The homological algebra of 2d integrable field theories

TL;DR

This work develops a homological, BV-formalism approach to derive 2d integrable field theories from 4d semi-holomorphic Chern-Simons theory on $X=\Sigma\times C$. By constructing auxiliary, singular, and boundary $L_\infty$-algebras on $X$ and applying homotopy transfer, the authors produce two intrinsic 2d models on $\Sigma$: a sigma-model-type theory $ (\mathcal{F}(\Sigma),\ell') $ and a Lax-connection theory $ (\mathcal{L}(\Sigma),\ell') $. A canonical $L_\infty$-morphism between these models sends on-shell 2d fields to their corresponding Lax connections, thereby revealing the integrable structure of the reduced theory. The framework also carries a cyclic structure that induces action functionals on both the 4d and 2d theories. The analysis is extended to higher genus by incorporating a holomorphic adjoint bundle to kill cohomological obstructions, which yields refined pictures of Lax connections (meromorphic vs flat) and preserves the possibility of defining conserved charges via Wilson loops locally. Overall, the paper provides a rigorous, transferable algebraic mechanism to connect topological-holomorphic 4d theories with their 2d integrable reductions and lays groundwork for generalizations to higher dimensions.

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.

Theorems & Definitions (22)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4
• Proposition 2.5
• Definition 3.1
• Proposition 3.2
• proof
• Definition 3.3
• Proposition 3.4