Higher genus twistor spaces and the celestial torus

This paper studies novel four-dimensional integrable field theories that are deformations of self-dual Yang-Mills. They are engineered by considering holomorphic Chern-Simons and BF type theories on covers of twistor space obtained by pulling back the vector bundle $\mathcal{O}(1)^2\to\mathbb{CP}^1$ to hyperelliptic or elliptic curves. Compactifying to 4d yields an integrable theory, which in the examples I study, are determined to leading order. The form of the higher-order corrections are bootstrapped, and I argue that the index structure and coefficients of these terms are fixed by integrability. The celestial chiral algebras of these theories are shown to live on hyper-elliptic and elliptic curves, respectively. Symmetry reducing these integrable deformations to 2d yields an example of a hyperelliptic and elliptic integrable model governing a deformation of Hitchin's equations.