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

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 continuous symmetry can be only U(1) and SU(2), specific to (hyper-)Kähler 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 and . The briefly presented examples are the Euclidean Taub–NUT space and the Minkowski spacetime.