Wavefront sets and polarizations on supermanifolds

Claudio Dappiaggi, Heiko Gimperlein, Simone Murro, Alexander Schenkel
December 24, 2015
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.
open access link Journal of Mathematical Physics 58(2), 023504 (2017)
@article{Dappiaggi:2015jze, author = "Dappiaggi, Claudio and Gimperlein, Heiko and Murro, Simone and Schenkel, Alexander", title = "{Wavefront sets and polarizations on supermanifolds}", journal = "J. Math. Phys.", volume = "58", year = "2017", number = "2", pages = "023504", doi = "10.1063/1.4975213", eprint = "1512.07823", archivePrefix = "arXiv", primaryClass = "math-ph", reportNumber = "EMPG-15-27, SOAR-GMJT-01", SLACcitation = "%%CITATION = ARXIV:1512.07823;%%" }

Keywords: 
supergeometry, pseudodifferential operators, polarized wavefront sets, microlocal analysis, propagation of singularities