Higher Gauge Theory and Integrability

     Integrable field theories are remarkable for possessing an infinite number of conserved quantities, which often allow for their exact solvability. In two dimensions, this structure is elegantly captured by the existence of a Lax connection, whose path ordered exponential allows for the systematic construction of an infinite number of conserved quantities. In 2019, Costello, Witten and Yamazaki introduced a four-dimensional holomorphic extension of Chern-Simons theory that provides the first attempt at explaining the appearance of the Lax connection, whose origin had remained somewhat mysterious until then.

     In this talk, we present a generalization of these ideas to three-dimensional field theories, guided by the so-called “categorical ladder = dimensional ladder” principle. The central idea is that conserved quantities arise from surface-ordered exponentials of higher-rank tensors, defining a higher categorical notion of the Lax connection. We show that such a structure naturally emerges from a five-dimensional holomorphic extension of higher Chern-Simons theory. This work, carried out in collaboration with Hank Chen, provides a framework that enables the systematic construction of integrable field theories in three dimensions.