is0; also from Lemma 4.16 We can prove similarly that the map (4.7) is an isomorphism.
with respect to the ideal m (0,0) generated by χ 1 , . . . , χ n and m Σ . Therefore the left-hand side of (4.7) is the m (0,0) -adic completion of GM is the ideal of0. This shows that the map (4.7) is an isomorphism. The lemma is proved.
. The conclusion follows from the following (probably) well-known fact: for a Noetherian local ring (A, m) and a finite A-module M , M is a free A-module of rank r if and only if its m-adic completion M = M ⊗ A A is a free A-module of rank r. This follows, for instance, by combining the fact that a finite flat module over a local ring is free, [81,Theorem 22.4 (1)] and M/mM ∼ = M /m M . 11 Here we used the following fact. Let I ⊂ J be ideals of a ring R, let M be an R-module and assume that I

4.3.2.
Analytification of D-modules. Next we construct an extension of the completed Gauss-Manin system GM T (F ) Σ to an analytic neighbourhood of (0, 0) ∈ M T = M × Lie T. Let U ⊂ M T and B ⊂ Y denote analytic open neighbourhoods of (0, 0) and0 respectively as in Corollary 4.19. Recall that pr * O an B is a locally free O an U -module by Corollary 4.19. Definition 4.20 (cf. Definition 4.1). Let Σ ∈ Fan(S) be a stacky fan adapted to S.
(1) The analytified equivariant Gauss-Manin system around the limit point 0 is the sheaf over U GM an , together with the usual multiplication of functions in q and z (where χ i is a co-ordinate on Lie T and q is a co-ordinate on M, see §3.5 and §4.1). A flat connection (2) The (non-equivariant) analytified Gauss-Manin system GM an (F ) Σ around the limit point 0 is defined to be the restriction of GM an (independent of the choice of a splitting) and the grading operator Gr ∈ End C (GM an (F ) Σ ). The connection ∇ and Gr together give a flat connection in the z-direction as in Remark 4.5.
Remark 4.21. The overline for GM an T (F ) Σ , GM an (F ) Σ indicates that they are completed in the z-adic topology. Note also that these analytified systems depend on the choice of Σ ∈ Fan(S).
A more precise definition of the module structure is described as follows: where in the right-hand side we expand h(q, The action is well-defined, since we allow any formal power series in z.
] follows from the following fact. Let K be a ring with an ideal m and let M be a K-module. Suppose that K is m-adically complete and M is Hausdorff with respect to the m-adic topology, that is, ]. The next proposition shows that GM an T (F ) Σ is an analytification of GM T (F ) Σ (and thus justifies the name).
] are isomorphisms, where the first map is induced by the map ]. That the maps in (4.9) are isomorphisms follows from the following two facts: (a) all three modules in (4.9) are finite and free as O U ,(0,0) [[z]]-modules, and (b) the maps in (4.9) are isomorphisms modulo z. We already know that (b) holds by Lemma 4.17. In fact, the maps in (4.9) reduces to the isomorphisms in (4.5) modulo z. Thus we only need to show (a). Proposition 4.23 implies that (GM an  By restricting the above isomorphism to V = U ∩ (M × {0}) and using Proposition 4.10, we have

Note that the second and the third term is supported on
This is a multi-valued function on Y with parameter χ ∈ Lie T. For a fixed (q, χ) ∈ M T = M × Lie T, (logarithmic) critical points of F T | pr −1 (q) are solution to the equation: Thus we can regard pr : B → U (4.3) as a family 13 of critical points of F T . It follows from the study [27, Lemma 6.2] of critical points near 0 Σ that the fibre of the finite morphism pr| B : B → U at a generic point consists of dim H * CR (X Σ ) many reduced points. We write where "ss" means semisimplicity. The complement of U ss in U (called caustic) is an analytic . For (q, λ) ∈ U ss and a critical point p ∈ pr −1 (q, χ) ∩ B, we can define the formal asymptotic expansion of the oscillatory integral (see [27, §6.2]): We obtain the right-hand side by expanding the integrand e F T /z φ(x, q, λ) in Taylor series at p (with respect to the logarithmic co-ordinates log x 1 , . . . , log x n ) and performing termwise 13 More precisely, B is the union of branches of critical points that tend to0 as q → 0 = 0 Σ .
(Gaussian) integration. More precisely, we have where s 1 , . . . , s n are the logarithmic co-ordinates centred at p so that x j = x j (p)e s j , is the Hessian matrix at p, (h i,j ) are the coefficients of the matrix inverse to (h i,j ) and is the truncated Taylor expansion of F T at the critical point p.
(1) When the critical point p does not lie in the logarithmic locus of Y and φ(x, q, λ) is a polynomial, the above formal asymptotic expansion makes sense as an actual asymptotic expansion: for this we choose the integration cycle Γ(p) ⊂ Y q := pr −1 (q) to be a stable manifold for the Morse function x → (F T (x, q)) associated with p and assume that z approaches zero from the negative real axis.
(2) More precisely, the above formal asymptotic expansion depends on the choice of the square root of the Hessian. This corresponds to the choice of an orientation of the Morse cycle Γ(p) and a branch of (−2πz) n/2 . Definition 4.29. The higher residue pairing of two sections s 1 , s 2 ∈ GM an T (F ) Σ are defined as: where (q, χ) ∈ U ss .
The higher residue pairing gives a bilinear pairing ] which satisfies the following properties [27, Proposition 6.8]: (a) P is O an U ss -bilinear, non-degenerate, z-sesquilinear and symmetric: where E is the Euler vector field (4.2). (d) along z = 0, P equals the Grothendieck residue pairing.   Theorem 6.11], therefore it has poles along χ = 0 when X Σ is noncompact. When X Σ is compact, it becomes holomorphic and non-degenerate in a neighbourhood of (0 Σ , 0).

4.4.
Analytic mirror isomorphism. Using the convergence result [27, Theorem 7.2] (which generalizes [61, Theorem 1.2]), we show that the mirror isomorphism in Theorem 4.7 extends to a neighbourhood of 0 = 0 Σ as an isomorphism between the analytified Gauss-Manin system GM an T (F ) Σ and the analytic quantum D-module. Let U ⊂ M T be an open neighbourhood of (0 Σ , 0) as in Corollary 4.19. Recall that GM an T (F ) Σ was defined on U. The mirror isomorphism in Theorem 4.7 induces, via the isomorphism in Proposition 4.24, the following isomorphism: This isomorphism extends to an analytic neighbourhood of (0 Σ , 0). We recall the following facts from [27, §7]:  Theorem 7.1]; (c) the mirror map mir is also analytic in a neighbourhood of (0 Σ , 0) ∈ M T (ibid.). By part (a), the equivariant quantum D-module QDM T (X Σ ) (see (2.10)) extends to a small analytic neighbourhood U of q = τ = χ = 0 in the equivariant Kähler moduli space [M A,T (X Σ )/ Pic st (X Σ )] (see (2.5)). We denote it by    As explained before Theorem 4.7, in the above theorem, we choose a splitting N → O Σ of the refined fan sequence (3.12), which simultaneously defines a partial connection ∇ on GM an T (F ) Σ and the equivariant quantum D-module.
The analytic mirror theorem above shows that the analytified Gauss-Manin system can be further analytified in the z-direction (since QDM an T (X Σ ) is analytic in z). We can regard this as a solution to the Birkhoff problem (i.e. finding a normal form of the Gauss-Manin connection) which has been studied extensively in the construction of K. Saito's flat structure [96,97,11,92,37,38].

Discrepant wall-crossings
We study the change of quantum cohomology of smooth toric DM stacks under a "discrepant" wall crossing. We show a decomposition of formal (i.e. completed in the variable z) quantum D-modules under discrepant wall-crossings. We work in the set-up of §3.1 and fix the data (N, Π, S) as usual.
Recall from §3.1.1 that the toric stacks X Σ with Σ ∈ Fan(S) arise as the GIT quotients of C S by the torus L C × . These toric stacks are birational to each other since they contain the (stacky) torus [(C × ) S /L C × ] as an open dense subset. We can regard L R as the space of GIT stability conditions for the L C × -action on C S ; if we choose a stability condition from the interior of the maximal cone cpl(Σ) of the secondary fan Ξ (see Definition 3.6), then the corresponding GIT quotient is X Σ . We choose two adjacent maximal cones cpl(Σ + ), cpl(Σ − ) of Ξ which are separated by a hyperplane wall W ⊂ L R . Here we assume that W ∩ cpl(Σ + ) = W ∩ cpl(Σ − ) is a common codimension-one face of cpl(Σ ± ). Let w ∈ L be a primitive integral vector which is perpendicular to the wall W ⊂ L R and is non-negative on the chamber cpl(Σ + ). By the definition of L (see (3.3)), the vector w ∈ L gives rise to a linear relation (3.4)). This linear relation defines a circuit {b ∈ S : D b · w = 0} in the terminology of Gelfand-Kapranov-Zelevinsky [45], where a 'circuit' means a minimal linearly dependent set. The transition between Σ + and Σ − can be described in terms of the circuit ('modification along a circuit' [45]).
For I ⊂ S, we write 14 14 Note that σI is a closed cone, whereas ∠I is a relatively open cone.
- Figure 5. Modification along a circuit. The signs ± mean rays belonging to The set of rays of Σ ± is given by Here yields a simplicial cone σ I ∈ Σ 0 belonging to both Σ + and Σ − , and I ∈ S circ 0 yields a (not necessarily simplicial) cone σ I ∈ Σ 0 containing the circuit M + ∪ M − ; the cone σ I with I ∈ S circ 0 is subdivided into simplicial cones σ I\{v} , v ∈ M ± in the fans Σ ± . See Figure  5. We also remark that Let X ± denote the toric stack corresponding to Σ ± . As discussed in [15, §5], [31, §6.3], the toric birational map ϕ : X + X − fits into a commutative diagram where X 0 is the toric variety associated with Σ 0 , X is another smooth toric DM stack and f ± : X → X ± , g ± : X ± → X 0 are projective birational toric morphisms. Defineb ∈ N ∩ Π to be the vector: The smooth toric DM stack X is given by a stacky fan Σ adapted to S ∪ {b} (in the sense of Definition 3.3): the set of rays of Σ is R = (R + ∩ R − ) ∪ {b} where R ± := R(Σ ± ); the fan Σ underlying Σ is a simultaneous subdivision of Σ + and Σ − given by The toric morphisms f ± : X → X ± are induced by natural maps Σ → Σ ± of stacky fans. We refer the reader to [31, §6.3] for a description of X and f ± in terms of GIT quotients.
Lemma 5.1. Let K ± = K X ± denote the canonical class of X ± and E ⊂ X denote the toric divisor of X corresponding to the rayb. Then we have Proof. This follows immediately from [31,Proposition 6.21].
In view of the lemma above, we say that the birational transformation is on the wall W ) and discrepant otherwise. We shall restrict ourselves to the case where the transformation is discrepant. By exchanging X + and X − if necessary, we may assume: Remark 5.4. There are three types of discrepant wall-crossings: (I) X + and X − are isomorphic in codimension one ("flip"), (II) the birational map induces a map (i) X + → X − or (ii) X − → X + contracting a divisor to a toric subvariety, where X ± is the coarse moduli space of X ± ("discrepant resolution") and (III) X + = X − but the stack structures of X + and X − differ along a divisor. In terms of stacky fans, we have . This is similar to the classification of crepant transformations given in [31,Propositions 5.4,5.5] and can be shown by a parallel argument.
Example 5.6. A blow-up along a toric subvariety is an example of type (II) discrepant transformation.
Example 5.7. A root construction [21] along a toric divisor is an example of type (III) discrepant transformation.

The
LG model along a curve. Consider the partially compactified LG model (pr : Y → M, F ) from §3.2. Recall that M is defined in terms of the secondary fan Ξ consisting of maximal cones cpl(Σ), Σ ∈ Fan(S). Let C ⊂ M denote the 1-dimensional toric substack corresponding to the codimension-1 cone cpl(Σ + ) ∩ cpl(Σ + ). The curve C lies in the boundary of M and connects the large radius limit points 0 Σ + and 0 Σ − .
We cover C by the two open sets ; in these local charts, the embedding C ⊂ M is given by the C-algebra homomorphism: Lemma 5.8. The inverse image pr −1 (C) is covered by two charts ,v) to pr −1 (C ∩ M ± ) and Ψ ± := Ψ Σ ± (see Notation 3.10). The two charts are glued by 3). The conclusion follows from Remark 3.17 and the description of cones of Σ 0 , Σ ± in terms of the circuit (5.1).
Remark 5.9. We describe how the curve C looks like in the Kähler moduli space. Using the equation (5.3) for C and the asymptotics of the mirror map (Remark 4.9), we find that the image of C under the (non-equivariant) mirror map for X + is asymptotically close to, near the large radius limit point 0 Σ + , type (I) or (II-i) case: the curve given by τ = 0 and q d = 0 for all d ∈ Λ Σ + + \ Q ≥0 w (i.e. the curve corresponding to the extremal class w ∈ H 2 (X + , Q)); type (II-ii) or (III) case: the curve given by τ ∈ CD b + and q d = 0 for all d ∈ Λ Σ + + \ {0}, where b + is the unique element of M + and D b + ∈ H <2 CR (X + ) is as in Remark 4.9.
Here we use the notation on the Kähler moduli space from §2.2 and the classification in Remark 5.4. Similarly, near 0 Σ − , the image of C under the mirror map for X − is asymptotically close to type (I) or (II-ii) case: the curve given by τ = 0 and q d = 0 for all d ∈ Λ Σ − + \ Q ≥0 (−w) (i.e. the curve corresponding to the extremal class −w ∈ H 2 (X − , Q)); type (II-i) or (III) case: the curve given by τ ∈ CD b − and q d = 0 for all d ∈ Λ CR (X − ) is as in Remark 4.9. The above classes D b ± are supported on the image of the exceptional divisor. We note that they can be zero, and in that case we need to examine the higher-order terms in the mirror map to see the asymptotic behaviour of C.
Remark 5.10. Note that pr −1 (C) ⊂ Y is a possibly reducible toric substack and its components are in one-to-one correspondence with maximal cones of the fan Σ 0 . The LG potential restricted to pr −1 (C) is of the form F = b∈R + ∪R − u b (here u b with b ∈ S \(R + ∪R − ) vanishes on pr −1 (C)).

