Symmetry operators of Killing spinors and superalgebras in AdS5

Abstract

We construct the first-order symmetry operators of Killing spinor equation in terms of odd KillingYano forms. By modifying the Schouten-Nijenhuis bracket of Killing-Yano forms, we show that the symmetry operators of Killing spinors close into an algebra in AdS5 spacetime. Since the symmetry operator algebra of Killing spinors corresponds to a Jacobi identity in extended Killing superalgebras, we investigate the possible extensions of Killing superalgebras to include higherdegree Killing-Yano forms. We found that there is a superalgebra extension but no Lie superalgebra extension of the Killing superalgebra constructed out of Killing spinors and odd Killing-Yano forms in AdS5 background.

Extended superalgebras from twistor and Killing spinors

“…(22) reduces to (21) for p = 1. However, different from L K case, (22) is a symmetry operator for Killing spinor equation only for odd p and in constant curvature manifolds [12]. It can also be written in the following form by using the Killing spinor equation (9) L…”

“…The odd-odd bracket which takes two Killing spinors and gives a Killing vector field is the metric dual of the 1-form part of the squaring map defined in (12) which is the Dirac current of the Killing spinor ( ) 1 :…”

“…In constant curvature manifolds, the symmetry operators for the Killing spinor equation can be constructed from odd degree KY forms and for twistor equation, they are constructed out of CKY forms. Moreover, symmetry operators of the twistor equation can also be constructed from normal CKY forms in Einstein manifolds [12,14].…”

Gauged twistor spinors and symmetry operators

“…The symmetry operators of massless and massive Dirac equations are constructed out of CKY forms and KY forms, respectively [10][11][12][13][14][15][16]. Similarly, symmetry operators of geometric Killing spinors are written in terms of odd degree KY forms in constant curvature backgrounds [17]. CKY forms are used in the construction of the symmetry operators of twistor spinors in constant curvature backgrounds and normal CKY forms play the same role in Einstein manifolds [18].…”

“…CKY forms are used in the construction of the symmetry operators of twistor spinors in constant curvature backgrounds and normal CKY forms play the same role in Einstein manifolds [18]. Moreover, the symmetry operators of Killing and twistor spinors are also used in the definitions of more general structures such as extended Killing superalgebras and extended conformal superalgebras [17,18].In this paper, we consider the gauged twistor equation and find its integrability conditions in general n dimensions. We write the spinor bilinears of gauged twistor spinors and show that they correspond to gauged CKY forms which are generalizations of CKY forms with respect to a gauged covariant derivative.…”