Metrics with Galilean conformal isometry

Abstract

The Galilean conformal algebra (GCA) arises in taking the nonrelativistic limit of the symmetries of a relativistic conformal field theory in any dimensions. It is known to be infinite dimensional in all spacetime dimensions. In particular, the 2d GCA emerges out of a scaling limit of linear combinations of two copies of the Virasoro algebra. In this paper, we find metrics in dimensions greater than 2 which realize the finite 2d GCA (the global part of the infinite algebra) as their isometry by systematically looking at a construction in terms of cosets of this finite algebra. We list all possible subalgebras consistent with some physical considerations motivated by earlier work in this direction and construct all possible higher-dimensional nondegenerate metrics. We briefly study the properties of the metrics obtained. In the standard one higher-dimensional ``holographic'' setting, we find that the only nondegenerate metric is Minkowskian. In four and five dimensions, we find families of nontrivial metrics with a rather exotic signature. A curious feature of these metrics is that all but one of them are Ricci-scalar flat.