Carroll theories from Lorentzian light-cone theories

TL;DR

This work derives $d$-dimensional Carroll-invariant field theories by null reduction of $(d+1)$-dimensional Lorentzian light-cone actions, introducing a Bargmann-invariant deformation with parameter $\alpha$ to access both magnetic and electric Carroll sectors. Magnetic Carroll theories arise directly from the Lorentzian light-cone framework, while the electric sector requires the deformation, yielding complementary Carroll dynamics via two distinct scaling limits. A thorough canonical analysis clarifies constraint structures, showing the absence of second-class constraints in the deformed Bargmann framework and detailing the Carroll generators and transformation laws. The approach is extended to scalars, electromagnetism, Yang–Mills, and $p$-form fields, with a consistent Lagrangian and Hamiltonian treatment, including the role of the light-cone gauge. The results connect light-cone dynamics near null hypersurfaces to Carrollian physics and suggest extensions to fermions, gravity, and curved Bargmann geometries, with potential implications for flat space holography and double-null gravity.

Abstract

We derive Carrollian field theories via null reduction from Lorentzian light-cone actions in Minkowski spacetime. By suitably deforming the light-cone action, we reduce the Poincaré invariance to a Bargmann subgroup, from which both magnetic and electric Carroll actions can be obtained in one lower dimension. Through a canonical analysis, we show that the second-class constraints usually found in Lorentzian light-cone theories are absent for these deformed Bargmann-invariant actions. We demonstrate the procedure for theories with and without gauge symmetry. Notably, while the magnetic Carroll sector can be directly derived from the original Lorentzian action, the deformation is essential to obtain the electric Carroll sector. We further argue that magnetic Carroll solutions in $d$ dimensions represent a consistent truncation of the solutions of the $(d+1)$-dimensional Lorentzian parent theory, providing an effective description of light-cone dynamics near a null hypersurface. For gauge theories, we also highlight the role of the light-cone gauge condition in deriving Carrollian theories.

Figures (3)

• Figure 1: Null reduction to hypersurface $\Sigma: x^- = x^-_0$
• Figure 2: Two routes for deriving Carroll actions from Lorentzian light-cone actions
• Figure 3: Light-cone gauge condition for deriving Carrollian actions