$L^2$-index of the Dirac operator of generalized Euclidean Taub-NUT metrics

TL;DR

This paper calculates the axial anomaly for generalized Euclidean Taub-NUT metrics, revealing finiteness despite the Dirac operator not being Fredholm and the spectrum covering the entire real line.

Contribution

It provides the first computation of the axial anomaly for a broad class of generalized Taub-NUT metrics, highlighting spectral properties and anomaly finiteness.

Findings

Axial anomaly for generalized Taub-NUT metrics is finite.

Dirac operator's spectrum covers the entire real line.

Dirac operator is not Fredholm despite finite anomaly.

Abstract

We compute the axial anomaly for the Taub-NUT metric on $R^4$. We show that the axial anomaly for the generalized Taub-NUT metrics introduced by Iwai and Katayama is finite, although the Dirac operator is not Fredholm. We show that the essential spectrum of the Dirac operator is the whole real line.