The homological algebra of 2d integrable field theories

TL;DR

This paper explores the homological structure of 2d integrable field theories derived from 4d semi-holomorphic Chern-Simons theories, revealing how Lax connections naturally emerge via $L_$-algebras.

Contribution

It introduces a homological framework using $L_$-algebras to understand the emergence of integrable structures and Lax connections from 4d theories.

Findings

Homological description of 2d integrable field theories

Derivation of Lax connections from 4d theories

Application of homotopy transfer techniques

Abstract

This article provides a detailed and rigorous study of $4d$ semi-holomorphic Chern-Simons theories and their associated $2d$ integrable field theories from the homological perspective of $L_\infty$-algebras. Through the use of homotopy transfer techniques, it is shown precisely how both the integrable field theory and its corresponding Lax connection emerge from the $4d$ theory, which results in a novel perspective on Lax connections in terms of $L_\infty$-morphisms.