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Superization of homogeneous spin manifolds andgeometry of homogeneous supermanifolds

Superization of homogeneous spin manifolds andgeometry of homogeneous supermanifolds Let M 0=G 0/H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition $\mathfrak {g}_{0}=\mathfrak {h}+\mathfrak {m}$ and let S(M 0) be the spin bundle defined by the spin representation $\tilde{ \operatorname {Ad}}:H\rightarrow \mathrm {GL}_{\mathbb {R}}(S)$ of the stabilizer H. This article studies the superizations of M 0, i.e. its extensions to a homogeneous supermanifold M=G/H whose sheaf of superfunctions is isomorphic to the sheaf of sections of Λ(S *(M 0)). Here G is the Lie supergroup associated with a certain extension of the Lie algebra of symmetry $\mathfrak {g}_{0}$ to an algebra of supersymmetry $\mathfrak {g}=\mathfrak {g}_{\overline {0}}+\mathfrak {g}_{\overline {1}}=\mathfrak {g}_{0}+S$ via the Kostant-Koszul construction. Each algebra of supersymmetry naturally determines a flat connection $\nabla^{\mathcal {S}}$ in the spin bundle S(M 0). Killing vectors together with generalized Killing spinors (i.e. $\nabla^{\mathcal {S}}$ -parallel spinors) are interpreted as the values of appropriate geometric symmetries of M, namely even and odd Killing fields. An explicit formula for the Killing representation of the algebra of supersymmetry is obtained, generalizing some results of Koszul. The generalized spin connection $\nabla^{\mathcal {S}}$ defines a superconnection on M, via the super-version of a theorem of Wang. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Superization of homogeneous spin manifolds andgeometry of homogeneous supermanifolds

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References (60)

Publisher
Springer Journals
Copyright
Copyright © 2009 by Mathematisches Seminar der Universität Hamburg and Springer
Subject
Mathematics; Geometry ; Topology; Number Theory; Combinatorics; Differential Geometry; Algebra
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-009-0031-2
Publisher site
See Article on Publisher Site

Abstract

Let M 0=G 0/H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition $\mathfrak {g}_{0}=\mathfrak {h}+\mathfrak {m}$ and let S(M 0) be the spin bundle defined by the spin representation $\tilde{ \operatorname {Ad}}:H\rightarrow \mathrm {GL}_{\mathbb {R}}(S)$ of the stabilizer H. This article studies the superizations of M 0, i.e. its extensions to a homogeneous supermanifold M=G/H whose sheaf of superfunctions is isomorphic to the sheaf of sections of Λ(S *(M 0)). Here G is the Lie supergroup associated with a certain extension of the Lie algebra of symmetry $\mathfrak {g}_{0}$ to an algebra of supersymmetry $\mathfrak {g}=\mathfrak {g}_{\overline {0}}+\mathfrak {g}_{\overline {1}}=\mathfrak {g}_{0}+S$ via the Kostant-Koszul construction. Each algebra of supersymmetry naturally determines a flat connection $\nabla^{\mathcal {S}}$ in the spin bundle S(M 0). Killing vectors together with generalized Killing spinors (i.e. $\nabla^{\mathcal {S}}$ -parallel spinors) are interpreted as the values of appropriate geometric symmetries of M, namely even and odd Killing fields. An explicit formula for the Killing representation of the algebra of supersymmetry is obtained, generalizing some results of Koszul. The generalized spin connection $\nabla^{\mathcal {S}}$ defines a superconnection on M, via the super-version of a theorem of Wang.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Oct 24, 2009

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