Conformal symmetry superalgebras

Abstract

ABSTRACT. We show how the rigid conformal supersymmetries associated with a certain class of pseudo-Riemannian spin manifolds define a Lie superalgebra. The even part of this superalgebra contains conformal isometries and constant R-symmetries. The odd part is generated by twistor spinors valued in a particular R-symmetry representation. We prove that any manifold which admits a conformal symmetry superalgebra of this type must generically have dimension less than seven. Moreover, in dimensions three, four, five and six, we provide the generic data from which the conformal symmetry superalgebra is prescribed. For conformally flat metrics in these dimensions, and compact R-symmetry, we identify each of the associated conformal symmetry superalgebras with one of the conformal superalgebras classified by Nahm. We also describe several examples for Lorentzian metrics that are not conformally flat.

$${{SO(d,1)}}$$ S O ( d , 1 ) -Invariant Yang–Baxter Operators and the dS/CFT Correspondence

“…They are built from certain generalized creation/annihilation operators Table 1. Real Lie superalgebras with compact R-symmetry part whose "spacetime part" is the de Sitter algebra in d ≥ 4 dimensions [dMH13]. Here H refers to the quaternions, so u(1,…”

“…This super Lie algebra should be a) real, b) have a bosonic part so(d, 1)⊕r, where the R-symmetry part r is compact (since there are no unitary representations otherwise). The possibilities have been classified, see [dMH13]. For d ≥ 4, there are only two, displayed in Table 1.…”

Twistor spinors and extended conformal superalgebras

“…where we have used the expansion of the Clifford product in terms of the wedge product and interior derivative. For the right hand side of (15), we can write from (13) and the definition of the Dirac operator in (1) as…”

Extended superalgebras from twistor and Killing spinors

“…The tensor product of the spinor space and the dual spinor space gives the algebra of endomorphisms over the spinor space S ⊗ S * = EndS and it corresponds to the Clifford algebra of relevant dimension. So, the spinor bilinears which are the tensor products of a spinor with its dual can be written as a sum of different degree differential forms ψ ⊗ ψ = (ψ, ψ) + (ψ, e a .ψ)e a + (ψ, e ba .ψ)e ab + ... + (ψ, e ap...a2a1 .ψ)e a1a2...ap + ... + (−1) ⌊n/2⌋ (ψ, z.ψ)z (11) where e a1a2...ap = e a1 ∧ e a2 ∧ ... ∧ e ap , ⌊⌋ is the floor function that takes the integer part of the argument and z is the volume form. ( , ) is the spinor inner product an (11) is called as Fierz identitiy.…”