View source for Super-manifold

A generalization of the concept of a [[Manifold|manifold]], on which the functions take values in a commutative [[Superalgebra|superalgebra]]. The structure of a super-manifold on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911801.png" /> with structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911802.png" /> is defined by a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911803.png" /> of commutative superalgebras over the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911804.png" />, whereby any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911805.png" /> possesses a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911806.png" /> such that the ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911807.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911808.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s0911809.png" /> is the exterior algebra with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118010.png" /> 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 (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118011.png" />) is called the dimension of the super-manifold. A super-manifold of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118013.png" /> is an open submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118014.png" />, is called a super-domain of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118015.png" />. Every super-manifold is locally isomorphic to a super-domain.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118016.png" /> is a vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118017.png" />, then the sheaf of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118018.png" /> of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118019.png" /> defines the structure of a super-manifold on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118020.png" />. Every differentiable super-manifold is isomorphic to a super-manifold of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118021.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118022.png" /> can be defined by a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118023.png" /> from the category of commutative superalgebras into the category of sets; this functor assigns to the superalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118024.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118026.png" /> is the set of prime ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091180/s09118027.png" />, endowed with the natural sheaf of superalgebras (see [[Representable functor|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|Bose–Einstein statistics]] and [[Fermi–Dirac statistics|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.A. Berezin,   "Introduction to superanalysis" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.A. Leites (ed.) , ''Seminar on supermanifolds'' , Kluwer  (1990)</TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Rothstein,   "The axioms of supermanifolds and a new structure arising from them"  ''Trans. Amer. Math. Soc.'' , '''297'''  (1986)  pp. 159–180</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.J. Seiert (ed.)  C.J.S. Clarke (ed.)  A. Rosenblum (ed.) , ''Mathematical aspects of superspace'' , Reidel  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Cl. Bartocci,   U. Bruzzo,   D. Hernández-Ruipérez,   "The geometry of supermanifolds" , Kluwer  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. DeWitt,   "Supermanifolds" , Cambridge Univ. Press  (1984)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  F.A. Berezin,   M.A. Shubin,   "The Schrödinger equation" , Kluwer  (1991)  (Translated from Russian)  (Supplement 3: D.A. Leites, Quantization and supermanifolds)</TD></TR></table>