Hiroshi Iritani

Etienne Mann
email: etienne.mann@math.univ-montp2.fr

Thierry Mignon

Quantum Serre Theorem as a Duality Between Quantum D-Modules

We give an interpretation of quantum Serre theorem of Coates and Givental as a duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E ∨ → X, and (3) the quantum D-module of a submanifold Z ⊂ X cut out by a regular section of E. When E is the anticanonical line bundle K -1 X , we identify these twisted quantum D-modules with second structure connections with different parameters, which arise as Fourier-Laplace transforms of the quantum D-module of X.

In this case, we show that the duality pairing is identified with Dubrovin's second metric (intersection form).

Introduction

Genus-zero Gromov-Witten invariants of a smooth projective variety X can be encoded in different mathematical objects: a generating function that satisfies some system of

2 H. Iritani et al. partial differential equations (Witten-Dijkgraaf-Verlinde-Verlinde equations), an associative and commutative product called quantum product, the Lagrangian cone L X of Givental [START_REF] Givental | Symplectic Geometry of Frobenius Structures[END_REF] or in a meromorphic flat connection called quantum connection. These objects are all equivalent to each other; in this paper, we focus on the realization of Gromov-Witten invariants as a meromorphic flat connection.

Encoding Gromov-Witten invariants in a meromorphic flat connection defines the notion of quantum D-module [START_REF] Givental | Homological Geometry and Mirror Symmetry[END_REF], denoted by QDM(X), that is a tuple (F, ∇, S) consisting of a trivial holomorphic vector bundle F over H ev (X) × C z with fibre H ev (X), a meromorphic flat connection ∇ on F given by the quantum connection:

∇ = d + s α=0 (T α • τ )dt α + - 1 z (E•) + deg 2 dz z
and a flat non-degenerate pairing S on F given by the Poicar é pairing (see Definition 2.1 and Remark 2.4). These data may be viewed as a generalization of a variation of Hodge structure (see [START_REF] Katzarkov | Hodge theoretic aspects of mirror symmetry[END_REF]).

Quantum Serre theorem of Coates and Givental [4, Section 10] describes a certain relationship between twisted Gromov-Witten invariants. The data of a twist are given by a pair (c, E) of an invertible multiplicative characteristic class c and a vector bundle E over X. Since twisted Gromov-Witten invariants satisfy properties similar to usual Gromov-Witten invariants, we can define twisted quantum product, twisted quantum D-module QDM (c,E) (X) and twisted Lagrangian cone L (c,E) associated to the twist (c, E). Let c * denote the characteristic class satisfying c(V)c * (V ∨ ) = 1 for any vector bundle V. Quantum Serre theorem (at genus zero) gives the equality of the twisted Lagrangian cones: (c,E) .

L (c * ,E ∨ ) = c(E)L
(1.1)

Quantum Serre theorem of Coates and Givental was not stated as a duality. An observation in this paper is that this result can be restated as a duality between twisted quantum D-modules.

Theorem 1.1 (see Theorem 2.11 for more precise statements). There exists a (typically non-linear) map f : H ev (X) → H ev (X) (see (2.7)) such that the following holds:

(1) The twisted quantum D-modules QDM (c,E) (X) and f * QDM (c * ,E ∨ ) (X) are dual to each other; the duality pairing S QS is given by the Poincar é pairing.

(2) The map QDM (c,E) (X) → f * QDM (c * ,E ∨ ) (X) sending α to c(E) ∪ α is a morphism of quantum D-modules. Genus-zero twisted Gromov-Witten invariants were originally designed to compute Gromov-Witten invariants for Calabi-Yau hypersurfaces or non-compact local Calabi-Yau manifolds [START_REF] Chiang | Local mirror symmetry: calculations and interpretations[END_REF][START_REF] Givental | Equivariant Gromov-Witten Invariants[END_REF][START_REF] Kontsevich | Enumeration of Rational Curves via Torus Actions[END_REF]. Suppose that E is a convex vector bundle and c is the equivariant Euler class e λ . In this case, non-equivariant limits of (c, E)-twisted Gromov-Witten invariants yield Gromov-Witten invariants of a regular section Z ⊂ X of E and non-equivariant limits of (c * , E ∨ )-twisted Gromov-Witten invariants yield Gromov-Witten invariants for the total space E ∨ . The original statement (1.1) of quantum Serre theorem does not admit a non-equivariant limit since the non-equivariant Euler class is not invertible. We see however that our restatement above passes to the non-equivariant limit as follows:

Corollary 1.2 (Theorem 3.14, Corollary 3.17). Let E be a convex vector bundle and let e denote the (non-equivariant) Euler class. Let h: H ev (X) → H ev (X) be the map given by h(τ ) = τ + π √ -1c 1 (E) and let f : H ev (X) → H ev (X) denote the non-equivariant limit of the map f of Theorem 1.1 in the case where c = e λ . We have the following:

(1) The quantum D-modules QDM (e,E) (X) and (h • f) * QDM(E ∨ ) are dual to each other.

(2) Let Z be the zero-locus of a regular section of E and suppose that Z satisfies one of the conditions in Lemma 3.15. Denote by ι : Z → X the inclusion.

Then the morphism e(E) : QDM (e,E) (X) → (h • f) * QDM(E ∨ ) factors through the ambient part quantum D-module QDM amb (Z ) of Z as:

QDM (e,E) (X)

4 H. Iritani et al. isomorphic to the K-group K(X) of vector bundles. We show in Propositions 4.4 and 4.5 that the duality pairing S QS is identified with the Euler pairing on the K-groups, and that the maps appearing in the diagram (1.2) are induced by natural functorial maps between K-groups.

In Section 5, we consider the case where E = K -1 X and study quantum cohomology of a Calabi-Yau hypersurface in X and the total space of K X . We show that the small quantum D-modules SQDM (e,K -1 X ) (X) and SQDM(K X ) are isomorphic to the second structure connections of Dubrovin [START_REF] Dubrovin | Geometry of 2D Topological Field Theories[END_REF] (Here small quantum D-modules are the restriction of quantum D-modules to the H 2 (X)-parameter space). The second structure connections are meromorphic flat connections ∇(σ) on the trivial vector bundle F over H ev (X) × C x with fibre H ev (X), which is obtained from the quantum connection of X via the Fourier-Laplace transformation with respect to z -1 (see (5.5)):

∂ z -1
x, z -1 -∂ x .

The second structure connection has a complex parameter σ ; we will see that the two small quantum D-modules correspond to different values of σ .

Theorem 1.3 (see Theorems 5.16 and 5.19 for more precise statements). Suppose that the anticanonical class -K X of X is nef. Let n be the dimension of X:

(1) There exist maps π eu , π loc : H 2 (X) × C x → H 2 (X) and isomorphisms of vector bundles with connections:

ψ eu : ( F , ∇( n+1 2 ) )| H 2 (X)×C x -→ π * eu SQDM (e,K -1 X ) (X)| z=1 ψ loc : ( F , ∇(-n+1 2 ) )| H 2 (X)×C x -→ π * loc SQDM(K X )| z=1
which are defined in a neighbourhood of the large radius limit point and for sufficiently large |x|.

(2) The duality pairing S QS between π * eu SQDM (e,K -1 X ) (X) and π * loc SQDM(K X ) is identified with the second metric ǧ:

(O( F ), ∇( n+1 2 ) ) × (O( F ), ∇(-n+1 2 ) ) → O given by ǧ(γ 1 , γ 2 ) = X γ 1 ∪ (c 1 (X) • τ -x) -1 γ 2 over H 2 (X) × C x .
Combined with the commutative diagram (1.2), this theorem gives an entirely algebraic description of the ambient part quantum D-module of a Calabi-Yau hypersurface Z in a Fano manifold X (Corollary 5.20). We will also describe the A-model Hodge filtration for these small quantum D-modules in terms of the second structure connection in Section 5.8. These results are illustrated for a quintic threefold in P 4 in Section 6. Note that the second structure connection is closely related to the almost dual Frobenius manifold of Dubrovin (see [START_REF] Dubrovin | On Almost Duality for Frobenius Manifolds[END_REF]Proposition 3.3]) and our result may be viewed as a generalization of the example in [START_REF] Dubrovin | On Almost Duality for Frobenius Manifolds[END_REF]Section 5.4].

This paper arose out of our previous works [START_REF] Iritani | Quantum cohomology and periods[END_REF][START_REF] Mann | Quantum D-modules for toric nef complete intersections[END_REF] on quantum D-modules of (toric) complete intersections. The embedding of QDM amb (Z ) into QDM(E ∨ ) appeared in [START_REF] Iritani | Quantum cohomology and periods[END_REF]Remark 6.14] in the case where X is a weak Fano toric orbifold and E ∨ = K X ; in [START_REF] Mann | Quantum D-modules for toric nef complete intersections[END_REF]Theorem 1.1], QDM amb (Z ) was presented as the quotient QDM (e,E) (X) by Ker(e(E)∪) when E is a direct sum of ample line bundles. We would also like to draw attention to a recent work of Borisov and Horja [START_REF] Borisov | Applications of homological mirror symmetry to hypergeometric systems: duality conjectures[END_REF] on the duality of better behaved Gelfand Kapranov Zelevinsky systems. The conjectural duality in their work should correspond to a certain form of quantum Serre duality generalized to toric Deligne-Mumford stacks.

We assume that the reader is familiar with Givental's formalism and quantum cohomology Frobenius manifold. As preliminary reading for the reader, we list [4, 13; 27, Chapter I,II]. Notation 1.4. We use the following notation throughout the paper.

X

a smooth projective variety of complex dimension n. E a vector bundle over X of rank r with E ∨ the dual vector bundle.

(T 0 , . . . , T s ) an homogeneous basis of H ev (X) = n p=0 H 2 p (X, C) such that T 0 = 1 and {T 1 , . . . , T r } form a nef integral basis of H 2 (X).

(t 0 , . . . , t s ) the linear coordinates dual to the basis (T 0 , . . . , T s ); we write τ := s α=0 t α T α and ∂ α := (∂/∂t α ). (T 0 , . . . , T s ) the Poincar é dual basis such that X T α ∪ T β = δ β α . Eff(X) the set of classes in H 2 (X, Z) represented by effective curves. In this section, we reformulate quantum Serre theorem of Coates and Givental [START_REF] Coates | Quantum Riemann-Roch, Lefschetz and Serre[END_REF] as a duality of quantum D-modules. After reviewing twisted Gromov-Witten invariants and twisted quantum D-modules, we give our reformulation in Theorem 2.11.

γ (d) the pairing d γ between γ ∈ H 2 (X) and d ∈ H 2 (X).
6 H. Iritani et al.

Notation

We introduce the notation we use throughout the paper. Let {T 0 , . . . , T s } be a homogeneous basis of the cohomology group H ev (X) = n p=0 H 2 p (X; C) of even degree. We assume T 0 = 1 and T 1 , . . . , T r form a nef integral basis of H 2 (X) for r = dim H 2 (X) ≤ s.

Let {t 0 , . . . , t s } denote the linear co-ordinates on H ev (X) dual to the basis {T 0 , . . . , T s } and write τ = s α=0 t α T α for a general point of H ev (X). We write ∂ α = ∂/∂t α for the partial derivative.

Let Eff(X) denote the set of classes of effective curves in H 2 (X; Z). Let K be a commutative ring. For d ∈ Eff(X), we write Q d for the corresponding element in the group

ring K[Q] := K[Eff(X)]. The variable Q is called the Novikov variable. We write K[[Q]] for the natural completion of K[Q].
For an infinite set s = {s 0 , s 1 , s 2 , . . .} of variables, we define the formal power series ring

K[[s]] = K[[s 0 , s 1 , s 2 , . . .]] to be the (maximal) completion of K[s 0 , s 1 , s 2 , . . .] with respect to the additive valuation v defined by v(s k ) = k + 1. We write C[[Q, τ ]], C[[Q, s, τ ]], and C[z][[Q, s, τ ]] for the completions of C[[Q]][t 0 , . . . , t s ], C[[Q]][t 0 , . . . , t s , s 0 , s 1 , s 2 , . . .], and C[z][[Q]][t 0 , . . . , t s , s 0 , s 1 , s 2 , .
. .], respectively. We write τ 2 = r i=1 t i T i for the H 2 (X)-component of τ and set τ = τ 2 + τ . Because of the divisor equation in Gromov-Witten theory, the Novikov variable Q and τ 2 often appear in the combination (d) . Therefore we can also work with the subring

(Q e τ 2 ) d = Q d e t 1 T 1 (d)+•••+t r T r
C[[Q e τ 2 , τ ]] = C[[Q e τ 2 ]][[t 0 , t r+1 , . . . , t s ]] ⊂ C[[Q, τ ]]. The subrings C[[Q e τ 2 , s, τ ]] ⊂ C[[Q, s, τ ]], C[z][[Q e τ 2 , s, τ ]] ⊂ C[z][[Q, s, τ ]] are defined similarly.

Twisted quantum D-modules

Coates and Givental [START_REF] Coates | Quantum Riemann-Roch, Lefschetz and Serre[END_REF] introduced Gromov-Witten invariants twisted by a vector bundle and a multiplicative characteristic class. We consider the quantum D-module defined by genus-zero twisted Gromov-Witten invariants.

Twisted Gromov-Witten invariants and twisted quantum product

Let X be a smooth projective variety and let E be a vector bundle on X. Denote by Eff(X) the subset of H 2 (X, Z) of classes of effective curves. For d ∈ Eff(X) and g, ∈ N, we denote by Mg, (X, d) the moduli space of genus g stable maps to X of degree d and with marked points. Recall that Mg, (X, d) is a proper Deligne-Mumford stack and is equipped with a virtual fundamental class [ Mg, (X,

d)] vir in H 2D ( Mg, (X, d), Q) with D = (1 -g)(dim X -3) + (c 1 (X) • d) + .
In this paper, we only consider the genus-zero moduli spaces. The universal curve of M0, (X, d) is M0, +1 (X, d):

M0, +1 (X, d) π ev +1 / / X M0, (X, d)
where π is the map that forgets the ( + 1)th marked point and stabilizes, and ev +1 is the evaluation map at the ( + 1)th marked point.

The vector bundle

E defines a K-class E 0, ,d := π ! e * +1 E ∈ K 0 ( M0, (X, d))
on the moduli space, where π ! denotes the K-theoretic push-forward. The restriction to a point

( f : C → X) ∈ M0, (X, d) gives E 0, ,d | ( f:C →X) = [H 0 (C , f * E)] -[H 1 (C , f * E)].
For i ∈ {1, . . . , }, let L i denote the universal cotangent line bundle on M0, (X, d) at the ith marking. The fibre of

L i at a point (C , x 1 , . . . , x , f : C → X) is the cotangent space T * x i C at x i . Put ψ i := c 1 (L i ) in H 2 ( M0, (X, d), Q).
The universal invertible multiplicative characteristic class c(•) is given by:

c(•) = exp ∞ k=0 s k ch k (•)
with infinitely many parameters s 0 , s 1 , s 2 , . . .. In the discussion of Coates-Givental's quantum Serre theorem, we treat s 0 , s 1 , s 2 , . . . as formal infinitesimal parameters. On the other hand, the result we obtain later sometimes makes sense for non-zero values of the parameters. For example, we will use the equivariant Euler class for c(•) in Section 3. The genus-zero (c, E)-twisted Gromov-Witten invariants are defined by the following formula. For any γ 1 , . . . , γ ∈ H ev (X) and any k 1 , . . . , k ∈ N, we put:

γ 1 ψ k 1 , . . . , γ ψ k (c,E) 0, ,d := [ M0, (X,d)] vir i=1 ψ k i i ev * i γ i c(E 0, ,d ).
We also use the following notation:

γ 1 ψ k 1 , . . . , γ ψ k (c,E) τ := d∈Eff(X) k≥0 Q d k! γ 1 ψ k 1 1 , . . . , γ ψ k , τ, . . . , τ (c,E) 0, +k,d
The genus-zero twisted Gromov-Witten potential is

F 0 (c,E) (τ ) = (c,E) τ = d∈Eff(X) k≥0 Q d k! τ, . . . , τ (c,E) 0,k,d .
The genus-zero-twisted potential

F 0 (c,E) (τ ) lies in C[[Q, s, τ ]].
(•, •) (c,E) on H ev (X) ⊗ C[[s]] by (γ 1 , γ 2 ) (c,E) = X γ 1 ∪ γ 2 ∪ c(E).
The (c, E)-twisted quantum product • (c,E) τ is defined by the formula:

(T α • (c,E) τ T β , T γ ) (c,E) = ∂ α ∂ β ∂ γ F 0 (c,E) (τ ). (2.1)
The structure constants lie in

C[[Q e τ 2 , s, τ ]] ⊂ C[[Q, s, τ ]]. The product is extended bilin- early over C[[Q, s, τ ]] and defines the (c, E)-twisted quantum cohomology (H ev (X) ⊗ C[[Q, s, τ ]], • (c,E) τ
). It is associative and commutative, and has T 0 = 1 as the identity.

