Topological recursion for open intersection numbers

TL;DR

This paper develops a topological recursion formula for open intersection numbers on moduli spaces of open Riemann surfaces, extending techniques used for closed surfaces and incorporating features from $eta$-deformed models.

Contribution

It introduces a novel variation of the Eynard-Orantin recursion tailored for open Riemann surfaces, enabling refined calculations of intersection numbers with different boundary components.

Findings

Recursion formula for open intersection numbers derived

Spectral curve matches that of closed surfaces, but recursion differs

Conjectural refinement distinguishes boundary components in moduli spaces

Abstract

We present a topological recursion formula for calculating the intersection numbers defined on the moduli space of open Riemann surfaces. The spectral curve is $x = \frac{1}{2}y^2$, the same as spectral curve used to calculate intersection numbers for closed Riemann surfaces, but the formula itself is a variation of the usual Eynard-Orantin recursion. It looks like the recursion formula used for spectral curves of degree 3, and also includes features present in $\beta$-deformed models. The recursion formula suggests a conjectural refinement to the generating function that allows for distinguishing intersection numbers on moduli spaces with different numbers of boundary components.