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 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.