Carrollian Lie Algebroids: Taming Singular Carrollian Geometries

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.

💡 Research Summary

The paper addresses a pressing gap in the mathematical formulation of Carrollian geometry that arises in modern applications such as Carrollian gravity, holography, and extreme gravitational phenomena. Traditional Carrollian manifolds are defined by a degenerate metric whose kernel is spanned by a nowhere‑vanishing, globally defined Carroll vector field. However, many physically relevant situations—null boundaries of black holes, mixed null‑spacelike hypersurfaces, and certain holographic setups—feature Carroll vector fields that become singular (they vanish on a non‑empty set) or change rank. These singularities cannot be accommodated within the standard framework.

To resolve this, the authors introduce Carrollian Lie algebroids, a construction that lifts the Carrollian structure from the tangent bundle to a more flexible Lie algebroid (A\to M). A Lie algebroid consists of a vector bundle (A), a Lie bracket on its sections, and an anchor map (\rho:A\to TM) that intertwines the bracket with the usual Lie bracket of vector fields. On (A) they place a possibly degenerate metric (g) whose kernel is a rank‑1 sub‑algebroid (L\subset A). The key object is the Carroll distribution \