Super-manifold

A generalization of the concept of a manifold, on which the functions take values in a commutative superalgebra. The structure of a super-manifold on a differentiable manifold $ M $ with structure sheaf $ {\mathcal O} _ {M} $ is defined by a sheaf $ {\mathcal O} _ {\mathcal M} $ of commutative superalgebras over the sheaf $ {\mathcal O} _ {M} $, whereby any point $ p \in M $ possesses a neighbourhood $ U $ such that the ringed space $ ( U, {\mathcal O} _ {\mathcal M} \mid _ {U} ) $ is isomorphic to $ ( U, ( {\mathcal O} _ {M} \mid _ {U} ) \otimes \Lambda ( \mathbf R ^ {m} )) $, where $ \Lambda ( \mathbf R ^ {m} ) $ is the exterior algebra with $ m $ odd generators. Analytic super-manifolds are defined in the same way. The differentiable (or analytic) super-manifolds form a category whose morphisms are the morphisms of ringed spaces that are even on the structure sheaves. The pair ( $ \mathop{\rm dim} M, m $) is called the dimension of the super-manifold. A super-manifold of the form $ ( U, {\mathcal O} _ {U} \otimes \Lambda ( \mathbf R ^ {m} )) $, where $ ( U, {\mathcal O} _ {U} ) $ is an open submanifold in $ \mathbf R ^ {n} $, is called a super-domain of dimension $ ( n, m) $. Every super-manifold is locally isomorphic to a super-domain.

If $ E $ is a vector bundle over $ M $, then the sheaf of sections $ L _ {\Lambda E } $ of the bundle $ \Lambda E $ defines the structure of a super-manifold on $ M $. Every differentiable super-manifold is isomorphic to a super-manifold of the form $ ( M, L _ {\Lambda E } ) $; in the complex analytic case this is not true. At the same time there are more morphisms in the category of super-manifolds than in the category of vector bundles.

A super-manifold $ {\mathcal M} $ can be defined by a functor $ \underline {\mathcal M} $ from the category of commutative superalgebras into the category of sets; this functor assigns to the superalgebra $ C $ the set $ {\mathcal M} ( C) = \mathop{\rm Mor} ( \mathop{\rm Spec} C, {\mathcal M} ) $, where $ \mathop{\rm Spec} C $ is the set of prime ideals in $ C $, endowed with the natural sheaf of superalgebras (see Representable functor).

The basic concepts of analysis on differentiable manifolds are also applied to super-manifolds.

The concept of a super-manifold also arose in theoretical physics; it enables one to join particles with Bose–Einstein statistics and Fermi–Dirac statistics into single multiplets, and also enables one to join the internal and dynamic symmetries of gauge theories in a single super-group.

References

[1]	F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian)
[2]	D.A. Leites (ed.) , Seminar on supermanifolds , Kluwer (1990)

Comments

As noted above, part of the motivation for the study of super-manifolds comes from theoretical physics, in particular supersymmetry and supergravity, [a4]. Not all authors agree that the definitions given above are the best for these purposes, cf. [a2], [a3]. "Desirability axioms" for a "well-behaved" category of super-manifolds are discussed in [a1]. Some definitions of super-manifolds satisfy these axioms, for instance the one above, and some others do not.

References

[a1]	M. Rothstein, "The axioms of supermanifolds and a new structure arising from them" Trans. Amer. Math. Soc. , 297 (1986) pp. 159–180
[a2]	H.J. Seiert (ed.) C.J.S. Clarke (ed.) A. Rosenblum (ed.) , Mathematical aspects of superspace , Reidel (1984)
[a3]	Cl. Bartocci, U. Bruzzo, D. Hernández-Ruipérez, "The geometry of supermanifolds" , Kluwer (1991)
[a4]	B. DeWitt, "Supermanifolds" , Cambridge Univ. Press (1984)
[a5]	D.A. Leites, "Introduction to the theory of supermanifolds" Russian Math. Surveys , 35 : 1 (1980) pp. 1–64 Uspekhi Mat. Nauk , 35 : 1 (1980) pp. 3–58
[a6]	F.A. Berezin, M.A. Shubin, "The Schrödinger equation" , Kluwer (1991) (Translated from Russian) (Supplement 3: D.A. Leites, Quantization and supermanifolds)