B.1
Riemannian
When
is Riemannian, there exists a natural embedding of the Clifford algebra
into
:
This embedding depends on a sign
(for “embedding”) and has the property that
. Thus, it
embeds
into
. When
is odd, the dimension of the minimal spinor
representation in
is double the one in
. In this case, the Clifford-valued volume form
in both
manifolds is mapped as follows
Thus, spinors in
will be mapped to spinors of a definite chirality in
.
Plugging this embedding into the expression for
, one sees that a
-parallel spinor
in the
cone, restricts to
to a geometric Killing spinor
obeying
This is the defining equation for a geometric Killing spinor. Furthermore,
- if
is even: there exists a one-to-one correspondence between parallel spinors
in
and geometric Killing spinors
in
; and
- if
is odd: there exists a one-to-one correspondence between parallel spinors
in
of
definite chirality
eigenvalues and geometric Killing spinors
in
.