Basic and Equivariant Cohomology in Balanced Topological Field Theory

We present a detailed algebraic study of the N=2 cohomological set--up describing the balanced topological field theory of Dijkgraaf and Moore. We emphasize the role of N=2 topological supersymmetry and $sl(2,R)$ internal symmetry by a systematic use of superfield techniques and of an $sl(2,R)$ covariant formalism. We provide a definition of N=2 basic and equivariant cohomology, generalizing Dijkgraaf's and Moore's, and of N=2 connection. For a general manifold with a group action, we show that: $i$) the N=2 basic cohomology is isomorphic to the tensor product of the ordinary N=1 basic cohomology and a universal $sl(2,R)$ group theoretic factor: $ii$) the affine spaces of N=2 and N=1 connections are isomorphic.