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

TL;DR

This paper investigates the spectrum of the Dirac operator on pseudo-Riemannian spin manifolds, providing criteria for spectral invariance under different subbundles and explicit computations for specific flat and product manifolds.

Contribution

It establishes conditions under which the Dirac spectrum is independent of the choice of time-like subbundle and computes the spectrum explicitly for certain flat and product manifolds.

Findings

Spectra are equal for different maximal time-like subbundles under certain conditions.

On compact manifolds, the spectrum is independent of the choice of subbundle.

Explicit spectrum calculations are provided for b5^{p,q}, b5^{p,q} torus, and b5^{1,1} imes F.

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.