5.3.
Decomposition of the Gauss-Manin system. In this section, we show that the analytified Gauss-Manin system associated with Σ − is a direct summand of that associated with Σ + in a neighbourhood of C. For this, we study the family The C × -action generated by the Euler vector field (4.2) plays an important role in the following discussion. Consider the elements b∈S e b ∈ (Z S ) and b∈S D b ∈ L ; they define C × -actions, respectively, on Y and M such that pr : Y → M is C × -equivariant. In terms of the co-ordinates (u b ) b∈S , the C × -action is given by with s ∈ C × and the potential function F = b∈S u b is of weight 1 with respect to the action. Introduce the C × -action on Lie T given by the scalar multiplication; then the map pr : Y → M × Lie T is C × -equivariant. Let 0 ± := 0 Σ ± = {q ±w = 0} ∈ C denote the large radius limit points of X ± (see Definition 3.7) and let0 ± =0 Σ ± ∈ pr −1 (C) denote the torus-fixed points such that pr(0 ± ) = 0 ± as in §4.3.1. The C × -action on the family pr −1 (C) → C is given by (with notation as in Lemma 5.8) By Assumption 5.2, we have that lim s→0 s·x = 0 + , lim s→∞ s·x = 0 − for every x ∈ C \{0 + , 0 − } and lim s→0 s · y =0 + for every y ∈ pr −1 (C \ {0 − }).
We choose analytic open sets B ± ⊂ Y, U ± ⊂ M T with0 ± ∈ B ± , (0 ± , 0) ∈ U ± such that the conclusion of Corollary 4.19 holds. Since pr is C × -equivariant, even after replacing B ± and we have that the conclusion of Corollary 4.19 still holds. We henceforth assume that B ± , U ± are preserved by the C × -action. Since every point in pr Proof. First note that U + ∩U − contains (C \{0 + , 0 − })×{0}. Since pr : Take any point y from the complement, and choose a compact neighbourhood K of pr(y) in U 0 . Since pr : We consider the analytified equivariant Gauss-Manin system GM an T (F ) ± := GM an T (F ) Σ ± from Definition 4.20. Note that it is defined over the U ± above, since the only properties we need in the construction are those in Corollary 4.19.
Corollary 5.12. Let U 0 , R be as in Lemma 5.11.
(1) We have a direct sum decomposition of O an U 0 -algebras: (2) We have a direct sum decomposition of O an U 0 [[z]]-modules: Under this decomposition, the Gauss-Manin connection ∇, the grading operator Gr and the higher residue pairing P split into the direct sum.
For a generic (q, χ) ∈ U 0 , pr −1 (q, χ) ∩ R consists of finitely many reduced points, and the asymptotic expansion in Definition 4.27 defines an isomorphism Asym : Remark 5.14. Since rank pr * (O an  Figure 6. Family of critical points. Out of d critical points, d − 2 points go to infinity at the large radius limit of O P 1 (−d) (d = 5 in the picture).
Example 5.15. Consider the example of the LG model in §3.6.3. This corresponds to a discrepant transformation between X + = [C 2 /µ d ] (of type 1 d (1, 1)) and its minimal resolution LG model was given by the potential function d), {t = 0} is the large radius limit point of X + and {q = 0} is the large radius limit point of X − . The relative critical scheme of F on the chart (x 1 , x 2 , t) is given by This is schematically depicted in Figure 6. Mir an ± : GM an over an open neighbourhood U ± ⊂ U ± of (0 ± , 0) ∈ M T = M × Lie T, where mir ± : U ± → U ± denotes the mirror map and the overline · · · means the z-adic completion. Let us observe that the analytic mirror isomorphism extends to a domain which is closed under the C × -action (as discussed in §5.3). We have already seen in §5.3 that GM an T (F ) ± extends to C × U ± = s∈C × s· U ± . Introduce the C × -action on the equivariant Kähler moduli space [M A,T (X ± )/ Pic st (X ± )] generated by the Euler vector field (2.7); since the mirror map mir ± intertwines the Euler vector fields, it can be extended to a unique C × -equivariant map mir ± : C × U ± → C × U ± . We may assume (by shrinking U if necessary) that GM an T (F ) ± | U is generated by Gr-homogeneous sections Ω ± 0 , . . . , Ω ± s over O an where QDM an T (X ± ) denotes the z-adic completion of the equivariant quantum D-module of X ± (see §4.4) and R is as in Definition 5.13. Under this decomposition, the flat connection, the grading operator and the pairing split as follows: Gr ± , P ± denote respectively the quantum connection, the grading operator (2.8) and the pairing (2.9) on the quantum D-module of X ± .
Remark 5.17. The one-form α arises from the difference of the splittings of the extended divisor sequence (3.4) over Q, one chosen for X + and the other for X − (see Remark 4.33). It is of the form α(ξq ∂ ∂q ) = α(ξ) for some α ∈ Hom(L Q , M Q ); in particular α vanishes in the non-equivariant limit. Recall that the Gauss-Manin connection in the equivariant case depends on the choice of a splitting (see §4.1).
Remark 5.18. By construction, the above decomposition (5.5) preserves the additional structure given by multiplication by the mirror co-ordinates x 1 , . . . , x n . They correspond to the equivariant shift operators (Seidel representaion), see Remark 4.2.

5.5.
Comparison of Gromov-Witten theories in all genera. Using a formula due to Givental [50,49] and Teleman [103] for higher genus Gromov-Witten potentials, we show that the ancestor Gromov-Witten potential of X + is decomposed into the ancestor potential of X − and a product of Witten-Kontsevich tau functions, under the action of a quantized symplectic operator. We use Givental's formula for orbifolds studied by Brini-Cavalieri-Ross [19] and Zong [108]. 5.5.1. Ancestor potentials. Let X be a smooth DM stack equipped with a T-action satisfying the assumptions in §2.1; we use the notation there. Consider the forgetful morphism p : X g,l+m,d → M g,l+m → M g,l which forgets the map and the last m marked points, and let ψ i ∈ H 2 (X g,l+m,d ), 1 ≤ i ≤ l denote the pull-back of the universal cotangent class ψ i ∈ H 2 (M g,l ) by p. We define the ancestor Gromov-Witten invariants as the T-equivariant integral: where α i , β j ∈ H * CR,T (X). When the moduli stack X g,m+l,d is not proper, the right-hand side is defined by the virtual localization formula and lies in S T = Frac(R T ) as before. We choose a homogeneous R T -basis {φ i } N i=0 of H * CR,T (X) satisfying (2.4) and introduce the infinite set y = {y i k } k≥0,0≤i≤s of co-ordinates on H * CR,T (X)[[z]] given by The ancestor potential of X is the following generating function of the ancestor Gromov-Witten invariants: which we regard as a function in the formal neighbourhood of y = 0. Introduce another set of variables x = {x i k } k≥0,0≤i≤N that are related to y by the formula (Dilaton shift): Using the co-ordinate x(z), we shall regard A X,τ as a formal function on the formal neighbourhood of ]. As in §2.2, we can specialize Q to one in the ancestor potential by using the divisor equation for τ . We have 2). Henceforth we assume that Q is specialized to one in the ancestor potential.
Let V be a finite-dimensional C-vector space equipped with a symmetric non-degenerate pairing (·, ·) V . Then V ((z)) has the following symplectic form Ω V : where ψ k 1 , . . . , ψ k l g,l = M g,l ψ k 1 · · · ψ k l , ∞ k=0 x k z k denotes the co-ordinate on C[[z]] and y k = x k + δ k,1 . We used the dilaton equation   . . , k) are given, we write A 1 ⊗ · · · ⊗ A k for the function ). Theorem 5. 19. Let A ±,τ denote the ancestor potential of X ± . For (q, χ) ∈ U 0 ss , we have Remark 5.20. Implicit in the above theorem is the convergence and analyticity of A ±,τ with respect to τ . This follows from the Givental formula. We refer to [29, Theorem 1.4, Definition 3.13] for the discussion on the convergence of ancestor potentials.
Remark 5.21. The ancestor potentials of X ± define sections of the Fock sheaves [30] associated with the quantum D-modules of X ± , and the above relationship can be interpreted in this language.
Remark 5.22. We can state the relationship between the Gromov-Witten theories of X + and X − in terms of Cohomological Field Theory (CohFT): the CohFT of X + transforms into the product of the CohFT of X − and the trivial semisimple CohFT under the Givental action of U q,χ . We refer the reader to [103,101,87] for the Givental group action on CohFTs.
where mir : U → [M A,T (X Σ )/ Pic st (X Σ )] is the mirror map. We flip the sign of z and set R q,χ := R q,χ | z→−z .
Proof. The mirror isomorphism Mir an intertwines the Gauss-Manin connection with the quantum connection, and it was shown in [27, Lemma 6.7] that e F T (p)/z Asym p gives a solution to the Gauss-Manin connection for each critical branch p of F T . Let U denote the diagonal matrix whose entries are relative critical values of F T . It follows that R q,χ e −U/z gives a formal fundamental solution for mir * ∇ in the sense of [47, Proposition 1.1], [108, Theorem 5.1] (our R q,χ corresponds to ΨR in [47,108]; see also Proposition 6.1 below). We need to flip the sign of z because we use the sign convention for the quantum connection opposite to [47,108]. It now suffices to check that R q,χ satisfies the classical limit condition at the large radius limit q = 0 Σ given in [ Remark 5.24. It has been observed by Givental [49] that the operator R can be constructed by equivariant mirrors (for Fano toric manifolds).
Remark 5.25. Recall from Remark 4.28 that the asymptotic expansion is ambiguous up to sign. Therefore R q,χ is ambiguous up to the right multiplication by a signed permutation matrix. The right-hand side of the Givental formula is, however, independent of the choice (see, e.g. [29,Proposition 4.3]).
Proof of Theorem 5. 19. Let R ± q,χ denote the R-operators for X ± introduced above. By the construction of U q,χ in §5.5.3, we have the following commutative diagram for (q, χ) ∈ U 0 ss (when we order the critical points of F T appropriately): where N ± = dim H * CR (X ± ). The conclusion follows by Proposition 5.23 and the 'chain rule' of the Givental quantization (see, e.g. [29,Remark 3.22]).

Formal decomposition and analytic lift
We discuss the formal decomposition and its analytic lift for quantum D-modules or Gauss-Manin systems. This is known as the Hukuhara-Turrittin theorem for irregular differential equations. For mirrors of the small quantum cohomology of weak Fano compact toric stacks, the analytic lift is described explicitly in terms of oscillatory integrals. In this section, we restrict ourselves to the non-equivariant quantum cohomology and Gauss-Manin system, and assume that the toric stacks are compact.
6.1. Hukuhara-Turrittin type result. Recall that the non-equivariant quantum connection (or, equivalently, the non-equivariant Gauss-Manin connection) can be extended in the direction of z. The connection in the z-direction is of the form (see (2.11)): When the quantum product τ is semisimple, ∇ decomposes over where ∇ is understood as being extended in the z-direction by (2.11) and V is the preimage [z] is a Pic st (X Σ )-equivariant sheaf, which shall be regarded as a sheaf on the stack V = [ V / Pic st (X Σ )]. We omitted the grading operator from the data because it can be recovered from the other data as Gr = ∇ E + ∇ z ∂ ∂z + 1 2 dim X Σ . We also write ], ∇, P ) for the quantum D-module completed in z. We set When where u 1 , . . . , u N are eigenvalues of E τ with N = dim H * CR (X Σ ) and P std is the pairing given by the multiplication P std (f, g) = f (−z)g(z) with f, g ∈ O an [[z]]. Once we fix the ordering of u 1 , . . . , u N , the isomorphism Φ is unique up to a right multiplication by diag (±1, . . . , ±1). Remark 6.2. In this proposition, we do not require that eigenvalues of E τ are mutually distinct. Note however that u 1 , . . . , u N form a co-ordinate system in a neighbourhood of a semisimple point τ 0 [40, Lecture 3], and thus they are generically mutually distinct.

