Remarks on the spectrum of the Dirac operator of pseudo-Riemannian spin manifolds

Abstract

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_\xi(S)$. The definition of $L^2_\xi(S)$ involves the choice of a $p$-dimensional time-like subbundle $\xi\subset TM$. We establish a sufficient criterion for the spectra of $D$ induced by two maximal time-like subbundles $\xi_1,\xi_2\subset TM$ to be equal. If the base manifold $M$ is compact, the spectrum does not depend on $\xi$ at all. We then proceed by explicitely computing the full spectrum of $D$ for $\mathbb R^{p,q}$, the flat torus $\mathbb T^{p,q}$ and products of the form $\mathbb T^{1,1}\times F$ with $F$ being an arbitrary compact, even-dimensional Riemannian spin manifold.