Les Houches lecture notes on moduli spaces of Riemann surfaces

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.

💡 Research Summary

These lecture notes, based on the 2024 Les Houches summer school “Quantum Geometry”, give a comprehensive introduction to the moduli space of Riemann surfaces and its many connections to two‑dimensional quantum gravity, topological string theory, matrix models, and recent developments such as Jackiw–Teitelboim (JT) gravity.

The authors begin by recalling the basic definition of the moduli space (\mathcal{M}{g,n}) of smooth, compact, connected complex curves of genus (g) with (n) marked points. They emphasize the stability condition (2g-2+n>0), which guarantees that the automorphism group of each curve is finite. Under this condition (\mathcal{M}{g,n}) acquires the structure of a smooth complex orbifold of dimension (3g-3+n). Concrete examples, such as (\mathcal{M}{0,4}) (the cross‑ratio) and (\mathcal{M}{1,1}) (the upper half‑plane modulo (SL(2,\mathbb{Z}))), illustrate how the orbifold picture resolves the issue of infinite automorphism groups.

The notes then introduce the Deligne–Mumford compactification (\overline{\mathcal{M}}{g,n}). The boundary consists of stable nodal curves obtained by pinching cycles; each boundary stratum is indexed by a dual graph encoding the combinatorics of the degeneration. This recursive stratification is the geometric backbone for many later formulas. The authors compute the orbifold Euler characteristic, recalling the Harer–Zagier formula (\chi(\mathcal{M}{g,n})=(1-2g)^{n-1}\zeta(1-2g)), and remark that the original proof used matrix‑model techniques.

Section 3 develops the intersection theory on (\overline{\mathcal{M}}_{g,n}). The tautological classes (\psi_i), (\kappa_m), and (\lambda_j) are defined, and their basic relations are listed. The central object is the intersection number
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