Remark 6.3. The isomorphism Φ modulo z is given by a normalized idempotent basis
In plain words, Proposition 6.1 means that the differential equation ∇ z ∂ ∂z f = 0 for a cohomology-valued function f = f (z) admits a formal power series solution with Ψ i,n ∈ H * CR (X Σ ), where Ψ i,0 = Ψ i is the normalized idempotent in Remark 6.3. These formal power series are typically divergent. Over an appropriate angular sector in the z-plane, however, we can find an actual analytic solution whose asymptotic expansion is given by (6.3); moreover the actual solution with prescribed asymptotics is unique if the angle of the sector is bigger than π.
To state this analytic lift in a sheaf-theoretic language, we follow Sabbah [94] and introduce a sheaf A of "holomorphic" functions on the real blowup of C. An (open) sector is a subset of C of the form {z ∈ C × : φ 1 < arg z < φ 2 , |z| < δ} for some φ 1 , φ 2 ∈ R and δ ∈ (0, ∞]. A holomorphic function f (z) on the sector I = {z ∈ C × : φ 1 < arg z < φ 2 , |z| < δ} is said to have the asymptotic expansion f ∼ ∞ k=0 a k z k as z → 0 along I if for every > 0 and for every m ≥ 0, there exists a constant C ,m > 0 such that for all z with φ 1 + ≤ arg z ≤ φ 2 − and |z| ≤ δ/2. Definition 6.4 ([94, Chapter II, §5.c]). Let C := [0, ∞) × S 1 denote the oriented real blowup of C at the origin. This is a smooth manifold with boundary and is equipped with the map π : C → C, π(r, e iθ ) = re iθ . Let C ∞ C denote the sheaf of complex-valued C ∞ -functions on C and define A C to be the subsheaf of C ∞ C of germs annihilated by the Cauchy-Riemann operator z ∂ ∂z = 1 2 (r ∂ ∂r +i ∂ ∂θ ). Sections of A C over the open set {(r, e iθ ) ∈ C : 0 ≤ r < δ, φ 1 < θ < φ 2 } are precisely those holomorphic functions f (z) on the sector {z ∈ C × : φ 1 < arg z < φ 2 , |z| < δ} which admit asymptotic expansions f (z) ∼ ∞ k=0 a k z k along this sector. The same construction works in families: for a complex manifold M , we similarly define the sheaf A M × C over M × C to be the subsheaf of C ∞ M × C of germs annihilated by the Cauchy-Riemann operators z ∂ ∂z , ∂ M . The asymptotic expansion yields a map: for any ξ ∈ S 1 , where i ξ : M ∼ = M × {(0, ξ)} → M × C is the inclusion. This map is known to be surjective (the Borel-Ritt Lemma; see, e.g. [106]). The natural map π : M × C → M ×C is a map of ringed spaces, i.e. we have a map π −1 O an M ×C → A M × C of sheaves of rings. In particular, we can define the pull-back of an O an M ×C -module F to be π * F := π −1 F ⊗ π −1 O an M ×C A M × C . For a multi-set {u i } = {u 1 , . . . , u N } of complex numbers, we say that a direction e iφ ∈ S 1 or a phase φ is admissible for {u i } if e iφ is not parallel to any non-zero difference u i − u j , i.e. e iφ / ∈ R(u i − u j ) for all i, j.
Proposition 6.5. Let τ 0 ∈ V ss be a semisimple point and let e iφ ∈ S 1 be an admissible direction for the spectrum {u 0 1 , . . . , u 0 N } of E τ 0 . Let π : V ss × C → V ss × C denote the oriented real blowup along V ss × {0}. There exist an open neighbourhood B of τ 0 in V ss , a positive number > 0, and an isomorphism over the sector I φ = {(r, e iθ ) : |θ − φ| < π 2 + } such that Φ φ induces, via (6.4), the formal decomposition Φ in Proposition 6.1. Here we exclude the data of the pairing from QDM an (X Σ ). Moreover such a Φ φ is unique. We call Φ φ the analytic lift of Φ.
Remark 6.6. The uniqueness of the analytic lift Φ φ is ensured by the fact that the angle of the sector I φ is bigger than π: see [10, Remark 1.4]. The lift Φ φ depends on τ 0 and e iφ , and it depends continuously on (τ 0 , e iφ ) unless (τ 0 , e iφ ) crosses the locus where e iφ is non-admissible for the spectrum of E τ 0 . Remark 6.7. The analytic lift Φ φ preserves the pairing in the following sense: consider the analytic lift Φ φ+π associated with the opposite direction −e iφ , then for sections s − , s + of π * QDM an (X Σ ), respectively, over B × I φ+π , B × I φ . This follows from the fact that the asymptotic expansions of both sides coincide by Proposition 6.5, and that the pairings are flat.
6.2. Asymptotic basis and marked reflection system. The sectorial decomposition in Proposition 6.5 gives rise to a linear algebraic data which we call the marked reflection system where (·, ·) is the orbifold Poincaré pairing (2.2).
Recall the fundamental solution L(τ, z)z −µ z c 1 (X Σ ) of the quantum connection introduced in §2.4.
Let τ 0 ∈ V ss be a semisimple point and let φ be an admissible phase for the eigenvalues of E τ 0 . Let Φ φ be the sectorial decomposition associated with τ 0 and φ as in Proposition 6.5: We choose a base point τ ∈ V × corresponding to a real cohomology class and choose a path connecting τ and τ 0 in V × . Let s i be the flat section of QDM an (X Σ ) on a neighbourhood of (τ 0 , e iφ ) satisfying Φ φ (s i ) = e −u i /z e i , where e i is the ith standard basis of A ⊕N B×I φ . Then we have vectors v i ∈ H * CR (X Σ ) such that Note that there is no natural ordering of the asymptotic basis and also that each v i is determined up to sign v i → ±v i (because of the ambiguity of Φ in Proposition 6.1); we treat {v 1 , . . . , v N } as an unordered basis. When we order {v i } in such a way that (e −iφ u 1 ) ≥ (e −iφ u 2 ) ≥ · · · ≥ (e iφ u N ), the Gram matrix ([v i , v j )) is upper-triangular with diagonal entries all equal to one, and gives the Stokes matrix of the quantum connection at the irregular singular point z = 0 (see [40, Lecture 4], [44, Proposition 2.6.4]). When τ 0 and φ vary and cross the locus where the corresponding eigenvalues {u i } become non-admissible for φ, the corresponding marked reflection system undergoes mutation. We refer to [44, §4.2] for the full details of the deformation theory of marked reflection systems. We illustrate an example of mutation in Figure 7; the figure describes a typical procedure where {u i } varies in the configuration space of N points in C and crosses the wall of nonadmissible configurations. In the picture, we drew the half-ray u j + R ≥0 e iφ from each u j to show the direction e iφ . Suppose that the asymptotic basis {v 1 , . . . , v N } is marked by {u 1 , . . . , u N } in the leftmost picture, i.e. u j = m(v j ). We assume that the basis is ordered so that (e −iφ u 1 ) > (e −iφ u 2 ) > · · · > (e −iφ u N ). After passing through the non-admissible configuration in the middle picture, the vector v i marked by u i is transformed into and the other vectors remain the same (i.e. the marking is given by v i → u i and v j → u j for j = i in the rightmost picture). This is called the right mutation of v i with respect to v i+1 . The inverse procedure is the left mutation of v i with respect to v i+1 : The two operations (6.6), (6.7) are inverse to each other because of the semiorthogonality condition (6.5).
for some classes V i ∈ K(X Σ ) in the K-group, i.e. the corresponding flat section s i lies in the Γ-integral structure (see also Proposition 7.8). The Gamma conjecture II [44, §4.6] (recently proved by [41] for Fano toric manifolds) says that V i comes from a full exceptional collection in the derived category of coherent sheaves. In this situation, mutation of asymptotic basis corresponds to that of full exceptional collections. stack X Σ is weak Fano, the Gauss-Manin system GM(F ) over the small quantum cohomology locus of X Σ has the expected rank equal to dim H * CR (X Σ ) and gives a fully analytic (i.e. analytic both in the M direction and in the z-direction) D-module mirror to the small quantum D-module of X Σ . In this case, we can describe the analytic lift of the formal decomposition using oscillatory integrals. 6.3.1. Formal decomposition of the Gauss-Manin system. Let X Σ be a toric stack with Σ ∈ Fan(S) and let (pr : Y → M, F ) denote the LG model from Definition 3.6. Let B ⊂ Y, U ⊂ M T = M×Lie T be open neighbourhoods of (respectively)0 Σ and (0 Σ , 0) as in Corollary 4.19. Recall that the analytified equivariant Gauss-Manin system GM an T (F ) Σ is defined over U and its non-equivariant version GM an (F ) Σ is defined over V = U ∩(M×{0}) (see Definition 4.20). When X Σ is compact, the family of relative critical points of the LG potential F is a finite morphism whose generic fibre is reduced (see [63, Proposition 3.10]), i.e. there exists an open dense subset V ss of V such that for any q ∈ V ss , F | B∩pr −1 (q) has only non-degenerate critical points. We set Cr ss := Cr ∩ pr −1 (V ss ). By definition, pr : Cr ss → V ss is a finite covering. The following result describes a formal decomposition parallel to Proposition 6.1 for the analytified Gauss-Manin system.
Proof. As we remarked in Remark 4.28, we need to choose a square root of the Hessian when defining the formal asymptotic expansion; hence we need the µ 2 -local system ori. Along z = 0, the map Asym is given by φ · ω → φ| Cr ss /(|N tor | det(h i,j )). On the other hand, Remark 6.11. The eigenvalues u 1 , . . . , u N of E τ in Proposition 6.1 correspond to critical values of F . 6.3.2. Gauss-Manin system over the small quantum cohomology locus. To discuss an analytic lift of the above formal decomposition, we restrict the Gauss-Manin system to the small quantum cohomology locus. We assume that X Σ is a weak Fano (i.e. −K X Σ is nef) and compact toric stack. Furthermore, we assume that (6.8) where ∆ ⊂ N R denotes the convex hull of ray vectors {b : b ∈ R(Σ)}. The compactness implies that ∆ contains the origin in its interior and the weak-Fano condition implies that all rays b ∈ R(Σ) lie in the boundary of ∆. By replacing S with S − = S ∩ ∆ in the construction of the (partially compactified) LG model in §3.2, we obtain an LG model which we call the LG model mirror to the small quantum cohomology of X Σ . We also call M sm the small quantum cohomology locus of X Σ . Under the mirror map, M sm maps to H ≤2 CR (X Σ ). Lemma 6.12. The total space Y sm is a closed toric substack of Y corresponding to the cone (R ≥0 ) S\S − of Ξ; similarly M sm is a closed toric substack of M corresponding to the cone D((R ≥0 ) S\S − ) ∈ Ξ. Moreover, we have the pull-back diagram: Proof. Let Ξ − denote the fan defining Y sm . Recall that maximal cones of Ξ are in one-to-one correspondence with stacky fans adapted to S; likewise, maximal cones of Ξ − are in one-toone correspondence with stacky fans adapted to S − . Thus the set of maximal cones of Ξ − can be identified with a subset of the set of maximal cones of Ξ. We can see that this subset consists of maximal cones CPL + (Σ) of Ξ that contain (R ≥0 ) S\S − as a face; moreover the corresponding maximal cone of Ξ − is given by the image of CPL + (Σ) under the projection (R S ) → (R S − ) . Therefore Ξ − is a fan obtained as the star of the cone (R ≥0 ) S\S − in Ξ. This shows the first statement. A similar argument shows that M sm is a closed toric substack corresponding to D((R ≥0 ) S\S − ). To see the pull-back diagram, we recall the description of the uniformizing chart in (3.21). When Σ is adapted to S − , G(Σ) contains S \ S − . In the local chart associated with Σ, the diagram (6.10) is of the form: which is clearly a pull-back diagram.
Remark 6.13. The small quantum cohomology locus M sm ⊂ M depends on the choice of Σ ∈ Fan(S).
Remark 6.14. The condition (6.8) ensures that Assumption 3.1 holds for S − = S ∩ ∆. This condition is, however, not necessary at this point; we can define Y sm (or M sm ) as the substack corresponding to (R ≥0 ) S\S − (resp. to D((R ≥0 ) S\S − )). We shall need this condition later when we apply the results on the Γ-integral structure for toric stacks in [63] (see §3.1.4 ibid where the same assumption was made).
We observe that the non-equivariant Gauss-Manin system GM(F ) over M sm already has the expected rank; hence the completion and the analytification studied in § §4.2-4.3 are unnecessary over M sm in the weak-Fano case. Proof. This essentially follows from a result of Mann-Reichelt [80, Theorem 4.10] on the GKZ system; we give a proof of a more general statement (including the equivariant case and without assumption (6.8)) in Appendix A.
Let V ⊂ M denote the base of the analytified Gauss-Manin system GM an (F ) Σ as before. This is an analytic open neighbourhood of 0 Σ .
is an isomorphism.
(2) The mirror isomorphism Mir | χ=0 in Proposition 4.10 extends to an isomorphism GM(F )| V sm ∼ = mir * QDM an (X) over an open neighbourhood V sm ⊂ V sm of 0 Σ , where mir : V sm → V is the analytic mirror map (see §4.4 for the convergence of the mirror map).
Proof. For Part (1), it suffices to show that the map is an isomorphism along z = 0. As discussed in the proof of Proposition 6.10, the analytified Gauss-Manin system along z = 0 is isomorphic to pr * O an Cr . Thus the natural map in (1) is surjective along z = 0; it is an isomorphism since the ranks are the same. Part (2) follows from the convergence of the I-function in the weak-Fano case, see, e.g. [63, Proposition 4.8].
6.3.3. Analytic lift of the formal decomposition. We continue to assume that X Σ is a compact, weak-Fano toric stack and that the condition (6.8) holds. Consider the LG model (6.9) over 2). In fact, since ∆ contains the origin in its interior, we have a linear relation b∈S − λ b b = 0 with λ b ∈ Z >0 ; this shows that b∈S − u λ b b = q λ = 0 on the fibre at q ∈ M sm,× and in particular that u b = 0 for all b ∈ S − . 16 The following discussion works if we replace M sm,× with the slightly bigger subspace {q ∈ M sm Σ : q λ = 0 for all λ ∈ Λ Σ + }: this bigger space parametrizes Laurent polynomials with Newton polytope ∆.
The locally freeness of GM(F )| M sm,lf in Proposition 6.15 implies (by the restriction to z = 0) that the family of relative critical points of F (6.11) Cr sm := p ∈ Y sm : For q ∈ M sm,ss , let {c 1 , . . . c N } be the set of (mutually distinct) critical points of F q := F | pr −1 (q) , and let u i = F q (c i ) be the critical value. For an admissible phase φ for {u 1 , . . . , u N }, let Γ φ i ⊂ pr −1 (q) denote the Lefschetz thimble of F q emanating from the critical point c i whose image under F q is the half-line u i − R ≥0 e iφ ; it is given as the stable manifold of the Morse function (e −iφ F q ) : pr −1 (q) → R: where ϕ t is the upward gradient flow 17 of (e −iφ F q ) with respect to the complete Kähler metric i  Remark 6.17. For a non-admissible phase φ, the Lefschetz thimble Γ φ i is not always defined because Γ φ i may hit other critical points c j with u j ∈ u i − R >0 e iφ . On the other hand, when some of the critical values u 1 , . . . , u N coalesce at q 0 ∈ M sm,ss and φ is an admissible phase for the critical values of F q 0 , the Lefschetz thimbles Γ φ 1 , . . . , Γ φ N are well-defined in a neighbourhood of q = q 0 despite the possibility that φ can be non-admissible at a nearby point. This is because different Lefschetz thimbles associated with the same critical value do not intersect each other, and no non-trivial Picard-Lefschetz transformations occur among these thimbles around q = q 0 . Proposition 6.18. Let q 0 be a point in M sm,ss . Choose an admissible phase φ for the critical values {u 1,0 , . . . , u N,0 } of F q 0 . Choose a sufficiently small open neighbourhood B ⊂ M sm,ss of q 0 and a sufficiently small number > 0 such that e −iφ (u i − u j ) / ∈ R whenever q ∈ B, |φ − φ | < and u i,0 = u j,0 . Let π : M sm,ss × C → M sm,ss × C denote the oriented real blowup along M sm,ss × {0}. Define the map : |θ−φ| < π 2 + } and we choose δ ∈ (− , ) depending on the argument of z so that the integral converges. Then Φ φ is an isomorphism that intertwines the Gauss-Manin connection ∇ with N i=1 (d + d(u i /z)) and induces the formal decomposition in Proposition 6.10 (combined with Proposition 6.16) if B ⊂ V ss : Proof. First observe that the oscillatory integral Φ φ (s) converges for a suitable choice of δ ∈ (− , ), and does not depend on δ as far as it converges. If | arg(z) − φ| < π 2 , we can Remark 6.19. The choice of an orientation of Γ φ+δ i and the choice of a branch of (−2πz) −n/2 in (6.13) together give rise to a section of the µ 2 -local system ori in Proposition 6.10.

