How to Find the Holonomy Algebra of a Lorentzian Manifold
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- 作者：Anton S. Galaev (1)
1. Faculty of Science ; University of Hradec Kr谩lov茅 ; Jana Koziny 1237 ; 500 03 ; Hradec Kr谩lov茅 ; Czech Republic - 关键词：53C29 ; 53C25 ; 53C50 ; 81T30 ; Lorentzian manifold ; holonomy group ; holonomy algebra ; de Rham ; Wu decomposition
- 刊名：Letters in Mathematical Physics
- 出版年：2015
- 出版时间：February 2015
- 年：2015
- 卷：105
- 期：2
- 页码：199-219
- 全文大小：256 KB
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- 刊物主题：Physics
Mathematical and Computational Physics
Statistical Physics
Geometry
Group Theory and Generalizations - 出版者：Springer Netherlands
- ISSN：1573-0530
文摘
Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de Rham and Wu decompositions, this problem is reduced to the case of locally indecomposable manifolds. In the case of locally indecomposable Riemannian manifolds, it is known that the holonomy algebra can be found from the analysis of special geometric structures on the manifold. If the holonomy algebra \({\mathfrak{g}\subset\mathfrak{so}(1,n-1)}\) of a locally indecomposable Lorentzian manifold (M, g) of dimension n is different from \({\mathfrak{so}(1,n-1)}\) , then it is contained in the similitude algebra \({\mathfrak{sim}(n-2)}\) . There are four types of such holonomy algebras. Criterion to find the type of \({\mathfrak{g}}\) is given, and special geometric structures corresponding to each type are described. To each \({\mathfrak{g}}\) there is a canonically associated subalgebra \({\mathfrak{h} \subset\mathfrak{so}(n-2)}\) . An algorithm to find \({\mathfrak{h}}\) is provided.