Twisted quantum D-module and fundamental solution

Definition 2.1. The (c, E)-twisted quantum D-module is a triple

QDM (c,E) (X) = (H ev (X) ⊗ C[z][[Q, s, τ ]], ∇ (c,E) , S (c,E) )
where ∇ (c,E) is the connection defined by and S (c,E) is the "z-sesquilinear" pairing on

∇ (c,E) α : H ev (X) ⊗ C[z][[Q, s, τ ]] → z -1 H ev (X) ⊗ C[z][[Q, s, τ ]] ∇ (c,E) α = ∂ α + 1 z (T α • (c,E) τ ), α = 0, . . . ,
H ev (X) ⊗ C[z][[Q, s, τ ]] defined by S (c,E) (u, v) = (u(-z), v(z)) (c,E) for u, v ∈ H ev (X) ⊗ C[z][[Q, s, τ ]]. The connection ∇ (c,E) is called the quantum connection.
When c = 1 and E = 0, the triple

QDM(X) = (H ev (X) ⊗ C[z][[Q, τ ]], ∇ = ∇ (c=1,E=0) , S = S (c=1,E=0) ) is called the quantum D-module of X. Remark 2.2. The module H ev (X) ⊗ C[z][[Q, s, τ ]
] should be viewed as the module of sections of a vector bundle over the formal neighbourhood of the point

Q = s = τ = z = 0. Since the connection ∇ (c,E) does not preserve H ev (X) ⊗ C[z][[Q, s, τ ]], quantum D-module
is not a D-module in the traditional sense. It can be regarded as a lattice in the D-module [34, p. 18]).

H ev (X) ⊗ C[z ± ][[Q, s, τ ]] (see, e.g.
Remark 2.3. As discussed, structure constants of the quantum product belong to the

subring C[[Q e τ 2 , s, τ ]].
Therefore, the twisted quantum D-modules can be defined over

C[z][[Q e τ 2 , s, τ ]
]. This will be important when we specialize Q to one in Section 3.

Remark 2.4. For c = 1 and E = 0, we can complete the quantum connection ∇ in the z-direction as a flat connection. We define

∇ z∂ z = z∂ z - 1 z (E• τ ) + deg 2 where E = s α=0 (1 -1 2 deg T α )t α T α + c 1 (T X) is the Euler vector field.
The quantum connection ∇ (c,E) is known to be flat and admit a fundamental solution. The fundamental solution of the following form was introduced by Givental [11,

Corollary 6.2]. We define L (c,E) (τ, z) ∈ End(H ev (X)) ⊗ C[z -1 ][[Q, s, τ ]
] by the formula:

L (c,E) (τ, z)γ = γ - s α=0 γ z + ψ , T α (c,E) τ T α c(E)
where γ /(z + ψ) in the correlator should be expanded in the geometric series

∞ n=0 γ ψ n (-z) -n-1 .
Proposition 2.5 (see, e.g. [18, Proposition 2.1; 33, Section 2]).

The quantum connection ∇ (c,E) is flat and L (c,E) (τ, z) gives the fundamental solution for ∇ (c,E) . Namely we have

∇ (c,E) α (L (c,E) (τ, z)γ ) = 0, α = 0, . . . , s
for all γ ∈ H ev (X). Moreover, S (c,E) is flat for ∇ (c,E) and L (c,E) (τ, z) is an isometry for S (c,E) :

dS (c,E) (u, v) = S (c,E) (∇ (c,E) u, v) + S (c,E) (u, ∇ (c,E) v) S (c,E) (u, v) = S (c,E) (L (c,E) u, L (c,E) v) where u, v ∈ H ev (X) ⊗ C[z][[Q, s, τ ]].
From the last point of Proposition 2.5, one can deduce that the inverse of

L (c,E)
is given by the adjoint of L (c,E) (τ, -z). Explicitly,

L (c,E) (τ, z) -1 γ = γ + s α=0 T α z -ψ , γ (c,E) τ T α c(E) . (2.2)
Definition 2.6. The (c, E)-twisted J-function is defined to be

J (c,E) (τ, z) := zL (c,E) (τ, z) -1 1 = z + τ + s α=0 T α z -ψ (c,E) τ T α c(E) (2.3)
We deduce the following equality for α = 0, . . . , s:

L (c,E) (τ, z) -1 T α = L (c,E) (τ, z) -1 z∇ (c,E) α 1 = ∂ α J (c,E) (τ, z). (2.4)
Remark 2.7. When c = 1 and E = 0, we can complete the quantum connection ∇ in the z-direction as in Remark 2.4. The fundamental solution for flat sections, including in the z-direction, is given by L(τ, z)z -deg /2 z c 1 (T X) ; see [START_REF] Iritani | Quantum cohomology and periods[END_REF]Proposition 3.5].

Remark 2.8. The divisor equation for descendant invariants shows that

L (c,E) (τ, z)γ = e -τ 2 /z γ + (d, ) =(0,0) d∈Eff(X), ≥0 Q d e τ 2 (d) ! e -τ 2 /z γ -z -ψ , τ , . . . , τ , T α (c,E) 0, +2,d T α c(E) . ( 2
c * (•) = exp k≥0 (-1) k+1 s k ch k (•) . (2.6)
The class c * corresponds to the choice of parameters s * = (s * 0 , s * 1 , s * 2 , . . .) with s * k = (-1) k+1 s k . For any vector bundle G, we have

c * (G ∨ )c(G) = 1.
Definition 2.9. Define the map f : H ev (X) → H ev (X) by the formula:

f(τ ) = s α=0 T α , c * (E ∨ ) (c,E) τ T α .
(2.7)

More precisely, the formula defines a morphism

f : SpfC[[Q, s, τ ]] → SpfC[[Q, s, τ ]] of for- mal schemes.
Definition 2.10. The quantum Serre pairing S QS is the z-sesquilinear pairing on

H ev (X) ⊗ C[z][[Q, s, τ ]
] defined by:

S QS (u, v) = X u(-z) ∪ v(z) for u, v ∈ H ev (X) ⊗ C[z][[Q, s, τ ]].
Theorem 2.11. [START_REF] Borisov | Applications of homological mirror symmetry to hypergeometric systems: duality conjectures[END_REF] The twisted quantum D-modules QDM (c,E) (X) and

f * QDM (c * ,E ∨ ) (X)
are dual to each other with respect to S QS , that is,

∂ α S QS (u, v) = S QS (∇ (c,E) α u, v) + S QS (u, ( f * ∇ (c * ,E ∨ ) ) α v) (2.8) for u, v ∈ H ev (X) ⊗ C[z][[Q, s, τ ]].
(2) The isomorphism of vector bundles

c(E) : QDM (c,E) (X) → f * QDM (c * ,E ∨ ) (X) α → c(E) ∪ α intertwines the connections ∇ (c,E) , f * ∇ (c * ,E ∨ )
and the pairings S (c,E) , f * S (c * ,E ∨ ) .

(3) The fundamental solutions satisfy the following properties:

c(E)L (c,E) (τ, z) = L (c * ,E ∨ ) ( f(τ ), z)c(E), S QS (u, v) = S QS (L (c,E) u, ( f * L (c * ,E ∨ ) )v).
We will give a proof of this theorem in Section 2.5.

Remark 2.12. [START_REF] Borisov | Applications of homological mirror symmetry to hypergeometric systems: duality conjectures[END_REF] The pull back f * ∇ (c * ,E ∨ ) is defined to be

( f * ∇ (c * ,E ∨ ) ) α = ∂ α + 1 z s β=0 ∂ f β (τ ) ∂t α (T β • (c * ,E ∨ ) f(τ ) ).
where we set f(τ ) = s α=0 f α (τ )T α . The flatness (2.8) of S QS implies a certain complicated relationship between the quantum products

• (c,E) τ , • (c * ,E ∨ ) f(τ ) . (2)
The map c(E) in the above theorem is obtained as the composition of the quantum Serre duality and the self-duality: Coates and Givental [START_REF] Coates | Quantum Riemann-Roch, Lefschetz and Serre[END_REF] stated quantum Serre theorem as an equality of Lagrangian cones. We review the language of Lagrangian cones and explain quantum Serre theorem.

(-) * ( f * QDM (c * ,E ∨ ) (X)) ∨ S (c * ,E ∨ ) ∼ = ) ) S S S S S S S S S S S S S S QDM (c,E) (X) S QS ∼ = 6 6 l l l l l l l l l l l l l c(E) / / f * QDM (c * ,E ∨ ) (X)

Givental's symplectic vector space

Givental's symplectic vector space for the (c, E)-twisted Gromov-Witten theory is an

infinite dimensional C[[Q, s]]-module: H = H ev (X) ⊗ C[z, z -1 ][[Q, s]]
equipped with the anti-symmetric pairing The space H has a standard polarization

Ω (c,E) ( f, g) = Res z=0 ( f(-z), g(z)) (c,E) dz.
H = H + ⊕ H -
where H ± are Ω (c,E) -isotropic subspaces:

H + = H ev (X) ⊗ C[z][[Q, s]] H -= z -1 H ev (X) ⊗ C[z -1 ][[Q, s]].
This polarization identifies H with the total space of the cotangent bundle T * H + . A general element on H can be written in the form (note that the dual basis of {T α } s α=0 with respect to the pairing

(•, •) (c,E) is {c(E) -1 T α } s α=0 ): ∞ k=0 s α=0 q α k T α z k + ∞ k=0 s α=0 p k,α c(E) -1 T α 1 (-z) k+1 with p k,α , q α k ∈ C[[Q, s]].
The coefficients p k,α , q α k here give Darboux co-ordinates on H in the sense that Ω (c,E) = k,α dp k,α ∧ dq α k .

Twisted Lagrangian cones

The genus-zero gravitational descendant Gromov-Witten potential is a function on the formal neighbourhood of -z1 in H + defined by the formula:

F 0,grav (c,E) (-z + t(z)) = (c,E) t(ψ) = d∈Eff(X) ∞ k=0 Q d k! t(
p k,α = ∂F 0,grav (c,E) ∂q α k .
In other words, it consists of points of the form:

-z + t(z) + s α=0 T α -z -ψ (c,E) t(ψ) T α c(E) (2.9)
with t(z) ∈ H + .

Givental [START_REF] Givental | Symplectic Geometry of Frobenius Structures[END_REF] showed that the submanifold L (c,E) is in fact a cone (with vertex at the origin of H). Moreover, he showed the following geometric property of L (c,E) . For every tangent space T of L (c,E) (T is a linear subspace of H),

• zT = T ∩ L (c,E) ;

• the tangent space of L (c,E) at any point in zT ⊂ L (c,E) is T.

Note that the twisted J-function (2.3) is a family of elements lying on L (c,E) :

J (c,E) (τ, -z) = -z + τ + s α=0 T α -z -ψ (c,E) τ T α c(E)
obtained from (2.9) by setting t(z) = τ . L (c,E) . An R-valued point on L (c,E) is a point of the form (2.9) with

t α k ∈ R such that t α k are topologically nilpotent, that is, lim n→∞ (t α k ) n = 0. A C[[Q, s]]-valued point is given by t α k ∈ C[[Q, s]] with t α k | Q=s=0 = 0. The J-function is a C[[Q, s, τ ]]-valued point on L (c,E) .
In what follows we mean by a point (2.9) on L (c,E) a C[[Q, s]]-valued point, but the discussion applies to a general R-valued point.

Tangent space to the twisted Lagrangian cone

Let g = g(t) denote the point on L (c,E) given in Equation (2.9). Differentiating g(t) in t α k , we obtain the following tangent vector:

∂g(t) ∂t α k = T α z k + s β=0 T α ψ k , T β -z -ψ (c,E) t(ψ) T β c(E) in T g L (c,E) .
(2.10)

The tangent space T g L (c,E) is spanned by these vectors. Since T g L (c,E) is complementary to H -, T g L (c,E) intersects with 1 + H -at a unique point. The intersection point is the one (2.10) with k = α = 0:

(1 + H -) ∩ T g L (c,E) = ∂g(t) ∂t 0 0 = 1 - τ (t) z + O(z -2 )
where

τ (t) = s α=0 1, T α (c,E) t(z)
T α c(E) .

(2.11)

Givental [START_REF] Givental | Symplectic Geometry of Frobenius Structures[END_REF] observed that each tangent space to the cone is uniquely parametrized by the value τ (t), that is, the tangent spaces at g(t

equals the tangent space at g( τ (t)) = J (c,E) ( τ (t), -z).
Proposition 2.15 ([13], see also [START_REF] Coates | Computing genus-zero twisted Gromov-Witten invariants[END_REF]Proposition B.4]).

Let g = g(t) denote the point of L (c,E) given in Equation (2.9). The tangent space

T g L (c,E) is a free C[z][[Q, s]
]-module generated by the derivatives of the twisted J-function:

∂ J (c,E) ∂t α (τ, -z) τ = τ (t)
.

Proof. As discussed, T g L (c,E) equals the tangent space of L (c,E) at J(τ, -z) with τ = τ (t).
On the other hand, the tangent space at J (c,E) (τ, -z) is freely generated by the derivatives

∂ α J (c,E) (τ, -z) [3, Lemma B.5],
and the result follows.

Relations between Lagrangian cones and QDM (c,E) (X)

Proposition 2.15 means that the quantum D-module can be identified with the family of tangent spaces to the Lagrangian cone

L (c,E) at the J-function J (c,E) (τ, -z). (-) * QDM (c,E) (X) τ ∼ = T J (c,E) (τ,-z) L (c,E) T α → ∂ α J (c,E) (τ, -z) = L (c,E) (τ, -z) -1 T α
where (-) * denotes the pull back by the sign change z → -z and we used (2.4). This identification preserves the pairing S (c,E) and intertwines the quantum connection ∇ on QDM (c,E) (X) with the trivial differential d on H: this follows from the properties of L (c,E) in Proposition 2.5.

Quantum Serre theorem of Coates-Givental

c(E)L (c,E) = L (c * ,E ∨ ) .
For any γ = s α=0 γ α T α ∈ H ev (X), we write ∂ γ := s α=0 γ α ∂ α for the directional derivative.

Corollary 2.17. Let T τ denote the tangent space of L (c,E) at J (c,E) (τ, -z) and let T * τ denote the tangent space of L (c * ,E ∨ ) at J (c * ,E ∨ ) (τ, -z). Then we have the following equations:

(1) c(E)T τ = T * f(τ ) , where f was defined in (2.7); (2) c(E)∂ α J (c,E) (τ, -z) = (∂ c(E)∪T α J (c * ,E ∨ ) )(τ * , -z)| τ * = f(τ ) for α = 0, 1, . . . , s.
Remark 2.18. Exchanging the twist (c, E) with (c * , E ∨ ) in the above Corollary, we obtain

c * (E ∨ )z∂ c(E) J (c * ,E ∨ ) (τ, z) = J (c,E) ( f(τ ), z) where f(τ ) = s α=0 T α , c(E) (c * ,E ∨ ) τ
T α . This is exactly [4, Corollary 10].

