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- W753784171 abstract "In Physics and in Mathematics Z2 -gradings, n ≥ 2, do appear quite frequently. The corresponding sign rules are determined by the ‘scalar product’ of the involved Z2 -degrees. The present paper is the first of a series on Z2 -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: ‘Z2-Supergeometry (classical differential Geometry) contains the germ of Z2-Supergeometry (standard Supergeometry), which in turn contains the sprout of Z2-Supergeometry, etc.’ The Z2 -supergeometric viewpoint provides deeper insight and simplified solutions; interesting relations with Quantum Field Theory and Quantum Mechanics are expected. In this article, we define Z2 -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 Z2 -supermanifolds is closed with respect to the tangent and cotangent functors. The fundamental theorem describing supermorphisms in terms of coordinates is extended to the Z2 -context. MSC 2010: 17A70, 58A50, 13F25, 16L30" @default.
- W753784171 created "2016-06-24" @default.
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- W753784171 date "2014-01-01" @default.
- W753784171 modified "2023-09-27" @default.
- W753784171 title "Z_2^n-Supergeometry I: Manifolds and Morphisms" @default.
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