5d 2-Chern-Simons theory and 3d integrable field theories

The $4$-dimensional semi-holomorphic Chern-Simons theory of Costello and Yamazaki provides a gauge-theoretic origin for the Lax connection of $2$-dimensional integrable field theories. The purpose of this paper is to extend this framework to the setting of $3$-dimensional integrable field theories by considering a $5$-dimensional semi-holomorphic higher Chern-Simons theory for a higher connection $(A,B)$ on $\mathbb{R}^3 \times \mathbb{C}P^1$. The input data for this theory are the choice of a meromorphic $1$-form $\omega$ on $\mathbb{C}P^1$ and a strict Lie $2$-group with cyclic structure on its underlying Lie $2$-algebra. Integrable field theories on $\mathbb{R}^3$ are constructed by imposing suitable boundary conditions on the connection $(A,B)$ at the $3$-dimensional defects located at the poles of $\omega$ and choosing certain admissible meromorphic solutions of the bulk equations of motion. The latter provides a natural notion of higher Lax connection for $3$-dimensional integrable field theories, including a $2$-form component $B$ which can be integrated over Cauchy surfaces to produce conserved charges. As a first application of this approach, we show how to construct a generalization of Ward's $(2+1)$-dimensional integrable chiral model from a suitable choice of data in the $5$-dimensional theory.