Andrzej Trautman
The Dirac Operator on Pin Manifolds
The paper is published: as "The Dirac Operator on Hypersurfaces", Acta Physica Polonica B 26 (1995) 1283-1310

MSC:

15A66 Clifford algebras, spinors

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

83C60 Spinor and twistor methods

Abstract: Odd-dimensional Riemannian spaces that
are non-orientable, but have a pin structure, require the
consideration of the twisted adjoint representation of the
corresponding pin group. It is shown here how the Dirac
operator should be modified, also on even-dimensional
spaces, to make it equivariant with respect to the action
of that group when the twisted adjoint representation is
used in the definition of the pin structure. An explicit
description of a pin structure on a hypersurface, defined
by its immersion in a Euclidean space, is used to derive a
simple formula for the Dirac operator in that case.

Keywords: pin structures, Dirac operator