---
title: Carroll-Weyl Scalings in Carrollian Geometry
url: https://www.emergentmind.com/topics/carroll-weyl-scalings
type: topic
---

# Carroll-Weyl Scalings in Carrollian Geometry

Carroll-Weyl scalings are conformal rescalings adapted to Carrollian geometry, where the basic structure is a degenerate spatial metric together with a distinguished temporal direction rather than a non-degenerate Lorentzian metric. In recent work, they appear in several closely related settings: as anisotropic conformal transformations of Carroll field theories with dynamical exponent $z$, as local gauge symmetries of Carrollian worldsheets in null string theory, and as Weyl rescalings of conformal Carroll structures compatible with projective data and torsion constraints [2505.23755], [2605.25817], [2606.00799]. Their common role is to organize how temporal and spatial data scale in the ultra-relativistic regime, but the precise implementation depends on context.

## 1. Geometric definition and scaling laws

A Carroll manifold is a smooth manifold equipped with a degenerate two-tensor $h_{\mu\nu}$ whose kernel is spanned by a nowhere-vanishing vector $\tau^\mu$; in flat Carrollian structures one may take
$$
\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).
$$
Conformal Carroll symmetries preserve these data up to Weyl rescaling. In the anisotropic formulation, the defining equations are
$$
\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},
$$
with exponent $z\equiv 2/k$ [2505.23755].

The associated global dilatation is generated by
$$
D^{(z)} = z\, t\,\partial_t + x^i \partial_i,
$$
so that
$$
t\to \lambda^z t,\qquad x^i\to \lambda x^i.
$$
In the classification of conformal Carroll algebras, this same anisotropic scaling is the defining action of the dilatation generator, and under spatial conformal maps with $dx'^2=\Omega^2(x)\,dx^2$ one has $\Phi\to \Omega(x)^{-\Delta}\Phi$, $t\to \Omega(x)^{2z} t$, and $\delta(x)\to \Omega(x)^{-d}\delta(x)$ for primary scalars and spatial delta functions [2409.19953].

The case $z=0$ is structurally special. In the field-theoretic formulation it corresponds to $k\to\infty$, so the Carrollian time direction is not rescaled while the spatial metric scales as $h_{\mu\nu}\to \lambda h_{\mu\nu}$. In this limit the Weyl Ward identity reduces to the tracelessness of the spatial stress,
$$
T^i{}_i=0,
$$
rather than a full spacetime trace condition [2505.23755].

A more intrinsic geometric formulation replaces $(\tau^\mu,h_{\mu\nu})$ by a Carroll structure $(v,\gamma)$ with $v^\mu \gamma_{\mu\nu}=0$. A conformal Carroll structure is then the equivalence class
$$
(v,\gamma)\sim \bigl(e^{-\frac12\lambda}v,\ e^\lambda \gamma\bigr),
$$
and a compatible connection satisfies
$$
\nabla v=-\tfrac12\,\varphi\otimes v,\qquad \nabla \gamma=\varphi\otimes \gamma,
$$
with Weyl one-form transforming as $\varphi\mapsto \varphi+d\lambda$ [2606.00799]. This formulation makes clear that Carroll-Weyl scaling is not merely a coordinate dilation; it is a rescaling of the defining degenerate geometric data.

## 2. Symmetry algebras and Ward identities

In two spacetime dimensions, the $z$-conformal Carroll algebra is generated by modes $L_n$ and $M_n$ with
$$
[L_n,L_m]=(n-m)L_{n+m},\qquad
[L_n,M_m]=((n+1)z-m)M_{m+n},\qquad
[M_n,M_m]=0.
$$
In three spacetime dimensions one introduces $L_n$, $\bar L_n$, and $M_{r,s}$, obtaining two Witt algebras acting on supertranslations with $z$-dependent weights [2505.23755]. The general infinite-dimensional structure was systematized as $\widetilde{\mathfrak{confcarr}_z(d+1)}=\mathrm{conf}(S^d)\ltimes_z C^\infty(S^d)$, with finite-dimensional truncations for integer or half-integer $z$ [2409.19953].

The $z=0$ algebra is closely related to warped conformal symmetry, but the equivalence is only algebraic. In two dimensions, the $z=0$ Carrollian algebra coincides as an infinite-dimensional Lie algebra with the Warped Conformal algebra, yet the global subgroups differ: the WCFT vacuum respects $SL(2,\mathbb{R})\times U(1)$, whereas the Carrollian vacuum respects a four-generator subgroup $\{L_0,L_{-1},M_0,M_1\}$ in $d=2$ and a seven-generator type-D subgroup in $d=3$. The distinction propagates to normal ordering, spectra, and correlators [2505.23755].

