Fourier analysis of equivariant quantum cohomology
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.
Keywords
Cite
@article{arxiv.2501.18849,
title = {Fourier analysis of equivariant quantum cohomology},
author = {Hiroshi Iritani},
journal= {arXiv preprint arXiv:2501.18849},
year = {2025}
}
Comments
42 pages, 7 figures, a partly expository paper for the volume on "Advances in Complex Lagrangians, Integrable systems and Quantization", v2: minor revision, errors corrected, reference to the work of Cassia-Longhi-Zabzine added