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Les Houches lecture notes on moduli spaces of Riemann surfaces
by Alessandro Giacchetto, Danilo Lewański
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Submission summary
| Authors (as registered SciPost users): | Alessandro Giacchetto |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202504_00028v1 (pdf) |
| Date submitted: | April 19, 2025, 2:42 p.m. |
| Submitted by: | Giacchetto, Alessandro |
| Submitted to: | SciPost Physics Lecture Notes |
| for consideration in Collection: | |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| 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.
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Strengths
The lecture note gives a great introduction to the theory of the moduli space of complex curves, focusing especially on CohFT and its interrelation to topological recursion. This topic is of ongoing interest in different research communities. The author gives, instead of heavy technical definitions, a good intuitive explanation of the different objects. A lot of examples give the reader the possibility to understand the different constructions and inspire further exploration.
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Requested changes
At the beginning of Section 2: the word "structure" is used several times (e.g., in lines 79, 83, 87) with different meanings. If referring to \emph{complex structure}, please specify it.
\begin{itemize}
\item l.~156: What is \( X \)?
\item l.~176: Leave a space after the full stop.
\item l.~254: What is Figure 14?
\item Eq.~(2.33): Swap \( M \) and \( N \).
\item l.~382: To my knowledge, the \(\kappa\)-classes are usually called \textbf{Morita--Miller--Mumford}, and the boundary classes \textbf{Arbarello--Cornalba}.
\item Fig.~7: Is it possible to provide the reader with the partition of the different families of dots?
\item Eq.~(3.1): \( 1_{g-1,n+1} \to 1_{g-1,n+2} \)
\item Eq.~(3.24): What is \( q \)?
\item Title of Section 4: The title seems to suggest a discussion of open problems, rather than further known results from the literature. The authors could (if they wish) consider renaming the section.
\item l.~857: \( B_n \to B_m \)
\end{itemize}
\paragraph{General Comment:} At the beginning, both 2D gravity and 2D quantum gravity are mentioned. Later, JT gravity also appears. The authors might want to clarify whether these theories are related or not.
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Publish (surpasses expectations and criteria for this Journal; among top 10%)