Les Houches lecture notes on topological recursion

TL;DR

These lecture notes provide an accessible introduction to topological recursion, a framework used across various fields like matrix models, string theory, and gauge theories, aiming to clarify its core concepts and applications.

Contribution

The notes offer a clear, beginner-friendly explanation of topological recursion, bridging the gap between complex applications and foundational understanding.

Findings

Clarifies the core principles of topological recursion

Connects topological recursion to multiple mathematical physics fields

Provides accessible educational material for newcomers

Abstract

You may have seen the words "topological recursion" mentioned in papers on matrix models, Hurwitz theory, Gromov-Witten theory, topological string theory, knot theory, topological field theory, JT gravity, cohomological field theory, free probability theory, gauge theories, to name a few. The goal of these lecture notes is certainly not to explain all these applications of the topological recursion framework. Rather, the intention is to provide a down-to-earth (and hopefully accessible) introduction to topological recursion itself, so that when you see these words mentioned, you can understand what it is all about. 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.