The construction of Dirac operators on orientifolds

Abstract

Motivated by Wigner's theorem, a canonical construction is described that produces an Atiyah-Singer Dirac operator with both unitary and anti-unitary symmetries. This Dirac operator includes the Dirac operator for KR-theory as a special case, filling a long-standing gap in the literature. In order to make the construction, orientifold Spinc-structures are defined and classified using semi-equivariant Dixmier-Douady theory, and analogues of several standard theorems on the existence of Spinc-structures are proved.