SciPost Submission Page
Les Houches lecture notes on moduli spaces of Riemann surfaces
by Alessandro Giacchetto, Danilo Lewański
Submission summary
| Authors (as registered SciPost users): | Alessandro Giacchetto |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202504_00028v2 (pdf) |
| Date accepted: | Aug. 21, 2025 |
| Date submitted: | Aug. 8, 2025, 2:35 p.m. |
| Submitted by: | Alessandro Giacchetto |
| Submitted to: | SciPost Physics Lecture Notes |
| for consideration in Collection: |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
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.
Author comments upon resubmission
List of changes
- Correction of minor typos
- Added references [55-59] to recent applications of the topological recursion/CohFT correspondence
- Updated bibliography
Referee 1
- Line 156: X -> Sigma
- Line 176: noted
- Line 254: corrected to "figure 5"
- Equation (2.33): noted
- Line 382: called kappa-classes also Morita–Miller–Mumford; removed Arbarello–Cornalba, as we were not able to find a consistent reference to the boundary classes as such.
- Figure 7: the difference in the first subleading is between correlators with and without a tau_0. Modified the figure and the caption as: "The $2$-point correlators, normalised by their leading asymptotics: note the convergence to $1 + \bigO(g^{-1})$. Also observe the differing convergence behaviour of the correlators with or without a $\tau_0$ (in green and blue, respectively); this suggests that the subleading terms do depend on the partition $(d_1,\dots,d_n)$. This is indeed the case, and it can be proved via resurgence."
- Equation (3.1): noted
- Equation (3.24): added explanation: "Here, $q$ is a formal variable, known as the Novikov variable and defined as $q^\beta = e^{- \int_\beta \omega}$, used to grade contributions by the curve class $\beta \in H_2(X,\Z)$ according to its symplectic area."
- Section 4: agree, modified to "Further directions"
- Line 857: noted
Referee 2
- Line 80: add "without boundary"
- Line 125: add "After identifying the arcs $AB \sim AB'$ and the half-lines $BC \sim B'C'$"
- Line 129: noted
- Line 149: "smooth" is meant in the orbifold sense (which, in this context, is equivalent to smooth as a DM stack). But to avoid confusion, we incorporated the suggestions in the form of short remark below the theorem.
- Line 161: noted
- Line 178: noted
- Line 179: noted
- Line 196: noted
- Line 247: noted
- Line 290: modified to: "pairs $e = (h, h')$ of half-edges forming an edge $e$ of $\Gamma$."
- Line 305: noted
- Line 307: noted
- Line 351: noted
- Line 422: agree and noted
- Line 612: noted
- Line 658: agree and noted
- Line 799: agree. Added reference to 1104.0176. Added a new exercise (now 3.4) with the Weil–Petersson CohFT for pedagogical reasons. Added a reference to 1104.0176 there too. Exercise 4.1 adjusted accordingly.
Current status:
Editorial decision:
For Journal SciPost Physics Lecture Notes: Publish
(status: Editorial decision fixed and (if required) accepted by authors)