Carrollian Physics and Holography

TL;DR

This work surveys Carrollian physics as the $c\to0$ contraction of the Poincaré group and develops a comprehensive framework for flat space holography. It builds Carrollian geometry, frames, and connections, then derives Carrollian field theories and Carrollian CFTs, including primary operators, Ward identities, and OPEs. A central contribution is the Carrollian holography program: gravity in asymptotically flat spacetimes is dual to a 3D Carrollian CFT living at null infinity, with Carrollian amplitudes encoding the bulk massless S-matrix and connecting to celestial holography through a flat-space limit of AdS/CFT. The work further outlines the flat space/Carrollian limit correspondence, concrete examples of Carrollian amplitudes, and the soft/collinear structures that underlie holographic dualities, while highlighting open questions in the quantization and bootstrap of Carrollian CFTs.

Abstract

This report reviews key developments in Carrollian physics with an emphasis on their role in the emerging framework of holography in asymptotically flat spacetimes. We begin by introducing the Carrollian limit, understood as the non-relativistic contraction of the Poincaré group obtained by formally taking the speed of light to zero. The geometric structures associated with this limit are described and argued to arise naturally on null hypersurfaces, most notably on null infinity, as well as black hole and cosmological horizons. Building on this, we examine the relation between the Bondi-Metzner-Sachs symmetries governing asymptotically flat gravity and the conformal Carrollian symmetries. Explicit examples of Carrollian field theories are constructed by implementing the limit on well-known relativistic field theories, with particular attention to Carrollian CFTs. We then present the Carrollian holography proposal, according to which gravity in asymptotically flat spacetimes is dual to a Carrollian CFT living at null infinity in one lower dimension. In this framework, the massless $\mathcal{S}$-matrix written in position space at null infinity is naturally reinterpreted in terms of boundary Carrollian CFT correlators, called Carrollian amplitudes. We highlight their relation to celestial amplitudes and show how they naturally emerge from holographic CFT correlators through a correspondence between the flat space limit in the bulk and the Carrollian limit at the boundary. Using this correspondence, we provide strong evidence that flat space holography arises from a controlled and consistent limiting procedure applied to both sides of the AdS/CFT duality. We conclude by outlining future directions and open questions in the program.

Figures (15)

• Figure 1: The causal structure can be represented by light cones. The effect of taking the Galilean limit is to open the light cones. In the limit, all spacetime events become causally connected.
• Figure 2: The effect of taking the Carrollian limit is to close the light cones. In the limit, all spacetime events become causally disconnected unless they are at the same spatial point. This reflects the ultra-local nature of Carrollian physics.
• Figure 3: This figure illustrates the fiber bundle structure $\pi : \mathscr{S} \to \mathcal{S}$ of a Carrollian manifold $(\mathscr{S}, q_{ab}, n^a)$. The fibers $\ker \pi$ are generated by the vector field $n^a$, which lies in the kernel of the rank $d-1$ metric $q_{ab}$. The Ehresmann connection $k_a$ fixes a choice of horizontal bundle $H_k$, and the horizontal frame is denoted by $\{ m^a_I \}$. If the distribution $H_k$ is integrable, then the spacetime admits a foliation by co-dimension $1$ leaves.
• Figure 4: The green dashed lines represent null directions. For a timelike/spacelike/null hypersurface $\mathcal{N}$, the normal vector $n^\mu$ is spacelike/timelike/null, respectively. In the null case, the normal is also tangent since $n^\mu n_\mu = 0$, and it is then necessary to introduce a null transverse vector $\ell^\mu$ in order to project onto $\mathcal{N}$.
• Figure 5: This figure illustrates the membrane paradigm. The null hypersurface $\mathscr{H}$ represents the horizon, while the green hypersurfaces are timelike slices approaching it. As these slices move closer to the horizon, their induced geometry and dynamics undergo a Carrollian limit: the Lorentzian structure degenerates into a Carrollian one, and the induced Einstein equations reduce accordingly to the Carrollian fluid conservation equations.