7 research outputs found
How to find the holonomy algebra of a Lorentzian manifold
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
of a locally indecomposable
Lorentzian manifold of dimension is different from
, then it is contained in the similitude algebra
. There are 4 types of such holonomy algebras. Criterion
how to find the type of are given, and special geometric
structures corresponding to each type are described. To each
there is a canonically associated subalgebra
. An algorithm how to find
is provided.Comment: 15 pages; the final versio
Irreducible holonomy algebras of Riemannian supermanifolds
Possible irreducible holonomy algebras \g\subset\osp(p,q|2m) of Riemannian
supermanifolds under the assumption that \g is a direct sum of simple Lie
superalgebras of classical type and possibly of a one-dimensional center are
classified. This generalizes the classical result of Marcel Berger about the
classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.Comment: 27 pages, the final versio
Superization of Homogeneous Spin Manifolds and Geometry of Homogeneous Supermanifolds
Let M_0=G_0/H be a (pseudo)-Riemannian homogeneous spin manifold, with
reductive decomposition g_0=h+m and let S(M_0) be the spin bundle defined by
the spin representation Ad:H->\GL_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
Lambda(S^*(M_0)). Here G is the Lie supergroup associated with a certain
extension of the Lie algebra of symmetry g_0 to an algebra of supersymmetry
g=g_0+g_1=g_0+S via the Kostant-Koszul construction. Each algebra of
supersymmetry naturally determines a flat connection nabla^{S} in the spin
bundle S(M_0). Killing vectors together with generalized Killing spinors (i.e.
nabla^{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^{S} defines a superconnection on M, via the super-version of a theorem of
Wang.Comment: 50 page