Semisimple Frobenius structures at higher genus

TL;DR

This paper presents a formula for higher genus potentials of semisimple Frobenius structures, linking them to genus 0 invariants and supporting a conjecture in Gromov-Witten theory with proven cases.

Contribution

It introduces a new formula for genus g>1 potentials in Frobenius manifolds, connecting higher genus Gromov-Witten invariants to genus 0 data.

Findings

Formula for genus g>1 potentials in Frobenius structures

Supports conjecture relating higher and genus 0 Gromov-Witten invariants

Proven case for equivariant Gromov-Witten invariants of tori

Abstract

We describe genus g>1 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In Gromov-Witten theory, it becomes a conjecture expressing higher genus GW-invariants in terms of genus 0 GW-invariants of symplectic manifolds with generically semisimple quantum cup-product. The conjecture is supported by the corresponding theorem about equivariant GW-invariants of tori actions with isolated fixed points. The parallel theory of gravitational descendents is also presented.