Proof of Corollary 2.17. (1) Theorem 2.16 implies that c(E)T τ is a tangent space to the cone L (c * ,E ∨ ) . Therefore, by the discussion in the previous section Section 2.4.3, c(E)T τ equals T * σ with σ given by the intersection point:

(1 + H -) ∩ c(E)T τ = 1 - σ z + O(z -2 ) .
(2.12)

Note that

∂ c * (E ∨ ) J (c,E) (τ, -z) = c * (E ∨ ) - 1 z s α=0 T α , c * (E ∨ ) τ T α c(E) + O(z -2 )
lies in T τ . Multiplying this by c(E), we obtain the intersection point in (2.12) and we have

σ = f(τ ) as required (recall that c(E)c * (E ∨ ) = 1).
(2) By part (1), the vector c(E)∂ α J (c,E) (τ, -z) belongs to the tangent space T * f(τ ) of L (c * ,E ∨ ) . It has the following asymptotics:

c(E)∂ α J (c,E) (τ, -z) = c(E) ∪ T α + O(z -1 ).
By the description of tangent spaces in Section 2.4.3, a tangent vector in T * f(τ ) with this asymptotics is unique and is given by

(∂ c(E)∪T α J (c * ,E ∨ ) )(τ * , -z) with τ * = f(τ ).

A proof of Theorem 2.11

We use the correspondence in Section 2.4.4 between quantum D-module and tangent spaces to the Givental cone. Then Corollary 2.17 implies that the map (2) of the theorem. Also part (2) of Corollary 2.17 implies, in view of (2.4)

c(E) : QDM (c,E) (X) → f * QDM (c * ,E ∨ ) (X)
c(E)L (c,E) (τ, -z) -1 T α = L (c * ,E ∨ ) ( f(τ ), -z) -1 (c(E) ∪ T α )
for α = 0, 1, . . . , s. This implies the first equation of part [START_REF] Coates | Computing genus-zero twisted Gromov-Witten invariants[END_REF]. To see the second equation of part (3), we calculate:

S QS (u, v) = S (c,E) (u, c(E) -1 v) = S (c,E) (L (c,E) u, L (c,E) e(E) -1 v) = S (c,E) (L (c,E) u, c(E) -1 ( f * L (c * ,E ∨ ) )v) = S QS (L (c,E) , ( f * L (c * ,E ∨ ) )v)
where we used Proposition 2.5. Part (1) of the theorem is equivalent to part (2), as explained in Remark 2.12.

Quantum Serre Duality for Euler-Twisted Theory

In this section we apply Theorem 2.11 to the equivariant Euler class e λ and a convex vector bundle E. By taking the non-equivariant limit, we obtain a relationship among the quantum D-module twisted by the Euler class and the bundle E, the quantum D-module of the total space of E ∨ , and the quantum D-module of a submanifold Z ⊂ X cut out by a regular section of E.

To ensure the well-defined non-equivariant limit, we assume that our vector bundle E → X is convex, that is, for every genus-zero stable map f : C → X we have

H 1 (C , f * E) = 0.
The convexity assumption is satisfied, for example, if O(E) is generated by global sections.

Equivariant Euler class

In this section, we take c to be the C × -equivariant Euler class e λ . Given a vector bundle G, we let C × act on G by scaling the fibres and trivially on X. With respect to this

C × -action we have e λ (G) = rkG i=0 λ r-i c i (G)
where λ is the C × -equivariant parameter: the C × -equivariant cohomology of a point is

H * C × (pt) = C[λ]
. Choosing e λ means the following specialization:

s k := ⎧ ⎪ ⎨ ⎪ ⎩ log λ if k = 0 (-1) k-1 (k -1)!λ -k if k > 0
Although the parameters s k contain log λ and negative powers of λ, we will see that the (e λ , E)-twisted theory and (e -1 λ , E ∨ )-twisted theory are defined over the polynomial ring in λ, and hence admit the non-equivariant limit λ → 0. Here the convexity of E plays a role.

Specialization of the Novikov variable

We henceforth specialize the Novikov variable Q to one. By Remark 2.3, the specialization Q = 1 is well defined: one has

T α • (c,E) τ T β | Q=1 ∈ H ev (X) ⊗ C[[e τ 2 , s, τ ]] L (c,E) (τ, z)| Q=1 ∈ End(H ev (X)) ⊗ C[z -1 ][[e τ 2 , s, τ ]][τ 2 ] f(τ )| Q=1 ∈ H ev (X) ⊗ C[[e τ 2 , s, τ ]] see (2.7)
where C[[e τ 2 , s, τ ]] is the completion of C[e t 1 , . . . , e t s , t 0 , t r+1 , . . . , t s , s 0 , s 1 , s 2 , . . .]. Since we chose T 1 , . . . , T r to be a nef integral basis of H 2 (X, Z), we have only nonnegative integral powers of e t 1 , . . . , e t r in the structure constants of quantum cohomology. The Eulertwisted quantum D-module will be defined over

C[z][[e τ 2 , τ ]]. In what follows, we shall omit (• • • )| Q=1 from the notation.

Non-Equivariant limit of QDM (e λ ,E) (X)

Let e = lim λ→0 e λ denote the non-equivariant Euler class. We first discuss the nonequivariant limit of QDM (e λ ,E) (X). Recall the K-class E 0, ,d on the moduli space M0, (X, d) introduced in Section 2.2.1. The convexity assumption for E implies that E 0, ,d is represented by a vector bundle. Moreover, the natural evaluation morphism E 0, ,d → ev * j E at the jth marking is surjective for all j ∈ {1, . . . , }. Define E 0, ,d ( j) by the exact sequence:

0 / / E 0, ,d ( j) / / E 0, ,d / / ev * j E / / 0 . (3.1)
We use the following variant of (e λ , E)-twisted invariants (see [START_REF] Pandharipande | Rational curves on hypersurfaces (after A. Givental)[END_REF]). For any γ 1 , . . . , γ ∈ H ev (X) and any k 1 , . . . , k ∈ N, we put:

γ 1 ψ k 1 1 , . . . , γ j ψ k j j , . . . , γ ψ k (e λ ,E) 0, ,d := [ M0, (X,d)] vir i=1 ψ k i i ev * i γ i e λ (E 0, ,d ( j)).
This lies in the polynomial ring C[λ]. Lemma 3.1 ([33]). Suppose that E is convex. The (e λ , E)-twisted quantum product

T α • (e λ ,E) τ T β lies in H ev (X) ⊗ C[λ][[e τ 2 , τ ]
] and admits the non-equivariant limit

T α • (e,E) τ T β := lim λ→0 (T α • (e λ ,E) τ T β ).
Proof. Recall that the twisted quantum product (2.1) is given by:

γ 1 • (e λ ,E) τ γ 2 = s α=0 γ 1 , γ 2 , T α e λ (E) (e λ ,E) τ T α .
From the exact sequence (3.1), we deduce that

e λ (E 0, ,d ) ev * 3 e λ (E) = e λ (E 0, ,d (3)).
Therefore, we have

γ 1 • (e λ ,E) τ γ 2 = s α=0 γ 1 , γ 2 , Tα (e λ ,E) τ T α
and this lies in

H ev (X) ⊗ C[λ][[e τ 2 , τ ]]. Lemma 3.2. Suppose that E is convex. For (c, E) = (e λ , E), the map f(τ ) in (2.7) lies in H ev (X) ⊗ C[λ][[e τ 2 , τ ]
] and admits the non-equivariant limit f(τ

) := lim λ→0 f(τ ). Proof. Note that c * (E ∨ ) = e * λ (E ∨ ) in (2.7
) equals e λ (E) -1 . Arguing as in Lemma 3.1, we have

f(τ ) = s α=0 T α , 1 (e λ ,E) τ T α .

The conclusion follows.

By Lemma 3.1, we deduce that the non-equivariant limit ∇ (e,E) = lim λ→0 ∇ (e λ ,E) of the quantum connection exists. Moreover, it can be completed in the z-direction in a flat connection:

∇ (e,E) z∂ z = z∂ z - 1 z (E (e,E) • (e,E) τ ) + deg 2
where E (e,E) is the Euler vector field:

E (e,E) = s α=0 1 - deg T α 2 t α T α + c 1 (T X) -c 1 (E). (3.
2)

The non-equivariant limit S (e,E) (u, v) := X u(-z) ∪ v(z) ∪ e(E) of the pairing S (e λ ,E) becomes degenerate. The pairing S (e,E) is not flat in the z-direction, but satisfies the following equation:

z∂ z S (e,E) (u, v) -S (e,E) (∇ (e,E) z∂ z u, v) -S (e,E) (u, ∇ (e,E) z∂ z v) = -(dim X -rkE)S (e,E) (u, v).
We refer to this by saying that S (e,E) is of weight

-(dim X -rkE). Note that z dim X-rkE S (e,E)
is flat in both τ and z.

Definition 3.3 (cf. Definition 2.1). We call the triple

QDM (e,E) (X) := (H ev (X) ⊗ C[z][[e τ 2 , τ ]], ∇ (e,E) , S (e,E) )
the (e, E)-twisted quantum D-module.

Remark 3.4. By a similar argument, the fundamental solution L (e λ ,E) in Proposition 2.5 can be written as:

L (e λ ,E) (τ, z)γ = γ - s α=0 γ z + ψ , Tα (e λ ,E) τ T α
and therefore admits the non-equivariant limit L (e,E) . The fundamental solution for E) , including in the z-direction, is given by E) . All the properties of Proposition 2.5 are true for the limit (see [START_REF] Mann | Quantum D-modules for toric nef complete intersections[END_REF]Section 2] for a more precise statement).

∇ (e,
L (e,E) (τ, z)z -deg 2 z c 1 (T X)-c 1 (

Quantum D-module of a section of E

In this section, we describe a relationship between the (e, E)-twisted quantum D-module and the quantum D-module of a submanifold Z ⊂ X cut out by a regular section of E.

Let ι : Z → X denote the natural inclusion. The functoriality of virtual classes

[20] [ M0, (X, d)] vir ∩ e(E 0, ,d ) = ι * (d )=d ι * [ M0, (Z , d )] vir
together with the argument in [START_REF] Iritani | Quantum cohomology and periods[END_REF]Corollary 2.5;[START_REF] Pandharipande | Rational curves on hypersurfaces (after A. Givental)[END_REF] shows that

ι * (γ 1 • (e,E) τ γ 2 ) = (ι * γ 1 ) • Z ι * τ (ι * γ 2 ) (3.3)
for γ 1 , γ 2 ∈ H ev (X). Define the ambient part of the cohomology of Z by

H * amb (Z ) = Im(ι : H * (X) → H * (Z )
(Z ) ⊗ C[z][[e τ 2 , τ ]], ∇ Z , S Z )
where the parameter τ = τ 2 + τ is restricted to lie in the ambient part H ev amb (Z ) and S Z (u, v) = Z u(-z) ∪ v(z). We complete the quantum connection ∇ Z in the z-direction as in Remark 2.4; then S Z is of weight (dim Z ). Equation (3.3) proves the following proposition.

Proposition 3.6. The restriction map ι * : H ev (X) → H ev amb (Z ) induces a morphism between the quantum D-modules

ι * : QDM (e,E) (X) → (ι * ) * QDM amb (Z )
which is compatible with the connection and the pairing.

Quantum D-module of the total space of E ∨

We explain that the non-equivariant limit of the (e -1

λ , E ∨ )-twisted quantum D-module is identified with the quantum D-module of the total space of E ∨ .

The (e -1 λ , E ∨ )-twisted Gromov-Witten invariants admit a non-equivariant limit under the concavity (a bundle E ∨ is said to be concave if for every non-constant genuszero stable map f : C → X, one has H 0 (C , f * E ∨ ) = 0) assumption for E ∨ and they are called local Gromov-Witten invariants [START_REF] Chiang | Local mirror symmetry: calculations and interpretations[END_REF][START_REF] Givental | Equivariant Gromov-Witten Invariants[END_REF][START_REF] Givental | Elliptic Gromov-Witten Invariants and the Generalized Mirror Conjecture[END_REF]. In this paper, we only impose the weaker assumption that E is convex (see Remark 3.9). In this case, a non-equivariant limit of (e -1 λ , E ∨ )-twisted invariants may not exist, but a non-equivariant limit of the twisted quantum product is still well-defined.

The virtual localization formula [START_REF] Graber | Localization of virtual classes[END_REF] gives the following proposition. Proposition 3.7. For γ 1 , . . . , γ ∈ H ev (X) and non-negative integers k 1 , . . . , k , we have

γ 1 ψ k 1 , . . . , γ ψ k (e -1 λ ,E ∨ ) 0, ,d = γ 1 ψ k 1 , . . . , γ ψ k E ∨ ,C × 0, ,d
where the right-hand side is the C × -equivariant Gromov-Witten invariant of E ∨ with respect to the C × -action on E ∨ scaling the fibres.

The non-equivariant Gromov-Witten invariants for E ∨ are ill-defined in general because the moduli space M0, (E ∨ , d) can be non-compact. The following lemma, however, shows the existence of the non-equivariant quantum product of E ∨ . Lemma 3.8. Let E be a vector bundle on X such that f * E is generated by global sections for any stable maps f : C → X of genus g. Then the evaluation map ev

i : Mg, (E ∨ , d) → E ∨ is proper for all i ∈ {1, . . . , }. In particular, when E is convex, ev i : M0, (E ∨ , d) → E ∨ is proper.
Proof. The convexity of E implies that, for any map u:

P 1 → X, u * E is isomorphic to r i=1 O(k i ) with k i ≥ 0.
Thus the latter statement follows from the former.

Let us prove the former statement. We start with the remark that, for every stable map f : C → X, the evaluation map ev i :

H 0 (C , f * E ∨ ) → E ∨ f(x i ) at the ith marking x i ∈ C is injective. Suppose that a section s ∈ H 0 (C , f * E ∨ ) vanishes at x i , that is, ev i (s) = 0. For every u∈ H 0 (C , f * E), the pairing s, u is a global section of O C which vanishes at x i .
Then s, u must be identically zero on C . Since f * E is generated by global sections, this implies that s = 0. Hence we have shown that the evaluation map

H 0 (C , f * E ∨ ) → E ∨ f(x i ) is injective.
Giving a stable map to E ∨ is equivalent to giving a stable map f : C → X and a section of H 0 (C , f * E ∨ ). Therefore, by the preceding remark, the moduli functor Mg,

(E ∨ , d) is a subfunctor of Mg, (X, d) × E ∨ via the natural projection Mg, (E ∨ , d) → Mg, (X, d) and the evaluation map ev i : Mg, (E ∨ , d) → E ∨ . Since Mg, (X, d) is proper, it suffices to show that the map Mg, (E ∨ , d) → Mg, (X, d) × E ∨ is proper.
We use the valuative criterion for properness (see [START_REF] Deligne | The irreducibility of the space of curves of given genus[END_REF]Theorem 4.19]). Let R be a DVR. Suppose that we are given a stable map f : C R → X over Spec(R) and an R-valued point v ∈ E ∨ (R). These data ( f, v) give a map Spec(R) → Mg, (X, d) × E ∨ . Suppose moreover that there exists a

section s ∈ H 0 (C K , f * E ∨ ) over the field K of fractions of R such that ev i (s) = v in E ∨ (K),
where

C K = C R × Spec(R) Spec(K). Then ( f, s) defines a map Spec(K) → Mg, (E ∨ , d) such
that the following diagram commutes:

Mg, (E ∨ , d) / / Mg, (X, d) × E ∨ Spec(K) / / ( f,s) O O Spec(R). ( f,v) O O g g O O O O O O
We will show that there exists a morphism Spec(R) → Mg, (E ∨ , d) which commutes with the maps in the above diagram. Since Mg, (E ∨ , d) is a subfunctor of Mg, (X, d) × E ∨ , it suffices to show the existence of a morphism Spec(R) → Mg, (E ∨ , d) which makes the upper-right triangle commutative, that is, v is the image of a section in H 0 (C R , f * E ∨ ). Let π : C R → Spec(R) denote the structure map and x i : Spec(R) → C R denote the ith marking. Note that the composition Spec(R) v -→ E ∨ → X coincides with f • x i , since the two maps coincide when we compose them with Spec(K) → Spec(R) (by the existence of s) and by the separatedness of X. Thus v defines a section of x * i f * E ∨ , which we denote again by v. We need to show that v is in the image of R 0 π * f * E ∨ → x * i f * E ∨ . Let p ∈ Spec(R) denote the unique closed point and let k( p) be the residue field at p. We claim that the maps

R 0 π * f * E ∨ R k( p) → H 0 (C p , f * E ∨ ), H 0 (C p , f * E ∨ ) → (x * i f * E ∨ ) R k( p) are injective.
The injectivity of the latter map has been shown. To see the injectivity of the former, we take the so-called Grothendieck complex [31, Section 5, p. 46]: a complex G 0 → G 1 of finitely generated free R-modules such that the sequences

0 ----→ R 0 π * f * E ∨ ----→ G 0 d 0 ----→ G 1 0 ----→ H 0 (C p , f * E ∨ ) ----→ G 0 R k( p) ----→ G 1 R k( p)
are exact. Since R is a PID, the image of d 0 is a free R-module. Therefore

Tor R 1 (Imd 0 , k( p)) = 0 and we obtain the exact sequence:

0 ----→ (R 0 π * f * E ∨ ) R k( p) ----→ G 0 R k( p) ----→ (Imd 0 ) R k( p) ----→ 0. Now the claim follows. The claim implies that R 0 π * f * E ∨ → x * i f * E ∨ is injective at the fibre of p. Then it follows that the cokernel M of R 0 π * f * E ∨ → x * i f * E ∨ is a free R-module. In fact, let N be the image of R 0 π * f * E ∨ → x * i f * E ∨ ; then the inclusion N ⊂ x * i f * E ∨ induces an injection N R k( p) → (x * i f * E ∨ ) R k( p). Because x * i f * E ∨ is a free R-module, we have the exact sequence 0 / / Tor R 1 (M, k( p)) / / N R k( p) / / x * i f * E ∨ R k( p) / / M R k( p) / / 0.
Therefore, Tor R 1 (M, k( p)) = 0; thus M is free. We know by assumption that the image of v in M R K vanishes. Thus v has to vanish in M. The conclusion follows.

