Wavefront sets and polarizations on supermanifolds

TL;DR

This paper develops a microlocal analysis framework for supermanifolds, introducing super wavefront sets that capture polarization information of singularities, with applications to supersymmetric field theories.

Contribution

It generalizes Dencker's polarization sets to supergeometry and establishes a refined pullback theorem for superdistributions.

Findings

Super wavefront sets detect polarization of singularities.

Refined pullback theorem for superdistributions.

Application to singularities in supersymmetric field solutions.

Abstract

In this paper we develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super wavefront set for superdistributions which generalizes Dencker's polarization sets for vector-valued distributions to supergeometry. In particular, our super wavefront sets detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application of our framework, we study the singularities of distributional solutions of a supersymmetric field theory.