Open topological recursion relations in genus $1$ and integrable systems

TL;DR

This paper establishes explicit formulas and PDE systems for open topological recursion relations in genus 1, advancing the understanding of open Gromov-Witten invariants and their governing equations.

Contribution

It provides an explicit formula for solutions in genus 1 and proves that the open descendent potential's exponent satisfies a linear PDE system.

Findings

Derived an explicit formula analogous to Dijkgraaf-Witten for genus 1

Proved the PDE system governs the open descendent potential at genus 1

Established a connection between open Gromov-Witten invariants and integrable systems

Abstract

The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov-Witten invariants in genus 1. We find an explicit formula for any solution analogous to the Dijkgraaf-Witten formula for a descendent Gromov-Witten potential in genus 1. We then prove that at the approximation up to genus 1 the exponent of an open descendent potential satisfies a system of explicitly constructed linear evolutionary PDEs with one spatial variable.