Symmetries of the Dirac operators associated with covariantly constant Killing-Yano tensors

TL;DR

This paper investigates the symmetries of Dirac operators linked to covariantly constant Killing-Yano tensors in various manifolds, revealing specific continuous and discrete symmetry groups and their relation to special geometries.

Contribution

It identifies the continuous and discrete symmetry groups of Dirac operators generated by particular Killing-Yano tensors in arbitrary manifolds, including cases beyond special geometries.

Findings

Continuous symmetries are limited to U(1) and SU(2) groups.

Discrete symmetries include groups Z_4 and Q.

Symmetry groups are present even outside special geometries like hyper-Kahler spaces.

Abstract

The continuous and discrete symmetries of the Dirac-type operators produced by particular Killing-Yano tensors are studied in manifolds of arbitrary dimensions. The Killing-Yano tensors considered are covariantly constant and realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. The Dirac operators are related among themselves through continuous or discrete transformations. It is shown that the groups of the continuous symmetry can be only U(1) and SU(2), specific to (hyper-)Kahler spaces, but arising even in cases when the requirements for these special geometries are not fulfilled. The discrete symmetries are also studied obtaining the discrete groups Z_4 and Q. The briefly presented examples are the Euclidean Taub-NUT space and the Minkowski spacetime.