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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.
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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.
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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]]).
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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  $  M $
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with structure sheaf  $  {\mathcal O} _ {M} $
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is defined by a sheaf  $  {\mathcal O} _  {\mathcal M}  $
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of commutative superalgebras over the sheaf  $  {\mathcal O} _ {M} $,
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whereby any point  $  p \in M $
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possesses a neighbourhood  $  U $
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such that the ringed space  $  ( U, {\mathcal O} _  {\mathcal M}  \mid  _ {U} ) $
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is isomorphic to  $  ( U, ( {\mathcal O} _ {M} \mid  _ {U} ) \otimes \Lambda ( \mathbf R  ^ {m} )) $,
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where  $  \Lambda ( \mathbf R  ^ {m} ) $
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is the exterior algebra with  $  m $
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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 $)
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is called the dimension of the super-manifold. A super-manifold of the form  $  ( U, {\mathcal O} _ {U} \otimes \Lambda ( \mathbf R  ^ {m} )) $,
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where  $  ( U, {\mathcal O} _ {U} ) $
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is an open submanifold in  $  \mathbf R  ^ {n} $,
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is called a super-domain of dimension  $  ( n, m) $.
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Every super-manifold is locally isomorphic to a super-domain.
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If  $  E $
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is a vector bundle over  $  M $,
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then the sheaf of sections  $  L _ {\Lambda E }  $
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of the bundle  $  \Lambda E $
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defines the structure of a super-manifold on  $  M $.  
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Every differentiable super-manifold is isomorphic to a super-manifold of the form  $  ( M, L _ {\Lambda E }  ) $;
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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.
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A super-manifold  $  {\mathcal M} $
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can be defined by a functor $  \underline {\mathcal M}  $
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from the category of commutative superalgebras into the category of sets; this functor assigns to the superalgebra $  C $
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the set $  {\mathcal M} ( C) = \mathop{\rm Mor} (  \mathop{\rm Spec}  C, {\mathcal M} ) $,  
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where $  \mathop{\rm Spec}  C $
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is the set of prime ideals in $  C $,  
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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 basic concepts of analysis on differentiable manifolds are also applied to super-manifolds.
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====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 08:24, 6 June 2020


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)
How to Cite This Entry:
Super-manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-manifold&oldid=48909
This article was adapted from an original article by D.A. Leites (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article