Topological-Holomorphic Field
Theories and Integrability

Abstract

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