Superalgebras of Dirac operators on manifolds with special Killing-Yano tensors

TL;DR

This paper explores new Dirac-type operators generated by special Killing-Yano tensors on manifolds, revealing their algebraic structures and symmetries, including connections to superalgebras and specific Lie groups.

Contribution

It introduces a class of Dirac-type operators linked to covariantly constant Killing-Yano tensors and analyzes their algebraic and symmetry properties, including superalgebra formations.

Findings

Dirac operators form superalgebras with automorphisms combining isometries and SU(2) transformations.

The continuous transformation group of these operators is U(1) or SU(2).

On Minkowski spacetime, the automorphisms of these superalgebras are explicitly studied.

Abstract

We present the properties of new Dirac-type operators generated by real or complex-valued special Killing-Yano tensors that are covariantly constant and represent roots of the metric tensor. In the real case these are just the so called complex or hyper-complex structures of the K\" ahlerian manifolds. 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. In this way the Dirac operators are related among themselves through continuous transformations associated with specific discrete ones. We show that the group of these continuous transformations can be only U(1) or SU(2). It is pointed out that the Dirac and Dirac-type operators can form N=4 superalgebras whose automorphisms combine isometries with the SU(2) transformation generated by the Killing-Yano tensors. As an example we study the automorphisms of the superalgebras of Dirac operators on Minkowski spacetime.