5d 2-Chern-Simons Theory and 3d Integrable Field Theories

Abstract 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 $$\mathbb {R}^3 \times \mathbb {C}P^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> . The input data for this theory are the choice of a meromorphic 1-form $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> on $$\mathbb {C}P^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> and a strict Lie 2-group with cyclic structure on its underlying Lie 2-algebra. Integrable field theories on $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> are constructed by imposing suitable boundary conditions on the connection ( A , B ) at the 3-dimensional defects located at the poles of $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> 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 $$(2+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional integrable chiral model from a suitable choice of data in the 5-dimensional theory.