Towards integration on colored supermanifolds

Abstract

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 involved Z n 2 -degrees. This Z n 2 -Supergeometry exhibits interesting dierences with classical Supergeometry, provides a sharpened viewpoint, and has better categorical properties. Further, it is closely related to Cliord calculus: Cliord 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-Gawedski theorem. However, the main focus is on the Z n 2 -Berezinian and on rst steps towards the corresponding integration theory, which is related to an algebraic variant of the multivariate residue theorem.

Riemannian Structures on Z 2 n -Manifolds

“…The notions of the gradient of a function, the covariant divergence of a vector field and the connection Laplacian all generalize to the setting of Z n 2 -manifolds. We restrict attention to the covariant divergence and the connection Laplacian (with respect to the Levi-Civita connection) to avoid subtleties of introducing Berezin volumes on Z n 2 -manifolds (see [12]). The reader should note that unless the Riemannian metric is of degree zero, there is no canonical Berezin volume.…”

“…Recently, there has been a series of papers developing Z n 2 -geometry (Z n 2 := Z 2 × • • • × Z 2 ) and related algebraic questions, including [6][7][8][9][10][11][12]. Alongside this, there has been a renewed interest in the physical applications of Z n 2 -gradings, see for example [13][14][15][16][17][18][19][20][21].…”

The graded differential geometry of mixed symmetry tensors

“…In other words, one can work locally on Z n 2 -manifolds in more-or-less the same way as one works on classical manifolds and indeed, supermanifolds. The glaring exception here is the theory of integration on Z n 2 -manifolds which is expected to be quite involved (see Poncin [19] for work in this direction).…”

Linear Z₂ⁿ-Manifolds and Linear Actions

“…Loosely speaking, Z n 2 -manifolds (Z n 2 = Z ×n 2 ) are "manifolds" for which the structure sheaf has a Z n 2 -grading and the commutation rules for the local coordinates comes from the standard scalar product (see [11,13,14,15,18,19,20,21,37] for details). This is not just a trivial or straightforward generalization of the notion of a standard supermanifold, as one has to deal with formal coordinates that anticommute with other formal coordinates, but are themselves not nilpotent.…”