Simple proof of Weil triviality in supersymmetric gauge theories.

Abstract

Using methods of supermanifold cohomology we give a simple proof of Weil triviality in supersymmetric gauge theories, generalizing previous results of Bonora, Pasti, and Tonin which are relevant to the solution of the anomaly problem for such theories. By considering a supersymmetric gauge theory over an (m,n)-dimensional supermanifold with trivial topology in the odd directions, we prove without imposing constraints on the supercurvature and supertorsion forms the exactness of the forms P(${F}^{k}$), where F is the curvature form of a connection on the relevant principal super fiber bundle and P is an invariant polynomial on the (super) Lie algebra of the structure (super)group, of degree k>m/2. In order to prove the locality of the form X such that P(${F}^{k}$)=dX we have to use the constraints, but the necessary calculations turn out to be rather easy for any space-time dimension.