Twistor spinors and extended conformal superalgebras

Abstract

We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in constant curvature backgrounds and from normal conformal Killing-Yano forms in Einstein manifolds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing-Yano forms and the symmetry operators of twistor spinors. The even part of the extended conformal superalgebra corresponds to conformal Killing-Yano forms and the odd part consists of twistor spinors.

Spin raising and lowering operators for Rarita-Schwinger fields

“…The second integrability condition given in (6) is the standard Weitzenbock identity. One can see from the third integrability condition in (7) that all conformally flat manifolds can have solutions of the twistor equation (4).…”

“…It can easily be seen that by using the defining equations of ϕ and u given by (1) and (4) and the integrability conditions (5)- (7), one can find the result Dψ = 0. A reverse procedure of obtaining a solution ϕ of massless spin-0 equation from a solution ψ of the massless Dirac equation and a twistor spinor u can also be constructed [13,14].…”

“…where we have used the identity e b .R ab = −P a for zero torsion, e b .F.e b = 0 and the integrability condition (7). Then, for Ψ to satisfy (21), we obtain the condition…”

“…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.…”

Harmonic spinors from twistors and potential forms

“…The spinor bilinears of twistor spinors correspond to conformal Killing-Yano (CKY) forms which are antisymmetric generalizations of conformal Killing vector fields to higher degree forms [5]. CKY forms are also used in the construction of symmetry operators of harmonic and twistor spinors [6][7][8]. Symmetry operators of an equation are defined as the operators that take any solution of the equation and give another solution.…”