On the structure of higher-dimensional integrable field theories

TL;DR

This paper introduces a universal framework for higher-dimensional integrable field theories using $L_ ablafty$-algebras, connecting topological-holomorphic Chern-Simons theories to higher Lax connections and conserved charges.

Contribution

It develops a general $L_ ablafty$-algebra-based approach for constructing integrable field theories in arbitrary dimensions, extending known 2D models.

Findings

Constructed models describe $(d+1)$-dimensional theories on manifold $M$.

Established a link between these theories and higher Lax connections.

Recovered the Costello-Yamazaki construction in 2D as a special case.

Abstract

We propose a general framework for integrable field theories in arbitrary spacetime dimension $d+1$ which is based on $d$-term $L_\infty$-algebras. Specifically, we introduce cyclic $L_\infty$-algebras describing topological-holomorphic higher Chern-Simons theories on $M \times \mathbb{C}P^1$ with suitable singularity structures and boundary conditions, controlled by a meromorphic $1$-form on $\mathbb{C}P^1$. Using homological perturbation theory and homotopy transfer, we construct weakly equivalent models describing $(d+1)$-dimensional field theories on $M$. Their integrability is witnessed by a natural map to an $L_\infty$-algebra describing higher Lax connections, yielding conserved charges associated with higher-dimensional cycles in $M$. The resulting theories admit natural action functionals and recover the Costello-Yamazaki construction in $2$ dimensions.