Homotopical Analysis of 4d Chern-Simons Theory and Integrable Field Theories

This paper provides a detailed study of 4-dimensional Chern-Simons theory on $$\mathbb {R}^2\times \mathbb {C}P^1$$ for an arbitrary meromorphic 1-form $$\omega $$ on $$\mathbb {C}P^1$$ . Using techniques from homotopy theory, the behaviour under finite gauge transformations of a suitably regularised version of the action proposed by Costello and Yamazaki is investigated. Its gauge invariance is related to boundary conditions on the surface defects located at the poles of $$\omega $$ that are determined by isotropic Lie subalgebras of a certain defect Lie algebra. The groupoid of fields satisfying such a boundary condition is proved to be equivalent to a groupoid that implements the boundary condition through a homotopy pullback, leading to the appearance of edge modes. The latter perspective is used to clarify how integrable field theories arise from 4-dimensional Chern-Simons theory.