Towards integration on colored supermanifolds
The aim of the present text is to describe a generalization of superalgebra and supergeometry to $\mathbb Z_2^n$-gradings, $n \gt 1$. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the $\mathbb Z_2^n$-degrees involved. This $\mathbb Z_2^n$-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 $\mathbb Z_2^n$-gradings has never been investigated. More precisely, we discuss the geometry of $\mathbb Z_2^n$-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 $\mathbb Z_2^n$-Berezinian and on first steps towards the corresponding integration theory, which is related to an algebraic variant of the multivariate residue theorem.