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 R 3 × C P 1 . The input data for this theory are the choice of a meromorphic 1-form ω ω on CP¹ C P 1 and a strict Lie 2-group with cyclic structure on its underlying Lie 2-algebra. Integrable field theories on R³ R 3 are constructed by imposing suitable boundary conditions on the connection ( A , B ) at the 3-dimensional defects located at the poles of ω ω 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 $$(2+1)$$ ( 2 + 1 ) -dimensional integrable chiral model from a suitable choice of data in the 5-dimensional theory.