Remark 3.9. The concavity of E ∨ implies the convexity of E. This can be proved as follows. For a stable map f : C → X of genus zero, we have

H 1 (C , f * E) = H 0 (C , f * E ∨ ⊗ ω C ) ∨ by
Serre duality, where ω C is the dualizing sheaf on C . Suppose that E ∨ is concave.

Since the degree of ω C on a tail component of C is negative, a section of f * E ∨ ⊗ ω C has to vanish on tail components and defines a section of f * E ∨ ⊗ ω C where C is obtained from C by removing all its tail components. By induction on the number of components, we can see that a section of

f * E ∨ ⊗ ω C vanishes and H 0 (C , f * E ∨ ⊗ ω C ) = 0.
By the above lemma, we can define the quantum product of E ∨ using the pushforward along the evaluation map ev 3 :

T α • E ∨ τ T β = d∈Eff(E ∨ ) ∞ =0 1 ! ev 3 * ⎛ ⎝ ev * 1 (T α )ev * 2 (T β ) +3 j=4 ev * j (τ ) ∩ [ M0,3+ (E ∨ , d)] vir ⎞ ⎠ . (3.4)
The quantum product • E ∨ τ defines a flat quantum connection ∇ E ∨ as in Definition 2.1.

Definition 3.10. The (non-equivariant) quantum D-module of E ∨ is a pair

QDM(E ∨ ) = (H ev (X) ⊗ C[z][[e τ 2 , τ ]], ∇ E ∨ )
where the connection ∇ E ∨ is completed in the z-direction as in Remark 2.4:

∇ E ∨ z∂ z = z∂ z - 1 z (E E ∨ • τ ) + deg 2
where

E E ∨ := s α=0 (1 -1 2 deg T α )t α T α + c 1 (T X) -c 1 (E)
is the Euler vector field (note that this is the same as E (e,E) in (3.2)). Here the standard identification

H ev (E ∨ ) ∼ = H ev (X) is understood.
We conclude the following proposition. Proposition 3.11. Suppose that E is convex. Define the map h: H ev (X) → H ev (X) by

h(τ ) = τ + π √ -1c 1 (E) (3.5)
Then we have

(1) The non-equivariant limit of ∇ (e -1 λ ,E ∨ ) exists and coincides with ∇ E ∨ . (2) The non-equivariant limit of ∇ (e * λ ,E ∨ ) exists and coincides with h * ∇ E ∨ .

Proof. Part (1) follows from Proposition 3.7 and the existence of the non-equivariant quantum product for E ∨ . To see part [START_REF] Chiang | Local mirror symmetry: calculations and interpretations[END_REF], notice a small difference between e * λ and e -1 λ : for a vector bundle G we have Since the virtual rank of (E ∨ ) 0, ,d equals rkE

e * λ (G) = 1 e λ (G ∨ ) = (-
+ c 1 (E)(d), we have ∞ n=0 1 n! T α , T β , T γ , τ, . . . , τ (e * λ ,E ∨ ) 0,n+3,d T γ e * λ (E ∨ ) = ∞ n=0 1 n! (-1) rkE+c 1 (E)(d) T α , T β , T γ , τ, . . . , τ (e -1 -λ ,E ∨ ) 0,n+3,d T γ e * λ (E ∨ ) = ∞ n=0 1 n! T α , T β , T γ , h(τ ), . . . , h(τ ) (e -1 -λ ,E ∨ ) 0,n+3,d e -λ (E ∨ )T γ
where we used the divisor equation in the second line. This implies that the (e * λ , E ∨ )twisted quantum product is the pull-back of the (e -1 -λ , E ∨ )-twisted quantum product by h. The conclusion follows by taking the non-equivariant limit λ → 0.

Remark 3.12. The pairing S (e * λ ,E ∨ ) does not have a non-equivariant limit.

Remark 3.13. We can define the fundamental solution for the quantum connection of E ∨ using the push-forward along an evaluation map similarly to (3.4). Therefore, the fundamental solution L (e -1 λ ,E ∨ ) admits a non-equivariant limit L E ∨ . Using the formula (2.5) and an argument similar to Proposition 3.11, we find that

L (e -1 -λ ,E ∨ ) (h(τ ), z) = L (e * λ ,E ∨ ) (τ, z) • e -π √ -1c 1 (E)/z
and thus

L E ∨ (h(τ ), z) = lim λ→0 L (e * λ ,E ∨ ) (τ, z) • e -π √ -1c 1 (E)/z . (3.6) Then L E ∨ (τ, z)z -deg 2 z c 1 (T X)-c 1 (E)
is a fundamental solution of ∇ E ∨ including in the z-direction.

Non-equivariant limit of quantum Serre duality

We will state a non-equivariant limit of Theorem 2.11 when c is e λ and E is a convex vector bundle. From Sections 3.3 and 3.5, the quantum D-modules QDM (e λ ,E) (X) and QDM (e * λ ,E ∨ ) (X) have non-equivariant limits, and the limits are, respectively, QDM (e,E) (X) and h * QDM(E ∨ ). The map f in (2.7) also admits a non-equivariant limit by Lemma 3.2.

The quantum Serre pairing in Definition 2.10 has an obvious non-equivariant limit:

S QS : QDM (e,E) (X) × (h • f) * QDM(E ∨ ) → C[z][[e τ 2 , τ ]] defined by S QS (u, v) = X u(-z) ∪ v(z).
Theorem 3.14. Let E be a convex vector bundle on X. Let h be the map in (3.5) and let f be the map in Lemma 3.2.

(1) The pairing S QS is flat in the τ -direction and is of weight (dim X).

(2) The map e(E)∪ : QDM (e,E) (X) → (h • f) * QDM(E ∨ ), α → e(E) ∪ α respects the quantum connection in the τ -direction and is of weight rkE, that is, E) .

∇ α e(E) = e(E)∇ α ∇ z∂ z e(E) = e(E)∇ z∂ z + rk(E)e(E) for ∇ = (h • f) * ∇ E ∨ and ∇ = ∇ (e,
(3) The fundamental solutions in Remarks 3.4, 3.13 satisfy the following relations:

e(E) • L (e,E) (τ, z) = L E ∨ (h • f(τ ), z) e π √ -1c 1 (E)/z • e(E) (γ 1 , e -π √ -1c 1 (E)/z γ 2 ) = (L (e,E) (τ, -z)γ 1 , L E ∨ (h( f(τ )), z)γ 2 ).
where E) .

(u, v) = X u ∪ v
(h • f) * E (e,E) = E E ∨ . Therefore, gr := ∇ (e,E) z∂ z + ∇ (e,E) E (e,E) = z∂ z + E (e,E) + deg 2 = ((h • f) * ∇ E ∨ ) z∂ z + ((h • f) * ∇ E ∨ ) E (e,
On the other hand, we can check that

(z∂ z + E (e,E) )S QS (u, v) -S QS (gru, v) -S QS (u, grv) = -(dim X)S QS (u, v).
The flatness of S QS in the E (e,E) -direction shows that S QS is of weightdim X. The discussion for e(E) is similar.

Let Z ⊂ X be the zero-locus of a transverse section of E and let ι : Z → X be the inclusion map. We consider the following conditions for Z . (1) the Poincar é pairing on H ev amb (Z ) = Im(ι * : H ev (X) → H ev (Z )) is nondegenerate;

(2) we have the decomposition H ev (Z ) = Kerι * ⊕ Imι * ;

(3) ι * induces an isomorphism H ev (X)/Ker(e(E)∪) ∼ = H ev amb (Z ). In this section, we study a relation between quantum Serre duality for the Euler-twisted theory and the Γ -integral structure studied in [START_REF] Iritani | An integral structure in quantum cohomology and mirror symmetry for toric orbifolds[END_REF][START_REF] Iritani | Quantum cohomology and periods[END_REF][START_REF] Katzarkov | Hodge theoretic aspects of mirror symmetry[END_REF][START_REF] Mann | Quantum D-modules for toric nef complete intersections[END_REF]. The Γ -integral structure is a lattice in the space of flat sections for the quantum connection, which is isomorphic to the Grothendieck group K(X) of vector bundles on X. After introducing a similar integral structure in the Euler-twisted theory, we see that the quantum Serre pairing is identified with the Euler pairing on K-groups, and that the morphisms of flat connections in Corollary 3.17 are induced by natural maps between K-groups. In this section, the Novikov variable Q is specialized to one (see Section 3.2).

(ι * α, ι * β) = (α, ι * ι * β) = (α, e(E) ∪ β),
Recall 4.1. Recall the classical self-intersection formula in K-theory. Let j : X → Y be a closed embedding with normal bundle N between quasi-compact and quasi-separated schemes. In Theorem 3.1 of [START_REF] Thomason | Les K-groupes d'un sch éma éclat é et une formule d'intersection exc édentaire[END_REF], Thomason proves that we have for any [V] ∈ K(X)

j * j * [V] = [λ -1 N ∨ ] • [V] (4.1)
where

[λ -1 N ∨ ] := k≥0 (-1) k [∧ k N ∨ ] ∈ K(X).
Definition 4.2. For a vector bundle G with Chern roots δ 1 , . . . , δ r , we define the Γ -class to be

Γ (G) = r i=1 Γ (1 + δ i ).
We also define a (2π √ -1)-modified Chern character by

Ch(G) = (2π √ -1) deg 2 ch(G) = r i=1 e 2π √ -1δ i .
Suppose that E is a convex vector bundle on X. Let Z ⊂ X be a submanifold cut out by a transverse section s of E. For the (twisted) quantum connection ∇, we write Ker∇ for the space of flat sections:

Ker∇ (e,E) = {s ∈ H ev (X) ⊗ C[z ± ][[e τ 2 , τ ]][log z] : ∇ (e,E) s = 0}, Ker∇ Z = {s ∈ H ev amb (Z ) ⊗ C[z ± ][[e τ 2 , τ ]][log z] : ∇ Z s = 0}, Ker∇ E ∨ = {s ∈ H ev (X) ⊗ C[z ± ][[e τ 2 , τ ]][log z] : ∇ E ∨ s = 0}.
Definition 4.3. The K-group framing is a map from a K-group to the space of flat sections defined as follows:

(1) for the twist (e, E), the K-group framing Z (e,E) : K(X) → Ker∇ (e,E) is:

Z (e,E) (V) = 1 (2π √ -1) dim X-rkE L (e,E) (τ, z)z -deg 2 z c 1 (T X)-c 1 (E) Γ (T X) Γ (E) Ch(V);
(2) for a smooth section Z ⊂ X of E, the K-group framing

Z amb : K amb (Z ) → Ker∇ Z is: Z amb (V) = 1 (2π √ -1) dim Z L Z (τ, z)z -deg 2 z c 1 (T Z) Γ (T Z)Ch(V);
(3) for the total space E ∨ , the K-group framing Z E ∨ : K(X) → Ker∇ E ∨ is:

Z E ∨ (V) = 1 (2π √ -1) dim E ∨ L E ∨ (τ, z)z -deg 2 z c 1 (T X)-c 1 (E) Γ (T E ∨ )Ch(V).
where K amb (Z ) = Im(ι * : K(X) → K(Z )) and L Z , L E ∨ are the fundamental solutions for Z and E ∨ , respectively. Recall from Remarks 2.7, 3.4, and 3.13 that these formula define a section which is flat in both τ and z.

Proposition 4.4. For any vector bundles V, W on X, we have

χ(V ⊗ W ∨ ) = (-2π √ -1z) dim X S QS (Z (e,E) (V)(τ, e π √ -1 z), Z E ∨ (W)(h • f(τ ), z))
where

χ(V ⊗ W ∨ ) = dim X i=0 (-1) i dim Ext i (W, V) is the holomorphic Euler characteristic.
Proof. This is analogous to [17, Proposition 2.10]. Since the pairing z dim X S QS is flat, the right-hand side is constant with respect to τ and z. Evaluating the right-hand side at

z = 1, we obtain 1 (-2π √ -1) dim X L (e,E) (τ, -1) e -π √ -1 deg 2 e π √ -1(c 1 (T X)-c 1 (E)) γ 1 , L E ∨ (h • f(τ ), 1)γ 2 with γ 1 = Γ (T X) Γ (E) Ch(V), γ 2 = Γ (T X) Γ (E ∨ )Ch(W)
, where (•, •) is the Poincar é pairing on X. By Theorem 3.14(3), we find that this equals

1 (-2π √ -1) dim X e -π √ -1 deg 2 e π √ -1(c 1 (T X)-c 1 (E)) γ 1 , e -π √ -1c 1 (E) γ 2 . Since the adjoint of deg 2 is dim X -deg 2 , this is 1 (2π √ -1) dim X e π √ -1c 1 (T X) γ 1 , e π √ -1 deg 2 γ 2 .
Using the following identities:

e π √ -1 deg 2 Γ (1 + δ) = Γ (1 -δ) and e π √ -1 deg 2 ch(W) = ch(W ∨ ) (2π √ -1) -dim X X γ = X (2π √ -1) -deg 2 γ (2π √ -1) -deg 2 Γ (1 + δ) = (2π √ -1) deg 2 Γ 1 + δ 2 √ -1π
with δ a degree-two cohomology class, we deduce that the right-hand side of the proposition is

X ch(V ⊗ W ∨ )e ρ/2 n i=1 Γ 1 + ρ i 2 √ -1π Γ 1 - ρ i 2 √ -1π
where ρ 1 , . . . , ρ n are the Chern roots of T X and ρ = c 1 (T X) = ρ 1 + • • • + ρ n . Finally, we use

Γ (x)Γ (1 -x) = π/ sin(π x) to get e ρ/2 n i=1 Γ 1 + ρ i 2 √ -1π Γ 1 - ρ i 2 √ -1π = Td(T X). (4.2)
We conclude the proposition by the theorem of Hirzebruch-Riemann-Roch.

The following proposition shows that the integral structures are compatible with the diagram in Corollary 3.17.

Proposition 4.5. Let E be a convex vector bundle and Z ⊂ X be a submanifold cut out by a regular section of E. Let ι : Z → X, j : X → E ∨ denote the natural inclusions. Assume

ι * & & Z (e,E) % % K K K K K K K K K K j * j * / / K(X) Z E ∨
x x r r r r r r r r r r

Ker∇ (e,E) ι * c(z)e(E)∪ / / Ker( f • h) * ∇ E ∨ Ker(ι * ) * ∇ amb 9 × c(z)ι * 5 5 j j j j j j j j j j j j j j j K amb (Z ) Z amb O O (-1) rkE det(E)⊗ι * H H where c(z) = 1/(-2π √ -1z) rkE .
Proof. We first prove that the top square is commutative. Recall the following equation from part (3) of Theorem 3.14:

e(E)L (e,E) (τ, z) = L E ∨ (h • f(τ ), z) e π √ -1c 1 (E)/z e(E).
So it remains to prove that for any V ∈ K(X), we have

(-2π √ -1) rkE e π √ -1c 1 (E) e(E) Γ (T X) Γ (E) -1 Ch(V) = Γ (T X) Γ (E ∨ )Ch( j * j * V)
This follows from (4.2) applied to the vector bundle E and from

ch j * j * V = e(E ∨ )Td(E ∨ ) -1 ch(V), see (4.1).
The commutativity of the left square follows from the properties of the Γ -class and the following facts (see [START_REF] Iritani | Quantum cohomology and periods[END_REF]Proposition 2.4] for the second property):

0 / / T Z / / ι * T X / / ι * E / / 0 is exact; L Z (ι * τ, z)ι * γ = ι * (L (e,E) (τ, z)γ ) ∀γ ∈ H ev (X).
The identity j * j * = (-1) rkE det(E) ⊗ ι * ι * implies that the right square is commutative.

