Developments in Topological Gravity

TL;DR

This paper introduces recent advances in two-dimensional topological gravity, highlighting Mirzakhani's proof of intersection formulas and recent generalizations to Riemann surfaces with boundary, emphasizing their significance in mathematical physics.

Contribution

It provides an accessible overview of Mirzakhani's work and recent developments in intersection theory on moduli spaces with boundaries, connecting mathematical proofs to physical models.

Findings

Mirzakhani's proof of intersection formulas for moduli space

Generalization of intersection theory to Riemann surfaces with boundary

Resolution of construction difficulties in extending formulas

Abstract

This note aims to provide an entr\'ee to two developments in two-dimensional topological gravity -- that is, intersection theory on the moduli space of Riemann surfaces -- that have not yet become well-known among physicists. A little over a decade ago, Mirzakhani discovered \cite{M1,M2} an elegant new proof of the formulas that result from the relationship between topological gravity and matrix models of two-dimensional gravity. Here we will give a very partial introduction to that work, which hopefully will also serve as a modest tribute to the memory of a brilliant mathematical pioneer. More recently, Pandharipande, Solomon, and Tessler \cite{PST} (with further developments in \cite{Tes,BT,STa}) generalized intersection theory on moduli space to the case of Riemann surfaces with boundary, leading to generalizations of the familiar KdV and Virasoro formulas. Though the existence of such a generalization appears natural from the matrix model viewpoint -- it corresponds to adding vector degrees of freedom to the matrix model -- constructing this generalization is not straightforward. We will give some idea of the unexpected way that the difficulties were resolved.