Spectral Properties of Schwarzschild Instantons

TL;DR

This paper investigates the spectral characteristics of Dirac and scalar Laplace operators on Euclidean Schwarzschild space, revealing simplified zero-mode forms and numerical spectrum approximations linked to Taub-NUT space.

Contribution

It provides new insights into zero-mode structures and demonstrates accurate numerical spectrum approximations using models from Euclidean Taub-NUT space.

Findings

Zero-modes of the gauged Dirac operator are simple in terms of Euclidean time radius.

Numerical spectra of the gauged Laplace operator closely match those of the Taub-NUT space.

Spectral properties can be interpreted in geometric models of matter.

Abstract

We study spectral properties of the Dirac and scalar Laplace operator on the Euclidean Schwarzschild space, both twisted by a family of abelian connections with anti-self-dual curvature. We show that the zero-modes of the gauged Dirac operator, first studied by Pope, take a particularly simple form in terms of the radius of the Euclidean time orbits, and interpret them in the context of geometric models of matter. For the gauged Laplace operator, we study the spectrum of bound states numerically and observe that it can be approximated with remarkable accuracy by that of the exactly solvable gauged Laplace operator on the Euclidean Taub-NUT space.