A representation formula for maps on supermanifolds | Zendy

Frédéric Hélein | Zendy

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A representation formula for maps on supermanifolds

Author(s) -

Frédéric Hélein

Publication year - 2008

Publication title -

journal of mathematical physics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.708

H-Index - 119

eISSN - 1089-7658

pISSN - 0022-2488

DOI - 10.1063/1.2840464

Subject(s) - supermanifold , morphism , mathematics , superspace , uniqueness , superalgebra , pure mathematics , manifold (fluid mechanics) , representation (politics) , algebra over a field , simple (philosophy) , iterated function , space (punctuation) , mathematical analysis , supersymmetry , computer science , mechanical engineering , philosophy , epistemology , politics , political science , law , engineering , mathematical physics , operating system

In this paper we analyze the notion of morphisms of rings of superfunctionswhich is the basic concept underlying the definition of supermanifolds asringed spaces (i.e. following Berezin, Leites, Manin, etc.). We establish arepresentation formula for all morphisms from the algebra of functions on anordinary manifolds to the superalgebra of functions on an open subset ofR^{p|q}. We then derive two consequences of this result. The first one is thatwe can integrate the data associated with a morphism in order to get a (nonunique) map defined on an ordinary space (and uniqueness can achieved byrestriction to a scheme). The second one is a simple and intuitive recipe tocompute pull-back images of a function on a manifold by a map defined on asuperspace.Comment: 23 page

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