2014/08/12 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.1408.2755
29 pages, added references, more context in the introduction and in section 3.2
arxiv created 2014/11/10 · arxiv updated 2014/11/11
In Physics and in Mathematics ℤ2n-gradings, n ≥ 2, do appear quite frequently. The corresponding sign rules are determined by the `scalar product' of the involved ℤ2n-degrees. The present paper is the first of a series on ℤ2n-Supergeometry. 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). It is based on the hierarchy: ` ℤ20-Supergeometry (classical differential Geometry) contains the germ of ℤ21-Supergeometry (standard Supergeometry), which in turn contains the sprout of ℤ22-Supergeometry, etc.' 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 define ℤ2n-supermanifolds and provide examples in the atlas, the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the category of ℤ2n-supermanifolds is closed with respect to the tangent and cotangent functors. The fundamental theorem describing supermorphisms in terms of coordinates is extended to the ℤ2n-context.