The Hilb/Sym correspondence for C2: descendents and Fourier-Mukai

TL;DR

This paper explores the deep connections between the crepant resolution correspondence in Gromov-Witten theory of Hilb(C2) and Sym(C2), the Fourier-Mukai equivalence of derived categories, and symplectic transformations, providing explicit computations and theoretical insights.

Contribution

It explicitly computes the symplectic transformation K and links the Fourier-Mukai equivalence to this transformation via Iritani's integral structure, advancing understanding of crepant resolutions.

Findings

Explicit computation of the symplectic transformation K.

Establishment of a relationship between Fourier-Mukai equivalence and symplectic transformation.

Alignment of results with Iritani's crepant resolution perspective.

Abstract

We study here the crepant resolution correspondence for the torus equivariant descendent Gromov-Witten theories of Hilb(C2) and Sym(C2).The descendent correspondence is obtained from our previous matching of the associated CohFTs by applying Givental's quantization formula to a specific symplectic transformation K. The first result of the paper is an explicit computation of K. Our main result then establishes a fundamental relationship between the Fourier-Mukai equivalence of the associated derived categories (by Bridgeland, King, and Reid) and the symplectic transformation K via Iritani's integral structure. The results use Haiman's Fourier-Mukai calculations and are exactly aligned with Iritani's point of view on crepant resolution.