Basic cohomology and topological quantum field TH

A particularly interesting question is connected with the observables of these models: on the one hand we have a fermionic nilpotent B.R.S. symmetry and the fact that the classical action and the energy-momentum tensor are B.R.S. variations insures, as expected, that there are no local observables in the theory. On the other hand the presence of a nilpotent operator suggests that the observable sector should be linked with some cohomology space of the same operator, but a naive computations quickly shows that the unconstrained cohomology of the B.R.S. operator of the topological models is empty. Thus we are led to consider a constrained version of the cohomology space; the basic cohomology is one such possibility where the constraints are induced by an additional symmetry which nicely fits into a Cartan algebra structure with the original one. In this paper, the author analyzes first the basic cohomology in the case of the non linear sigma models; in particular discusses the characterization of the basic cohomology by means of a unique nilpotent operator which summarizes all the relevant symmetries of these models