2014/08/13 by Tiffany Covolo, Covolo, Tiffany, Janusz Grabowski +3
Mathematics · Physics and Astronomy · #13F25 #16L30 #17A70 #58A50 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #math-ph #math.AG #math.DG #math.MP #math.QA #msc:13F25 #msc:16L30 #msc:17A70 #msc:58A50
paper · pdf · doi:10.48550/arxiv.1408.2939
20 pages, more context in the introduction
arxiv created 2014/11/10 · arxiv updated 2014/11/11
Quite a number of ℤ2n-gradings, n≥ 2, appear in Physics and in Mathematics. The corresponding sign rules are given by the `scalar product' of the involved ℤ2n-degrees. The new theory exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise (the parity is the parity of the total degree). Formal series are the appropriate substitute for nilpotency; the category of ℤ2^\bullet-manifolds is closed with respect to the tangent and cotangent functors. The ℤ2n-supergeometric viewpoint provides deeper insight and simplified solutions; interesting relations with Quantum Field Theory and Quantum Mechanics are expected. In this article, we introduce split ℤ2n-manifolds as intrinsic superizations of ℤ2n∖\0\-graded vector bundles and prove that, conversely, any ℤ2n-manifold is noncanonically split. We thus provide a complete proof of the ℤ2n-extension of the so-called Batchelor-Gawedzki Theorem.