The construction of Dirac operators on orientifolds (Q2237960)

A basic principle of quantum mechanics (Wigner's theorem) is that symmetries of quantum mechanical systems are represented by unitary/anti-unitary operators on Hilbert spaces. So the aim of this paper is to take anti-unitary symmetries into account in the construction of Dirac operators and their index theory. This motivates the author's definition of orientifolds, orientifold bundles, and orientifold Dirac operators. Here an orientifold is a manifold \(M\) equipped with an action of a group \(\Gamma\) and a homomorphism \(\epsilon\colon \Gamma\to \mathbb Z_2\), so that elements of \(\Gamma\) in the kernel of \(\epsilon\) act on vector bundles by linear symmetries, while elements of \(\Gamma\) not in the kernel of \(\epsilon\) act by anti-linear symmetries. The author gives necessary and sufficient conditions for such an orientifold to admit an orientifold Dirac operator, and defines the index for such an operator, in a way generalizing the known conditions when \(\epsilon\) is trivial.