Submaximal conformal symmetry superalgebras for Lorentzian manifolds of low dimension

TL;DR

This paper classifies and analyzes conformal symmetry superalgebras on low-dimensional Lorentzian manifolds with specific geometric structures, identifying maximal and submaximal cases and their algebraic properties.

Contribution

It introduces a classification of conformal symmetry superalgebras for certain Lorentzian manifolds, detailing their structure and conditions for maximal and submaximal symmetry.

Findings

Maximal conformal symmetry superalgebras occur only in conformally flat geometries.

Explicit submaximal superalgebras are determined for specific classes of geometries.

Symmetry superalgebras are computed for Ricci-flat geometries with null Killing vectors.

Abstract

We consider a class of smooth oriented Lorentzian manifolds in dimensions three and four which admit a nowhere vanishing conformal Killing vector and a closed two-form that is invariant under the Lie algebra of conformal Killing vectors. The invariant two-form is constrained in a particular way by the conformal geometry of the manifold. In three dimensions, the conformal Killing vector must be everywhere causal (or null if the invariant two-form vanishes identically). In four dimensions, the conformal Killing vector must be everywhere null and the invariant two-form vanishes identically if the geometry is everywhere of Petrov type N or O. To the conformal class of any such geometry, it is possible to assign a particular Lie superalgebra structure, called a conformal symmetry superalgebra. The even part of this superalgebra contains conformal Killing vectors and constant R-symmetries while the odd part contains (charged) twistor spinors. The largest possible dimension of a conformal symmetry superalgebra is realised only for geometries that are locally conformally flat. We determine precisely which non-trivial conformal classes of metrics admit a conformal symmetry superalgebra with the next largest possible dimension, and compute all the associated submaximal conformal symmetry superalgebras. In four dimensions, we also compute symmetry superalgebras for a class of Ricci-flat Lorentzian geometries not of Petrov type N or O which admit a null Killing vector.