Quantum Serre theorem as a duality between quantum -modules
We give an interpretation of quantum Serre theorem of Coates and Givental as a duality of twisted quantum -modules. This interpretation admits a non-equivariant 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 Fourier-Laplace 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, Gromov-Witten invariants, quantum differential equation, Fourier-Laplace transformation, quantum Serre, second structure connection, Frobenius manifold
2010 Mathematics Subject Classification:14N35, 53D45, 14F10
Genus-zero Gromov-Witten 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 Gromov-Witten invariants as a meromorphic flat connection.
Encoding Gromov-Witten 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 non-degenerate pairing on given by the Poicaré pairing (see Definition LABEL:def:quantumD-mod and Remark LABEL:rem:connection_z). These data may be viewed as a generalization of a variation of Hodge structure (see [Katzarkov-Pantev-Kontsevich-ncVHS]).
Quantum Serre theorem of Coates and Givental [Givental-Coates-2007-QRR, §10] describes a certain relationship between twisted Gromov-Witten invariants. The data of a twist is given by a pair of an invertible multiplicative characteristic class and a vector bundle over . Since twisted Gromov-Witten invariants satisfy properties similar to usual Gromov-Witten 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:
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 non-linear) 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.
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 [Kontsevich:enumeration, Givental-Equivariant-GW, LocalMirrorSymmetry-Chiang-Klemm-Yau-Zaslow-99]. Suppose that is a convex vector bundle and is the equivariant Euler class . In this case, non-equivariant limits of -twisted Gromov-Witten invariants yield Gromov-Witten invariants of a regular section of and non-equivariant limits of -twisted Gromov-Witten invariants yield Gromov-Witten invariants for the total space . 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 LABEL:thm:euler,quantum,Serre,QDM, Corollary LABEL:cor:Z).
Let be a convex vector bundle and let denote the (non-equivariant) Euler class. Let be the map given by and let denote the non-equivariant 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 zero-locus 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: