B Cone Construction and Supersymmetry
It is well known that Sd and AdSd 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
.