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arXiv · 2604.24864

On the structure of higher-dimensional integrable field theories

Marco Benini·Ryan A. Cullinan·Alexander Schenkel·Benoit Vicedo
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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.

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Marco Benini, Ryan A. Cullinan, Alexander Schenkel, Benoit Vicedo. 2026-04-27. On the structure of higher-dimensional integrable field theories. https://arxiv.org/abs/2604.24864

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