Functoriality under toric birational morphisms
We study the analytic lift of the formal decomposition of the quantum D-modules in Theorem 5.16 in the case where the birational map X + X − extends to a morphism. We show that the anlalytic lift associated with a certain deformation parameter τ + and a phase is induced by the pull-back between the K-groups via the Γ-integral structure. Moreover, the sectorial decomposition of the quantum D-module of X + at some τ + corresponds to an Orlov-type semiorthogonal decomposition of the K-group. We assume that both X + and X − are compact weak-Fano smooth toric DM stacks and restrict ourselves to the non-equivariant quantum D-modules. 7.1. Notation and assumption. Consider a discrepant transformation X + X − arising from a codimension-one wall crossing as in §5.1. Let Σ ± be the stacky fan of X ± . We assume that X + X − extends to a birational morphism ϕ : X + → X − . In this case, the common blowup X in (5.2) is isomorphic to X + , and ϕ is necessarily a type (II-i) or (III) discrepant transformation in the classification of Remark 5.4, i.e. ϕ is a divisorial contraction or a root construction. We further assume that • X + and X − are compact weak Fano toric stacks; we write ∆ ± ⊂ N R for the fan polytopes of X ± ; they are convex polytopes containing the origin in their interiors and we have ∆ − ⊂ ∆ + ; Here the fan polytope ∆ ± means the convex hull of ray vectors of the stacky fan Σ ± and S is a finite subset of N as in §3.1 such that both Σ + and Σ − are adapted to S in the sense of Definition 3.3. We need these assumptions so that we can apply the results [63] on the Γ-integral structure for toric stacks, where the same assumptions were made (see §3.1.4 ibid ). As in §5.1, let W denote the hyperplane wall between the maximal cones cpl(Σ + ) and cpl(Σ − ) of the secondary fan Ξ, and let w ∈ L denote the primitive normal vector of the wall W pointing towards cpl(Σ + ). Set M ± = {b ∈ S : ±D b · w > 0} as before. When the wall-crossing induces a morphism X + → X − , M − is a singleton {b} with Db · w = −1 and the corresponding circuit is (see ( Assumption 5.2 gives b∈M + k b > 1. In the type (II-i) case, we have M + ≥ 2 and the stacky fan Σ + is obtained from Σ − by adding the new rayb; the cone σ M + = b∈M + R ≥0 b of Σ − is subdivided into cones σ M + ∪{b}\{v} with v ∈ M + . In the type (III) case, M + is also a singleton {b} and Σ + is obtained from Σ − by replacingb withb = kbb: see Figure 9. We write R ± = R(Σ ± ) for the set of rays. Then R + = R − {b} in the type (II-i) case and R + {b} = R − {b} in the type (III) case. We also note that M + ⊂ R − . For simplicity of notation, we assume that S is chosen to be a minimal extension of S − : We do not lose any generality by this assumption: the base M of the LG model for a larger S always contains the locus corresponding to S − ∪ {b}.
The smooth toric DM stacks X + , X − are the GIT quotients of C S . The toric birational morphism ϕ : X + → X − is induced by the self-map It is easy to check that this sends the stable locus for cpl(Σ + ) to the stable locus for cpl(Σ − ). The map ϕ contracts the divisor {zb = 0} onto the toric substack Z = v∈M + {z v = 0}. In the type (II-i) case, ϕ is a weighted blowup along the codimension ≥ 2 substack Z; in the type (III) case where M + is a singleton {b}, ϕ exhibits X + as a root stack of X − with respect to the divisor Z = {zb = 0}. We write (pr : Y → M, F ), (pr : Y sm → M sm , F ) for the LG models associated with S and S − = S ∩ ∆ − respectively (see Definition 3.6). These two LG models are related by the pull-back (Lemma 6.12):  Notation 3.10).
In this section, we fix an isomorphism N ∼ = Z n × N tor and a splitting ς : N → O Σ − of the refined fan sequence (3.12) for Σ − and use the associated co-ordinates x 1 , . . . , x n along the fibres of the LG model as introduced in §3.5.  Lemma 5.8). Recall also that F = b∈R + ∪R − u b on pr −1 (C). We study the family of relative critical points (7.2) x 1 ∂F ∂x 1 = · · · = x n ∂F ∂x n = 0 over the curve C.
Proposition 7.2. A relative critical point of the LG potential F over C is given by an assignment of a complex number w v to every v ∈ N satisfying w 0 = 1, The corresponding critical value of F is given by Jγ.
Proof. Take a relative critical point and let w v denote the value of the function w v at that point. We have w 0 = 1. The equation (7.2) for relative critical points reads and v belong to a common cone of Σ − = Σ 0 0 otherwise where note that Σ 0 in Lemma 5.8 coincides with the underlying fan Σ − of Σ − in the current setting. Therefore there exists a cone σ ∈ Σ − such that {v : In the former case, we have σ = 0; this corresponds to the case where γ = 0 in the proposition. In the latter case, we have σ ⊃ σ M + ; the linear relation b = b∈M + k b b together with (7.3) implies: This proves the proposition.
We write Y q for the fibre of pr : Y → M at q ∈ M; and Γ R = Γ R (q) for the fibre of Y R → M R at q ∈ M R : Consider the restriction of F q = F | Yq to the real positive locus Γ R . Then • it is a strictly convex function since the Hessian • it is proper and bounded from below since 0 is in the interior of the convex hull ∆ + of {b : b ∈ S}. Therefore, F q | Γ R attains a global minimum at a unique critical point cr R = cr R (q) ∈ Γ R (q); the point cr R is called the conifold point [43,44]. Since Γ R is preserved by the gradient flow of (F q ) (with respect to the Kähler metric i (1) For q ∈ V + ∩ M R , we have an isomorphism which varies locally constantly in q such that , Ω ∈ GM(F ) and z > 0, where s V (τ, z) is the flat section of the Γ-integral structure (see Definition 2.7).
In other words, the integral structure of the Gauss-Manin system GM(F ) dual to the lattice H n (Y q , { (F q /z) 0}; Z) corresponds to the Γ-integral structure of the quantum D-module under the mirror isomorphism.
0}; Z) is the parallel translate of Γ 1 in the local system from θ = 0 to θ = π, and #(e −πi Γ 1 · Γ 2 ) denotes the algebraic intersection number. Remark 7.7 (cf. Remark 2.8). We need to specify a branch of the multi-valued section s V in the above theorem. We have a standard choice for the branch of s V (mir + (q), z) with V ∈ K(X + ) when q ∈ M R , and the above theorem holds for this choice. By the argument preceding [63, Proposition 4.8], the fundamental solution L(mir(q), z) can be obtained from the I-function via the Birkhoff factorization; the I-function has a standard determination on the positive real locus (by requiring log q a ∈ R in the formula [63, (59)] of the I-function). We also have a standard determination of z −µ z c 1 (X) given by log z ∈ R for z > 0. Hence we obtain a standard identification of the space of flat sections of GM(F ) over (q, z) ∈ (M R ∩ V + ) × R >0 with the K-group K(X + ) ⊗ C.
Choose an admissible phase φ ∈ (−α 0 , α 0 ) for {u 1 (q 0 ), . . . , u N + (q 0 )} and let Φ + φ , Φ + φ be the associated analytic lifts over a neighbourhood B × I φ of (q 0 , (0, e iφ )) as above. Let ϕ i denote a section of π * mir * + QDM an (X + )| B×I φ such that Φ + φ (ϕ i ) = e i . Take a local section Ω = Ω q,z of GM(F ) near q = q 0 . By the definition (6.13) of Φ + φ , for q ∈ B and z ∈ C × with | arg(z) − φ| < π/2, we have 18 where P std is the diagonal pairing as in Proposition 6.10 and we used Remark 6.7 in the last step. By Theorem 7.5(1), the left-hand side of the above equation equals Since the above equation holds for all Ω, we have ϕ i = e u i (q)/z s V i (mir + (q), z). We now set i = 1. By the choice of α 0 and φ, the cycle Γ φ+π 1 (q 0 ) (and hence Γ φ+π Remark 7.9. Recall from Propositions 6.1, 6.10 that the formal decomposition and its analytic lift Φ + φ are ambiguous up to multiplication by diag(±1, . . . , ±1), and this ambiguity is fixed once we trivialize the µ 2 -local system ori. We note that there is a standard trivialization of ori at the conifold point cr R since the Hessian of F q at cr R is positive-definite; hence the component of the analytic lift Φ + φ corresponding to cr R (q) (which is the first component) is unambiguous.
Remark 7.10. Proposition 7.8 says that s O is characterized by the exponential asymptotics s O ∼ e −u 1 (q)/z Ψ R (q) as z → 0 along the sector arg z ∈ (− π 2 − α 0 , π 2 + α 0 ) when q lies in a neighbourhood of the positive real locus, where Ψ R (q) is the normalized idempotent (see Remark 6.3) corresponding to the conifold point cr R (q). In terms of the marked reflection system in §6.2, it also says that Γ X + ∪(2πi) deg 0 /2 inv * ch(V + i ), 1 ≤ i ≤ N + give the asymptotic basis at q 0 and φ.
Remark 7.11. At various places in §7, we work around for the fact that u 1 (q) + R ≥0 may contain other critical values; the argument would become much simpler if otherwise. 7.3.2. The structure sheaf of X − . We repeat the same discussion for X − . The difference is that we consider the analytic lift over a region V − which protrudes from the small quantum cohomology locus M sm of X − .
Let M sm,× := L ⊗ C × , Y sm,× := (C × ) S − denote the open dense tori in M sm and Y sm respectively and set: ) (since the Gauss-Manin system GM sm (F ) has the rank N − by Proposition 6.15). We define −,R . Let cr 1 (q), . . . , cr N − (q) be all branches of critical points of F q contained in B − and defined in a neighbourhood of q = q 0 in V ss − . We may assume that cr 1 (q 0 ) = cr R (q 0 ) and write u i (q) = F q (cr i (q)) as before. By combining the mirror isomorphism (7.4) and the formal decomposition of the analytified Gauss-Manin system in Proposition 6.10, we obtain a formal decomposition over a neighbourhood of q 0 in V ss − . By Proposition 6.5, for an admissible phase φ for {u 1 (q 0 ), . . . , u N − (q 0 )}, we have a connected open neighbourhood B of q 0 in V ss − , a sector I φ = {(r, e iφ ) : |θ − φ| < π 2 + } (with small > 0) and an analytic lift of Φ: where π : V ss − × C → V ss − × C is the oriented real blow-up. By the uniqueness of the analytic lift, this coincides with the analytic lift of the Gauss-Manin system from Proposition 6.18 over the small quantum cohomology locus B ∩ M sm , via the mirror isomorphism (7.5). Each component of Φ, Φ − φ is ambiguous up to sign, but recall from Remark 7.9 that the sign of the first component (corresponding to the conifold point cr R (q)) is determined canonically. We have the following result for X − parallel to Proposition 7.8: Proposition 7.13. For q 0 ∈ V sm,ss −,R , there exists α 0 ∈ (0, π/2) such that the following holds. For every admissible phase φ ∈ (−α 0 , α 0 ) for {u 1 (q 0 ), . . . , u N − (q 0 )}, there exists a basis Proof. The same argument as Proposition 7.13 (using Theorem 7.12 in place of Theorem 7.5) shows that there exist z). It follows from the flatness of s i and e −u i (q)/z e i that Φ − φ (e u i (q)/z s i ) = e i holds over the whole B.

7.4.
Inclusion of the local systems of Lefschetz thimbles. The Gauss-Manin system over the small quantum cohomology locus is underlain by a local system of Lefschetz thimbles.
In this section, we observe an inclusion of the local system over M sm in a neighbourhood of 0 − = 0 Σ − (mirror to X − ) to the local system over M (mirror to X + ) under a slide in the 'positive real' direction. The inclusion shall be identified with the pull-back ϕ * : K(X − ) → K(X + ) in K-theory in §7.5.

Convergent and divergent critical branches.
Let V ± be (sufficiently small) analytic open neighbourhoods of 0 ± ∈ M as in the previous section §7. 3. Recall the C × -action on pr : Y → M generated by the Euler vector field considered in § §5.3-5.4. As discussed there, we may assume that V ± is C × -invariant, because the mirror map mir ± , the mirror isomorphism Mir ± and the analytified Gauss-Manin system can be extended to the orbit C × V ± (V ± is the intersection of U ± ⊂ M × Lie T in §5.4 with M × {0}). Let Cr ± → V ± denote the (finite, flat) family of relative critical points over V ± : where B − is the subset of Y appearing in the construction of GM an (F ) Σ − in §4.3 and pr is the map in (4.3). Since all the relative critical points over V + are contained in B + , we do not need to take the intersection with B + in the first formula. By Lemma 5.11, we have an open such that the ramified covering Cr + | V 0 → V 0 decomposes as: where D ⊂ Cr + is an open subset giving a subcover of Cr + | V 0 . By taking the C × -orbit, we may assume that V 0 is also C × -invariant. The subcover D → V 0 consists of branches of critical points that diverge at 0 − . We call critical points corresponding to D divergent and those corresponding to Cr − convergent. Among the critical points over C described in Proposition 7.2, those corresponding to γ = 0 are divergent.

7.4.2.
Local co-ordinate system around 0 − = 0 Σ − . Recall from Proposition 3.15 that the local chart of M around 0 − is given by Recall that Pic st (X − ) acts on the chart M − (7.6) by the age pairing (3.18); since tb = q −w and w ∈ L, Pic st (X − ) acts trivially on the last co-ordinate tb.
On the chart (7.6), the C × -action generated by the Euler vector field is given by: where J = ( b∈M + D b · w) − 1 (as in Proposition 7.2). Note that the C × -weight of the co-ordinate q λ with λ ∈ Λ (Σ − ) + is non-negative by Lemma A.5.
denotes open disc of radius ρ centered at z ∈ C, and K, J are as in Proposition 7.2. The last two conditions imply that convergent critical values over D are contained in {|u| < ρ 0 } and divergent critical values over D are contained in {|u| > 2ρ 0 } and away from the sector − π 2J ≤ arg(u) ≤ π 2J with vertex at the origin, see Figure 10. See also Figure 11. Definition 7.14. Define the sliding map s t : D sm → M − with 0 ≤ t ≤ 1 by s t (q) = (q, t).
Note that the map D sm → M − , q → (q, t) between the uniformizing charts is Pic st (X − )equivariant and thus descends to the map s t . Note also that s 0 = id Dsm .  (2) For 0 < t ≤ 1, we have Im(s t ) ⊂ t −1/J · D ⊂ V 0 , where t −1/J · (−) denotes the C × -action generated by the Euler vector field.

