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

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.