Gauging the Carroll Algebra and Ultra-Relativistic Gravity

TL;DR

This work constructs a gauging framework for the Carroll algebra to define Carrollian space-times and their geometry, highlighting their realization on null hypersurfaces and their duality with Newton–Cartan structures. It introduces a covariant Carrollian connection, augmented by a vector $M^ u$, and develops a consistent set of invariants ($ar{ au}_ u$, $ar{h}^{ u ho}$, $ar{ ext{Φ}}$) to realize local Carrollian symmetries. The authors then formulate ultra-relativistic (Carrollian) gravity theories in 2+1 dimensions with dynamical exponent $z<1$, including an anisotropic Weyl-invariant case at $z=0$, providing HL-like kinetic terms and curvature potentials. These developments extend non-relativistic holography concepts beyond Bargmann/HL gravity, offering tools to study flat-space holography, BMS symmetries, and WCFTs in Carrollian backgrounds. Potential applications include higher-dimensional Carrollian gravity, holographic duals of flat spacetimes, and systematic coupling of field theories to Carrollian geometries.

Abstract

It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincare algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Horava-Lifshitz gravity. Here we consider the case where we contract the Poincare algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z<1 including cases that have anisotropic Weyl invariance for z=0.