Submaximal conformal symmetry superalgebras for Lorentzian manifolds of low dimension

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.