Spectral Geometry of Nuts and Bolts

Abstract

We study the spectrum of Laplace operators on a one-parameter family of gravitational instantons of bi-axial Bianchi IX type coupled to an abelian connection with self-dual curvature. The family of geometries includes the Taub-NUT, Taub-bolt and Euclidean Schwarzschild geometries and interpolates between them. The interpolating geometries have conical singularities along a submanifold of co-dimension two, but we prove that the associated Laplace operators have natural self-adjoint extensions and study their spectra. In particular, we determine the essential spectrum and prove that its complement, the discrete spectrum, is infinite. We compute these eigenvalues numerically and compare the numerical results with an analytical approximation derived from the asymptotic Taub-NUT form of each of the geometries in our family.

arXiv comment27 pages, 2 figures and 3 tables; minor corrections throughout the text

References in corpus (7)

• Curvature, Cones, and Characteristic Numbers
• Dirac operators on the Taub-NUT space, monopoles and SU(2) representations
• Taub-NUT Dynamics with a Magnetic Field
• Spectral Properties of Schwarzschild Instantons
• Harmonic Forms and Spinors on the Taub-bolt Space
• $\mathbb{C}{P}^2$ and $\mathbb{C}{P}^1$ Sigma Models in Supergravity: Bianchi type IX Instantons and Cosmologies
• Quantum Bound States in Yang-Mills-Higgs Theory

Cited by in corpus (1)

• Geodesic motion in Euclidean Schwarzschild geometry