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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]].
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| [[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.
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| 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]].
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| ==See also==
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| *[[Supermanifold]]
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| *[[Graded manifold]]
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| *[[Supersymmetry]]
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| *[[Connection (algebraic framework)]]
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| *[[Supermetric]]
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| ==References==
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| *{{Citation
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| | last = Bartocci
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| | first = C.
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| | last2 = Bruzzo
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| | first2 = U.
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| | last3 = Hernandez Ruiperez
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| | first3 = D.
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| | year = 1991
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| | title = The Geometry of Supermanifolds
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| | publisher = Kluwer
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| | isbn = 0-7923-1440-9
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| }}.
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| *{{Citation
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| | last = Rogers
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| | first = A.
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| | year = 2007
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| | title = Supermanifolds: Theory and Applications
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| | publisher = World Scientific
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| | isbn = 981-02-1228-3
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| }}.
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| *{{Citation
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| | last = Mangiarotti
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| | first = L.
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| | last2 = [[Gennadi Sardanashvily|Sardanashvily]]
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| | first2 = G.
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| | year = 2000
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| | title = Connections in Classical and Quantum Field Theory
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| | publisher = World Scientific
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| | isbn = 981-02-2013-8
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| }}.
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| ==External links==
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| *[[Gennadi Sardanashvily|G. Sardanashvily]], Lectures on supergeometry, [http://xxx.lanl.gov/abs/0910.0092 arXiv: 0910.0092].
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| [[Category:Supersymmetry]]
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| [[Category:Differential geometry|*]]
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