Supergeometry

Supergeometry is the differential geometry of modules over $\mathbb {Z} _{2}$ ${\ displaystyle \ mathbb {Z} _ {2}}$ $\ mathbb Z_2$ - graded algebras on supermanifolds and graded varieties . Supergeometry is an integral part of many classical and quantum field models involving odd fields , for example, supersymmetric field theory, BRST theory , supergravity .

Supergeometry is formulated in terms of $\mathbb {Z} _{2}$ ${\ displaystyle \ mathbb {Z} _ {2}}$ $\ mathbb Z_2$ -graded modules and beams over $\mathbb {Z} _{2}$ ${\ displaystyle \ mathbb {Z} _ {2}}$ $\ mathbb Z_2$ -graded commutative algebras. In particular, superconnections are defined as connections on these modules and sheaves. However, supergeometry is not a special case of non-commutative geometry due to different definitions of differentiation .

Graded varieties and supermanifolds are described in terms of pencils of graded commutative algebras. Graded manifolds are characterized by sheaves on smooth manifolds , while supermanifolds are determined by gluing together sheaves of supervector spaces. There are several types of supermanifolds: smooth supermanifolds (including $H^{\infty }$ ${\ displaystyle H ^ {\ infty}}$ $H ^ {\ infty}$ -, $G^{\infty }$ ${\ displaystyle G ^ {\ infty}}$ $G ^ {\ infty}$ -, $GH^{\infty }$ ${\ displaystyle GH ^ {\ infty}}$ $Gh ^ {\ infty}$ supermanifolds), $G$ ${\ displaystyle G}$ $G$ - supermanifolds and supermanifolds of Devitt . In particular, supervector bundles and principal superbundles are considered in the category $G$ ${\ displaystyle G}$ $G$ supermanifolds. Moreover, the main superbundles and superconnections on them are defined similarly to the smooth main bundles and connections on them. It is worth noting that the main bundles are also considered in the category of supermanifolds.

Literature

Bartocci, C., Bruzzo, U., Hernandez Ruiperez, D. , The Geometry of Supermanifolds (Kluwer Academic Publ., 1991) ISBN 0-7923-1440-9
Rogers, A. , “Supermanifolds: Theory and Applications” (World Scientific, 2007) ISBN 981-02-1228-3
Mangiarotti, L., Sardanashvily, G. , Connections in Classical and Quantum Field Theory (World Scientific, 2000) ISBN 981-02-2013-8
Sardanashvili G. A. , “Modern methods of field theory. 4. Geometry and quantum fields ”(URSS, 2000) ISBN 5-88417-221-4 .

Links

G. Sardanashvily , Lectures on supergeometry, arXiv: 0910.0092

Supergeometry

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Literature

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