On pseudo-Riemannian manifolds with many Killing spinors
- University of Edinburgh and Maxwell Institute for Mathematical Sciences JCMB, The King's Buildings, Edinburgh, EH9 3JZ (United Kingdom)
- Department Mathematik und Zentrum fuer Mathematische Physik Universitaet Hamburg, Bundesstrasse 55, D-20146 Hamburg (Germany)
Let M be a pseudo-Riemannian spin manifold of dimension n and signature s and denote by N the rank of the real spinor bundle. We prove that M is locally homogeneous if it admits more than (3/4)N independent Killing spinors with the same Killing number, unless n {identical_to} 1(mod 4) and s {identical_to} 3(mod 4). We also prove that M is locally homogeneous if it admits k{sub +} independent Killing spinors with Killing number {lambda} and k{sub -} independent Killing spinors with Killing number -{lambda} such that k{sub +}+k{sub -}>(3/2)N, unless n {identical_to} s {identical_to} 3(mod 4). Similarly, a pseudo-Riemannian manifold with more than (3/4)N independent conformal Killing spinors is conformally locally homogeneous. For (positive or negative) definite metrics, the bounds (3/4)N and (3/2)N in the above results can be relaxed to (1/2)N and N, respectively. Furthermore, we prove that a pseudo-Riemannnian spin manifold with more than (3/4)N parallel spinors is flat and that (1/4)N parallel spinors suffice if the metric is definite. Similarly, a Riemannnian spin manifold with more than (3/8)N Killing spinors with the Killing number {lambda}(set-membership sign)R has constant curvature 4{lambda}{sup 2}. For Lorentzian or negative definite metrics the same is true with the bound (1/2)N. Finally, we give a classification of (not necessarily complete) Riemannian manifolds admitting Killing spinors, which provides an inductive construction of such manifolds.
- OSTI ID:
- 21260245
- Journal Information:
- AIP Conference Proceedings, Vol. 1093, Issue 1; Conference: Workshop on geometry and physics: Special metrics and supersymmetry, Bilbao (Spain), 29-31 May 2008; Other Information: DOI: 10.1063/1.3089206; (c) 2009 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0094-243X
- Country of Publication:
- United States
- Language:
- English
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