Conformal symmetry superalgebras

TL;DR

This paper characterizes conformal symmetry superalgebras on pseudo-Riemannian spin manifolds, linking geometric structures with Lie superalgebras, and classifies these in certain dimensions and cases.

Contribution

It establishes the structure of conformal symmetry superalgebras, relates them to Nahm's classification, and explores their existence in various dimensions and metric types.

Findings

Superalgebras contain conformal isometries and R-symmetries.

Manifolds with such superalgebras are typically under seven dimensions.

Explicit classifications are provided in dimensions three to six.

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.