7.4.4.
Kouchnirenko's condition and a local system of Lefschetz thimbles. Let (q, t) = (q, tb) be the local co-ordinates on M − in §7.4.2. By abuse of notation, we mean by q (resp. (q, t)) either a point on the uniformizing chart M sm − (resp. M − ) or its image in M sm − (resp. M − ) depending on the context. We write Y q,t := pr −1 (q, t) ⊂ Y, Y sm q := pr −1 (q) ⊂ Y sm for the fibres of the LG model; note that Y sm q = Y q,0 . The LG potential F q,t = F | Yq,t with q ∈ M sm,× is a Laurent polynomial of the form (cf. (3.22)) We are interested in the case where T is S or S − , and f is F q,t with (q, t) ∈ M × or F q,0 with q ∈ M sm,× . The compactness of X ± implies that the Newton polytope contains the origin in its interior in these cases. We define M sm,nd := {q ∈ M sm,× : F q,0 is Newton non-degenerate}, We define a function H(x) of (x 1 , . . . , x n ) ∈ (C × ) n as: where · is the norm with respect to the Kähler metric i 2 n j=1 d log x j ∧ dlog x j , i.e. dF q,0 (x) = ( n j=1 |∂F q,0 /∂ log x j | 2 ) 1/2 . (2) The same estimate holds for dF q,t by replacing M sm,nd with M nd , Y sm with Y, and Proof. This is a refinement of [63, Lemma 3.11], which says that dF q,0 (x) is proper on pr −1 (K). In fact, a slight modification of the argument there yields a proof of the proposition.
It suffices to show that there exists > 0 such that {x ∈ pr −1 (K) : dF q,0 (x) ≤ H(x)} is compact. Suppose on the contrary that {x ∈ pr −1 (K) : dF q,0 (x) ≤ H(x)/k} is non-compact for all k ≥ 1. Then we can find q(k) ∈ K and x(k) ∈ Y sm q(k) such that dF q(k),0 (x(k)) ≤ H(x(k))/k and H(x(k)) ≥ k. Passing to a subsequence we may assume that q(k) converges in K; we may also assume that we can label elements of S − as {b (1) (1) The family {F −1 q,0 (u) ⊂ Y sm q } q,u of affine varieties is a locally trivial family of C ∞ manifolds over {(q, u) : q ∈ M sm,nd , u is a regular value of F q,0 }.
(2) The same conclusion holds for F −1 q,t (u) ⊂ Y q,t when we replace M sm,nd with M nd . Proof. Take (q 0 , u 0 ) such that q 0 ∈ M sm,nd and u 0 is a regular value of F q 0 ,0 . Choose a sufficiently small co-ordinate neighbourhood (B; q 1 , . . . , q r , u) of (q 0 , u 0 ) which does not intersect the discriminant locus. The ambient family (q,u)∈B Y sm q is trivial over B, and is identified with B × Hom(N, C × ) through the co-ordinates x 1 , . . . , x n . It suffices to shows that the coordinate vector fields 19 α∂/∂q i , α∂/∂u with α ∈ C on B can be lifted to integrable vector fields tangent to the family (q,u)∈B F −1 q,0 (u). Lifts of α∂/∂q i , α∂/∂u are given under the trivialization (q,u)∈B Y sm Since the potential F q,0 is of the form (7.8), we have |∂F q,0 /∂q i | ≤ C · H(x) for some constant C > 0 over B. Thus the estimate in Proposition 7.19 shows that these lifts are bounded on the family (q,u)∈B F −1 q,0 (u). Therefore the flows of these vector fields exist as long as the corresponding flows on the base B exists. Proposition 7.19 implies that the improper function (F q,0 (x)) satisfies the so-called Palais-Smale condition 20 when q ∈ M sm,nd , and hence the usual Morse theory can be applied to it. It 19 Here we identify (real) vector fields of a complex manifold with (1, 0) vector fields. 20 dFq,0(x) is bounded away from zero outside a neighbourhood of the critical set.
follows (see [63, §3.3.1]) that the relative homology group H n (Y sm q , {x : (F q,0 (x)) ≥ M }; Z) is a free Z-module of rank dim H * CR (X − ) when M is large enough so that all critical values of F q,0 are contained in { (z) < M }. This group is independent of the choice of sufficiently large M , and we denote it by H n (Y sm q , {x : (F q,0 (x)) 0}; Z). By the local triviality in Corollary 7.21, we have q,0 (u); Z) for any u > 0 such that all critical values of F q,0 are contained in {z : (z) < u}, and this forms a local system over M sm,nd . Similarly, the relative homology groups (7.10) Lef q,t := H n (Y q,t , {x : (F q,t (x)) 0}; Z) ∼ = H n (Y q,t , F −1 q,t (u); Z) form a local system of rank dim H * CR (X + ) over M nd , where u > 0 is such that all critical values of F q,t are contained in {z : (z) < u}. These two local systems have different ranks and we will relate them below. 7.4.5. Inclusion of the local systems: statement and proof. Let s t be the sliding map in Definition 7.14 and set D × sm := D sm ∩ M sm,× . Let s >0 denote the map (0, 1] × D × sm → M nd given by s >0 (t, q) = s t (q). Since (0, 1] is contractible, we have where p : (0, 1] × D × sm → D × sm is the projection to the second factor. Let j : We have an inclusion of the local systems: The map ι maps the positive real Lefschetz thimble (introduced in §7.3) to the positive real one, i.e. ι(Γ R (q)) = Γ R (s t (q)) when q ∈ D × sm ∩ M sm R . For (q, t) ∈ M − and η > 0, we set where H(x) is as in (7.9). When η > min x∈Yq,t H(x), A q,t (η) is a compact region such that the inclusion A q,t (η) → Y q,t is a homotopy equivalence. Indeed, via the Hom(N, S 1 )-action, we have A q,t (η) ∼ = Hom(N, S 1 ) × {(x 1 , . . . , x n ) ∈ (R >0 ) n : H(x) ≤ η} and the second factor is contractible since H| (R >0 ) n is strictly convex. For any (q, u) ∈ M sm,× × C, the real algebraic function H(x) 2 restricted to F −1 q,0 (u) has finitely many critical values by [83,Corollary 2.8], and thus there exists η 0 > 0 such that F −1 q,0 (u) and ∂A q,0 (η) intersect transversally for all η with η ≥ η 0 . We will show in the following lemma that such an η 0 can be chosen independently of (q, u) as far as (q, u) varies in a compact subset of M sm,nd × C. We note that Nemethi-Zaharia [84] and Parusinski [88] have obtained analogous results for a single polynomial function on C n (and the proof is similar). Lemma 7.24. (1) For a compact subset K ⊂ M sm,nd × C, there exists η 0 such that for all η ≥ η 0 and all (q, u) ∈ K, F −1 q,0 (u) and ∂A q,0 (η) intersect transversally.
(2) The same result on the transversality of F −1 q,t (u) and ∂A q,t (η) holds when we replace M sm,nd with M nd .
The proof of Lemma 7.24 will be given in Appendix B. In the situation of Lemma 7.24(1), A q,0 (η) ∩ F −1 q,0 (u) is a deformation retract of F −1 q,0 (u) for η ≥ η 0 ; we can see this using the Morse flow for the function H(x) on F −1 q,0 (u). In particular, the inclusion of pairs (A q,0 (η), F −1 q,0 (u) ∩ A q,0 (η)) → (Y sm q , F −1 q,0 (u)) induces an isomorphism of relative homology. This fact will be used in the following proof.
Proof of Theorem 7.22. We shall construct an inclusion of local systems: Note that the stalk of i −1 j * s −1 >0 Lef at q 0 ∈ D × sm consists of a Gauss-Manin flat family of relative homology classes in Lef st(q) with t > 0 sufficiently small and q in a small contractible neighbourhood of q 0 in M sm,nd . The construction of ι will be done in the following 5 steps.
(1) Construction of ι on the stalk at q 0 ∈ D × sm . Let ρ 2 > 0 be the constant in Lemma 7.15(3) and let u 0 > ρ 2 be such that all critical values of F q 0 ,0 are contained in {u : (u) < u 0 }. By Lemma 7.24, there exists η 0 > 0 such that F −1 q,0 (u) ∂A q,0 (η) for all η ≥ η 0 and (q, u) in a neighbourhood of (q 0 , u 0 ). Then by the remark preceding the proof, the inclusion induces an isomorphism of relative homology (we use Z coefficients unless otherwise mentioned): Since {∂A q,t (η 0 )} q,t is a proper family, F −1 q,t (u 0 ) and ∂A q,t (η 0 ) intersect transversally for (q, t) in a sufficiently small contractible neighbourhood B of (q 0 , 0) in M − . The Ehresman fibration theorem implies that F −1 q,t (u 0 ) ∩ A q,t (η 0 ) is a trivial family of C ∞ -manifolds (with boundary) when (q, t) varies in B. Therefore, whenever s t (q) lies in B, the inclusion of pairs induces an isomorphism of relative homology, where we set A B (η 0 ) := (q,t)∈B A q,t (η 0 ). Thus we obtain a Gauss-Manin flat isomorphism (7.12) : : for t > 0 sufficiently small and q in a small neighbourhood of q 0 . Composing (7.11), (7.12) and the natural map induced by the inclusion: Now recall from Lemma 7.15(3) that there are no critical values of F st(q) in the region {u ∈ C : |u| ≥ ρ 2 , arg(u) ∈ [− π 2J , π 2J ]}. Since u 0 > ρ 2 , the parallel transportation along a straight path defines a canonical isomorphism ) for all u 1 > u 0 . Thus (7.13) gives a map Lef sm q 0 → Lef st(q) for t > 0 sufficiently small and q in a small neighbourhood of q 0 . This map is Gauss-Manin flat as (t, q) varies and defines the map ι q 0 on the stalks.
(2) Independence of the choice of η 0 , u 0 . We show that the map ι q 0 is independent of the choices made. That replacing η 0 with a bigger η 1 > η 0 does not change the map ι q 0 follows from the commutative diagram: ). Next we show that the map is independent of u 0 . Suppose that we choose u 1 > u 0 in place of u 0 . By Lemma 7.24 again, we can choose η 0 > 0 such that F −1 q,0 (u) ∂A q,0 (η) for all η ≥ η 0 , all q in a neighbourhood of q 0 and all u in the interval [u 0 , u 1 ]. Then the family of C ∞manifolds F −1 q,t (u) ∩ A q,t (η 0 ) is trivial as u varies in [u 0 , u 1 ] and (q, t) varies in a contractible neighbourhood B of (q 0 , 0). The various maps in the construction of ι q 0 can be compared with the following sequence of maps: ) and the maps between the first four pairs induce isomorphisms in relative homology. It follows that the different choices u 0 , u 1 give the same map ι q 0 . is given by Lefschetz thimbles: setting u i = F q 0 ,0 (cr i ) with 1 ≤ i ≤ N − and choosing a system of mutually non-intersecting paths γ i connecting u i and u 0 , we have finite Lefschetz thimbles Γ i ∼ = D n emanating from the critical point cr i and fibred over the path γ i ; then Γ 1 , . . . , Γ N − define relative cycles in the pair (Y sm q 0 , F −1 q 0 ,0 (u 0 )) and form a basis of Lef sm q 0 . We choose η 0 > 0 big enough so that Γ i are contained in A q 0 ,0 (η 0 ) and that the conditions in (1) are satisfied. The critical points cr i belong to convergent critical branches, and as such, vary continuously in a neighbourhood of (q 0 , 0) in M − ; the path γ i can be also continuously deformed to a family of paths γ i (q, t) connecting the critical value F q,t (cr i ) and u 0 . Due to the compactness, the finite thimble Γ i can be continuously deformed 21 to a finite thimble Γ i (q, t) ⊂ A q,t (η 0 ) for F q,t fibred over γ i (q, t) when (q, t) is sufficiently close to (q 0 , 0); Γ i (q, t) gives a relative cycle of (A q,t (η 0 ), F −1 q,t (u 0 ) ∩ A q,t (η 0 )). As relative cycles in (Y st(q) , F −1 st(q) (u 0 )), Γ 1 (s t (q)), . . . , Γ N − (s t (q)) form part of a basis of the nth relative homology (corresponding to the convergent critical points) when t > 0 is sufficiently small and q is sufficiently close to q 0 . Thus ι q 0 sends a basis to part of a basis, and is injective.
where N ± = dim H * CR (X ± ) and u i are relative critical values of F . By Proposition 6.5, by shrinking B if necessary, we have analytic lifts of these formal decompositions over B × I φ , where I φ = {(r, e iθ ) ∈ C : |θ − φ| < π 2 + } (for some small > 0): where π : M × × C → M × × C denotes the oriented real blowup along M × {0}. Summands of this sectorial decomposition are indexed by relative critical points of F ; those for X + are indexed by all critical points and those for X − are indexed by the subset of convergent critical points. Combining these two decompositions, we get a sectorial decomposition: This gives an analytic lift of the formal decomposition in Theorem 5.16. We consider the inclusion (7.16) π * mir * − QDM an (X − ) B×I φ → π * mir * + QDM an (X + ) B×I φ induced by (7.15). Note that the maps (7.15), (7.16) are associated with a semisimple point q 0 and an admissible direction e iφ .
Theorem 7.25. Let ϕ : X + → X − be a toric birational morphism as in §7.1. For each point q * ∈ V sm,ss −,R ∩ D ×• sm ⊂ M sm R , there exists a contractible open neighbourhood W * of q * in V sm,ss −,R ∩ D ×• sm and positive numbers t * ∈ (0, 1), α 0 ∈ (0, π 2J ) such that the following holds. For each q 0 ∈ 0<t<t * s t (W * )∩V ss 0 and each φ ∈ (−α 0 , α 0 ) that is admissible for the critical values of F q 0 , the inclusion (7.16) associated with q 0 and e iφ is induced by the pull-back in K-theory via the Γ-integral structure, i.e. it sends s V to s ϕ * V for V ∈ K(X − ). Remark 7.26. As we explained in Remark 7.7, we have a standard identification between K-classes of X + (resp. X − ) and flat sections over . We use these identifications in the above theorem; we also use the analytic continuation along the sliding homotopy s t for X − . Remark 7.27. By the duality of the analytic lift discussed in Remark 6.7, if the inclusion (7.16) corresponds to the pull-back ϕ * : K(X − ) → K(X + ) in K-theory, the projection associated with the opposite direction −e iφ = e i(φ+π) π * mir * + QDM an (X + ) B×I φ+π π * mir * − QDM an (X − ) B×I φ+π corresponds to the push-forward ϕ * : K(X + ) → K(X − ) in K-theory.
Lemma 7.28. Let q * be a point in M sm R ∩ D × sm . We have a commutative diagram (7.17) where the vertical arrows are induced by the isomorphisms in Theorems 7.5, 7.12, the top horizontal arrow is the inclusion in Theorem 7.22.
Proof. Note that the left vertical arrow is induced, via the Γ-integral structure, by the isomorphism of local systems over D × sm : satisfying 1 (2π) n/2 Γ e −Fq Ω q,−1 = P (Mir(Ω), s Γ ) for every local section Ω q,z of GM sm (F ). Here (· · · ) ∇ denotes the local system defined by the kernel of ∇. Also, the right vertical arrow in (7.17) is induced by the isomorphism of local systems over Im(s >0 ): ∇ which is defined similarly. We conclude the theorem by studying the monodromy of these local systems. Let L = Ker(Z S − → N) be as in §6.3.3. The inclusion D × sm ⊂ M sm,× = (L ) ⊗ C × induces an isomorphism π 1 (D × sm ) ∼ = (L ) ; an element ξ ∈ (L ) corresponds to the loop [0, 2π] θ → e iθξ · q * based at q * ∈ D × sm . We claim that the monodromy of Lef sm around the loop e iθξ · q * corresponds to tensoring by L −1 ξ in K(X − ) on the Γ-integral structure under the isomorphism (7.18), where L ξ denotes the orbi-line bundle as in §3.1.2. We also claim that the monodromy of Lef around s t (e iθξ · q * ) = (e iθξ · q * , t) corresponds to tensoring by ϕ * L −1 ξ in K(X + ) under the isomorphism (7.19). These two claims prove the lemma. In fact, since the map ι is monodromy-equivariant, the composition intertwines tensoring by L ξ with tensoring by ϕ * L ξ ; also this map (7.20) sends the structure sheaf to the structure sheaf since ι sends the positive real Lefschetz thimble Γ R (q * ) to the positive-real one Γ R (s t (q * )) (see Theorem 7.22); since K(X − ) is generated by line bundles we used the formula (2.12). This proves the first claim. To prove the second claim, it suffices to show that the homotopy classξ ∈ L ∼ = π 1 (M × ) of the loop s t (e iθξ · q * ) defines a line bundle Lξ isomorphic to ϕ * L ξ . Observe that we have a direct sum decomposition 3 (see also §7.1) which corresponds to the co-ordinate t = tb. The classξ ∈ L is a unique lift of ξ ∈ (L ) such that ξ · δ Σ − b = 0. The map (7.1) inducing the birational morphism ϕ : X + → X − is equivariant with respect to the homomorphism L ⊗ C × → L ⊗ C × , exp(λ) → exp(λ + (Db · λ)w) (where λ ∈ L C ), which is dual to the above lift (L ) → L , ξ →ξ. In view of the definition of Lξ in §3.1.2, we see that Lξ ∼ = ϕ * L ξ . The lemma is proved.
We show that α 0 appearing in Proposition 7.8 can be chosen independently of q 0 ∈ V ss +,R if q 0 is close to a given point q * in V sm,ss −,R ∩ D ×• sm .
Let Γ φ i (q) denote the Lefschetz thimble (6.12) of F q associated with the critical point cr i (q) and the phase φ. Here q can be either in W * or in 0<t<t * s t (W * )-we have 1 ≤ i ≤ N − in the former case and 1 ≤ i ≤ N + in the latter case. In view of the proof of Proposition 7.8, it suffices to show that the relative homology class represented by the thimble Γ π+φ 1 (q) is constant as (q, φ) varies in 0<t<t * s t (W * ) × (−α 0 , α 0 ). The homology class [Γ π+φ 1 (q)] can jump by the Picard-Lefschetz transformation as (q, φ) varies (see, e.g. [7, Chapter I]). By our choice of W * , t * and α 0 , it suffices to check that the intersection numbers of vanishing cycles at cr 1 (q) and cr i (q) (with 2 ≤ i ≤ k) are zero at some q ∈ 0<t<t * s t (W * ); here we consider vanishing cycles associated with paths from u i (q) to a base point u 0 0 inside the sector − + I * . At (q, φ) = (q * , 0), the Lefschetz thimble Γ π 1 (q * ) = Γ R (q * ) lying over u 1 (q * ) + R ≥0 does not contain the critical points cr i (q * ), 2 ≤ i ≤ k, and therefore the vanishing cycles at cr 1 (q * ) and cr i (q * ) (with 2 ≤ i ≤ k) do not intersect. For 1 ≤ i ≤ k, choose a path γ i starting from u i (q * ) and ending at u 0 which avoids other critical points, and let Γ i denote the finite Lefschetz thimble (with boundary in F −1 q * (u 0 )) associated with cr i (q * ) and the path γ i . We choose a sufficiently big η 0 > 0 such that Γ i 's are contained in A q * (η 0 ) and that ∂A q * (η 0 ) F −1 q * (u 0 ). Arguing as in parts (1), (4) in the proof of Theorem 7.22, we see that Γ i can be continuously deformed to a relative cycle Γ i (q) of (A q (η 0 ), A q (η 0 ) ∩ F −1 q (u 0 )) as q varies in a small neighbourhood of q * in V ss − . This shows that the vanishing cycles ∂Γ 1 (q) and ∂Γ i (q) (with 2 ≤ i ≤ k) have zero intersection number. The lemma is proved.
Fix a sufficiently large u 0 0. Under the isomorphism H n (Y q * , { (F q * ) 0}) ∼ = H n (Y q * , F −1 q * (u 0 )), the class of Γ π+φ i (q * ) corresponds to the class of a finite Lefschetz thimble fibred over a bent ray as shown in Figure 13(ii). We move q * along the sliding map. For sufficiently small t > 0, these finite Lefschetz thimbles can be continuously deformed to finite thimbles for F st(q * ) with boundary in F −1 st(q * ) (u 0 ), as discussed in parts (1) and (4) in the proof of Theorem 7.22. By straightening these paths in the direction e iφ again, we see that these relative cycles in (Y st(q * ) , F −1 st(q * ) (u 0 )) corresponds to the Lefschetz thimbles Γ π+φ i (s t (q * )) over the straight ray u i (s t (q * )) + e iφ R ≥0 . Therefore, the map ι sends Γ π+φ i (q * ) to Γ π+φ i (s t (q * )) for sufficiently small t > 0. The assumptions (b), (c) above ensure that the homology class of Γ π+φ i (q) stays constant as q varies inside 0<t<t * s t (W * ); the Picard-Lefschetz transformation can only arise from the intersection of vanishing cycles at cr i (q) and cr j (q) with i = j, u i (q * ) = u j (q * ), i, j ∈ {1, . . . , N − }, but these intersection numbers vanish since the vanishing cycles do not intersect at q * . This proves the claim.
(iv) Figure 13. Deforming Lefschetz thimbles: (i) semi-infinite Lefschetz thimbles fibred over the ray u j (q * ) + R ≥0 e iφ ; (ii) bent rays passing through u 0 ; (iii) moving from q * to a nearby point in 0<t<t * s t (W * ); (iv) straightening the paths again. In (iii) and (iv), the paths from u 4 and u 5 overlaps, but it does not matter since the relevant vanishing cycles do not intersect.
The above discussion gives a commutative diagram where the vertical arrows are induced by the isomorphisms in Theorems 7.5, 7.12, the top horizontal arrow is the inclusion in Theorem 7.22 and the bottom arrow κ sends Comparing this with the commutative diagram in Lemma 7.28, we conclude that κ = ϕ * as required.
Finally, we explain that the conclusion of the theorem holds for every admissible phase φ ∈ (−α 0 , α 0 ) for {u 1 (q 0 ), . . . , u N + (q 0 )}. As we reviewed in §6.2, the analytic decompositions (7.14) for X ± change by mutation as the phase φ varies; see [44, §2.6, §4.2-4.3]. In the case at hand, the analytic decompositions are given by the basis {s ± j } of flat sections associated with the K-classes {V ± j }; they give rise to an asymptotic basis in the sense of §6.2. Their all possible mutations are completely determined by the configuration {u 1 (q 0 ), . . . , u N ± (q 0 )} of the critical values and the Euler pairings The theorem is proved.
Remark 7.30. We hope that we can analyze general discrepant transformations X + ← X → X − by using the functoriality under blowups (Theorem 7.25) twice. In the case of crepant toric wall crossings, we can see how a Fourier-Mukai transformation (as discussed in [15,31]) arises from this result, see the slides from [60]. 7.6. Orlov's decomposition and analytic lift. In the previous section, we have observed that the Lefschetz thimbles associated with 'convergent' critical points (at some point q 0 ∈ V ss 0 ∩ M R and for some phase φ) correspond to K-classes from ϕ * (K(X − )) ⊂ K(X + ). In this section, we see that the remaining 'divergent' critical points correspond to K-classes supported on the exceptional divisor of ϕ : X + → X − . We will see the correspondence between Orlov's semiorthogonal decomposition of the derived category D b (X + ) and the analytic decomposition (7.15) of the quantum D-module of X + . 7.6.1. Decomposition of the relative homology mirror to Orlov's decomposition. Recall from Lemma 7.15(3) that convergent critical values over s t (D sm ) are contained in B ρ 2 (0) and divergent critical values over s t (D sm ) are contained in the J 'satellite' discs B k (t) := t −1/J · B ρ 1 (3ρ 0 e (−2k−1)πi/J ), k ∈ Z/JZ (see Figure 10). We choose t 1 > 0 small enough so that B ρ 2 (0) + R ≥0 does not intersect any other satellite discs B k (t) when t is real and 0 < t ≤ t 1 . Then, if |φ| is sufficiently small, we have that does not intersect with any B k (t) when 0 < t ≤ t 1 . We choose a number h ∈ {0, 1, 2, . . . , J} and let S k ⊂ C, k = −h, −h + 1, . . . , J − h − 1 be mutually disjoint strip regions as in Figure  14; each S k is a closed region disjoint from S conv , emanating from B k (t), going around the origin by the angle (2k − 1)π/J + φ anticlockwise, and extending straight toward the direction e iφ near the end. These regions S conv , S k depend on t, φ and are defined when 0 < t ≤ t 1 and |φ| is sufficently small. Figure 14. Strip regions S conv and S k . In this picture, J = 5, h = 3 and φ = π 30 .
For sufficiently large M 0, The fibration given by F q,t is smoothly trivial outside this region by Corollary 7.21, and hence we get the decomposition of the relative homology Lef q,t (see (7.10)): for q ∈ D × sm and 0 < t ≤ t 1 , where we use the notation as in § §7.4.2-7.4.4 and Lef conv This decomposition is independent of φ (with |φ| sufficiently small) and is preserved by the Gauss-Manin connection. We say that a direct sum decomposition A = A 1 ⊕ · · · ⊕ A m is semiorthogonal with respect to a (not necessarily symmetric) pairing [·, ·) on A if [a i , a j ) = 0 for a i ∈ A i , a j ∈ A j whenever i > j. Note that the decomposition (7.22) is semiorthogonal with respect to the pairing #(e −πi Γ 1 · Γ 2 ) discussed in Remark 7.6.
Let E ⊂ X + be the exceptional divisor of the morphism ϕ : X + → X − ; this is the toric divisor corresponding tob ∈ R + . Let Z = ϕ(E) denote the image of E; this is a toric substack corresponding to the cone σ M + of Σ − (see (7.1)). We write ϕ E : E → Z for the restriction of ϕ to E and i E : E → X + for the inclusion.
Theorem 7.31. For q ∈ D × sm ∩ M sm R and 0 < t ≤ t 1 , the decomposition (7.22) of the relative homology corresponds to the decomposition of the K-group under the isomorphism Lef q,t ∼ = K(X + ) in Theorem 7.5(1), where K(Z) k denotes the subgroup O(−kE) ⊗ i E * ϕ * E K(Z) ⊂ K(X + ). Remark 7.32. The decomposition (7.23) is semiorthogonal with respect to the Euler pairing by Remark 7.6 and the above theorem. In view of this, we expect that D b (X + ) admits a semiorthogonal decomposition: . When X + is the blowup of a smooth variety X − along a smooth centre Z, this is the semiorthogonal decomposition proved by Orlov [86,Theorem 4.3]; Orlov stated the result for h = J, but the other cases 0 ≤ h ≤ J − 1 follow from this case by using the fact that Proof of Theorem 7.31. Observe that the subgroup Lef conv q,t ⊂ Lef q,t is the image of the map ι : Lef sm q → (s −1 t Lef) q in Theorem 7.22 (e.g. by considering the case φ = 0). Hence it corresponds to ϕ * (K(X − )) ⊂ K(X + ) by Lemma 7.28. It suffices to show that Lef q,t ⊂ Lef q,t . It suffices to show that (a) K(Z) k ⊂ K (k) ; (b) the decomposition (7.23) of the K-group holds true. As discussed, over s t (D sm ), convergent critical values are contained in B ρ 2 (0) and divergent critical values are contained in the J satellite discs B k (t), k ∈ Z/JZ; these statement still hold for a non-zero complex number t with |t| ≤ 1 by naturally extending the definition of s t and B k (t) to t ∈ C (see the proof of Lemma 7.15). Consider the class [Γ R ] ∈ Lef q,t of the positive real Lefschetz thimble. It corresponds to the class [O] ∈ K(X + ) of the structure sheaf. We study the monodromy action on Lef q,t along the loop [0, 2π] θ → (q, e iθ t). The monodromy corresponds to the tensor product by O(−E) on K(X + ); this follows from the same argument as in the proof of Lemma 7.28 and the fact that this loop corresponds to the element of L given by the second projection in the decomposition (7.21) (which equals Db ∈ L ). Under this monodromy, [Γ R ] undergoes the Picard-Lefschetz transformation. Among the satellite discs, only B −1 (t) passes through the strip region S conv as t rotates; here B −1 (t) becomes B −1 (e 2πi t) = B 1 (t) after rotation. Thus by the Picard-Lefschetz formula, the monodromy transform of [Γ R ] equals [Γ R ] − α 0 for some α 0 ∈ Lef and that β j corresponds to [O E (jE)]. Hence we have (7.24) [ Note that the decomposition (7.22) is invariant under monodromy in q ∈ D × sm . As discussed in the proof of Lemma 7.28, the monodromy around loops in D × sm corresdponds to the tensoring ϕ * L with L ∈ Pic(X − ). Since K(X − ) is generated by line bundles, we conclude from (7.24) that O(−kE) ⊗ O E ⊗ ϕ * V ∈ K (k) for all V ∈ K(X − ). Part (a) follows from the fact that K(Z) is also generated by line bundles and that the natural map Pic(X − ) → Pic(Z) is surjective. It remains to prove part (b). In view of part (a) and the decomposition (7.22), it suffices to show that K(X + ) is generated by the classes of O and O E (−kE), −h ≤ k ≤ J − h − 1 as a K(X − )-module, where the module structure is given by ϕ * : K(X − ) → K(X + ). Let L b = L −D b ∈ K(X + ) denote the class of the line bundle associated with −D b ∈ L for b ∈ S (see §3.1.2 for L −D b ); 1 − L b is the class of the structure sheaf of the toric divisor associated with b when b ∈ R + . Similarly, let L − b ∈ K(X − ) denote the class of the line bundle associated with −D b ∈ (L ) for b ∈ S − . We also set L := Lb = [O(−E)] ∈ K(X + ). Recall from the proof of Lemma 7.28 that the splitting (L ) → L , ξ →ξ induced by the decomposition (7.21) corresponds to the pull-back of line bundles, i.e. ϕ * L ξ ∼ = Lξ. Since the splitting (L ) → L as before), we have On the other hand, since the intersection of the toric divisors corresponding to rays b ∈ M + is empty in X + , we have the following relation in K(X + ) (see [16]): Combining the two relations (7.25), (7.26), we obtain: The left-hand side of the relation is a monic polynomial in L of degree J + 1 = b∈M + k b with coefficients in K(X − ) whose constant term is an invertible element in K(X − ). Note that Pic(X + ) is generated by ϕ * Pic(X − ) and L ± ; therefore any element in K(X + ) can be written as a Laurent polynomial of L with coefficients in K(X − ). Using the above relation (7.27), we find that every element of K(X + ) can be written in the form: which can also be rewritten as a K(X − )-linear combination of 1 and The theorem is proved. 7.6.2. Orlov's decomposition as a sectorial decomposition of the quantum D-module. Next we see that this Orlov-type decomposition actually arises from a sectorial decomposition of the quantum D-module of X + -as appears in Proposition 6.5 -associated with some τ + ∈ H * CR (X + ); the point τ + can be very far from the large radius limit point and is not explicit in general (see however Example 7.34).
The analytic quantum D-module (6.1) is originally defined in a neighbourhood of the large radius limit point. In the following theorem, we consider analytic continuation of the quantum D-module along certain paths in H * CR (X ± ); we choose a real class τ ,± ∈ H * CR (X ± ; R) which is sufficiently close to the large radius limit point as a base point (see Remark 2.8).
Theorem 7.33. There exist paths from τ ,± to τ ± ∈ H * CR (X ± ) and analytic continuation of the quantum D-modules QDM an (X ± ) along these paths with the following properties.
(2) Moreover, there exists a holomorphic submersion f : W → H * CR (X − ) with f (τ + ) = τ − such that we have the similar sectorial decomposition for X − where W and I are the same as part (1) and u 1 , . . . , u N − are the pull-backs along f of the eigenvalues of the Euler multiplication in the quantum cohomology of X − which form a subset of {u 1 , . . . , u N + } in part (1).
i is the flat section associated with V ± i via the Γ-integral structure (Definition 2.7) (which is analytically continued from τ ,± through the specified paths) and e i denotes the ith standard basis. Moreover, we have In particular, we have a sectorial decomposition π * QDM an (X + ) which corresponds to the decomposition (7.23) of the K-group via the Γ-integral structure, where R an Proof. Throughout §7, we have assumed that S − = S ∩ ∆ − and S = S − ∪ {b} so that M is the small quantum cohomology locus of X + . The construction of the sectorial decompositions (7.14) however, does not require this assumption because S can be arbitrarily large in the discussion of § §5-6. We choose a large enough finite set S ⊂ N ∩ Π containing S such that the corresponding mirror map mir + is generically submersive (this is possible by the argument in [27, §7.4]). Let M ⊃ M denote the base of the LG model corresponding to S. The discussion at the beginning of §7.5 yields, for any q 0 ∈ V ss 0 and an admissible phase φ for the critical values of F q 0 , the following decompositions (similarly to (7.14)) π * mir * ± QDM an (X ± ) over a neighbourhood B of q 0 in M. Here (u 1 , . . . , u N + ) are the eigenvalues of E τ in the quantum cohomology of X + at τ = mir + (q), and first N − (u 1 , . . . , u N − ) of them are the eigenvalues of E τ in the quantum cohomology of X − at τ = mir − (q), where q varies in B. By the choice of S, the eigenvalues u 1 , . . . , u N + are pairwise distinct at generic points in B (see Remark 6.2).
Suppose now that q 0 is a point from 0<t<t 1 s t (V sm,ss −,R ∩ D ×• sm ) ∩ V ss 0 and |φ| is sufficiently small so that the conclusion of Theorem 7.25 holds; then there exist K-classes V •± i ∈ K(X ± ) such that the associated flat sections s ± i (via the Γ-integral structure) correspond to e −u i /z e i under the decomposition (7.28) and that such that the corresponding eigenvalues u • 1 , . . . , u • N + are pairwise distinct, and set τ • ± = mir ± (q • ); we regard τ • ± as elements of H * CR (X ± ) (rather than their images in M A (X + )). Let C N denote the configuration space of distinct N points in C: Since the eigenvalues of E τ form a local co-ordinate system on H * (X ± ) near a semisimple point, we can identify a neighbourhood of (u • 1 , . . . , u • N ± ) in C N ± with a neighbourhood of τ • ± in H * CR (X ± ). Let C N denote the universal cover of C N . By isomonodromic deformation [40, Lemma 3.2, Exercise 3.3, Lemma 3.3], the quantum connection in a neighbourhood of τ ± ∈ H * CR (X ± ) can be extended to a meromorphic flat connection ∇ on the trivial bundle where A is an End(H * CR (X ± ))-valued 1-form on C N ± , U and V are End(H * CR (X ± ))-valued functions on C N ± and A, U, V are independent of z. Here the eigenvalues of U give the coordinates (u 1 , . . . , u N ± ) on C N ± . Moreover, this induces a Frobenius manifold structure on an open dense subset C • N ± ⊂ C N ± which is the complement of an analytic hypersurface. By choosing a basis {φ i } of H * CR (X ± ), we can determine the flat vector fields ∂ ∂τ i by A ∂/∂τ i 1 = φ i . In particular, we have a flat co-ordinate system ( C • N ± ) ∼ → H * CR (X ± ) on the universal cover ( C • N ± ) ∼ which equals τ • ± at (u • 1 , . . . , u • N ± ). We also have a submersion g : C N + → C N − sending (u 1 , . . . , u N + ) to (u 1 , . . . , u N − ); this induces a submersion g : is the complement of an analytic hypersurface in C • N + . We may assume that q • is sufficiently close to q 0 so that {u • 1 , . . . , u • N + } are contained in B ρ 2 (0) k∈Z/JZ B k (t), where 0 < t < t 1 is the number such that q 0 ∈ Im(s t ). Moving u • i along the strip regions S conv ∪S −h ∪· · ·∪S J−h−1 , we can connect (u • 1 , . . . , u • N + ) by a continuous path γ inside C •• N + with a point (u ♦ 1 , . . . , u ♦ N + ) ∈ C •• N + having mutually distinct imaginary parts: see Figure 15. The numbers u ♦ 1 , . . . , u ♦ N + are divided into J + 1 groups as in part (3) of the statement, depending on which strip regions they belong to. Let τ + ∈ H * CR (X + ) be the analytically continued (along the path γ) flat co-ordinate at the point (u ♦ 1 , . . . , u ♦ N + ) ∈ C •• N + and τ − ∈ H * CR (X − ) be the analytically continued (along the path g(γ)) flat co-ordinate at the point g(u ♦ 1 , . . . , u ♦ N + ) = (u ♦ 1 , . . . , u ♦ N − ) ∈ C • N − . By the above construction, the quantum D-modules of X ± are analytically continued to τ ± along certain paths, and the submersion g induces, when written in flat co-ordinates, a submersion f from a neighbourhood of τ + in H * CR (X + ) to a neighbourhood of τ − in H * CR (X − ) with f (τ + ) = τ − . We claim that the statement of the theorem holds for τ ± , f and a sufficiently small neighbourhood W of τ + . Parts (1), (2) follow from the construction and the Hukuhara-Turrittin decomposition (Proposition 6.5); note that 0 is an admissible phase for (u ♦ 1 , . . . , u ♦ N ± ). Part (3) has been already achieved. We show part (4). Recall that the decomposition (7.28) corresponds to the basis {V •± i } of K(X ± ) under the Γ-integral structure and it gives rise to the asymptotic basis (see §6.2) associated with τ • ± and φ. As we move (u 1 , . . . , u N ± ) along the path γ (or g(γ)) and change the phase from φ to zero, this asymptotic basis undergoes mutation as explained in §6.2. The basis {V ± i } ⊂ K(X ± ) in part (4) arises from {V •± i } by this mutation. Part (4-a) is obvious from this construction. By the same argument as in the last paragraph of the proof of Theorem 7.25, we see that the relation  Example 7.34. We give an example where the Orlov-type decomposition occurs at an explicit τ + . Set X − := P 4 and let X + be the blowup of X − along a line Z = P 1 ⊂ X − . Both X + and X − are Fano, and their small quantum cohomologies are defined over polynomial rings. Planes in X − = P 4 containing the line Z are parametrized by P 2 , and hence we have a natural projection X + → P 2 . Thus X + is a P 2 -bundle over P 2 : X + ∼ = P P 2 (O ⊕ O ⊕ O(−1)). Let p 1 be the pull-back of the ample class in H 2 (P 2 ; Z) and let p 2 = ϕ * (p) be the pull-back of the ample class p in H 2 (X − ; Z). They form a nef basis of H 2 (X + ; Z). The class of the exceptional divisor is given by [E] = p 1 − p 2 . The uncompactified LG mirror of X + is given by the family of Laurent polynomials parametrized by (q 1 , q 2 ) ∈ (C × ) 2 . The large radius limit (LRL) point for X + corresponds to q 1 = q 2 = 0. The mirror map for X + is trivial: mir + (q 1 , q 2 ) = p 1 log q 1 + p 2 log q 2 . On the other hand, the uncompactified LG mirror of X − is given by the family where the LRL point for X − is q = t = 0. The two families are related by the change of coordinates q = q 1 q 2 , t = q −1 1 . The small quantum cohomology locus for X − is t = 0. It is not easy to find a closed formula for the mirror map of X − , but we know that it has the asymptotic form mir − (q, t) ∼ p log q+t·p 3 as (q, t) → (0, 0) (see Remark 4.9) and that mir − (q, 0) = p log q. The Euler vector field gives a grading deg q 1 = 2, deg q 2 = 3 and deg t = −2, deg q = 5 (in complex unit). Define a dimensionless parameter λ := q − 3 2 1 q 2 = t 5 2 q. Critical points/values of F q 1 ,q 2 = F q,t are given by where x is a root of x 5 (x 2 + 1) 2 = λ. The discriminant locus (where x has multiple roots) is λ = 0 or ±400 √ 5i/3 9 . Therefore the quantum cohomology is semisimple over the positive real locus q 1 > 0, q 2 > 0. The critical values have the asymptotics as λ → 0 (or t → 0 with q fixed) and as λ → ∞ (or q 1 → 0 with q 2 fixed). The first line of (7.29) gives 5 'convergent' critical values (depending on the choice of q 1 5 ) and the next two lines give 4 'divergent' critical values around t = 0. They have mutually distinct imaginary parts when t > 0 is sufficiently close to zero and q > 0. We will identify the corresponding elements in the K-group for the phase φ = 0.
We consider a path where the parameter λ ∈ R increases from 0 to ∞. It is convenient to take t = t(λ) = λ 2 3 + λ 2 5 so that (q 1 , q 2 ) ∼ (λ − 3 2 , 1) as λ → ∞ and (q, t) ∼ (1, λ 2 5 ) as λ → 0. The move of the critical values is shown in Figure 16. When λ is large, say λ = 12.5, we can determine the mirror partners (in the K-group) of some Lefschetz thimbles by using the monodromy action on the (q 1 , q 2 )-space: see the rightmost picture of Figure 16. Here H 1 , H 2 denote the pull-backs of the hyperplanes on P 2 and P 4 respectively (so that p i is the Poincaré dual of H i ). A numerical calculation on computer shows that, as λ decreases from λ = 12.5 to λ = 0.0009, the thimble corresponding to O(−H 1 ) undergoes a Picard-Lefschetz transformation with respect to that corresponding to O(−H 2 ). As we saw in the proof of Theorem 7.33, this corresponds to mutation in K-group. After (left) mutation, O(−H 1 ) becomes Considering the monodromy action near t = 0 and performing further mutation, we find that the exceptional collection (adapted to the decomposition (7.23) with J = 2, h = 1) corresponds to the Lefschetz thimbles Γ π 1 , . . . , Γ π 9 (6.12) ordered in such a way that the imaginary parts of the corresponding critical values decrease (F (cr 1 )) > (F (cr 2 )) > · · · > (F (cr 9 )). These are the classes V + i in Theorem 7.33(4) in this case. Note that the corresponding sectorial decomposition occurs at a point τ + = [E] log t + p 2 log q ∈ H 2 (X + ), q, t > 0 such that t is sufficiently small (in the leftmost picture of Figure 16, t ≈ 0.0698, q ≈ 0.698).
Remark 7.35. We can use Theorem 7.33 to prove Gamma conjecture II [44, §4.6] in some cases. By applying Theorem 7.33 to the case where Z is a (non-stacky) point, we know that if the Gamma conjecture II holds for a weak-Fano compact toric stack X − , then it also holds for a weighted blowup X + of X − at a non-stacky torus-fixed point (as long as X + is weak-Fano).

