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 R³ CP¹. The input data for this theory are the choice of a meromorphic 1-form on CP¹ and a strict Lie 2-group with cyclic structure on its underlying Lie 2-algebra. Integrable field theories on R³ are constructed by imposing suitable boundary conditions on the connection (A, B) at the 3-dimensional defects located at the poles of 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.