Developments in Topological Gravity

TL;DR

The paper surveys two pivotal advances in 2D topological gravity: Mirzakhani’s program showing Weil–Petersson volumes encode intersection numbers and yield Virasoro/KdV structure, and the open-topological gravity framework with boundaries that introduces spin-structure data, boundary anomalies, and brane-based descriptions. It develops a comprehensive open-string theory T that cancels boundary orientation anomalies and connects to condensed-matter physics, while presenting two realizations (Majorana and twisted SUSY) and detailing boundary conditions, ζ-instanton equations, and disc amplitudes. It then translates these ideas into matrix-model language, showing how vector degrees of freedom and loop equations reproduce open/closed topological gravity via double-scaling limits and spectral-curve techniques, culminating in brane insertions tied to conformal primaries and Laplace-transformed open-closed partition functions. Together, the work bridges deep geometric structures on moduli spaces with exact matrix-model formulations, enriching both the mathematical understanding of moduli-space volumes and the physical modeling of open strings and branes in two-dimensional gravity.

Abstract

This note aims to provide an entrée 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.

Figures (11)

• Figure 1: (a) A marked point in a hyperbolic Riemann surface is treated as a cusp: it lies at infinity in the hyperbolic metric. (b) Instead of a cusp, a hyperbolic Riemann surface might have a geodesic boundary, with circumference any positive number $b$.
• Figure 2: (a) A three-holed sphere or "pair of pants." (b) A Riemann surface $\Sigma$, possibly with boundaries, that is built by gluing three-holed spheres along their boundaries. Each boundary of one of the three-holed spheres is either an external boundary -- a boundary of $\Sigma$ -- or an internal boundary, glued to a boundary of one of the three-holed spheres (generically a different one). The example shown has one external boundary and four internal ones.
• Figure 3: A "cut" of a Riemann surface with boundary along an embedded circle may be separating as in (a) or non-separating as in (b).
• Figure 4: Building a hyperbolic surface $\Sigma$ by gluing a hyperbolic pair of pants with geodesic boundary onto a simpler hyperbolic surface $\Sigma'$. $\Sigma$ and $\Sigma'$ both have geodesic boundary. (Shown here is the case that $\Sigma'$ is connected.)
• Figure 5: A disc with five boundary punctures. The spin bundle of the boundary circle $S$ is a real line bundle that is inevitably of NS type. This real line bundle is not trivial globally over $S$, but -- since the number of boundary punctures is odd -- it can be trivialized on the complement of the boundary punctures in such a way that the trivialization changes sign whenever one crosses a boundary puncture.