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. © 1990 Springer-Verlag.