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Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, BRST theory, or supergravity.

Supergeometry is formulated in terms of $\mathbb {Z} _{2}$ -graded modules and sheaves over $\mathbb {Z} _{2}$ -graded commutative algebras (supercommutative algebras). In particular, superconnections are defined as Koszul connections on these modules and sheaves. However, supergeometry is not particular noncommutative geometry because of a different definition of a graded derivation.

Graded manifolds and supermanifolds also are phrased in terms of sheaves of graded commutative algebras. Graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. There are different types of supermanifolds. These are smooth supermanifolds ( $H^{\infty }$ -, $G^{\infty }$ -, $GH^{\infty }$ -supermanifolds), $G$ -supermanifolds, and DeWitt supermanifolds. In particular, supervector bundles and principal superbundles are considered in the category of $G$ -supermanifolds. Definitions of principal superbundles and principal superconnections straightforwardly follow that of smooth principal bundles and principal connections. Principal graded bundles also are considered in the category of graded manifolds.

There is a different class of Quillen–Ne'eman superbundles and superconnections. These superconnections have been applied to computing the Chern character in K-theory, noncommutative geometry, and BRST formalism.

References

Bartocci, C.; Bruzzo, U.; Hernandez Ruiperez, D. (1991), The Geometry of Supermanifolds, Kluwer, ISBN 0-7923-1440-9.
Rogers, A. (2007), Supermanifolds: Theory and Applications, World Scientific, ISBN 981-02-1228-3.
Mangiarotti, L.; Sardanashvily, G. (2000), Connections in Classical and Quantum Field Theory, World Scientific, ISBN 981-02-2013-8.

External links

G. Sardanashvily, Lectures on supergeometry, arXiv:0910.0092.

@@ Line 1: / Line 1: @@
+{{Short description | Differential geometry of supermanifolds}}
-'''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
+'''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]].
-and quantum [[field theory|field theories]] involving odd [[field (physics)|field]]s, e.g., [[supersymmetry|SUSY]] field theory,
-[[BRST formalism|BRST theory]], [[supergravity]].
+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]].
-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. 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. 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.
-[[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.
-It should be mentioned a different class of [[Quillen]]-[[Yuval Ne'eman|Ne'eman]] superbundles and superconnections. These
+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]].
-superconnections have been applied to computing the [[Chern class|Chern character]] in [[K-theory]], [[noncommutative geometry]], [[BRST formalism]].
-== References ==
+==See also==
+*[[Supermanifold]]
+*[[Graded manifold]]
+*[[Supersymmetry]]
+*[[Connection (algebraic framework)]]
+*[[Supermetric]]
+==References==
 *{{Citation
  | last = Bartocci
@@ Line 42: / Line 26: @@
  | title = The Geometry of Supermanifolds
  | publisher = Kluwer
- | isbn =
+ | isbn = 0-7923-1440-9
 }}.
 *{{Citation
  | last = Rogers
  | first = A.
+ | authorlink = Alice Rogers
  | year = 2007
  | title = Supermanifolds: Theory and Applications
  | publisher = World Scientific
- | isbn = 9810212283
+ | isbn = 981-02-1228-3
 }}.
 *{{Citation
@@ Line 60: / Line 45: @@
  | title = Connections in Classical and Quantum Field Theory
  | publisher = World Scientific
- | isbn = 9810220138
+ | isbn = 981-02-2013-8
 }}.
-== See also ==
+==External links==
+*[[Gennadi Sardanashvily|G. Sardanashvily]], Lectures on supergeometry, {{arXiv|0910.0092}}.
-*[[Supermanifold]]
-*[[Graded manifold]]
-*[[Supersymmetry]]
+{{Supersymmetry topics}}
 [[Category:Supersymmetry]]
+[[Category:Differential geometry|*]]

v t e Supersymmetry
General topics	Supersymmetry Supersymmetric gauge theory Supersymmetric quantum mechanics Supergravity Superstring theory Super vector space Supergeometry
Supermathematics	Superalgebra Lie superalgebra Super-Poincaré algebra Superconformal algebra Supersymmetry algebra Supergroup Superspace Harmonic superspace Super Minkowski space Supermanifold
Concepts	Supercharge R-symmetry Supermultiplet Short supermultiplet BPS state Superpotential D-term FI D-term F-term Moduli space Supersymmetry breaking Konishi anomaly Seiberg duality Seiberg–Witten theory Witten index Wess–Zumino gauge Localization Mu problem Little hierarchy problem Electric–magnetic duality
Theorems	Coleman–Mandula Haag–Łopuszański–Sohnius Nonrenormalization
Field theories	Wess–Zumino N = 1 super Yang–Mills 4D N = 1 N = 4 super Yang–Mills Super QCD MSSM NMSSM 6D (2,0) superconformal ABJM superconformal
Supergravity	Pure 4D N = 1 supergravity 4D N = 1 supergravity N = 8 supergravity Higher dimensional 11D supergravity Type I supergravity Type IIA supergravity Type IIB supergravity Gauged supergravity
Superpartners	Axino Chargino Gaugino Goldstino Graviphoton Graviscalar Higgsino LSP Neutralino R-hadron Sfermion Sgoldstino Stop squark Superghost
Researchers	Affleck Bagger Batchelor Berezin Dine Fayet Gates Golfand Iliopoulos Montonen Olive Salam Seiberg Siegel Roček Rogers Wess Witten Zumino