Cech, Dolbeault and de Rham cohomologies in Chern-Simons and BF theories

TL;DR

This paper explores the relationships between Cech, de Rham, and Dolbeault cohomologies in topological and holomorphic Chern-Simons and BF theories, revealing hidden symmetries and connections to integrable systems.

Contribution

It demonstrates how Cech cohomology offers a clear framework for understanding symmetries in topological field theories and links these theories to integrable systems.

Findings

Cech cohomology clarifies hidden symmetries in non-Abelian CS and BF theories.

Connections between holomorphic BF theories and multidimensional integrable systems are established.

Dressing symmetries of integrable systems are briefly discussed.

Abstract

Topological Chern-Simons (CS) and BF theories and their holomorphic analogues are discussed in terms of de Rham and Dolbeault cohomologies. We show that Cech cohomology provides another useful description of the above topological and holomorphic field theories. In particular, all hidden (nonlocal) symmetries of non-Abelian CS and BF theories can be most clearly seen in the Cech approach. We consider multidimensional Manin-Ward integrable systems and describe their connections with holomorphic BF theories. Dressing symmetries of these generic integrable systems are briefly discussed.