Carrollian Lie Algebroids: Taming Singular Carrollian Geometries
Andrew James Bruce
TL;DR
This work introduces Carrollian Lie algebroids to model Carrollian geometries where the Carroll vector field can be singular, by making the kernel of the degenerate metric a trivial line subbundle and defining the Carroll distribution as the anchor image of that line. The framework treats the Carroll distribution as a (potentially singular) Stefan–Sussmann distribution, enabling a rigorous treatment of singular Carrollian vectors and their associated foliations. It provides a suite of examples, including invariant Carrollian structures on Atiyah algebroids and mixed null-spacelike hypersurfaces, and proves the existence of Carrollian connections that are compatible with the degenerate metric; in particular, torsion-free connections exist if and only if the algebroid is stationary. The results generalize standard Carrollian geometry and lay groundwork for Carrollian gravity, holography, and related physics with singular null directions.
Abstract
Developments in Carrollian gravity and holography necessitate the use of singular Carroll vector fields, a feature that cannot be accommodated within standard Carrollian geometry. We introduce Carrollian Lie algebroids as a framework to study such singular Carrollian geometries. In this approach, we define the Carroll distribution as the image of the kernel of the degenerate metric under the anchor map. The Carroll distribution is, in general, a singular Stefan--Sussmann distribution that will fluctuate between rank-1 and rank-0, and so captures the notion of a singular Carroll vector field. As an example, we show that an invariant Carrollian structure on a principal bundle leads to a Carrollian structure on the associated Atiyah algebroid that will, in general, have a singular Carroll distribution. Mixed null-spacelike hypersurfaces, under some simplifying assumptions, also lead to examples of Carrollian Lie algebroids. Furthermore, we establish the existence of compatible connections on Carrollian Lie algebroids, and as a direct consequence, we conclude that Carrollian manifolds can always be equipped with compatible affine connections.