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   » » Wiki: Supergeometry
Tag Wiki 'Supergeometry'.
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Supergeometry
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Supergeometry is differential geometry of modules over graded commutative algebras, and . Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., field theory, , or .

Supergeometry is formulated in terms of \mathbb Z_2- 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.

and also are phrased in terms of sheaves of graded commutative algebras. are characterized by sheaves on , while 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 and principal connections. Principal graded bundles also are considered in the category of .

There is a different class of –Ne'eman superbundles and superconnections. These superconnections have been applied to computing the in , noncommutative geometry, and .


See also

  • .
  • .
  • .


External links
  • G. Sardanashvily, Lectures on supergeometry, .

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