Basic and equivariant cohomology in balanced topological field theory (Q5931238)

In general, a cohomological topological field theory is characterized by a symmetry Lie algebra \({\mathfrak g}\), a graded algebra of fields \({\mathfrak f}\), and a set of graded derivations on \({\mathfrak f}\) generating a Lie algebra \({\mathfrak t}\). In turn, the topological algebra \({\mathfrak t}\) provides the algebraic and geometric framework for the definition of the topological observables within this field theory. NEWLINENEWLINENEWLINEA few years ago, R. Dijkgraaf and G. Moore showed that all known \(N=2\) topological models were examples of ``balanced topological field theories'', and they developed a cohomological framework suitable for their study [cf. \textit{R. Dijkgraaf} and \textit{G. Moore}, Commun. Math. Phys. 185, 411-440 (1997; Zbl 0888.58008)]. In the paper under review, the author provides a detailed algebraic study of the \(N=2\) cohomological set-up describing the balanced topological field theory of Dijkgraaf and Moore. More precisely, after a brief review of the basic facts of the theory of superalgebras and supermodules, the \(N=1\) and \(N=2\) topological algebras and Weil algebras are introduced, analyzed and compared. This is followed by the description of the corresponding \(N=1\) and \(N=2\) ``basic'' cohomology theories, the Weil superoperation and their (basic) cohomologies, and a detailed comparison of those \(N=1\) and \(N=2\) cohomologies. Then, after defining \(N=1\) and \(N=2\) abstract connections, equivariant cohomology and the related Weil homomorphism, the entire cohomological set-up developed so far is used to study the \(N=1\) and \(N=2\) (basic) cohomology of a smooth manifold equipped with a right group action. The main results consist in comparison theorems between \(N=1\) and \(N=3\) (basic) cohomologies.NEWLINENEWLINENEWLINEAltogether, by emphasizing the role of topological supersymmetry, throughout the paper, and by exhibiting the similarities and the differences of the \(N=1\) and \(N=2\) cases, the author has provided a very substantial contribution towards the better understanding and the further development of the fundamental ideas of Dijkgraaf and Moore in balanced topological quantum field theory.