Lie algebra of conformal Killing–Yano forms

Abstract

We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms. A new Lie bracket for conformal Killing-Yano forms that corresponds to slightly modified Schouten-Nijenhuis bracket of differential forms is proposed. We show that conformal Killing-Yano forms satisfy a graded Lie algebra in constant curvature manifolds. It is also proven that normal conformal Killing-Yano forms in Einstein manifolds also satisfy a graded Lie algebra. The constructed graded Lie algebras reduce to the graded Lie algebra of Killing-Yano forms and the Lie algebras of conformal Killing and Killing vector fields in special cases.

Twistor spinors and extended conformal superalgebras

“…We calculate the commutator of symmetry operators of twistor spinors and find the conditions to obtain mutually commuting symmetry operators. On the other hand, CKY forms in constant curvature backgrounds and normal CKY forms in Einstein manifolds satisfy a graded Lie algebra structure as proven in [17]. So, by using the generalized symmetry operators of twistor spinors, graded Lie algebra of CKY forms and the p-form Dirac currents of twistor spinors, we obtain the extended conformal superalgebas of CKY forms and twistor spinors in constant curvature and Einstein manifolds.…”

“…We will consider the integrability conditions of the CKY equation for that aim. In general, the integrability conditions of the CKY equation for a p-form ω can be written as follows [17,22]…”

“…They constitute a graded Lie algebra structure with respect to Schouten-Nijenhuis (SN) bracket in constant curvature spacetimes [25]. Extended conformal superalgebras defined for CKY forms and twistor spinors reduce to extended Killing superalgebras in which the even part corresponds to the Lie algebra of odd KY forms and odd part is the space of Killing spinors [17]. Indeed, the CKY bracket in (56) reduces to SN bracket for KY forms ω 1 and ω 2…”

Harmonic spinors from twistors and potential forms

“…(51) is a special case of (48) and all CKY forms are normal CKY forms in constant curvature manifolds. Indeed, the symmetry operator defined in (49) gives way to define the extended conformal superalgebras constructed out of twistor spinors and CKY forms by considering the graded Lie algebra structure of CKY forms and the higher-degree Dirac currents of twistor spinors as is shown in [7,8,18].…”

Gauged twistor spinors and symmetry operators

“…For p = 1, (27) reduces to the shear-free vector field equation which is the generalization of the conformal Killing equation and describes the vector fields that constitute shear-free congruences [20]. Integrability conditions of the gauged CKY equation can be calculated by taking second covariant derivatives of (26). After some manipulations, they can be obtained as follows…”