Topological-Holomorphic Field Theories and Integrability

Recent developments have revealed that various two-dimensional integrable and conformal field theories (CFTs) can be understood as descending from a common higherdimensional origin: holomorphic Chern-Simons theory in six dimensions. Building on foundational ideas by Costello, the work of Bittleston and Skinner, described how two distinct approaches of deriving integrable models, namely from defects in fourdimensional Chern-Simons theory or via symmetry reductions of four-dimensional antiself- dual Yang-Mills (ASDYM) equations, are in fact unified within a six-dimensional framework. This thesis provides a complete description of this framework for a broad class of deformed sigma-models, extending beyond previously studied Dirichlet boundary conditions. By formulating holomorphic Chern-Simons theory on twistor space with a meromorphic three-form, we construct novel four-dimensional integrable field theories whose equations of motion can be identified with ASDYM. Subsequent symmetry reduction yields rich families of two-dimensional integrable models, including multi-parameter deformations of sigma-models. Additionally, we show that performing the reduction in reverse order—first obtaining four-dimensional Chern-Simons theory with generalised boundary conditions, then constructing defect theories—recovers the same integrable models. Importantly, we extend this correspondence to include models realised through gaugings, thereby providing a higher-dimensional origin for coset CFTs and homogeneous sine-Gordon models. This expanded framework not only unifies known constructions but also uncovers novel classes of integrable theories, offering new directions in the study of integrable systems