Quantum Cohomology at Higher Genus: Topological Recursion Relations and Virasoro Conditions

TL;DR

This paper develops higher genus topological recursion relations and Virasoro constraints for 2D topological field theories with gravity, enabling the enumeration of higher genus holomorphic curves in Fano varieties, exemplified by results for CP^2.

Contribution

It introduces new higher genus topological recursion relations and Virasoro conditions applicable to general 2D topological field theories coupled to gravity.

Findings

Reproduces known genus 2 results for CP^2

Enables counting of higher genus holomorphic curves in Fano varieties

Provides a framework for higher genus enumerative geometry

Abstract

We construct topological recursion relations (TRR's) at higher genera $g\ge2$ for general 2-dimensional topological field theories coupled to gravity. These TRR's when combined with Virasoro conditions enable one to determine the number of higher genus holomorphic curves in any Fano varieties. In the case of $CP^2$ we reproduce the known results at genus $g=2$.