Conformal Carroll Scalars with Boosts

TL;DR

<3-5 sentence high-level summary>We construct two conformal Carroll scalar actions on curved backgrounds that are invariant under local Carroll boosts and Weyl transformations: a timelike action with time-derivative domination and a spacelike action with spatial-derivative domination that requires boost-consistent constraints. Both yield traceless energy-momentum tensors and satisfy boost, diffeomorphism, and Weyl Ward identities, making conserved currents for (conformal) Carroll isometries natural. Notably, the spacelike action can be dimensionally reduced to a lower-dimensional Euclidean CFT, highlighting a connection to embedding-space ideas and providing a concrete arena to study Carroll holography and related bootstrap or anomaly questions. The results offer a robust, boost-invariant action framework for Carrollian field theories with potential applications to flat-space holography and nonrelativistic/ultralocal limits of gravity.

Abstract

We construct two distinct actions for scalar fields that are invariant under local Carroll boosts and Weyl transformations. Conformal Carroll field theories were recently argued to be related to the celestial holography description of asymptotically flat spacetimes. However, only few explicit examples of such theories are known, and they lack local Carroll boost symmetry on a generic curved background. We derive two types of conformal Carroll scalar actions with boost symmetry on a curved background in any dimension and compute their energy-momentum tensors, which are traceless. In the first type of theories, time derivatives dominate and spatial derivatives are suppressed. In the second type, spatial derivatives dominate, and constraints are present to ensure local boost invariance. By integrating out these constraints, we show that the spatial conformal Carroll theories can be reduced to lower-dimensional Euclidean CFTs, which is reminiscent of the embedding space construction.

Figures (2)

• Figure 1: Foliation of spacetime in terms of \ref{['fig:galilean-foliation']}) spatial slices from $\tau_\mu$ in Newton--Cartan geometry, and \ref{['fig:carroll-foliation-curved']}) timelike curves from $v^\mu$ in Carroll geometry. In the latter case, using adapted coordinates as in \ref{['fig:carroll-foliation-adapted']}), we can still use $\tau_\mu$ to define coordinates in a fixed boost frame.
• Figure 2: Carroll structure on the light cone of the embedding space formalism.