Continuity of Dirac Spectra

TL;DR

This paper proves that the entire Dirac spectrum varies continuously with the Riemannian metric when viewed as a function from integers to reals, extending local spectral continuity to a global setting.

Contribution

It establishes the global continuity of Dirac spectra with respect to Riemannian metrics and constructs a family of functions representing the spectrum for all metrics.

Findings

Dirac spectrum depends continuously on the Riemannian metric.

Existence of a non-decreasing family of functions representing the spectrum.

Spectral flow prevents these functions from descending to the quotient space of metrics.

Abstract

It is a well-known fact that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function from the integers to the reals and endow the space of all spectra with an arsinh-uniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a non-decreasing family of functions on the space of all Riemannian metrics, which represents the entire Dirac spectrum at any metric. We also show that in general these functions do not descend to the space of Riemannian metrics modulo spin diffeomorphisms due to spectral flow.