Quantum anomalies for generalized Euclidean Taub-NUT metrics

The generalized Taub-NUT metrics exhibit in general gravitational anomalies. This is in contrast with the fact that the original Taub-NUT metric does not exhibit gravitational anomalies, which is a consequence of the fact that it admits Killing-Yano tensors forming Staeckel-Killing tensors as products. We have found that for axial anomalies, interpreted as the index of the Dirac operator, the presence of Killing-Yano tensors is irrelevant. In order to evaluate the axial anomalies, we compute the index of the Dirac operator with the APS boundary condition on balls and on annular domains. The result is an explicit number-theoretic quantity depending on the radii of the domain. This quantity is 0 for metrics close to the original Taub-NUT metric but it does not vanish in general