Conjecture and discussion
8.1. General conjecture. In view of our main results (Theorems 5.16, 5.19, 7.25, 7.31, 7.33), we conjecture the following phenomena for more general discrepant birational transformations. Suppose that we have a birational transformation ϕ : X + X − between smooth (not necessarily toric) DM stacks which fits into the diagram with X smooth and f ± projective birational morphisms, such that f * + K X + − f * − K X − is an effective divisor. We assume that the coarse moduli spaces of X ± are projective, but some of the discussions below can be also adapted to non-compact cases.
We choose a base point τ ,± ∈ H * CR (X ± ; R) which is real and sufficiently close to the large radius limit point.
Conjecture 8.1 (formal decomposition). There exist paths from τ ,± to τ ± ∈ H * CR (X ± ) and a holomorphic map f from a neighbourhood W of τ + in H * CR (X + ) to H * CR (X − ) with f (τ + ) = τ − such that the quantum D-modules QDM an (X ± ) are analytically continued along these paths, and that we have the decomposition of the quantum D-modules completed in z (see (6.2)): Here R is a locally free O W [[z]]-module equipped with a meromorphic flat connection ∇ R and a z-sesquilinear pairing P R , and the formal decomposition respects both the connection and the pairing (in particular it is orthogonal with respect to the pairing (2.9)).
Building on the above conjecture, we can also state a conjecture comparing higher genus Gromov-Witten theories. This involves quantization of the formal decomposition (8.1). See §5.5 for the notation.
Conjecture 8.2 (comparison in all genera). The ancestor potentials A ±,τ of X ± can be analytically continued along the paths from τ ,± to τ ± in Conjecture 8.1. There exist a family A τ ∈ AFock(V τ , D τ ) of tame functions such that T s U τ A +,τ = A −,f (τ ) ⊗ A τ for τ ∈ W , where s = (−zφ 0 , D τ ) + U τ (zφ 0 ), V τ ⊂ R τ is a C-vector subspace such that R τ = V τ [[z]] and that P R restricts to the C-valued pairing V τ × V τ → C, D τ ∈ zR τ and U τ : ] is the unitary isomorphism obtained from (8.1) by restricting to τ and flipping the sign of z.
In this conjecture we implicitly assume that the operator T s on U τ A +,τ is well-defined; this holds if A +,τ is rational (see §5.5.2). Note also that the space AFock(V τ , D τ ) itself depends only on R τ and D τ and does not depend on the choice of V τ , but that A τ depends on the choice of V τ .
Next we state a conjecture relating the analytic lift of the formal decomposition (8.1) with a semiorthogonal decomposition of the K-group. Recall from (2.11) that the action of −∇ z 2 ∂z on QDM an (X ± )| z=0 equals the Euler multiplication E τ . In particular, the formal decomposition (8.1) implies that E τ on H * CR (X + ) is conjugate to E f (τ ) ⊕(−∇ R z 2 ∂z ) on H * CR (X − ) ⊕ R| z=0 . Conjecture 8.3 (analytic lift). We can arrange τ ± and the paths from τ ,± to τ ± in Conjecture 8.1 so that (in addition to Conjecture 8.1) the following holds.
(2) Moreover, we have a sector I φ = {(r, e iθ ) ∈ C : |θ − φ| < π 2 + } with some > 0 and an analytic decomposition where π : W × C → W × C is the oriented real blow-up and R an i is a locally free A W ×I φ -module equipped with a flat connection, such that ], i = 1, 2 and that the formal decomposition (8.1) is induced by (8.2).
(3) Via the Γ-integral structure for X + and X − , the decomposition (8.2) is induced by a semiorthogonal decomposition of the topological K-groups: such that the associated inclusion K(X − ) → K(X + ) respects the Euler pairing.
Remark 8.4. We expect that the semiorthogonal decomposition (8.3) arises naturally from geometry. In our setting, for example, we could hope that there is a semiorthogonal decomposition of the bounded derived category of X + of the form (see [9,Conjecture 4.3.7]): When X ± are toric stacks arising from the variation of GIT quotients as in §5, Ballard-Favero-Katzarkov [9, Theorem 5.2.1] showed that such a semiorthogonal decomposition exists (see also [71,55]). A semiorthogonal summand of D b (X + ) gives a K-motive (see [52, §4]), which in turn defines a direct summand of the topological K-group of X +