5 Quantum Serre Duality and Abstract Fourier-Laplace Transform

In this section, we study quantum Serre duality with respect to the anticanonical line bundle K -1 X . We consider the (e, K -1 X )-twisted quantum D-module of X and the quantum D-module of the total space of K X . On the small quantum cohomology locus H 2 (X), we identify these quantum D-modules with Dubrovin's second structure connections with different parameters σ . We show that the duality between them is given by the second metric ǧ. Throughout the section, we assume that the anticanonical class -K X = c 1 (X) of X is nef and the Novikov variable is specialized to one (Section 3.2). We also set n:= dim C X and ρ := c 1 (X).

Convergence assumption

In this section, we assume certain analyticity of quantum cohomology of X. In Sections 5.2 and 5.3, we assume that the big quantum cohomology of X is convergent, that is, the quantum product (with Novikov variables specialized to one; see Section 3.2)

T α • τ T γ ∈ C[[e τ 2 , τ ]] = C[[e t 1 , . . . , e t r , t 0 , t r+1 , . . . , t s ]]
converges on a region U ⊂ H ev (X, C) of the form:

U = {τ ∈ H ev (X, C) : |e t i | < (1 ≤ i ≤ r), |t j | < (r + 1 ≤ j ≤ s)}.
For the main results in this section, we only need the convergence of the small quantum product. This means that the quantum product T α • τ T β restricted to τ = τ 2 to lie in

H 2 (X, C) converges on a region U sm ⊂ H 2 (X, C) of the form U sm = {τ 2 ∈ H 2 (X, C) : |e t i | < (1 ≤ i ≤ r)}. (5.1)
When X is Fano, that is, if -K X is ample, the convergence of small quantum cohomology is automatic because the structure constants are polynomials in e t 1 , . . . , e t r for degree reason.

Quantum connection with parameter σ

We introduce a variant of the quantum connection parametrized by a complex number σ . Consider the trivial vector bundle

F = H ev (X) × (U × C z )
∇ (σ ) α = ∂ α + 1 z (T α • τ ) ∇ (σ ) z∂ z = z∂ z - 1 z (E• τ ) + μ - 1 2 -σ
where μ is an endomorphism of H ev (X) defined by

μ(T α ) = |α| - n 2 T α with |α| = 1 2 deg T α , n= dim C X. Let (-) : U × C z → U × C z denote the map sending (τ, z) to (τ, -z).
We note the following facts.

g: (-) * (F, ∇ (σ ) ) × (F, ∇ (-1-σ ) ) → O U ×C z defined by g(T α , T β ) = X T α ∪ T β is flat.
Proposition 5.2 (see, e.g. [START_REF] Iritani | An integral structure in quantum cohomology and mirror symmetry for toric orbifolds[END_REF]Proposition 2.4]). Let L(τ, z) be the fundamental solution for the quantum connection of X from Proposition 2.5 (with c = 1, E = 0). We have that σ ) including in the z-direction.

L(τ, z)z -(μ-1 2 -σ ) z c 1 (T X) is a fundamental solution of ∇ (
Remark 5.3. The variable z in this paper corresponds to z -1 in Hertling's book [START_REF] Hertling | Frobenius Manifolds and Moduli Spaces for Singularities[END_REF]Section 9.3]. For convenience of the reader, we made a precise link of notation with the book of Hertling:

U = E• τ , D = 2 -n, V = -μ - n 2 . (5.2)
Using the divisor equation, the inverse of the fundamental solution L(τ, z) for X (see (2.2)) can be written in the form:

L(τ, z) -1 T α = e τ 2 /z ⎛ ⎜ ⎜ ⎝ T α + (d,l) =(0,0) β∈{0,...,s} T α , τ , . . . , τ , T β z -ψ 0,l+2,d e τ 2 (d) T β l! ⎞ ⎟ ⎟ ⎠ (5.3)
Denote by K (σ ) α the αth column of the inverse fundamental solution matrix for ∇ (σ ) :

K (σ ) α (τ, z) := z -c 1 (T X) z μ-1 2 -σ L(τ, z) -1 T α .
If we restrict τ to lie in H 2 (X), we have the following expression.

Lemma 5.4. For any α ∈ {0, . . . , s} and τ 2 ∈ H 2 (X), we have

K (σ ) α (τ 2 , z) = d∈Eff(X) N α,d (1) e τ 2 +τ 2 (d) z ρ+ρ(d)-|α|+ n+1 2 +σ
where ρ = c 1 (X) and

N α,d (z) := ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ s β=0 T α , T β z -ψ 0,2,d T β if d = 0; T α if d = 0.
(5.4)

Proof. Restricting to H 2 (X) means setting τ = 0 in (5.3). For d = 0 ∈ Eff(X), we have

N α,d (z) = s β=0 T α , T β z -ψ 0,2,d T β = k≥0 s β=0 T α , ψ k T β 0,2,d T β z k+1 .
By the degree axiom for Gromov-Witten invariants, only the term with

k = n -|β| - |α| + ρ(d) -1 contributes. Therefore, z μ N α,d (z) = N α,d (1)z -(ρ(d)+ n 2 -|α|) .
Using the fact that z μ • e τ 2 /z = e τ 2 • z μ , we deduce the formula of the lemma.

The second structure connection

We introduce the second structure connection [8, lecture 3; 9, Section 2.3; 15, Section 9.2; 27, II, Section 1]. Let x be the variable Laplace-dual to z -1 and let C x denote the complex plane with co-ordinate x. Consider the trivial vector bundle

F = H ev (X) × (U × C x ) over U × C x .
The second structure connection is a meromorphic flat connection on the bundle F defined by

∇(σ) α = ∂ α + μ - 1 2 -σ ((E• τ ) -x) -1 (T α • τ ) ∇(σ) ∂ x = ∂ x -μ - 1 2 -σ ((E• τ ) -x) -1 .
(5.5)

The connection has a singularity along the divisor Σ ⊂ U × C: The second structure connection has an invariant pairing called the second metric (or the intersection form).

Σ := {(τ, x) ∈ U × C x | det((E• τ ) -x) = 0}.
Proposition 5.5 [START_REF] Hertling | Frobenius Manifolds and Moduli Spaces for Singularities[END_REF]Theorem 9.4.c]). The O U ×C x -bilinear pairing

ǧ : ( F , ∇(σ) ) × ( F , ∇(-σ) ) → O U ×C x (Σ) defined by ǧ(T α , T β ) = X T α ∪ ((E• τ ) -x) -1
T β is flat. This is called the second metric.

We now explain how the second structure connection ∇(σ) arises from the Fourier-Laplace transformation of the quantum connection ∇ (σ -1) (see [7, 1.b; 34, V]).

Consider the module M = H ev (X) ⊗ O U [z] of sections of the trivial bundle F which are polynomials in z. The quantum connection ∇ (σ -1) equips M[z -1 ] with the structure of an O U ∂ α , z ± , ∂ z -module by the assignment:

∂ z → ∇ (σ -1) ∂ z , ∂ α → ∇ (σ -1) α .
Consider the isomorphism of the rings of differential operators:

O U ∂ α , z -1 , ∂ z -1 ∼ = O U ∂ α , x, ∂ x sending ∂ z -1 = -z 2 ∂ z to
O U [x]- submodule of M[z -1 ]. Note that M[z -1 ] is generated by M over O U x, ∂ x since z -1 = -∂ x .
Regard T α ∈ H ev (X) as an element of M. Under the abstract Fourier-Laplace transformation, we have

(∂ x x) • T α = ∇ (σ -1)
z∂ z T α = ∂ x • (E • τ T α ) + (μ + 1 2 -σ )T α ∂ β • T α = ∇ (σ -1) β T α = -∂ x • (T β • τ T α )
Regarding (E• τ ), μ, (T β • τ ) as matrices written in the basis {T α }, we obtain ) is not always an isomorphism. A sufficient condition for the map (5.6) to be an isomorphism is given by a result of Sabbah [START_REF] Sabbah | D éformations isomonodromiques et vari ét és de Frobenius[END_REF].

[∂ x T 0 , . . . , ∂ x T s ](x -E• τ ) = [T 0 , . . . , T s ](μ -1 2 -σ ) [∂ β T 0 , . . . , ∂ β T s ] = -[∂ x T 0 , . . . , ∂ x T s ](T β • τ ) Inverting (x -E• τ )
Proposition 5.6 ([34, Proposition V.2.10]). Suppose that σ / ∈ -n-1 2 + Z ≥0 with n= dim C X. Then the second structure connection ( F , ∇(σ) ) coincides with the abstract Fourier transform of the quantum connection (F, ∇ (σ -1) ), that is, the map (5.6) is an isomorphism.

Fundamental solution for the second structure connection

We will henceforth restrict ourselves to the small quantum cohomology locus H 2 (X).

We find an inverse fundamental solution for the second structure connection using a truncated Laplace transformation. Definition 5.7. Consider a cohomology-valued power series of the form:

K(z) = z -γ k a k z -k
with a k ∈ H ev (X) and γ ∈ H ev (X), where z -γ = e -γ log z . We assume that the exponent k ranges over a subset of C of the form {k 0 , k 0 + 1, k 0 + 2, . . .}. Let be a complex number such that

• -k 0 ∈ Z and, • 0 / ∈ {k 0 + 1, k 0 + 2, . . . , -1} if k 0 ≤ -2.
We define the truncated Laplace transform of K(z) to be

Lap ( ) (K)(x) := k a k x -γ -k-1 Γ (γ + k + 1) Γ (γ + )
where note that and the above condition for ensures that we do not have the division by γ when k ≤ -2 and that this expression is well defined.

Γ (γ + k + 1) Γ (γ + ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (γ + )(γ + + 1) • • • (γ + k) if k ≥ ; 1 i f k = -1; 1 (γ + k + 1)(γ + k + 2) • • • (γ + -1) if k ≤ -
The truncated Laplace transformation satisfies the following property:

Lap ( ) (z -1 K) = (-∂ x )Lap ( ) (K) Lap ( ) (-z 2 ∂ z K) = Lap ( ) (∂ z -1 K) = xLap ( ) (K) (5.7)
Remark 5.8. Suppose that K(z) is convergent for all z ∈ C × , (k 0 ) > -1 and that we have an estimate |K(z)| ≤ C e M/z over the interval z ∈ (0, 1) for some C , M > 0. Then we can write the truncated Laplace transform as the actual Laplace transform:

Lap ( ) (K)(x) = 1 Γ (γ + ) ∞ 0 K(z) e -x/z d(z -1 ).
Proposition 5.9. Let be a complex number such that

≡ n-1 2 + σ mod Z. Assume that we have either / ∈ Z >0 or σ / ∈ n-1 2 + Z ≤0 . Then (1)
The truncated Laplace transform

Ǩ(σ, ) α (τ 2 , x) := Lap ( ) (K (σ -1) α (τ 2 , •)) = d∈Eff(X) N α,d (1) e τ 2 +τ 2 (d) x ρ+ρ(d)-|α|+ n+1 2 +σ Γ (ρ + ρ(d) -|α| + n+1 2 + σ ) Γ (ρ + ) .
with τ 2 ∈ H 2 (X) is well-defined. Here ρ = c 1 (X), |α| = 1 2 deg T α and N α,d ( 1) is given in (5.4).

(2) Under the convergence assumption for the small quantum cohomology of X (see Section 5.1), Ǩ(σ, ) α (τ 2 , x) converges on a region of the form {(τ 2 , x) : τ 2 ∈ U sm , |x| > c} where U sm is a region of the form (5.1) and c ∈ R >0 .

(3) These Laplace transforms define a cohomology-valued solution to the second structure connection ∇(σ) , that is, the multi-valued bundle map

Ǩ(σ, ) : ( F , ∇(σ) ) -→ ( F , d), T α -→ Ǩ(σ, ) α defined over {(τ, x) ∈ U sm × C : |x| > c} intertwines ∇(σ)
|N α,d (1)| ≤ C 1 C |d|+ρ(d) 2 1 ρ(d)! (5.8)
for some constants C 1 , C 2 > 0 independent of α and d, where

| • | is a fixed norm on H ev (X)
and H 2 (X). The convergence of the series Ǩ(σ, ) α follows from this.

Next we show that Ǩ(σ, ) gives a solution to the second structure connection.

Since K (σ -1) α , α = 0, . . . , s are the columns of an inverse fundamental solution for ∇ (σ -1) , they satisfy the same differential relations as T α :

[z∂ z K (σ -1) 0 , . . . , z∂ z K (σ -1) s ] = [K (σ -1) 0 , . . . , K (σ -1) s ] - 1 z (E• τ ) + μ + 1 2 -σ [∂ β K (σ -1) 0 , . . . , ∂ β K (σ -1) s ] = [K (σ -1) 0 , . . . , K (σ -1) s ] 1 z (T β • τ )
where we regard (E• τ ), μ, (T β • τ ) as matrices written in the basis [T 0 , T 1 , . . . , T s ]. Applying the truncated Laplace transformation Lap ( ) to the above formulae and using (5.7), we find the following equations:

[∂ x x Ǩ(σ, ) 0 , . . . , ∂ x x Ǩ(σ, ) s ] = [∂ x Ǩ(σ, ) 0 , . . . , ∂ x Ǩ(σ, ) s ](E• τ ) + [ Ǩ(σ, ) 0 , . . . , Ǩ(σ, ) s ](μ + 1 2 -σ ), [∂ β Ǩ(σ-1) 0 , . . . , ∂ β Ǩ(σ-1) s ] = -[∂ x Ǩ(σ-1) 0 , . . . , ∂ x Ǩ(σ-1) s ](T β • τ ).
The first equation can be rewritten as

[∂ x Ǩ(σ, ) 0 , . . . , ∂ x Ǩ(σ, ) s ] = [ Ǩ(σ, ) 0 , . . . , Ǩ(σ, ) s ](μ -1 2 -σ )(x -E• τ ) -1 .
Together with the second equation, this implies that Ǩ(σ, ) α , α = 0, . . . , s define a solution to the second structure connection ∇(σ) .

Remark 5.10. Note that the convergence region U sm in the above proposition depends on c. The real positive number c can be chosen arbitrarily, but U sm becomes smaller if we choose a smaller c.

Small twisted quantum D-modules

In this section, we study the (e, K -1 X )-twisted quantum D-module QDM (e,K -1 X ) (X) and the quantum D-module QDM(K X ) of the total space of K X over the small quantum cohomology locus H 2 (X) using quantum Lefschetz theorem [START_REF] Coates | Quantum Riemann-Roch, Lefschetz and Serre[END_REF].

