Fourier analysis of equivariant quantum cohomology

TL;DR

This paper explores a Fourier duality in equivariant quantum cohomology linking the geometry of a space with its GIT quotient, introducing quantum volume and analyzing mirror symmetry through Fourier analysis.

Contribution

It introduces the concept of quantum volume and conjectural Fourier duality, connecting equivariant and non-equivariant quantum cohomology, and applies Fourier analysis to prove mirror symmetry results.

Findings

Proposes a conjectural Fourier duality between equivariant and non-equivariant quantum volumes.

Expresses the I-function of the GIT quotient as a Fourier transform of the equivariant J-function.

Demonstrates Fourier analysis applications in toric mirror symmetry and quantum cohomology decompositions.

Abstract

Equivariant quantum cohomology possesses the structure of a difference module by shift operators (Seidel representation) of equivariant parameters. Teleman's conjecture suggests that shift operators and equivariant parameters acting on QH_T(X) should be identified, respectively, with the Novikov variables and the quantum connection of the GIT quotient X//T. This can be interpreted as a form of Fourier duality between equivariant quantum cohomology (D-module) of X and quantum cohomology (D-module) of the GIT quotient X//T. We introduce the notion of "quantum volume," derived from Givental's path integral over the Floer fundamental cycle, and present a conjectural Fourier duality relationship between the T-equivariant quantum volume of X and the quantum volume of X//T. We also explore the "reduction conjecture," developed in collaboration with Fumihiko Sanda, which expresses the I-function of X//T as a discrete Fourier transform of the equivariant J-function of X. Furthermore, we demonstrate how to use Fourier analysis of equivariant quantum cohomology to observe toric mirror symmetry and prove a decomposition of quantum cohomology D-modules of projective bundles or blowups.