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., supersymmetry field theory, BRST formalism, or supergravity.
Supergeometry is formulated in terms of - and sheaves over -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 manifold, while are constructed by gluing of sheaves of supervector spaces. There are different types of supermanifolds. These are smooth supermanifolds (-, -, -supermanifolds), -supermanifolds, and DeWitt supermanifolds. In particular, supervector bundles and principal superbundles are considered in the category of -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 Daniel Quillen–Ne'eman superbundles and superconnections. These superconnections have been applied to computing the Chern class in K-theory, noncommutative geometry, and BRST formalism.
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