Towards Integration on Colored Supermanifolds

By Norbert Poncin. 2016

55 citations recorded by the source corpus.

. The aim of the present text is to describe a generalization of superalgebra and supergeometry to Z n 2 -gradings, n > 1. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the Z n 2 -degrees involved. This Z n 2 -supergeometry exhibits interesting differences with classical supergeometry, provides a sharpened viewpoint, and has better categorical properties. Further, it is closely related to Clifford calculus: Clifford algebras have numerous applications in physics, but the use of Z n 2 -gradings has never been investigated. More precisely, we discuss the geometry of Z n 2 -supermanifolds, give examples of such colored supermanifolds beyond graded vector bundles, and study the generalized Batchelor– Gawędzki theorem. However, the main focus is on the Z n 2 -Berezinian and on first steps towards the corresponding integration theory, which is related to an algebraic variant of the multivariate residue theorem.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4d5dce0d4b9b7a86ec9048bc1b77a9607625d4c6