Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

The aim of the talk is to describe a generalization of Superalgebra and Supergeometry to Z_2^n -gradings, n>1. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the involved Z_2^n -degrees. This Z_2^n -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_2^n -gradings has never been investigated. More precisely, we discuss the geometry of Z_2^n -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_2^n -Berezinian and on first steps towards the corresponding integration theory, which is related to an algebraic variant of the multivariate residue theorem.