Higher-degree Dirac currents of twistor and Killing spinors in supergravity theories

Abstract

We show that higher degree Dirac currents of twistor and Killing spinors correspond to the hidden symmetries of the background spacetime which are generalizations of conformal Killing and Killing vector fields respectively. They are the generalizations of 1-form Dirac currents to higher degrees which are used in constructing the bosonic supercharges in supergravity theories. In the case of Killing spinors, we find that the equations satisfied by the higher degree Dirac currents are related to Maxwell-like and Duffin-Kemmer-Petiau equations. Correspondence between the Dirac currents and harmonic forms for parallel and pure spinor cases is determined. We also analyze the supergravity twistor and Killing spinor cases in 10 and 11-dimensional supergravity theories and find that although different inner product classes induce different involutions on spinors, the higher degree Dirac currents still correspond to the hidden symmetries of the spacetime.

Lie algebra of conformal Killing–Yano forms

“…The construction of the graded Lie algebras of KY forms and CKY forms can be used to extend the Killing superalgebras to include higher-degree differential forms. It is known that the higher-degree Dirac currents of Killing spinors correspond to KY forms while for twistor spinors they are equal to CKY forms [10]. So, the odd parts of the superalgebras can generate the even parts of them.…”

Gauged twistor spinors and symmetry operators

“…where e a1a2...ap = e a1 ∧ e a2 ∧ ... ∧ e ap , ⌊⌋ is the floor function that takes the integer part of the argument, z is the volume form and ( , ) denotes the spinor inner product. Every p-form component on the right hand side of (20) is called the p-form Dirac current as the generalization of the Dirac current that corresponds to the metric dual of the 1-form part of the spinor bilinear [24]. p-form Dirac currents will be denoted as follows (ψψ) p = (ψ, e ap...a2a1 .ψ)e a1a2...ap .…”

Spin raising and lowering operators for Rarita-Schwinger fields

“…Supersymmetry generators of both superconformal field theories in curved backgrounds and conformal supergravity theories correspond to twistor spinors [2][3][4][5]. They are also used in the construction of extended conformal superalgebras and are related to the conformal hidden symmetries of a background that are conformal Killing-Yano forms [6][7][8][9]. Twistor spinors contain Killing spinors and parallel spinors as special cases which are supersymmetry generators of supersymmetric field theories and supergravity theories.…”