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.