Quantum Serre theorem as a duality between quantum modules
Abstract.
We give an interpretation of quantum Serre theorem of Coates and Givental as a duality of twisted quantum modules. This interpretation admits a nonequivariant limit, and we obtain a precise relationship among (1) the quantum module of twisted by a convex vector bundle and the Euler class, (2) the quantum module of the total space of the dual bundle , and (3) the quantum module of a submanifold cut out by a regular section of .
When is the anticanonical line bundle , we identify these twisted quantum modules with second structure connections with different parameters, which arise as FourierLaplace transforms of the quantum module of . In this case, we show that the duality pairing is identified with Dubrovin’s second metric (intersection form).
Key words and phrases:
quantum cohomology, GromovWitten invariants, quantum differential equation, FourierLaplace transformation, quantum Serre, second structure connection, Frobenius manifold2010 Mathematics Subject Classification:
14N35, 53D45, 14F10Contents
1. Introduction
Genuszero GromovWitten invariants of a smooth projective variety can be encoded in different mathematical objects: a generating function that satisfies some system of PDE (WDVV equations), an associative and commutative product called quantum product, the Lagrangian cone of Givental [Givental:symplectic] 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 GromovWitten invariants as a meromorphic flat connection.
Encoding GromovWitten invariants in a meromorphic flat connection defines the notion of quantum module [Givental:ICM], denoted by , that is a tuple consisting of a trivial holomorphic vector bundle over with fiber , a meromorphic flat connection on given by the quantum connection:
and a flat nondegenerate pairing on given by the Poicaré pairing (see Definition LABEL:def:quantumDmod and Remark LABEL:rem:connection_z). These data may be viewed as a generalization of a variation of Hodge structure (see [KatzarkovPantevKontsevichncVHS]).
Quantum Serre theorem of Coates and Givental [GiventalCoates2007QRR, §10] describes a certain relationship between twisted GromovWitten invariants. The data of a twist is given by a pair of an invertible multiplicative characteristic class and a vector bundle over . Since twisted GromovWitten invariants satisfy properties similar to usual GromovWitten invariants, we can define twisted quantum product, twisted quantum module and twisted Lagrangian cone associated to the twist . Let denote the characteristic class satisfying for any vector bundle . Quantum Serre theorem (at genus zero) gives the equality of the twisted Lagrangian cones:
(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 modules:
Theorem 1.1 (see Theorem LABEL:thm:quantum,Serre,QDM for more precise statements).
There exists a (typically nonlinear) map (see (LABEL:eq:f,map)) such that the following holds:

The twisted quantum modules and are dual to each other; the duality pairing is given by the Poincaré pairing.

The map sending to is a morphism of quantum modules.
Genuszero twisted GromovWitten invariants were originally designed to compute GromovWitten invariants for CalabiYau hypersurfaces or noncompact local CalabiYau manifolds [Kontsevich:enumeration, GiventalEquivariantGW, LocalMirrorSymmetryChiangKlemmYauZaslow99]. Suppose that is a convex vector bundle and is the equivariant Euler class . In this case, nonequivariant limits of twisted GromovWitten invariants yield GromovWitten invariants of a regular section of and nonequivariant limits of twisted GromovWitten invariants yield GromovWitten invariants for the total space . The original statement (1.1) of quantum Serre theorem does not admit a nonequivariant limit since the nonequivariant Euler class is not invertible. We see however that our restatement above passes to the nonequivariant limit as follows:
Corollary 1.2 (Theorem LABEL:thm:euler,quantum,Serre,QDM, Corollary LABEL:cor:Z).
Let be a convex vector bundle and let denote the (nonequivariant) Euler class. Let be the map given by and let denote the nonequivariant limit of the map of Theorem 1.1 in the case where . We have the following:

The quantum modules and are dual to each other.

Let be the zerolocus of a regular section of and suppose that satisfies one of the conditions in Lemma LABEL:lem:cond_Z. Denote by the inclusion. Then the morphism factors through the ambient part quantum module of as:
(1.2)