Aspects of Carrollian field theory from holography

Abstract: We study holography for three-dimensional asymptotically flat spacetimes in which the bulk gravitational dynamics is conjectured to be dual to a two-dimensional Carrollian (equivalently BMS) conformal field theory. Such theories are known to arise as ultrarelativistic ($c\rightarrow0$) contractions of relativistic 2d conformal field theories. Under this contraction, the Virasoro algebra becomes the conformal Carrollian algebra and the Brown-Henneaux central charges become $c_L=0$, $c_M=\frac{3}{G}$. In this article, we recover these central charges directly from holography. We first construct the holographic quasilocal stress tensor for asymptotically flat spacetimes carrying Bondi mass and angular momentum. Under BMS$_3$ transformations, the stress tensor acquires an inhomogeneous term, the ``BMS Schwarzian", and extracting the central charges from it reproduces the algebraic result exactly. We then obtain the action for the boundary BMS Schwarzian modes. Finally, we show that the holographic stress-tensor correlators satisfy the expected Ward identity, while stress-tensor conservation obeys flux-balance laws.

• The paper constructs a null-boundary holographic stress tensor for asymptotically flat three-dimensional gravity, matching ADM charges and deriving the BMS3 central charges c_L = 0 and c_M = 3/G_N without an AdS contraction.
• The resulting Carrollian Ward identities become flux-balance laws that reproduce Bondi mass and angular-momentum loss, while their anomalous transformations recover the centrally extended BMS3 current algebra.
• The on-shell null-boundary action yields the BMS Schwarzian on the vacuum coadjoint orbit BMS3/ISO(2,1), while unresolved issues include Weyl anomalies, other asymptotic regions, and the quantum Carrollian theory.

Overview

This paper constructs the holographic dictionary for three-dimensional asymptotically flat spacetimes (AFS) directly at null infinity, without passing through an ultra-relativistic contraction limit of AdS$_3$/CFT$_2$. The bulk dynamics is conjectured to be dual to a two-dimensional Carrollian (BMS) conformal field theory living on $\mathscr{I}^+$. The authors adapt the Balasubramanian–Kraus holographic renormalization program—previously formulated for timelike AdS boundaries—to the degenerate (Carrollian) geometry of null infinity, and extract the central charges of the boundary theory from the transformation properties of a holographically computed quasilocal stress tensor. The main quantitative result is the direct recovery of the Barnich–Compère central charges $c_L = 0$ and $c_M = 3/G_N$ from the gravitational side, together with a Ward identity expressed as flux-balance laws and an on-shell boundary action reproducing the BMS Schwarzian.

Holographic stress tensor at null infinity

The construction begins with the general vacuum AFS$_3$ solution in Bondi gauge,

$$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$

where the constraint equations leave only the mass aspect $M(\phi)$ and angular momentum aspect $N(\phi)$ as free data. The authors foliate the bulk by timelike hypersurfaces $r - \tfrac{1}{2}uM(\phi) = \text{const}$, whose normals become null as $_2$0. A null-tilted co-normal $_2$1 is introduced, together with the auxiliary rigging vector $_2$2 normalized so that $_2$3. The boundary stress tensor is then computed from the Weingarten map $_2$4 via $_2$5, following the null-boundary Brown–York formalism of Chandrasekaran et al. and Bhambure–Krishna.

The resulting components are

Component	Value
$_2$6	$_2$7
$_2$8	$_2$9
$\mathscr{I}^+$0	$\mathscr{I}^+$1
$\mathscr{I}^+$2	$\mathscr{I}^+$3

The boundary stress tensor is defined by stripping the universal $\mathscr{I}^+$4 factor, justified by the radial inertness of the Carrollian source $\mathscr{I}^+$5 and the linear scaling of the volume form—the exact analogue of the Fefferman–Graham definition of the CFT stress tensor in AdS. Contracting this tensor with BMS$\mathscr{I}^+$6 generators $\mathscr{I}^+$7 reproduces the ADM Bondi charges, $\mathscr{I}^+$8 and $\mathscr{I}^+$9, in agreement with the covariant phase-space results of Barnich–Compère and the Wald–Zoupas prescription.

Central charges from the BMS Schwarzian

Under a finite BMS$c_L = 0$0 transformation $c_L = 0$1, the two components of the Carrollian stress tensor acquire anomalous inhomogeneous terms controlled by $c_L = 0$2 (via the ordinary Schwarzian $c_L = 0$3 in the superrotation sector) and $c_L = 0$4 (via the BMS Schwarzian $c_L = 0$5 in the supertranslation sector). Evaluating the Schwarzian and BMS Schwarzian to linear order on the Minkowski vacuum, where the boundary currents take the values $c_L = 0$6 and $c_L = 0$7, and matching against the direct gravitational transformation of the holographic stress tensor fixes

$c_L = 0$8

This is the central result: the charges previously obtained algebraically from the coadjoint orbit analysis of BMS$c_L = 0$9 [Barnich–Compère] or by ultra-relativistic contraction of two Virasoro algebras with Brown–Henneaux charges are here recovered directly from the gravitational stress tensor, with no contraction limit taken. The vanishing of $c_M = 3/G_N$0 reflects parity invariance of pure 3d Einstein gravity and would be lifted by a gravitational Chern–Simons term. The paper notes a caveat: the comparison is performed on the Minkowski background for convenience, though the same values are claimed to follow from the general Bondi metric. The authors also observe that the bulk trace $c_M = 3/G_N$1 does not vanish, and its connection to the Carrollian Weyl anomaly remains unexplained.

