Localization of virtual classes

TL;DR

This paper establishes a localization formula for virtual fundamental classes in torus-equivariant settings and applies it to express higher genus Gromov-Witten invariants of projective space as graph sums, extending known genus 0 results.

Contribution

It introduces a new localization formula for virtual classes and generalizes Gromov-Witten invariants computations to higher genus cases.

Findings

Derived a localization formula for virtual classes

Expressed higher genus Gromov-Witten invariants as graph sums

Computed excess integrals over higher genus multiple covers

Abstract

We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.