2011/10/31 by Anton S. Galaev · 1 citation
Mathematics · Physics and Astronomy · #Algebra over a field #Algebra representation #Black Holes and Theoretical Physics #Holonomy #Homotopy and Cohomology in Algebraic Topology #Hyperkähler manifold #Indecomposable module #Manifold (fluid mechanics) #Noncommutative and Quantum Gravity Theories #Subalgebra #Supergravity #gr-qc #hep-th #math.DG #msc:53B30 #msc:53C29 #msc:53C50
paper · pdf · doi:10.1007/s11005-014-0741-y
published as Lett. Math. Phys. 105 (2015), no. 2, 199--219 · 15 pages; the final version
openalex publication_date 2014/12/15 · openalex created_date 2016/06/24 · arxiv created 2016/11/07 · arxiv updated 2016/11/08 · openalex updated_date 2026/08/05
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 \mathfrakg⊂\mathfrakso(1,n-1) of a locally indecomposable Lorentzian manifold (M,g) of dimension n is different from \mathfrakso(1,n-1), then it is contained in the similitude algebra \mathfraksim(n-2). There are 4 types of such holonomy algebras. Criterion how to find the type of \mathfrakg are given, and special geometric structures corresponding to each type are described. To each \mathfrakg there is a canonically associated subalgebra \mathfrakh⊂\mathfrakso(n-2). An algorithm how to find \mathfrakh is provided.