Flux-balance Ward identities

The generating functional $c_M = 3/G_N$2 is defined on the Carrollian data $c_M = 3/G_N$3 at $c_M = 3/G_N$4, with sources packaged into the frame deformation $c_M = 3/G_N$5. Diffeomorphism invariance yields the Ward identity

$c_M = 3/G_N$6

where $c_M = 3/G_N$7 is matter flux through null infinity. The stress tensor is therefore not conserved but obeys flux-balance laws: at vanishing sources these reduce to the Bondi mass-loss and angular-momentum-loss equations, and integrating them across $c_M = 3/G_N$8 produces the memory effect and the 3d soft graviton theorem. The appendix verifies explicitly that the projected Carrollian divergence of the finite-$c_M = 3/G_N$9 stress tensor yields $_3$0 and $_3$1 after renormalization, with all subleading $_3$2 terms cancelling. The anomalous transformation laws of the densities $_3$3 and $_3$4 are shown, via canonical smearing and inversion of delta-function derivatives, to fix the local Poisson brackets and hence the centrally extended BMS$_3$5 mode algebra—providing a consistency check that the holographic charges generate the expected current algebra.

Boundary graviton action and the BMS Schwarzian

The null-boundary contribution to the gravitational action, $_3$6, is evaluated on shell. For the Bondi family the asymptotic generator is affinely parametrized ($_3$7) and $_3$8, so the boundary term reduces to $_3$9 with $$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$0—the Hamiltonian of the supermomentum zero mode. Dressing the Minkowski vacuum ($$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$1) by a finite superrotation and using the chain-rule identity $$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$2 gives

$$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$3

the BMS Schwarzian action, with the superrotation mode living on $$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$4. The full boundary-graviton phase space is the vacuum coadjoint orbit $$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$5, which fibers as $$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$6. The Schwarzian alone is only the Hamiltonian piece; the complete first-order action $$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$7 includes a presymplectic term whose variation imposes the flat-space Bondi equations $$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$8 and $$ds^2 = M(\phi)\,du^2 - 2\,du\,dr + 2\Big[N(\phi) + \tfrac{u}{2}\,\partial_\phi M(\phi)\Big]du\,d\phi + r^2 d\phi^2,$$9. This agrees with prior results from Chern–Simons reduction [Barnich–Gonzalez–Salgado-Rebolledo, Merbis–Riegler] and the soft-graviton analysis of Cotler et al.

Carrollian scalar theories and the contraction limit

Two appendices supply context. The first reviews the timelike (electric), spacelike (magnetic), and mixed Carrollian scalar actions and shows that all three stress tensors transform homogeneously under BMS$M(\phi)$0, implying classical central charges $M(\phi)$1. The authors caution that this classical result does not fix the quantum central charges, which may depend on vacuum choice, normal ordering, and improvements; the quantum treatment of Hao et al. for the timelike theory finds $M(\phi)$2, $M(\phi)$3, and whether other Carrollian actions reproduce the gravitational values is left open. The second appendix reviews the ultra-relativistic contraction $M(\phi)$4, $M(\phi)$5, under which the Brown–Henneaux charges map as $M(\phi)$6 and $M(\phi)$7, recovering $M(\phi)$8, $M(\phi)$9 for $N(\phi)$0—consistent with the direct holographic extraction.

Limitations and open questions

The paper is explicit about several restrictions. The holographic dictionary is constructed only at $N(\phi)$1; the analogous constructions at spatial infinity $N(\phi)$2 (via dS$N(\phi)$3 slicing) and timelike infinity $N(\phi)$4 (via hyperboloidal slicing), and their unification into a single BMS structure across the five asymptotic regions as achieved in 4d by Compère et al., are deferred to future work. The identification of the bulk stress-tensor trace with the Carrollian Weyl anomaly is not understood. The Weyl and Carrollian boost Ward identities are only sketched, since the boost anomaly requires relaxing the Bondi gauge conditions so that $N(\phi)$5. Finally, the connection between the classical free Carrollian scalars (zero central charge) and the gravitational central charges remains unresolved at the quantum level.

Conclusion

The paper establishes a direct holographic route to 2d Carrollian CFT data from AFS$N(\phi)$6 gravity: a Weingarten-map stress tensor whose Brown–York charges match the ADM and covariant phase-space results, whose anomalous BMS$N(\phi)$7 transformation fixes $N(\phi)$8 and $N(\phi)$9 without any contraction limit, whose conservation law is the flux-balance/Bondi mass-loss identity, and whose on-shell null-boundary action is the BMS Schwarzian on the vacuum coadjoint orbit $r - \tfrac{1}{2}uM(\phi) = \text{const}$0. The remaining open questions—extension to the other asymptotic regions, the Weyl/boost anomalies beyond Bondi gauge, and the quantum Carrollian CFT realizing the gravitational charges—define the immediate continuation of this program.