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.