Michel Cahen, Simone Gutt, Andrzej Trautman
Pin Structures and the Modified Dirac Operator
The paper is published: J. Geom. Phys. 17, 3 (1995) 283-297

MSC:

15A66 Clifford algebras, spinors

57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)

83C60 Spinor and twistor methods

Abstract: General theorems on
pin structures on products of
manifolds and on homogeneous (pseudo-)Riemannian
spaces are given and used to find explicitly all such
structures on odd-dimensional real
projective quadrics, which are
known to be non-orientable
\cite{CaGuT}.
It is shown that the product of two manifolds
has a pin structure if, and only if, both are pin and at least
one of them is orientable. This general result is
illustrated by the example of the product of two real
projective planes.
It is shown how the Dirac
operator should be modified to make it equivariant with
respect to the twisted adjoint action of the Pin group.
A simple formula is derived for the spectrum of the
Dirac operator on the product of
two pin manifolds, one of which is orientable, in terms of the
eigenvalues of the Dirac operators on the factor spaces.

Keywords: pin structures, Dirac operator