22
. We expect that (8.3) arises from (8.4) in this way (cf. [76] for the discussion on Hochschild homology). On the other hand, in view of the deformation invariance of Gromov-Witten invariants, it is more natural to state conjecture in terms of topological K-groups instead of derived categories. 8.2. Functoriality and Riemann-Hilbert problem. We discuss how to recover the quantum cohomology of X − from the quantum cohomology of X + , assuming Conjecture 8.3. This involves solving a Riemann-Hilbert boundary value problem. Proposition 8.5. Assume that the quantum D-modules QDM an (X ± ) are of exponential type. Suppose that we are given the analytic continuation of the quantum D-module QDM an (X + ) to a neighbourhood W of τ + ∈ H * CR (X + ) and its formal decomposition QDM an (X + )| W = Q ⊕ R corresponding to the decomposition (8.1) for which Conjecture 8.3 holds. Then we can recover the map f in Conjecture 8.3 and the quantum D-module f * QDM an (X − ) of X − (trivialized as a vector bundle over W × C) together with an isomorphism f * QDM an (X − )| W ∼ = Q.
Before giving a proof of this proposition, we review the exponential type assumption (see [70,Definition 2.12]). This was originally introduced by Hertling-Sevenheck [57, Definition 8.1] under the name "require no ramification". We also review a mutation system of Sanda-Shamoto [100, Definition 2.30].
The quantum D-module of X = X ± is of exponential type if for each τ ∈ H * CR (X), we have a formal decomposition of the quantum D-module QDM an (X) τ := QDM an (X)| {τ }×C (see [57,Lemma 8.2]): where Spec(E τ ) denotes the set of (mutually distinct) eigenvalues of E τ , e u/z denotes the rank one connection (C{z}, d + d(u/z)) and F u is a free C{z}-module equipped with a regular singular meromorphic connection. The decomposition is (automatically) orthogonal with respect to the pairing P , and induces a z-sesquilinear pairing P u on each F u ; F u is called the regular singular piece in [57,Lemma 8.2]. Moreover, the Hukuhara-Turrittin theorem (see [57,Lemma 8.3] in this context) implies that, for each phase φ admissible for Spec(E τ ), the formal decomposition admits a unique analytic lift over the sector I φ = {(r, e iθ ) ∈ C : |θ − φ| < π 2 + } (for some > 0) (8.6) π * (QDM an (X) τ ) where π : C → C is the oriented real blowup. In §6.1, we discussed in details the special case of these decompositions where the quantum cohomology is semisimple.
The analytic germ at z = 0 of the quantum D-module QDM an (X) τ can be determined by the formal decomposition (8.5) and the Stokes data. Sanda-Shamoto [100, Definition 2.30] encoded the Stokes data in linear-algebraic data which they called a mutation system. This can be viewed as a generalization of a marked reflection system in §6.2. Let V denote the space of flat sections of QDM an (X) τ over the sector I × φ = {z ∈ C × : | arg z − φ| < π 2 + }. We define a pairing [·, ·) on V by (cf. §6.2) [s 1 , s 2 ) = P (s 1 (e −πi z), s 2 (z)) where s 1 (e −πi z) denotes the analytic continuation of s 1 (z) along the path [0, π] θ → e −iθ z. The mutation system (see [100, Proposition 2.5, §2.7]) for QDM an (X) τ associated with the admissible phase φ is given by the data 23 • the tuple (V, [·, ·)) of a vector space and a pairing; • the decomposition V ∼ = u∈Spec(E τ ) V u induced by the analytic lift (8.6), where V u is the space of flat sections of e u/z ⊗ F u over the sector I × φ satisfying the semiorthogonality: [v 1 , v 2 ) = 0 if v 1 ∈ V u 1 , v 2 ∈ V u 2 and (e −iφ u 1 ) < (e −iφ u 2 ).
The pairing [·, ·) restricted to V u is induced by the pairing P u on F u [57, Lemma 8.4]. It was shown [100, Proposition 2.5, §2.5] that a mutation system is equivalent to Stokes data (or a Stokes filtered local system) equipped with a pairing. By the Riemann-Hilbert correspondence (see [100, §2.7] in this context), the formal structure (8.5) and the mutation system together recover the analytic germ at z = 0 of QDM(X) τ .
Proof of Proposition 8.5. Fix τ ∈ W . We may assume that φ in Conjecture 8.3 is admissible for the set Spec(E τ ) of eigenvalues of E τ on H * CR (X + ), by perturbing φ if necessary. Under the assumption that QDM an (X + ) τ is of exponential type, the summands Q, R 1 , R 2 of QDM an (X + ) τ admit decompositions similar to (8.5). We write Spec(Q) (resp. Spec(R i )) for the set of the eigenvalues of the operators −∇ z 2 ∂z on Q| z=0 (resp. on R i | z=0 ). By Conjecture 8.3(1), the sets Spec(Q), Spec(R 1 ), Spec(R 2 ) are mutually distinct; therefore Q, R 1 , R 2 are partial sums of the right-hand side of the formal decomposition (8.5) for X = X + . Writing V ∼ = u∈Spec(E τ ) V u for the mutation system of QDM an (X + ) τ , we can decompose V as Since the inclusion K(X − ) → K(X + ) respects the Euler pairing, we see that the restriction of the pairing [·, ·) on V to V Q coincides with the pairing on V Q as a mutation system for QDM an (X − ) f (τ ) . Therefore the mutation system for QDM an (X − ) f (τ ) can be recovered as a summand of the mutation system for QDM an (X + ) τ . Thus we recover the analytic germ at z = 0 of QDM an (X − ) f (τ ) by the Riemann-Hilbert correspondence. We write Q an for the germ so reconstructed. We extend Q an to the trivial vector bundle QDM an (X − ) f (τ ) over C z and construct the fundamental solution L(f (τ ), z) for X − (see §2.4) and the map f . Consider the trivial bundle Q (∞) := H * (X − ) × (P 1 \ {0}) → (P 1 \ {0}) equipped with the meromorphic connection This has z −µ z c 1 (X) as a fundamental solution and the facts recalled in §2.4 imply that the quantum connection on {f (τ )} × C z is gauge equivalent to ∇ (∞) via the gauge transformation by L(f (τ ), z) (which is regular and the identity at z = ∞). We glue this trivial bundle Q (∞) with the germ Q an of vector bundle at z = 0 to get a vector bundle Q over P 1 . The gluing is given by sending a flat section s ∈ V Q over the sector I × φ to the flat section for ∇ (∞) : (2π) −n/2 z −µ z c 1 (X − ) Γ X − · (2πi) deg 0 /2 inv * ch(Φ(s)) with n = dim X − , where Φ is the isomorphism in (8.8). In view of the definition of the Γ-integral structure, this glued bundle Q must be isomorphic to the trivial extension of QDM an (X − ) f (τ ) to P 1 (with respect to the given trivialization). In particular, Q is trivial, and the identification Q (∞) | ∞ ∼ = H * CR (X − ) at infinity induces a global trivialization Q ∼ = H * CR (X − ) × P 1 . The trivial bundle Q equipped with a meromorphic connection gives the quantum D-module QDM an (X − ) f (τ ) . Moreover, the isomorphism Q| P 1 \{0} ∼ = Q (∞) written in the respective trivializations gives the fundamental solution L(f (τ ), z) as an End(H * CR (X − ))valued function. Varying τ in W , we recover the full quantum connection for f * QDM an (X − ) from L(f (τ ), z). We also recover f (τ ) from the expansion L(f (τ ), z) −1 1 = 1 + f (τ ) z + O(z −2 ).
The proposition is proved. Recall from §4.1 that, by choosing a splitting L C → (C S ) , ξ →ξ of (3.4), G has the structure of a module over the ring of differential operators: where χ i ∈ R T acts by zx i ∂ ∂x i + x i ∂F ∂x i and zξq ∂ ∂q acts by zξu ∂ ∂u +ξu ∂F ∂u . For convenience, we choose co-ordinates q 1 , . . . , q m ∈ C[Λ(Σ)] corresponding to a Z-basis of Λ(Σ) and write θ i = q i for every (ν b ) b∈S ∈ (Z ≥0 ) S and v ∈ N ∩ Π.
Proof. By definition, zD b q ∂ ∂q + χ(b) acts on G by zu b ∂ ∂u b + u b . The first equality follows from this and u b ∂ ∂u b w v = Ψ b (v)w v . The second equality is just by definition, see §3.5.
When b ∈ R(Σ) and v ∈ N ∩ Π lie in the same cone of Σ, we have a relation in G by Lemma A.2. From this we can see that G is generated by finitely many w v as a D-module. For example, the set {w v : v ∈ Box(Σ)} generates G. By Lemma A.2, w v is annihilated by P v,λ ∈ D (A.2) P v,λ := b∈S:λ b >0 for any λ ∈ L ∩ NE(X Σ ) ⊂ Z S , cf. [27, §5.1].
A.2. Characteristic variety and coherence. Define an increasing filtration F l (D) of D by the rank of differential operators, i.e. F l (D) consists of differential operators of the form k 1 +···+km≤l a k (q, χ, z)(zθ 1 ) k 1 · · · (zθ m ) km .
Choose generators w v 1 , . . . , w v k of G as a D-module and introduce a filtration on G by An easy argument shows that if gr F (G )| V is finitely generated as an O V -module for an open set V ⊂ B, then G | V is also finitely generated as an O V -module. We shall show that gr F (G ) is finitely generated on a neighbourhood of {0 Σ } × Lie T × C z . The associated graded module gr F (G ) is a gr F (D)-module generated by w v 1 , . . . , w v k . We Note that Ch(G ) induces a closed subset C ⊂ B × P m−1 and V is the complement of π(C), where π : B × P m−1 → B is the projection; thus V is Zariski-open. For a differential operator a ∈ F l (D) \ F l−1 (D), its principal symbol σ(a) is the image of a in gr F (D); explicitly σ(a) := k 1 +···+km=l a k (q, χ, z)ξ k 1 1 · · · ξ km m if a = k 1 +···+km≤l a k (q, χ, z)(zθ 1 ) k 1 · · · (zθ m ) km .
The principal symbol of the relation P v,λ in (A.2) is given by where D b (ξ) := σ(zD b q ∂ ∂q ) is a linear form in ξ 1 , . . . , ξ m . Because σ(P v,λ ) is independent of v, it is an annihilator of gr F (G ). Therefore Ch(G ) is contained in the closed subset of B × C m defined by σ(P v,λ ) = 0 for all λ ∈ L ∩ NE(X). Proof. Note that σ(P v,λ ) does not depend on (χ, z) ∈ Lie T × C z . Therefore it suffices to show that ξ = 0 if ξ ∈ C m satisfies (A. 4) σ(P v,λ ) q=0 Σ = 0 for all λ ∈ L ∩ NE(X).
Suppose that ξ ∈ C m satisfies (A.4). We first show that there exists a cone σ ∈ Σ such that {b ∈ S : D b (ξ) = 0} = R(Σ) ∩ σ. Let {b 1 , . . . , b s } be the set of b ∈ S such that D b (ξ) = 0. The relative interior of the convex hull of {b 1 , . . . , b s } intersects with the relative interior of some cone σ of Σ. Hence we get a relation of the form belong to L ∩ NE(X) (recall the definition of NE(X) around (3.14)). Note that b∈S λ b = l( s i=1 c i − b∈R(Σ)∩σ f b ) ≥ 0. Therefore, if λ = 0, we have the relation This contradicts the fact that D b 1 (ξ) = 0, . . . , D bs (ξ) = 0. Hence λ = 0 and we conclude that A.3. Locally freeness and rank. We complete the proof of Proposition A.1. Recall from Theorem 4.7 that the completed equivariant Gauss-Manin system is isomorphic to the quantum D-module of X. Thus we have where m Σ ⊂ C[Λ(Σ) + ] denotes the maximal ideal corresponding to 0 Σ and G denotes the m Σ -adic completion as discussed in §4.2. This implies that the restriction G | {0 Σ }×Lie T×Cz is free of rank dim H * CR (X). Hence by coherence, G is generated by dim H * CR (X) many sections in a neighbourhood of {0 Σ } × Lie T × C z . On the other hand, the localization map "Loc" appearing in [27, Definition 4.17] gives dim H * CR (X) many linearly independent solutions of G for a generic (χ, z) ∈ Lie T×C z ; the weak Fano condition ensures the convergence of the power series solution Loc. This implies that G is locally free of rank dim H * CR (X) in a neighbourhood of {0 Σ } × Lie T × C z .
To see that G is locally free on an open set of the form U × Lie T × C z (for some open set U ⊂ Spec C[Λ(Σ) + ] containing 0 Σ ), we use the grading operator (4.1). The grading operator makes G a C × -equivariant sheaf; the induced C × -action on the base is given by the same grading operator on C[Λ(Σ) + ][z] and the weight one C × -action on Lie T. Proof. It suffices to show that b∈S λ b ≥ 0 for any (λ, v) ∈ O(Σ) + . By the definition (3.15) of O(Σ) + , it suffices to show that b∈S λ b ≥ 0 for all maximal cones σ ∈ Σ and λ ∈ C Σ,σ (see (3.14)). Define a linear function h : N R → R by h(b) = 1 for all b ∈ R(Σ) ∩ σ. Then the weak Fano condition (i.e. the convexity of ∆) together with (A.1) implies that h(b) ≤ 1 for all b ∈ S. Hence, for λ ∈ C Σ,σ , we have 0 ≤ h(β(λ)) = b∈R(Σ)∩σ This proves the lemma.
In particular, C[Λ(Σ) + ] is non-negatively graded. Because the locus where G is locally free is preserved by the C × -action and contains a neighbourhood of (0 Σ , 0, 0) ∈ B, it follows that G is locally free on an open set of the form U × Lie T × C z . The proof of Proposition A.1 is now complete.
Remark A.6. The generic rank of the GKZ system has been studied by many people, notably by Gelfand-Kapranov-Zelevinsky [46], Adolphson [6], Matusevich-Miller-Walther [82] and has been identified with the volume 25 of ∆ (when χ is not special). Over the open torus (C × ) m contained in Spec C[Λ(Σ) + ], the Gauss-Manin system in this paper corresponds to the betterbehaved GKZ system of Borisov-Horja [17]; they showed that the generic rank of the betterbehaved GKZ system equals vol(∆) (independently of χ).

