Les Houches lecture notes on moduli spaces of Riemann surfaces

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.

46 pages; comments, suggestions and corrections are welcome and encouraged! v2: typos corrected

References in corpus (20)

• Remodeling the B-model
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• The Virasoro Minimal String
• The equivariant Gromov-Witten theory of P^1
• Strebel differentials on stable curves and Kontsevich's proof of Witten's conjecture
• Hurwitz numbers, matrix models and enumerative geometry
• Invariants of algebraic curves and topological expansion
• The complex Liouville string: the matrix integral
• Relations on ${\overline{\mathcal{M}}}_{g,n}$ and the negative $r$-spin Witten conjecture
• Virasoro constraints for target curves
• Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture
• Off-shell Partition Functions in 3d Gravity
• Les Houches lectures on two-dimensional gravity and holography
• Resurgent large genus asymptotics of intersection numbers
• Deriving the Simplest Gauge-String Duality -- I: Open-Closed-Open Triality
• Les Houches lectures on non-perturbative topological strings
• Les Houches lecture notes on topological recursion
• A Proof of a Conjecture of Marino-Vafa on Hodge Integrals
• Theta classes: generalized topological recursion, integrability and $\mathcal{W}$-constraints
• $(2,2p+1)$ minimal string and intersection theory I

Cited by in corpus (1)

• Les Houches lectures on two-dimensional gravity and holography