Higher Gauge Theory and Integrability

TL;DR

This paper develops a higher gauge theoretic framework for integrable systems using higher Chern-Simons theory, introducing new invariants and symmetries in higher dimensions that extend classical integrability concepts.

Contribution

It proposes a novel higher-dimensional approach to integrability via higher category theory and higher Chern-Simons theory, establishing new invariants and symmetry structures.

Findings

Construction of higher holonomies as homotopy invariants.

Identification of conserved quantities labeled by categorical characters.

Demonstration of infinite-dimensional symmetry algebra analogous to affine Lie 2-algebra.

Abstract

In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of $4$d Chern-Simons theory with defects, which provided a systematic framework for constructing two-dimensional integrable systems. In this article, we propose a novel approach to studying higher-dimensional integrable models employing techniques from higher category theory. Starting with higher Chern-Simons theory on the $4$-manifold $\mathbb{R}\times Y$, we complexify and compactify the real line to $\mathbb{C}P^1$ and introduce the disorder defect $\omega=z^{-1}\mathrm{d} z $. This procedure defines a holomorphic five-dimensional variant of higher Chern-Simons theory, which, when endowed with suitable boundary conditions, allows for the localisation to a three-dimensional theory on $Y$. The equations of motion of the resulting model are equivalent to the flatness of a $2$-connection $(L,H)$, that we then use to construct the corresponding higher holonomies. We prove that these are invariants of homotopies relative boundary, which enables the construction of conserved quantities. The latter are labelled by both the categorical characters of a Lie crossed-module and the infinite number of homotopy classes of surfaces relative boundary in $Y$. Moreover, we also demonstrate that the $3$d theory has left and right acting symmetries whose current algebra is given by an infinite dimensional centrally extended affine Lie 2-algebra. Both of these conditions are direct higher homotopy analogues of the properties satisfied by the 2d Wess-Zumino-Witten CFT, which we therefore interpret as facets of integrable structures.