# Integrability, Quantization and Moduli Spaces of Curves

## TL;DR

This paper introduces a new approach to integrable systems and their quantization using intersection theory on moduli spaces, offering an alternative to traditional frameworks and including quantum integrable systems.

## Contribution

It presents a novel method connecting intersection theory of moduli spaces with integrable systems and their quantization, expanding beyond the Witten-Kontsevich approach.

## Key findings

- Develops a new intersection-theoretic framework for integrable systems.
- Encompasses quantum integrable systems within the new approach.
- Provides alternative tools to traditional Witten-Kontsevich methods.

## Abstract

This paper has the purpose of presenting in an organic way a new approach to integrable (1+1)-dimensional field systems and their systematic quantization emerging from intersection theory of the moduli space of stable algebraic curves and, in particular, cohomological field theories, Hodge classes and double ramification cycles. This methods are alternative to the traditional Witten-Kontsevich framework and its generalizations by Dubrovin and Zhang and, among other advantages, have the merit of encompassing quantum integrable systems. Most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Gu\'er\'e.