Supersymmetric Field-Theoretic Models on a Supermanifold

TL;DR

This paper extends quantum field theory frameworks to supermanifolds, introducing new superdistribution constructions and a generalized spectral condition, bridging mathematical rigor with physical intuition in supersymmetric models.

Contribution

It develops a supermanifold framework compatible with quantum field theory, including new superdistribution constructions and a generalized spectral condition.

Findings

Supermanifolds with a smooth body structure are suitable for quantum field models.

A new construction of superdistributions and their wavefront sets is provided.

A generalized spectral condition for superdistributions is formulated.

Abstract

We propose the extension of some structural aspects that have successfully been applied in the development of the theory of quantum fields propagating on a general spacetime manifold so as to include superfield models on a supermanifold. We only deal with the limited class of supermanifolds which admit the existence of a smooth body manifold structure. Our considerations are based on the Catenacci-Reina-Teofillatto-Bryant approach to supermanifolds. In particular, we show that the class of supermanifolds constructed by Bonora-Pasti-Tonin satisfies the criterions which guarantee that a supermanifold admits a Hausdorff body manifold. This construction is the closest to the physicist's intuitive view of superspace as a manifold with some anticommuting coordinates, where the odd sector is topologically trivial. The paper also contains a new construction of superdistributions and useful results on the wavefront set of such objects. Moreover, a generalization of the spectral condition is formulated using the notion of the wavefront set of superdistributions, which is equivalent to the requirement that all of the component fields satisfy, on the body manifold, a microlocal spectral condition proposed by Brunetti-Fredenhagen-K\"ohler.