Les Houches lecture notes on moduli spaces of Riemann surfaces

TL;DR

These lecture notes introduce the moduli space of Riemann surfaces, exploring its boundary structure, cohomology, and connections to quantum gravity, string theory, and topological recursion, providing foundational insights for mathematical physics.

Contribution

The notes present a comprehensive overview of moduli spaces, including recent developments like Witten's conjecture and topological recursion, linking geometry with quantum theories.

Findings

Recursive boundary structure of moduli space analyzed

Witten's conjecture and its generalization discussed

Connections between hyperbolic geometry, JT gravity, and topological strings explored

Abstract

In these lecture notes, we provide an introduction to the moduli space of Riemann surfaces, a fundamental concept in the theories of 2D quantum gravity, topological string theory, and matrix models. We begin by reviewing some basic results concerning the recursive boundary structure of the moduli space and the associated cohomology theory. We then present Witten's celebrated conjecture and its generalisation, framing it as a recursive computation of cohomological field theory correlators via topological recursion. We conclude with a discussion of JT gravity in relation to hyperbolic geometry and topological strings. These lecture notes accompanied a series of lectures at the Les Houches school "Quantum Geometry (Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics)" in Summer 2024.