Appendix B. Proof of Lemma 7.24
We only prove part (1) of the lemma; the argument for part (2) is the same. It is easy to see that F −1 q,0 (u) and ∂A q,0 (η) intersect transversally at x ∈ F −1 q,0 (u) ∩ ∂A q,0 (η) if and only if grad F q,0 (x) and grad H(x) are linearly independent over C, where we set Suppose that the lemma is not true. Then we can find sequences (q(k), u(k)) ∈ K and x(k) ∈ Y sm q(k) such that the following holds: • η k = H(x(k)) → ∞ as k → ∞; • F q(k),0 (x(k)) = u(k); • grad F q(k),0 (x(k)) and grad H(x(k)) are linearly dependent over C, i.e. there exists α n ∈ C such that grad F q(k),0 (x(k)) = α n grad H(x(k)).
Then we have grad H(x) = v(x) 2H(x) . 25 The volume is normalized so that the standard simplex has volume 1. When we allow N to have torsions, the generic rank is |Ntor| × vol(∆).
Writing v · w = n i=1 v i w i for the C-bilinear scalar product, the third condition above can be written as grad F q(k),0 (x(k)) = grad F q(k),0 (x(k)) · v(x(k)) v(x(k)) 2 v(x(k)).
On the other hand, Proposition 7.19 and (B.2) give an estimate of the form grad F q(s),0 (x(s)) ≥ 1 H(x(s)) ≥ 2 s −m for sufficiently small s > 0 (for some 1 , 2 > 0). These two estimates contradict each other. Lemma 7.24 is proved.