Since c 1 (X) is assumed to be nef, the anticanonical line bundle K -1 X is convex. Therefore, by the results of Sections 3.3 and 3.5, the quantum D-modules QDM (e,K -1 X ) (X) and QDM(K X ) are well-defined. We shall see that, under the convergence assumption for the small quantum cohomology in Section 5.1, the quantum connections for these quantum D-modules are convergent on a region U sm ⊂ H 2 (X) of the form (5.1). Therefore, we have the following small quantum D-modules:

SQDM (e,K -1 X ) (X) := (H ev (X) ⊗ O U sm ×C z , ∇ eu , S eu ) SQDM(K X ) := (H ev (X) ⊗ O U sm ×C z , ∇ loc ) (5.9)
where

∇ eu = ∇ (e,K -1 X ) is the (e, K -1 X )-twisted quantum connection, S eu = S (e,K -1 X ) is the (e, K -1 X )-twisted pairing S eu (u, v) = X u(-z) ∪ v(z)
∪ ρ, and ∇ loc = ∇ K X is the quantum connection of K X . The superscript "eu" means "Euler" and "loc" means "local". We denote the fundamental solutions (in Proposition 2.5) for these quantum D-modules by

L eu (τ, z) = L (e,K -1 X ) (τ, z) (see Remark 3.4) L loc (τ, z) = L K X (τ, z) (see Remark 3.13)
where the Novikov variable is set to be one. For a smooth anticanonical hypersurface Z ⊂ X, we can similarly consider the small ambient part quantum D-module of Z (cf. Definition 3.5):

SQDM amb (Z ) := (H ev amb (Z ) ⊗ O U sm ×C z , ∇ Z , S Z ).
Definition 5.11. For α ∈ {0, . . . , s}, we put

I eu α (τ 2 , z) := e τ 2 /z d∈Eff(X) N α,d (z) e τ 2 (d) ρ(d) k=1 (ρ + kz) I loc α (τ 2 , z) := e τ 2 /z d∈Eff(X) N α,d (z) e τ 2 (d) ρ(d)-1 k=0 (-ρ -kz)
where recall that ρ = c 1 (X) and we set ρ(d)

k=1 (ρ + kz) = ρ(d)-1 k=0 (-ρ -kz) = 1 for d = 0. We call I eu α the (e, K -1 X )-twisted I -function of X and I loc α the I -function of K X .
Remark 5.12. (1) The large radius limit is the limit: With our choice of co-ordinates, the large radius limit corresponds to e t i → 0 for 1 ≤ i ≤ r.

(τ 2 (d)) → 0 for ∀d ∈ Eff(X) \ {0}.
The I -function satisfies the asymptotics

I eu α (τ 2 , z)
∼ lrl e τ 2 /z T α and I loc α (τ 2 , z) ∼ lrl e τ 2 /z T α under the large radius limit (the subscript "lrl" stands for the large radius limit). Therefore, they are linearly independent in a neighbourhood of the large radius limit.

(

) Put ∂ ρ := s β=0 ρ β ∂ β , where ρ = c 1 (T X) = s β=0 ρ β T β . We have e - √ -1πρ/z z∂ ρ I loc α (h(τ 2 ), z) = ρ I eu α (τ 2 , z) (5.10) 2
where h is the map in (3.5) with c

1 (E) = c 1 (K -1 X ) = ρ.
Lemma 5.13. Suppose that the small quantum product of X is convergent as in Section 5.1. There exists a region U sm ⊂ H 2 (X) of the form (5.1) such that the I -functions

I eu α (τ 2 , z), I loc α (τ 2 , z) are convergent and analytic on U sm × C × z .
Proof. This follows easily from the estimate (5.8) and [START_REF] Borisov | Applications of homological mirror symmetry to hypergeometric systems: duality conjectures[END_REF].

N α,d (z) = z -(ρ(d)-|α|) z -deg 2 N α,d
For α = 0, the I -functions I eu 0 and I loc 0 have the following z -1 -expansions:

I eu 0 (τ 2 , z) = F (τ 2 )1 + G(τ 2 )z -1 + O(z -2 ), I loc 0 (τ 2 , z) = 1 + H (τ 2 )z -1 + O(z -2 ),
where F (τ 2 ) is a scalar-valued function and G(τ 2 ) and H (τ 2 ) are H 2 (X)-valued functions.

We define the mirror maps by

m eu (τ 2 ) := G(τ 2 ) F (τ 2 ) , m loc (τ 2 ) := H (τ 2 ). ( 5.11)
Note that F (τ 2 ) is invertible in a neighbourhood of the large radius limit point and the mirror maps take values in H 2 (X). The mirror maps have the asymptotic m(τ 2 ) ∼ lrl τ 2 and thus induce isomorphisms between neighbourhoods of the large radius limit point.

Quantum Lefschetz Theorem of Coates and Givental [START_REF] Coates | Quantum Riemann-Roch, Lefschetz and Serre[END_REF] gives the following proposition:

Proposition 5.14. For any α ∈ {0, . . . , s}, there exist

v α (τ 2 , z), w α (τ 2 , z) ∈ H ev (X) ⊗ C[z][[e τ 2 ]]
such that Proof. We will just prove the equality for the (e, K -1 X )-twisted theory. The same argument applies to the other case. Coates and Givental [4, Theorem 2, see also p. 27 and 34] introduced the following "big" I -function:

I eu α (τ 2 , z) = L eu (m eu (τ 2 ), z) -1 v α (τ 2 , z), I loc α (τ 2 , z) = L loc (m loc (τ 2 ), z) -1 w α (τ 2 , z).
I(τ, z) := z1 + τ + s β=0 (d, ) =(0,0),(0,1) Q d ! T β z -ψ , τ, . . . , τ 0, +1,d T β ρ(d) k=1 (ρ + λ + kz)
and showed that I(τ, -z) lies in the Lagrangian cone L (e λ ,K -1 X ) of the (e λ , K -1 X )-twisted theory. This is related to our I -functions as

I eu α (τ 2 , z) = ∂ α I(τ, z)| τ =τ 2 ,Q=1,λ=0 .
(5.12)

Note that ∂ α I(τ 2 , -z) is a tangent vector to the cone L (e λ ,K -1 X ) at I(τ 2 , -z). Moreover, ∂ 0 I(τ 2 , -z) has the following expansion:

∂ 0 I(τ 2 , -z) = F(τ 2 ) -z -1 G(τ 2 ) + O(z -2 ) with F(τ 2 ) ∈ C[[Q, τ 2 ]] and G(τ 2 ) ∈ H ev (X) ⊗ C[λ][[Q, τ 2 ]
]. Therefore, ∂ 0 I(τ 2 , -z)/F(τ 2 ) gives the unique intersection point:

(1 + H -) ∩ T I(τ 2 ,-z) L (e λ ,K -1 X ) .
Set τ = G(τ 2 )/F(τ 2 ). The discussion in Section 2.4.3 shows that the tangent space at

I(τ 2 , -z) is generated by ∂ α J (e λ ,K -1 X ) ( τ , -z) = L (e λ ,K -1 X ) ( τ , -z) -1 T α over C[z, λ][[Q, τ 2 ]] (see (2.4)). Therefore, there exists v α (τ 2 , z) ∈ H ev (X) ⊗ C[z, λ][[Q, τ 2 ]] such that ∂ α I(τ 2 , z) = L (e λ ,K -1 X ) ( τ , z) -1 v α (τ 2 , z).
It is easy to check that τ -τ 2 and v α in fact belong to

H ev (X) ⊗ C[z, λ][[Q e τ 2 ]
]. Under the non-equivariant limit λ → 0 and the specialization Q = 1, τ becomes m eu (τ 2 ). Setting v α (τ 2 , z) = v α (τ 2 , z)| λ=0,Q=1 and using (5.12), we obtain the formula in the proposition. The asymptotics of v α (τ 2 , z) follows from the asymptotics of I eu α (τ 2 , z) in Remark 5.12. The homogeneity of v α (τ 2 , z) follows from the homogeneity of I eu α (τ 2 , z) and L eu (τ 2 , z).

We define the O-module map ψ eu by the formula (5.14) and show that it intertwines the connections. We have

L eu (π eu (τ 2 , x), 1) -1 ((-π * eu ∇ eu ) n-|α| ∂ x (x -1 v α (τ 2 -ρ log x, 1))) = (-∂ x ) n-|α| L eu (m eu (τ 2 -ρ log x), 1) -1 (x -1 v α (τ 2 -ρ log x, 1)) = (-∂ x ) n-|α| x -1 I eu α (τ 2 -ρ log x,
∼ lrl T α x ρ+n-|α|+1 (ρ + 1)(ρ + 2) • • • (ρ + n -|α|) ∈ H ≥2|α| (X, C)
(5.17)

From Proposition 5.9, we deduce that the expression in Formula (5.16) is exactly Ǩ( n+1 2 ,1) α (τ 2 , x). This implies that the morphism ψ eu is a morphism of vector bundles with connection. The asymptotics (5.17) at the large radius limit shows that it is an isomorphism.

Since {w α (τ 2 )} form a basis in a neighbourhood of the large radius limit, we can define an O-module map ψ -1 loc such that the formula (5.15) holds. Note that ∇(σ) ∂ x has no singularities on U sm × {|x| > c} if we take the neighbourhood U sm sufficiently small. We From Proposition 5.9, we deduce that the expression in Formula (5.18) is exactly

(-∂ x ) |α| Ǩ(-n+1 2 ,0) α (τ 2 , x).
This implies that the morphism ψ -1 loc is a morphism of vector bundle with connection. The asymptotics (5.19) at the large radius limit shows that it is an isomorphism. Remark 5.17. By construction in the proof, we have

ψ eu = L eu (π eu (τ 2 , x), 1) • Ǩ( n+1
Quantum Serre Theorem as a Duality Between Quantum D-Modules 45

ψ loc = L loc (π loc (τ 2 , x), 1) e π √ -1ρ • Ǩ(-n+1 2 ,0) .
with Ǩ(σ, ) in Proposition 5.9. In particular, we have the following formula for ψ loc (T α ): In this section, we see that the quantum Serre duality between QDM (e,K -1 X ) (X) and QDM(K X ) from Theorem 3.14 can be rephrased in terms of the second structure connections. As a corollary, we obtain a description of the quantum D-module of an anticanonical hypersurface Z ⊂ X in terms of the second structure connection. When X is Fano, this gives an entirely algebraic description of the quantum connection of Z .

ψ loc (T α ) = L loc (π loc (τ 2 , x), 1) e √ -1πρ d N α,d (1) e τ 2 +τ 2 (d) x ρ+ρ(d)-|α| ρ(d)-|α|-1 k=-∞ (ρ + k) -1 k=-∞ (ρ + k) . ( 5
We begin with the following lemma. Lemma 5.18. Let f be the map in Lemma 3.2 and h be the map in (3.5) in the case where

E = K -1
X . The map h • f relates the two mirror maps (5.11) as

(h • f)(m eu (τ 2 )) = m loc (h(τ 2 )).
In particular h • f| H 2 (X) is convergent and gives an isomorphism between neighbourhoods of the large radius limit point of the form (5.1).

We will postpone the proof of the lemma until the end of this section. Consider the quantum Serre pairing S QS from Theorem 3.14 in the case where E = K -1 X . By the above lemma, h • f preserves H 2 (X) and therefore S QS induces a flat pairing

S QS : SQDM (e,K -1 X ) (X)| z=-1 × (h • f) * SQDM(K X )| z=1 → O U sm .
Combined with the ∇ eu -flat shift (-1)

deg 2 : SQDM (e,K -1 X ) (X)| z=1 ∼ = SQDM (e,K -1 X ) (X)| z=-1
, we obtain a flat pairing

P : SQDM (e,K -1 X ) (X)| z=1 × (h • f) * SQDM(K X )| z=1 → O U sm defined by P (u, v) = X (-1) deg 2 u ∪ v.
(5.21)

The second structure connections satisfy a certain "difference equation" with respect to the parameter σ . It is easy to check that we have the following morphism of meromorphic Next, we prove that the morphism ρ corresponds to Δ, that is, ρ • ψ eu = ψ loc • Δ.

Recall from Remark 5.17 that ψ eu , ψ loc are given by [START_REF] Borisov | Applications of homological mirror symmetry to hypergeometric systems: duality conjectures[END_REF] ,

ψ eu = L eu (π eu (τ 2 , x), 1) • Ǩ( n+1 2 ,
ψ loc = L loc (π loc (τ 2 , x), 1) e π √ -1ρ • Ǩ(-n+1 2 ,0) .
Therefore, it suffices to prove the following formulae:

ρ • L eu (π eu (τ 2 , x), 1) = L loc (π loc (τ 2 , x), 1) e π √ -1ρ • ρ, ρ • Ǩ( n+1 2 ,1) = Ǩ(-n+1 2 ,0) • Δ.
The first equation follows from Theorem 3.14(3) in view of Lemma 5.18. To see the second equation, it suffices to prove

ρ • Ǩ( n+1 2 ,1) = Ǩ( n+1 2 ,0)
and Ǩ(σ+1,0) = -Ǩ(σ,0) • Δ σ .

The first formula is immediate from the definition. To see the second, we calculate:

Ǩ(σ,0) (Δ σ T α ) = Ǩ(σ,0) ( ∇(σ) ∂ x T α ) = ∂ x ( Ǩ(σ,0) T α ) = ∂ x Ǩ(σ,0) α = -Ǩ(σ+1,0) α .
The conclusion follows.

Combined with Corollary 3.17, the above theorem implies the following corollary.

Corollary 5.20. Let Z be a smooth anticanonical hypersurface of X that satisfies one of the conditions in Lemma 3.15. Then the small quantum D-module

(ι * • π eu ) * SQDM amb (Z )| z=1
of Z is isomorphic to the image ImΔ of the morphism:

Δ : ( F , ∇( n+1 2 ) )| U sm ×{|x|>c} → ( F , ∇(-n+1 2 ) )| U sm ×{|x|>c} .
where ι * • π eu is regarded as a map

ι * • π eu : U sm × {|x| > c} → ι * (U sm )/2π √ -1H 2 (Z , Z).
The isomorphism sends Δ(T α ) ∈ Im(Δ) to ι * ψ eu (T α ) ∈ (ι Proof of Lemma 5.18. We consider the following equivariant I -functions (cf.

Definition 5.11):

I eu,λ α (τ 2 , z) = e τ 2 /z d∈Eff(X) N α,d (z) e τ 2 (d) ρ(d) k=1 (ρ + λ + kz) I loc,λ α (τ 2 , z) = e τ 2 /z d∈Eff(X) N α,d (z) e τ 2 (d) ρ(d)-1 k=0 (-ρ -λ -kz)
and define the equivariant mirror maps m λ eu , m λ loc as in (5.11). By exactly the same argument as Proposition 5.14, we have that

I eu,λ α (τ 2 , z) = L eu,λ (m λ eu (τ 2 ), z) -1 v λ α (τ 2 , z), I loc,λ α (τ 2 , z) = L loc,λ (m λ loc (τ 2 ), z) -1 w λ α (τ 2 , z) (5.26) for some v λ α (τ 2 , z), w λ α (τ 2 , z) ∈ H ev (X) ⊗ C[λ, z][[e τ 2 ]
]. Here we set L eu,λ = L (e λ ,K -1 X ) and L loc,λ = L (e -1 -λ ,K X ) . Similarly to (5.10), we have the following relationship:

e -π √ -1ρ/z (z∂ ρ + λ)I loc,λ α (h(τ 2 ), z) = (ρ + λ)I eu,λ α (τ 2 , z). ( 5.27)
We compute both sides of this equation. By (5.26), the left-hand side equals

e -π √ -1ρ/z (z∂ ρ + λ)L loc,λ (m λ loc (h(τ 2 )), z) -1 w λ α (h(τ 2 ), z) = e -π √ -1ρ/z L loc,λ (m λ loc (h(τ 2 )), z) -1 wλ α (τ 2 , z) (5.28)
where wλ

α (τ 2 , z) is obtained from w α (h(τ 2 ), z) by applying z((m λ loc • h) * ∇ (e -1 -λ ,K X ) ) ρ + λ and is an element of H ev (X) ⊗ C[λ, z][[e τ 2 ]
]. On the other hand, by (5.26) again, the right-hand side of (5.27) is

(ρ + λ)L eu,λ (m λ eu (τ 2 ), z) -1 v λ α (τ 2 , z) = L (e * λ ,K X ) ( f(m λ eu (τ 2 )), z) -1 (ρ + λ)v λ α (τ 2 , z) (by Theorem 2.11(3)) = e -π √ -1ρ/z L loc,λ (h( f(m λ eu (τ 2 ))), z) -1 (ρ + λ)v λ α (τ 2 , z) (by Remark 3.13).
(5.29)

Comparing (5.28) and (5.29), we obtain

L loc,λ (τ 2 , z) -1 wλ α (τ 2 , z) = L loc,λ (τ 2 , z) -1 (ρ + λ)v λ α (τ 2 , z) with τ 2 = m λ loc (h(τ 2 )) and τ 2 = h( f(m λ eu (τ 2 ))). Since (ρ + λ)v λ α (τ 2 , z
), α = 0, . . . , s form a basis of H ev (X) and wλ α (τ 2 , z) does not contain negative powers of z, we find that

L loc,λ (τ 2 , z)L loc,λ (τ 2 , z) -1
does not contain negative powers in z. By the asymptotics L loc,λ (τ, z) = id + O(z -1 ), we

must have L loc,λ (τ 2 , z) -1 = L loc,λ (τ 2 , z) -1 . The asymptotics L loc,λ (τ, z) -1 1 = 1 + τ/z + O(z -2 )
shows that τ 2 = τ 2 . The conclusion follows by taking the non-equivariant limit.

