It is well known that S^{d} and AdS_{d} can be described as surfaces embedded in and . What is
less known, especially in the physics literature, is that geometric Killing spinors on the latter are
induced from parallel spinors on the former. This was proven by Bär [49] in the Riemannian
case and by Kath [336] in the pseudo-Riemannian case. In this appendix, I briefly review this
result.

Consider a Riemannian spin manifold having geometric Killing spinors satisfying the differential equation

is related to the curvature of the manifold and is a sign, to be spelled out below. From a physics point of view, the right-hand side of this equation is the remnant of the gravitino supersymmetry transformation in the presence of non-trivial fluxes proportional to the volume form of the manifold . Mathematically, it is a rather natural extension of the notion of covariantly constant Killing spinors. The statement that the manifold allows an embedding in a higher-dimensional Riemannian space corresponds, metrically, to considering the metric of a cone in with base space . Thus, where is the radius of curvature of . There exists a similar construction in the pseudo-Riemannian case in which the cone is now along a timelike direction. In the following, I will distinguish two different cases, though part of the analysis will be done simultaneously:- Riemannian with Riemannian cone , and
- Lorentzian with pseudo-Riemannian cone .

To establish an explicit map between Killing spinors in both manifolds, one needs to relate their spin connections. To do so, consider a local coframe for and for , defined as

The connection coefficients and satisfy the corresponding structure equations Given the relation between coframes, the connections are related as Let denote the spin connection on : where are the gamma-matrices for the relevant Clifford algebra. Plugging in the expression for the connection coefficients for the cone, one finds To continue we have to discuss the embedding of Clifford algebras in order to recognise the above connection intrinsically on . This requires distinguishing two cases, according to the signature of .

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
This work is licensed under a Creative Commons License. E-mail us: |