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

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 Kählerian 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.Pacs 04.62.+v *

Fermion on Curved Spaces, Symmetries, and Quantum Anomalies

“…The Dirac-type operators constructed with the aid of covariantly constant K-Y tensors are equivalent with the standard Dirac operator [37]. The non-covariantly constant K-Y tensors generates non-standard Dirac operators which are not equivalent to the standard Dirac operator and they are associated with the hidden symmetries of the space.…”

“…In the study of the Dirac equation in curved spaces, it has been proved that the K-Y tensors play an essential role in the construction of new Dirac-type operators. The Dirac-type operators constructed with the aid of covariantly constant K-Y tensors are equivalent with the standard Dirac operator [37]. The non-covariantly constant K-Y tensors generates non-standard Dirac operators which are not equivalent to the standard Dirac operator and they are associated with the hidden symmetries of the space.…”

f-SYMBOLS, KILLING TENSORS AND CONSERVED BEL-TYPE CURRENTS

“…They play an important role in the existence of the geodesic constants of motion [17] in curved spacetimes. There is a deep connection between Killing-Yano tensors, supersymmetries in pseudoclassical spinning particle models [11,24,25] (models which besides usual spacetimes coordinates includes a number of anticommuting ones to describe the spin degrees of freedom) and Dirac-type operators on curved spacetimes [26,27,28,29]. Killing-Yano tensors are directly related with the absence of gravitational quantum anomalies [30,27,31] (conservation laws valid at classical level ceasing to be true at quantum level).…”

Non-standard supersymmetries on spaces admitting Killing–Yano tensors

“…N f } having supplementary properties which should guarantee that (I) the linear space L f = {ρ|ρ = ρ i f i , ρ i ∈ R} is isomorphic with a real Lie algebra and (II) each element of L f − 0 is a root (of arbitrary norm). In these circumstances we have the following theorem [13]:…”