Genus expansion of open free energy in 2d topological gravity

TL;DR

This paper explores the genus expansion of open free energy in 2D topological gravity, providing explicit calculations and an efficient method for higher genus corrections based on Buryak's equation.

Contribution

It introduces a new method to compute higher genus corrections in open topological gravity using Buryak's equation, improving upon previous saddle point techniques.

Findings

Explicit genus zero free energy constructed.

Higher genus corrections computed efficiently up to high order.

Higher genus terms are polynomial in genus zero variables.

Abstract

We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy, the generating function for the open intersection numbers, obeys the open KdV equations and Buryak's differential equation and is related by a formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierarchy. Using these properties we study the genus expansion of the free energy in detail. We construct explicitly the genus zero part of the free energy. We then formulate a method of computing higher genus corrections by solving Buryak's equation and obtain them up to high order. This method is much more efficient than our previous approach based on the saddle point calculation. Along the way we show that the higher genus corrections are polynomials in variables that are expressed in terms of genus zero quantities only, generalizing the constitutive relation of closed topological gravity.