Congruences on K-theoretic Gromov--Witten invariants

TL;DR

This paper computes K-theoretic Gromov--Witten invariants for projective hypersurfaces, especially the quintic threefold, using virtual localization, providing extensive data up to genus 19 and degree 40, and extends methods to K-theoretic FJRW theory.

Contribution

It introduces a novel application of virtual localization to compute K-theoretic Gromov--Witten invariants and FJRW invariants modulo 41 for the quintic, covering high genus and degree.

Findings

Computed K-theoretic Gromov--Witten invariants of the quintic threefold up to genus 19 and degree 40.

Determined K-theoretic FJRW invariants modulo 41 for the quintic polynomial in all genera.

Provided explicit example in genus one and degree one for illustration.

Abstract

We study K-theoretic Gromov--Witten invariants of projective hypersurfaces using a virtual localization formula under finite group actions. In particular, it provides all K-theoretic Gromov--Witten invariants of the quintic threefold modulo 41, up to genus 19 and degree 40. As an illustration, we give an instance in genus one and degree one. Applying the same idea to a K-theoretic version of FJRW theory, we determine it modulo 41 for the quintic polynomial with minimal group and narrow insertions, in every genus.