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'''Supergeometry''' is [[differential geometry]] of [[module (mathematics)|module]]s over [[supercommutative algebra|graded commutative algebra]]s, [[supermanifold]]s and [[graded manifold]]s. Supergeometry is part and parcel of many classical and quantum [[field theory (physics)|field theories]] involving odd [[field (physics)|field]]s, e.g., [[supersymmetry|SUSY]] field theory, [[BRST formalism|BRST theory]], or [[supergravity]].
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Supergeometry is formulated in terms of <math>\mathbb Z_2</math>-graded [[module (mathematics)|module]]s and [[sheaf (mathematics)|sheaves]] over <math>\mathbb Z_2</math>-graded commutative algebras ([[supercommutative algebra]]s). In particular, superconnections are defined as [[Koszul connection]]s on these modules and sheaves. However, supergeometry is not particular [[noncommutative geometry]] because of a different definition of a graded [[derivation (abstract algebra)|derivation]].
 
[[Graded manifold]]s and [[supermanifold]]s also are phrased in terms of sheaves of graded commutative algebras. [[Graded manifold]]s are characterized by sheaves on [[manifold|smooth manifolds]], while [[supermanifold]]s are constructed by gluing of sheaves of [[supervector space]]s. Note that there are different types of supermanifolds. These are smooth supermanifolds (<math>H^\infty</math>-, <math>G^\infty</math>-, <math>GH^\infty</math>-supermanifolds), <math>G</math>-supermanifolds, and DeWitt supermanifolds. In particular, supervector bundles and principal superbundles are considered in the category of <math>G</math>-supermanifolds. Note that definitions of principal superbundles and principal superconnections straightforwardly follow that of smooth [[principal bundle]]s and [[connection (principal bundle)|principal connection]]s. Principal graded bundles also are considered in the category of [[graded manifold]]s.
 
There is a different class of [[Daniel Quillen|Quillen]]–[[Yuval Ne'eman|Ne'eman]] superbundles and superconnections. These superconnections have been applied to computing the [[Chern class|Chern character]] in [[K-theory]], [[noncommutative geometry]], and [[BRST formalism]].
 
==See also==
*[[Supermanifold]]
*[[Graded manifold]]
*[[Supersymmetry]]
*[[Connection (algebraic framework)]]
*[[Supermetric]]
 
==References==
*{{Citation
| last = Bartocci
| first = C.
| last2 = Bruzzo
| first2 = U.
| last3 = Hernandez Ruiperez
| first3 = D.
| year = 1991
| title = The Geometry of Supermanifolds
| publisher = Kluwer
| isbn = 0-7923-1440-9
}}.
*{{Citation
| last = Rogers
| first = A.
| year = 2007
| title = Supermanifolds: Theory and Applications
| publisher = World Scientific
| isbn = 981-02-1228-3
}}.
*{{Citation
| last = Mangiarotti
| first = L.
| last2 = [[Gennadi Sardanashvily|Sardanashvily]]
| first2 = G.
| year = 2000
| title = Connections in Classical and Quantum Field Theory
| publisher = World Scientific
| isbn = 981-02-2013-8
}}.
 
==External links==
*[[Gennadi Sardanashvily|G. Sardanashvily]], Lectures on supergeometry, [http://xxx.lanl.gov/abs/0910.0092 arXiv: 0910.0092].
 
 
 
 
[[Category:Supersymmetry]]
[[Category:Differential geometry|*]]

Latest revision as of 08:31, 21 October 2014

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