Quantum anomalies for generalized Euclidean Taub–NUT metrics

Abstract

The gravitational anomalies are investigated for generalized Euclidean Taub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz vector of the Kepler-type problem. In order to evaluate the axial anomalies, the index of the Dirac operator for these metrics with the APS boundary condition is computed. The role of the Killing-Yano tensors is discussed for these two types of quantum anomalies.

Finiteness of theL2-index of the Dirac operator of generalized Euclidean Taub–NUT metrics

“…In the case of the generalized Taub-NUT spaces the S-K tensors involved in the Runge-Lenz vector cannot be expressed as a product of K-Y tensors. The non-existence of the K-Y tensors on generalized Taub-NUT metrics leads to gravitational quantum anomalies proportional to a contraction of the S-K tensor with the Ricci tensor [13].…”

“…We conjecture that it equals 0 and hence, unlike on annular domain or balls, the axial anomaly is never present. Our guess is motivated by heuristically increasing the radius of a ball to infinity, and arguing that by [13], the index stabilizes at 0 for large radii. Such an argument is of course incomplete, and even dangerous in the light of the fact that the Dirac operator is not Fredholm.…”

Fermion on Curved Spaces, Symmetries, and Quantum Anomalies

“…In [22] we computed the index of the Dirac operator on annular domains and on disk, with the non-local APS boundary condition. For the generalized Taub-NUT metrics [19,20,21], we found that the index is a number-theoretic quantity which depends on the metrics.…”

“…We mentioned in [22] some open problems in connection with unbounded domains. The paper [35] brings new results in this direction.…”

“…In the case of the generalized Taub-NUT spaces [19,20,21] the S-K tensors involved in the Runge-Lenz vector cannot be expressed as a product of K-Y tensors. The non-existence of the K-Y tensors on generalized Taub-NUT metrics leads to gravitational quantum anomalies proportional to a contraction of the S-K tensor with the Ricci tensor [22].…”