Topological gauge theories and group cohomology

We show that three dimensional Chern-Simons gauge theories with a compact gauge group G (not necessarily connected or simply connected) can be classified by the integer cohomology group H⁴(BG, Z). In a similar way, possible Wess-Zumino interactions of such a group G are classified by H³(G, Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map from H⁴(BG, Z) to H³(G, Z). We generalize this correspondence to topological "spin" theories, which are defined on three manifolds with spin structure, and are related to what might be called Z₂ graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.