Remark 5.22. Consider the family of connection ∇( n+1 2 +k) for k ∈ Z. Via the morphisms Δ σ , we have:

• for k ∈ Z ≥0 , ∇( n+1 2 +k) is isomorphic to ∇( n+1 2 ) as meromorphic connections; • for k ∈ Z ≥0 , ∇(-n+1 2 -k) is isomorphic to ∇(- n+1
2 ) as meromorphic connections.

Theorem 5.16 above gives a geometric interpretation of these two connections.

It would be interesting to understand the intermediate connections

∇(k) for k ∈ {-n-1 2 , . . . , n-1 2 }.

Hodge filtration for the second structure connection

The small quantum D-modules SQDM (e,K -1 X ) (X) and SQDM(K X ) restricted to z = 1 have a natural filtration, called the A-model Hodge filtration [5, Section 8.5.4; 30, Lecture 7], and these small quantum D-modules are variations of Hodge structure. In this section, we identify the corresponding filtration on the second structure connection. See [START_REF] Konishi | Local B-model and mixed Hodge structure[END_REF][START_REF] Konishi | Mixed Frobenius structure and local A-model[END_REF][START_REF] Konishi | Local quantum cohomology and mixed Frobenius structure[END_REF][START_REF] Stienstra | Resonant Hypergeometric Systems and Mirror Symmetry[END_REF] for related studies on the Hodge structure for local quantum cohomology.

We follow the notation in Theorem 5.16 and write U sm and U sm for large radius limit neighbourhoods in H 2 (X) on which (respectively) the second structure connection ( F , ∇(σ) ) and our small quantum D-modules are convergent. Definition 5.23. We define the subbundle F p of the trivial bundle H ev (X) × U sm → U sm by F p := H ≤2n-2 p (X) × U sm and call it the A-model Hodge filtration. Because the small quantum product preserves the degree: the filtration satisfies Griffiths transversality with respect to the small quantum connections:

deg(T α • (e,K -1 X ) τ 2 T β ) = deg(T α ) + deg(T β ) deg(T α • K X τ 2 T β ) = deg(T α ) + deg(T α )
∇ eu α (F p ) ⊂ F p-1 and ∇ loc α (F p ) ⊂ F p-1
for α with |α| = 1. The Hodge filtration also satisfies the following orthogonality:

P (F p , F n-p+1 ) = 0
with respect to the pairing P in (5.21); in other words, the A-model Hodge filtrations on SQDM(K X )| z=1 and SQDM (e,K -1 X ) (X)| z=1 are annihilators of each other.

Next we introduce a filtration on the second structure connection.

Definition 5.24. Consider the second structure connection ( F , ∇(-n+1 2 ) ) restricted to the small parameter space U sm × C x . Define F p loc to be the

O U sm ×C x ( * Σ)-submodule of O( F )( * Σ) generated by {( ∇(-n+1 2 ) ∂ x ) k T α : |α| ≤ k ≤ n -p}.

Define

F p eu to be the ǧ-orthogonal of F n-p+1 loc , that is,

F p eu := {s ∈ O( F )( * Σ) : ǧ(s, γ ) = 0, ∀γ ∈ F n-p+1 loc }.
These are decreasing filtrations.

Lemma 5.25. The filtrations F p loc , F p eu satisfy the Griffiths transversality:

∇(-n+1 2 ) F p loc ⊂ Ω 1 U sm ×C x ⊗ F p-1 loc and ∇( n+1 2 ) F p eu ⊂ Ω 1 U sm ×C x ⊗ F p-1 eu .
Proof. It suffices to prove the Griffiths transversality for F p loc . We write ∇ for ∇(-n+1 2 ) to save notation. The inclusion ∇∂

x F p loc ⊂ F p-1 loc is obvious. We prove ∇β F p loc ⊂ F p-1 loc for β with |β| = 1. Take α and k ∈ Z ≥0 satisfying |α| ≤ k ≤ n -p. We have ∇β ( ∇∂ x ) k T α = ( ∇∂ x ) k ∇β T α = ( ∇∂ x ) k μ + n 2 ((E• τ ) -x) -1 T β • τ T α = -( ∇∂ x ) k+1 T β • τ T α . ( 5
ψ loc (( ∇∂ x ) k T α ) = (-1) |α| ((π * loc ∇) ∂ x ) k-|α| w α (τ 2 -ρ log x + π √ -1ρ, 1)
where

∇ = ∇(- n+1
2 ) . This belongs to F p by the Griffiths transversality for the A-model Hodge filtration. Considering the case k = |α|, we can see that these sections span F p . Remark 5.27. It follows from the above theorem that the filtrations F p loc , F p eu are subbundles over U sm × {|x| > c} with U sm sufficiently small. It would be interesting to study where they are not subbundles, and how we can extend them along the singularity Σ.

Let Z ⊂ X be a smooth anticanonical hypersurface. The small ambient part quantum D-module SQDM amb (Z ) also admits the A-model Hodge filtration

F p = H ≤2(n-1)-2 p amb (Z ) × U sm .
Combined with Corollaries 3.17 and 5.20, we obtain the following corollary.

Corollary 5.28. Suppose that an anticanonical hypersurface Z of X satisfies one of the conditions in Lemma 3.15. Under the isomorphism

(ι * • π eu ) * SQDM amb (Z ) ∼ = Im(Δ : ( F , ∇( n+1 2 ) ) → ( F , ∇(-n+1 2 ) ))
in Corollary 5.20, the A-model Hodge filtration F p on SQDM amb (Z ) corresponds to

Δ( F p+1 eu ), which is contained in F p loc .
6 Quintic in P 4

In this section, we make our result explicit in the case of X = P 4 and E = O(5 ). The small quantum connection of P 4 is given by

∇ (σ -1) ∂ t = ∂ t + z -1 (H • t ), ∇ (σ -1) z∂ z = z∂ z -z -1 5(H • t ) + (μ + 1 2 -σ )
where

H • t = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 0 e t 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , μ= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -2 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 6.

Fourier-Laplace transformation

We illustrate the Fourier-Laplace transformation in Section 5.3 for the small quantum connection of P 4 . We write ∂ t , z∂ z for the action of the small quantum connection ∇ (σ -1)

∂ t , ∇ (σ -1)
z∂ z

, respectively. We have

∂ t T 0 = z -1 T 1 , z∂ z T 0 = -5z -1 T 1 -(σ + 3 2 )T 0 ∂ t T 1 = z -1 T 2 , z∂ z T 1 = -5z -1 T 2 -(σ + 1 2 )T 1 ∂ t T 2 = z -1 T 3 , z∂ z T 2 = -5z -1 T 3 -(σ -1 2 )T 2 ∂ t T 3 = z -1 T 4 , z∂ z T 3 = -5z -1 T 4 -(σ -3 2 )T 3 ∂ t T 4 = e t z -1 T 0 , z∂ z T 4 = -5 e t z -1 T 0 -(σ -5 2 )T 4
Under the Fourier-Laplace transformation z∂ z = x∂ x + 1 and z -1 = -∂ x , we have The second structure connection has poles along the divisor Σ = {5 5 e tx 5 = 0}. We replace all the ∂ x -actions in the second column of (6.1) with the ∂ t -actions using the first column and deduce: We define, as in (5.13), π eu (t, x) = m eu (t -5 log x) = t -5 log x + g 1 (e t x -5 ) g 0 (e t x -5 ) , π loc (t, x) = m loc (t -5 log x + 5π √ -1) = t -5 log x + 5π √ -1 + g 2 (-e t x -5 ).

∂ t T 0 = (-∂ x )T 1 , x∂ x T 0 = 5∂ x T 1 -(σ + 5 2 )T 0 ∂ t T 1 = (-∂ x )T 2 , x∂ x T 1 = 5∂ x T 2 -(σ + 3 2 )T 1 ∂ t T 2 = (-∂ x )T 3 , x∂ x T 2 = 5∂ x T 3 -(σ + 1 2 )T 2 ∂ t T 3 = (-∂ x )T 4 , x∂ x T 3 = 5∂ x T 4 -(σ -1 2 )T 3 ∂ t T 4 = e t (
e -t
These maps converge when |e t x -5 | < 5 -5 . Theorem 5.16 and Lemma 6.1 together give the following isomorphisms: is a solution to the differential equation D loc ϕ = 0. These functions are the images of T 0 , respectively, under the maps Ǩ( 5 2 ,1) and e 5π √ -1H Ǩ(-5 2 ,0) in Proposition 5.9. In terms of the quantum D-modules, these solutions correspond, respectively, to L eu (π eu (t, x), 1) -1 and L loc (π loc (t, x), 1) -1 . Recall from Lemma 5.18 that the two mirror maps are related as follows:

m loc (t + 5π √ -1) = f(m eu (t)) + 5π √ -1 π loc (t, x) = f(π eu (t, x)) + 5π √ -1
where f is the map appearing in Lemma 3.2:

f(t) = t + ∞ d=1
e dt H 3 , 1 (e,K -1 X ) 0,2,d .

Consider the exponentiated mirror maps and exp( f):

M eu (e t ) := exp(m eu (t)), M loc (e t ) := exp(m loc (t)), F (e t ) := exp( f(t)).

These maps are related by M loc (-q) = -F (M eu (q)). Surprisingly, they have Taylor expansions in q = e t with integral coefficients [START_REF] Lian | Integrality of Certain Exponential Series[END_REF][START_REF] Zhou | Integrality Properties of Mirror Maps[END_REF]:

M eu (q) = q + 770q 2 + 1, 014, 275q 3 + 1, 703, 916, 750q 4 + 3, 286, 569, 025, 625q 5 + • • • M loc (q) = q -120q 2 + 63, 900q 3 -63, 148, 000q 4 + 85, 136, 103, 750q 5 + • • • F (q) = q -650q 2 + 50, 625q 3 -5, 377, 000q 4 -49, 529, 975, 000q

F 0 loc = F , F 0 eu = F F 1 loc = T 0 , ∂ x T 0 , ∂ 2 x T 0 , ∂ 3 x T 0 , F 1 eu = ( F 4 loc ) ⊥ F 2 loc = T 0 , ∂ x T 0 , ∂ 2 x T 0 , F 2 eu = ( F 3 loc ) ⊥ F 3 loc = T 0 , ∂ x T 0 , F 3 eu = ( F 2 loc ) ⊥ F 4 loc = T 0 F 4 eu = ( F 1 loc ) ⊥
where ⊥ means the orthogonal with respect to the second metric ǧ(γ x T0 = 30x -2 T 0 -1, 030x -3 T 1 + 15, 500x -4 T 2 -115, 500x -5 T 3 + 300, 000x -6 T 4 , ∂ 3

x T0 = -210x -3 T 0 + 6, 610x -4 T 1 -95, 300x -5 T 2 + 697, 500x -6 T 3 -1, 800, 000x -7 T 4 , ∂ 4

x T0 = 1, 680x -4 T 0 -48, 680x -5 T 1 + 679, 000x -6 T 2 -4, 905, 000x -7 T 3 + 12, 600, 000x -8 T 4 .

One can check that T0 corresponds to a multiple of the twisted I -function I eu 0 under the solution in Section 6.4: we have Ǩ( 52 ,1) ( T0 ) = 24x -5 I eu 0 (t -5 log x, 1).

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8 H.

 8 Iritani et al.

  where (-) * means the pull back by the change z → -z of sign and (• • • ) ∨ means the dual as C[z][[Q, s, τ ]]-modules. Therefore, part (1) of the theorem is equivalent to part (2).

2. 4

 4 Quantum Serre theorem of Coates-Givental

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Remark 2 . 14 .

 214 In [3, Appendix B], L (c,E) is defined as a formal scheme over C[[Q, s]]. For a complete Hausdorff topological C[[Q, s]]-algebra R, we have the notion of R-valued points on

Theorem 2 . 16 (

 216 Coates-Givental [4, Corollary 9]). The multiplication by c(E) defines a symplectomorphism c(E) : (H, Ω c(E) ) → (H, Ω c * (E ∨ ) ) and identifies the twisted Lagrangian cones:

  respects the quantum connection. Also it is obvious that the map c(E) intertwines the pairings S (c,E) and f * S (c * ,E ∨ ) . This shows part by guest on September 8, 2015 http://imrn.oxfordjournals.org/ Downloaded from Quantum Serre Theorem as a Duality Between Quantum D-Modules 17

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27 Lemma 3 . 15 .

 27315 by guest on September 8, 2015 http://imrn.oxfordjournals.org/ Downloaded from Quantum Serre Theorem as a Duality Between Quantum D-Modules The following conditions are equivalent:

Proof. ( 1 )

 1 ⇒ (2): it suffices to see that Kerι * ∩ Imι * = {0}. Suppose that α ∈ Kerι * ∩ Imι * . Then for every ι * β ∈ H ev amb (Z ) we have (ι * β, α) = (β, ι * α) = 0. By assumption we have α = 0. (2) ⇒ (3): we have ι * α = 0 if and only if ι * ι * α = e(E) ∪ α = 0. Therefore, Ker(ι * ) = Ker(e(E)∪) and part (3) follows. (3) ⇒ (1): since

Proposition 5 . 1 (

 51 [START_REF] Hertling | Frobenius Manifolds and Moduli Spaces for Singularities[END_REF] Theorem 9.8(c)]). The O U ×C z -bilinear pairing

  x and z -1 to -∂ x . Via this isomorphism, we may regard M[z -1 ] as an O U [x] ∂ α , ∂ x -module. This is called the abstract Fourier-Laplace transform. The subset M ⊂ M[z -1 ] is closed under the action of x = -z 2 ∂ z , and thus becomes an

  in the first equation, we obtain the connection matrices for the second structure connection ∇(σ) . In other words, writing O( F ) for the sheaf of holomorphic sections of F which are polynomials in x, the natural O U [x]-module map O( F ) → M, T α → T α (5.6) intertwines the meromorphic connection ∇(σ) on F with the action of ∂ x , ∂ α on M after inverting det(x -E• τ ) ∈ O U [x]. On the other hand, {T α } does not always give an O U [x]basis of M and the map (5.6

  with the trivial connection d. Proof. The well-definedness of the truncated Laplace transforms Lap ( ) (K (σ -1) α ) follows easily from Lemma 5.4 by checking the conditions in Definition 5.7. The coefficients N α,d (1) satisfy the following estimate [16, Lemma 4.1]:

  by guest on September 8, 2015 http://imrn.oxfordjournals.org/ Downloaded from Quantum Serre Theorem as a Duality Between Quantum D-Modules 41Moreover, we have the asymptotics v α ∼ lrl T α , w α ∼ lrl T α under the large radius limit, and v α , w α are homogeneous of degree 2|α| = deg T α with respect to the usual grading on H ev (X), deg z = 2 and deg e τ 2 = 0.

e τ 2

 2 +τ 2 (d) x ρ+ρ(d)+n-|α|+1 ρ(d)+n-|α| k=1 (ρ + k) (5.16)

  loc (π loc (τ 2 , x), 1) -1 w α (τ 2 -ρ log x + √ -1πρ, 1) = e -π √ -1ρ I loc α (τ 2 -ρ log x + √ -1πρ, 1) (by Proposition 5.14)= d N α,d (1) e τ 2 +τ 2 (d) x ρ+ρ(d) ρ(d)-1 k=0 (ρ + k) (5.18)∼ lrl T α e τ 2 x -ρ .(5.[START_REF] Katzarkov | Hodge theoretic aspects of mirror symmetry[END_REF]

.20) 5 . 7

 57 Quantum Serre duality in terms of the second structure connections

∞e, g 2 (e t ) = 5 ∞e

 25 d=0 dt (5d)! (d!) 5 , g 1 (e t ) d=1 dt (-1) d (5d -1)! (d!) 5 .

