How to find the holonomy algebra of a Lorentzian manifold

Abstract

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 4 types of such holonomy algebras. Criterion how to find the type of $\mathfrak{g}$ are 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 how to find $\mathfrak{h}$ is provided.

15 pages; the final version

References in corpus (8)

• Very Special Relativity
• Metrics With Vanishing Quantum Corrections
• Gauduchon-Tod structures, Sim holonomy and De Sitter supergravity
• Supersymmetry, holonomy and Kundt spacetimes
• Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n-2)
• On the full holonomy group of special Lorentzian manifolds
• One component of the curvature tensor of a Lorentzian manifold
• On the de Rham-Wu decomposition for Riemannian and Lorentzian manifolds

Cited by in corpus (1)

• Parallel spinors on Lorentzian Weyl spaces