A Fock Sheaf For Givental Quantization

TL;DR

This paper develops a coordinate-free, global quantization formalism for Gromov-Witten invariants using a Fock sheaf, unifying various existing formalisms and revealing modularity properties and applications to crepant transformations.

Contribution

It introduces a novel Fock sheaf framework for Gromov-Witten invariants, generalizing previous quantization methods and establishing modularity and transformation properties.

Findings

Defines a Fock sheaf for Gromov-Witten invariants.

Shows the canonical section's modularity in semisimple cases.

Proves a higher-genus Ruan's Crepant Transformation Conjecture.

Abstract

We give a global, intrinsic, and co-ordinate-free quantization formalism for Gromov-Witten invariants and their B-model counterparts, which simultaneously generalizes the quantization formalisms described by Witten, Givental, and Aganagic-Bouchard-Klemm. Descendant potentials live in a Fock sheaf, consisting of local functions on Givental's Lagrangian cone that satisfy the (3g-2)-jet condition of Eguchi-Xiong; they also satisfy a certain anomaly equation, which generalizes the Holomorphic Anomaly Equation of Bershadsky-Cecotti-Ooguri-Vafa. We interpret Givental's formula for the higher-genus potentials associated to a semisimple Frobenius manifold in this setting, showing that, in the semisimple case, there is a canonical global section of the Fock sheaf. This canonical section automatically has certain modularity properties. When X is a variety with semisimple quantum cohomology, a theorem of Teleman implies that the canonical section coincides with the geometric descendant potential defined by Gromov-Witten invariants of X. We use our formalism to prove a higher-genus version of Ruan's Crepant Transformation Conjecture for compact toric orbifolds. When combined with our earlier joint work with Jiang, this shows that the total descendant potential for compact toric orbifold X is a modular function for a certain group of autoequivalences of the derived category of X.