  π * eu SQDM (e,K -1 P 4 ) (P 4 )| z=1 ∼ = (O( F ), ∇( 5 2 ) ) ∼ = O ∂ t /O ∂ t D eu , π * loc SQDM(K P 4 )| z=1 ∼ = (O( F ), ∇(-5 2 ) ) ∼ = O ∂ t /O ∂ t D locover the region {(t, x) ∈ C 2 : |e t x -5 | < 5 -5 }.

6. 4 5 √

 45 SolutionsFor the Euler-twisted theory (σ =5 2 ), the cohomology-valued functionϕ(t, x) = (-∂ x ) 4 x -1 I eu 0 (t -5 log x, 1) = ∞ d=0 e t(H +d) x 5H +5d+5 5d+4 k=1 (5H + k) d k=1 (H + k) 5is a solution to the differential equation D eu ϕ = 0; for the local theory (σ = -5 2 ), the cohomology-valued function ϕ(t, x) = I loc 0 (t -5 log x +

F 4 eu

 4 = T0 , F 3 eu = T0 , ∂ x T0 , F 2 eu = T0 , ∂ x T0 , ∂ 2 x T0 , F 1 eu = T0 , ∂ x T0 , ∂ 2 x T0 , ∂ 3 x T0where∂ x = ∇ T0 := T 0 -125 3 x -1 T 1 + 2125 3 x -2 T 2 -5, 625x -3 T 3 + 15, 000x -4 T 4 , ∂ x T0 = -5x -1 T 0 + 565 3 x -2 T 1 -8975 3 x -3 T 2 + 22, 875x -4 T 3 -60, 000x -5 T 4 , ∂ 2

  By the divisor equation, we see that it lies in the subring C[[Q e τ 2 , s, τ ]]. Introduce the symmetric bilinear pairing

Quantum Serre theorem in terms of quantum D-modules We formally associate to c(•) another multiplicative class c

	Quantum Serre Theorem as a Duality Between Quantum D-Modules 11
	2.3
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	.5)
	See, for example, [18, Section 2.5]. In particular, L (c,E) belongs to End(H ev (X)) ⊗
	C[z -1 ][[Q e τ 2 , s, τ ]][τ 2 ].

* (•) by the formula:

  the kernel of the Poincar é pairing on H ev amb (Z ) is ι Recall that for a general non-compact space, the Poincar é duality pairs the cohomology with the cohomology with compact support. This analogy leads us to think of QDM (e,E) (X) as the quantum D-module with compact support of the total space E ∨ .

	Remark 3.19. It would be interesting to study if ι * always defines a morphism of quan-
	tum D-modules without assuming the conditions in Lemma 3.15.
	4 Quantum Serre Duality and Integral Structures

*

(Ker(e(E)∪)), which is zero.

Remark 3.16. The conditions in Lemma 3.15 hold if E is the direct sum of ample line bundles by the Hard Lefschetz theorem. They also hold if X is a toric variety and Z is a regular hypersurface with respect to a semiample line bundle E on X by a result of Mavlyutov

[START_REF] Mavlyutov | On the chiral ring of Calabi-Yau hypersurfaces in toric varieties[END_REF]

.

Corollary 3.17. Suppose that E is a convex vector bundle on X and Z ⊂ X be the zeroset of a regular section of E satisfying one of the conditions in Lemma 3.15. Then the morphism e(E)∪ : QDM (e,E) (X) → (h • f) * QDM(E ∨ ) in Theorem 3.14 factors through QDM amb (Z ) as: QDM (e,E) (X) ι * e(E)∪ Remark 3.18.

  Quantum Serre Theorem as a Duality Between Quantum D-Modules 33 over U × C z , where U is the convergence domain of the big quantum product in Section 5.1, and define a meromorphic flat connection ∇ (σ ) of F by the formula (cf.

	Definition 2.1 and Remark 2.4)
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  2,

	Quantum Serre Theorem as a Duality Between Quantum D-Modules 37
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  • π eu ) * SQDM amb (Z )| z=1 . Recall that the conditions in Lemma 3.15 are satisfied for an anticanonical hypersurface if X is Fano.

	Quantum Serre Theorem as a Duality Between Quantum D-Modules 49
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	Remark 5.21. by guest on September 8, 2015

*

  There exists a small neighbourhood U sm of the form (5.1) such that we haveψ loc ( F p loc ) = F p , ψ eu ( F p eu ) = F pover U sm × {|x| > c}, where ψ loc , ψ eu are the isomorphisms in Theorem 5.16 and F p is the A-model Hodge filtration of SQDM (e,K -1 X ) (X)| z=1 or of SQDM(K X )| z=1 .

	Proof. Since the A-model Hodge filtration satisfies the orthogonality, it suffices to show
	that ψ loc (	F p loc ) = F

.30)

Since ρ = c 1 (X) is nef, the small quantum product T β • τ T α is a linear combination of classes of degree less than or equal to 2|α| + 2. Therefore, the expression (5.30) lies in F p-1 .

Theorem 5.26. p . When |α| ≤ k ≤ np, we have

  Quantum Serre Theorem as a Duality Between Quantum D-Modules 53 hyperplane class and let t denote the co-ordinate on H 2 (P 4 ) dual to H . We use the basis{T 0 , T 1 , T 2 , T 3 , T 4 } = {1, H, H 2 , H 3 , H 4 } of H ev (P 4

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). This example was also studied by Dubrovin

[START_REF] Dubrovin | On Almost Duality for Frobenius Manifolds[END_REF] Section 5.4]

. Let H = c 1 (O(

1

)) ∈ H 2 (P 4 ) be the

  -∂ x )T 0 , x∂ x T 4 = 5 e t ∂ x T 0 -(σ -3 2 )T 4 (6.1)These formulas define the second structure connection ∇(σ) : t 5 3 e t x 5 2 e t x 2 5 e t x 3 e t x 4x 4 5 4 e t 5 3 e t x 5 2 e t x 2 5 e t x 2

	∇(σ) ∂ t = ∂ t +	1 5 5 e t -x 5 μ -	1 2	-σ	⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 5 2 x 2 5 4 e 5x 3	x 4 5x 3	5 4 e t x 4	5 3 e t x 5 2 e t x 2 5 4 e t 5 3 e t x	⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
					5 3 x 5 2 x 2	5x 3	x 4	5 4 e t
	∇(σ) ∂ x = ∂ x -	1 5 5 e t -x 5 μ -	1 2	-σ	⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 5 2 x 2 5x 3 x 4 5 4 e t 5 3 e t x 5 2 e t x 2 5 e t x 3 5x 3 x 4 5 4 e t 5 3 e t x 5 2 e t x 2 x 4 5 4 e t 5 3 e t x 5 3 x 5 2 x 2 5x 3 x 4 5 4 e t	⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
					5 4	5 3 x 5 2 x 2	5x 3	x 4

-twisted and local (small) quantum D-modules

  x∂ t T 4 = (5∂ t + σ + 5 2 )T 0 x∂ t T 0 = (5∂ t + σ + 3 2 )T 1 x∂ t T 1 = (5∂ t + σ + 1 2 )T 2 x∂ t T 2 = (5∂ t + σ -1 2 )T 3 x∂ t T 3 = (5∂ t + σ -3 2 )T 4From this we find the following differential equation for T 0 :((x∂ t ) 5e t (5∂ t + σ + 13 2 )(5∂ t + σ + 11 2 )(5∂ t + σ + 9 2 )(5∂ t + σ + 7 2 )(5∂ t + σ + 5 2 ))T 0 = 0. (6.2)A direct computation on computer (we used Maple) shows the following lemma. Let F denote the trivial H * (P 4 )-bundle over C 2 = H 2 (P 4 ) × C x . Suppose that Then the second structure connection (O( F )( * Σ), ∇(σ) ) is generated by T 0 = 1 as an O( * Σ) ∂ t -module and is defined by the relation (6.2).Recall from Theorem 5.16 that the second structure connection corresponds to the (e, K -1 P 4 )-twisted theory for σ = 5 2 and to the local theory for σ = -5 2 . For these cases, Quantum Serre Theorem as a Duality Between Quantum D-Modules 55 the differential Equation (6.2) specializes, respectively, to D eu := (x∂ t ) 5e t (5∂ t + 9)(5∂ t + 8)(5∂ t + 7)(5∂ t + 6)(5∂ t + 5) (for σ = 5 2 ), D loc := (x∂ t ) 5e t (5∂ t + 4)(5∂ t + 3)(5∂ t + 2)(5∂ t + 1)(5∂ t ) (for σ = -5 2 ).

	The I -functions in Definition 5.11 are given by
	I eu 0 (t, z) =	∞ d=0	e (d+H/z)t	5d k=1 (5H + kz) d k=1 (H + kz) 5	,
	I loc 0 (t, z) =	∞ d=0	e (d+H/z)t	5d-1 k=0 (-5H -kz) d k=1 (H + kz) 5	Downloaded from
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	Lemma 6.1. σ / ∈ {-3 2 , -1 2 , 1 2 , 3 2 }. 6.2 Eulerby guest on September 8, 2015

.

The mirror maps

(5.11)

are given by

m eu (t) = t + g 1 (e t ) g 0 (e t ) , m loc (t) = t + g 2 (e t )

where we set g 0 (e t ) =

  ByTheorem 5.19, this corresponds to the map Δ between the second structure connections. Since Δ maps T 0 in ( F , ∇(52 ) ) to (-∂ x )5 T 0 = e -t ∂ 5 t T 0 in ( F , ∇(-5 2 ) ), the above morphism corresponds to the map:δ : O ∂ t /O ∂ t D eu → O ∂ t /O ∂ t D loc , [ f(t, x, ∂ t )] → [ f(t, x, ∂ t ) e -t ∂ 5 t ].This is well-defined since D eu e -t ∂ 5 t = ∂ 5 t e -t D loc . By Corollary 5.20, we have π * eu SQDM amb (Z ) ∼ = Im(δ). for a quintic hypersurface Z ⊂ P 4 . We can therefore view SQDM amb (Z ) either as a quotient of the Euler-twisted quantum D-module or as a sub-D-module of the local quantum D-module. The former viewpoint yields a presentation: π * eu SQDM amb (Z ) ∼ = O ∂ t /O ∂ t (x(x∂ t ) 4 -5 e t (5∂ t + 9)(5∂ t + 8)(5∂ t + 7)(5∂

6.3 The small quantum D-module of a quintic

Recall from Theorem 3.14 and (5.23) that we have a natural morphism:

5H : π * eu SQDM (e,K -1 P 4 ) (P 4 ) → π * loc SQDM(K P 4 ) t + 6))

and the latter yields a (more familiar) presentation:

π * eu SQDM amb (Z ) ∼ = O ∂ t /O ∂ t (x(x∂ t )

4

-5 e t (5∂ t + 4)(5∂ t + 3)(5∂ t + 2)(5∂ t + 1)).

Table 1 .

 1 Quantum Serre Theorem as a Duality Between Quantum D-Modules 57 Gromov-Witten Invariants N d = H 3 , 1 (e,K -1

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  5

  Recall from Section 5.8 that we have Hodge filtrations on the second structure connections ( F , ∇ (52 ) ) and ( F , ∇ (-5 2 ) ) denoted, respectively, by F p eu and F p loc . They are given by

	58 H. Iritani et al.
	6.6 Hodge filtration
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+ • • • .

We can also deduce the Gromov-Witten invariants N d := H 3 , 1 (e,K -1 X ) 0,2,d as in Table

1

.

  1 , γ 2 ) = P 4 γ 1 ∪ (5H • t -x) -1 γ 2 and ∂ x means ∇

	(-5 2 ) ∂ x	in the first column. Using Maple, we find that

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Acknowledgements

H.I. thanks John Alexander Cruz Morales and Anton Mellit for helpful discussion. We thank referees for suggesting several improvements on the statement and the proof of Lemma 3.8.

Funding

This work is supported by the grants ANR-13-IS01-0001-01 and ANR-09-JCJC-0104-01 of the Agence nationale de la recherche and JSPS KAKENHI Grant Number 25400069, 23224002, and 26610008. by guest on September 8, 2015 http://imrn.oxfordjournals.org/ Downloaded from Quantum Serre Theorem as a Duality Between Quantum D-Modules 59

Lemma 5.15. Suppose that the small quantum cohomology of X is convergent as in Section 5.1. The flat connections ∇ eu , ∇ loc for the small quantum D-modules SQDM (e,K -1 X ) (X), SQDM(K X ) are convergent over a region U sm ⊂ H 2 (X) of the form (5.1). Also the functions v α , w α in Proposition 5.14 are convergent over the same region.

Proof. We only discuss the convergence of the (e, K -1 X )-twisted theory. The other case is similar. From Lemma 5.13, it follows that the mirror map m eu (τ 2 ) is convergent on a region of the form (5.1). Recall that L eu (τ 2 , z) is homogeneous of degree zero and that L eu (τ 2 , z) = id + O(z -1 ). Recall also that v α (τ 2 , z) is homogeneous of degree 2|α| from Proposition 5.14. Therefore, L eu is lower-triangular and the matrix [v 0 , . . . , v s ] is upper-triangular with respect to the grading on H ev (X). Therefore, the matrix

in Proposition 5.14 can be viewed as the LU decomposition. Therefore, we can solve for L -1 eu and v α from [I eu 0 , . . . , I eu s ] by simple linear algebra, and Lemma 5.13 implies that both L eu (m eu (τ 2 ), z) and v α are convergent. The conclusion follows.

The above lemma justifies the definition (5.9) at the beginning of Section 5.5.

Second structure connections are twisted quantum connections

We show that the small quantum D-modules SQDM (e,K -1 X ) (X) and SQDM(K X ) correspond to the second structure connections ∇( n+12 ) and ∇(-n+1 2 ) , respectively. By the divisor equation, the quantum connections ∇ eu , ∇ loc are invariant under

Therefore, the small quantum D-modules descend to the quotient space

Since the I -functions I eu 0 , I loc 0 satisfy the equation

; therefore, the mirror maps descend to isomorphisms

(5.13)

Choosing smaller U sm if necessary, each of π eu and π loc defines a map

We need to choose sufficiently small large radius limit neighbourhoods of the form (5.1) which may vary in each case: we denote by U sm , U sm for such neighbourhoods.

Theorem 5.16. Suppose that the small quantum cohomology of X is convergent as in Section 5.1. We have the following isomorphisms of vector bundles with connections:

where, as discussed above, we regard SQDM (e,K -1 X ) (X), SQDM(K X ) as flat connections on the quotient space U sm /2π √ -1H 2 (X, Z). These maps are given by the following formulae:

Proof. To show that ψ eu/loc intertwines the connections, we compare the solution Ǩ(σ, )

to the second structure connection from Proposition 5.9 with the inverse fundamental solution L -1 eu/loc of the small quantum D-modules. More precisely, we check the commutativity of a diagram of the form:

for suitable ; we shall take = 1 for σ = n+1 2 and = 0 for σ = -n+1 2 .

flat connections [START_REF] Hertling | Frobenius Manifolds and Moduli Spaces for Singularities[END_REF]Theorem 9.4.b]:

This is an isomorphism over

Thus the pairing (5.21) induces the flat pairing

We also have the morphism of flat connections

induced from the morphism in Theorem 3.14 (2).

Theorem 5.19. Via the isomorphisms ψ eu , ψ loc in Theorem 5.16, the pairing P (5.22) coincides with (-1) n+1 ǧ, that is, P (ψ eu T α , ψ loc T β ) = (-1) n+1 ǧ(T α , T β ) and the morphism ρ (5.23) coincides with the composition:

Moreover, we have

for γ 1 , γ 2 ∈ H ev (X).

Proof. First, we prove that P corresponds to (-1) n+1 ǧ. Since both pairings are flat, it is enough to compare the asymptotics of P (ψ eu T α , ψ loc T β ) and ǧ(T α , T β ) at the large radius limit. Since the quantum product equals the cup product at the large radius limit, we

(5.24)

We then compute the asymptotics of P (ψ eu (T α ), ψ loc (T β )). By Remark 5.17, after some computation, we find

(5.25)

We have already found a similar formula (5.20) for ψ loc (T α ). In view of Lemma 5.18, Theorem 3.14 (3) gives the identity: We deduce the equality of the pairings.