The stress-tensor sector reflects the same anisotropic structure. On a Carroll manifold, local tangent symmetries imply
$$
T^u{}_i=0,\qquad T^i{}_j=T^j{}_i,
$$
so there is no energy flux and the spatial stress is symmetric. For general $z$, the Weyl Ward identity is
$$
z\,T^u{}_u+T^i{}_i=0,
$$
while for $z=0$ it becomes purely spatial tracelessness [2505.23755]. In $d=3$ and $z=0$, the local operators
$$
M(z,\bar z)\equiv T^u{}_u,\qquad T_z\equiv T^u{}_z,\qquad T_{\bar z}\equiv T^u{}_{\bar z}
$$
transform as Virasoro primaries with weights $(1,1)$, $(2,1)$, and $(1,2)$, respectively; for general $z$, $M$ acquires weights $(1+z/2,1+z/2)$ while $T_z$ and $T_{\bar z}$ remain unchanged [2505.23755].

At $z=1$, conformal Carroll theories on curved backgrounds admit explicit Weyl rules for the Carrollian fields:
$$
\delta_\omega v^\mu=-\omega v^\mu,\qquad
\delta_\omega \tau_\mu=\omega \tau_\mu,\qquad
\delta_\omega h^{\mu\nu}=-2\omega h^{\mu\nu},\qquad
\delta_\omega h_{\mu\nu}=2\omega h_{\mu\nu},
$$
with scalar weight
$$
\delta_\omega \phi=-\frac{d-2}{2}\,\omega\,\phi.
$$
These transformations support boost-invariant conformal Carroll scalar actions whose energy-momentum tensors are traceless on shell [2207.03468].

## 3. Correlators and analytic branches

Carroll-Weyl scaling controls correlators already at the level of Ward identities. In the algebraic classification, translation and boost invariance imply that a two-point function splits into a magnetic term and an electric, ultra-local term,
$$
G^{(2)}(x_{12},t_{12}) = G(|x_{12}|)+F(t_{12})\delta(x_{12}),
$$
and for finite $z\neq 0$ dilatation covariance gives
$$
G^{(2)} = C_1 |x_{12}|^{-(\Delta_1+\Delta_2)}
+ C_2 |t_{12}|^{(\Delta_1+\Delta_2-d)/z}\,\delta(x_{12}).
$$
When spatial special conformal invariance is imposed and $\Delta_1=\Delta_2=\Delta$, the result is
$$
G^{(2)}=C_1 |x_{12}|^{-2\Delta}+C_2 |t_{12}|^{(d-2\Delta)/z}\delta(x_{12}),
$$
whereas distinct dimensions are still allowed in the electric sector [2409.19953].

In the anisotropic conformal Carroll theories analyzed holographically, the $z=0$ case is vacuum-sensitive. In $d=3$, primaries are labeled by eigenvalues of $(L_0,\bar L_0,M_{0,0})$, and time evolution takes the form
$$
\Phi(u,z,\bar z)=e^{-iQu}\,\phi(z,\bar z).
$$
For the $SL\times SL\times U(1)$ vacuum $W$, the two-point function is the standard two-dimensional power law on the spatial slice multiplied by $e^{-iQ(u_1-u_2)}$. For the type-D Carroll vacuum $C$, the $Q=0$ branch has only the spatial power law, while the $Q\neq 0$ branch is ultra-local in space,
$$
G_C^{(2)} \propto e^{-iQ(u_1-u_2)}\,
\delta^{(2)}(z_1-z_2,\bar z_1-\bar z_2),
$$
together with dimension and charge constraints. For generic $z\neq 0$, the quoted two-point function is spatially ultra-local and scales as a power of $u_1-u_2$ [2505.23755].

Momentum space exhibits the same structure in a different analytic form. The momentum-space dilatation Ward identity is
$$
\sum_{a=1}^n\Big[\Delta_a-d-\omega_a\partial_{\omega_a}-p_a^i\partial_{p_a^i}\Big]\mathcal A=0,
$$
and for scalar two-point functions the general solution is
$$
\langle \mathcal O_\Delta(\omega_1,\mathbf p_1)\mathcal O_\Delta(\omega_2,\mathbf p_2)\rangle
=
\delta^{(d-1)}(\mathbf p_1+\mathbf p_2)\Big[
b_1\,\delta(\omega_1+\omega_2)\,\omega_1^{2\Delta-d}
+
b_2\,\delta(\omega_1)\delta(\omega_2)\,k_1^{2\Delta-d+1}
\Big].
$$
The first term is the electric branch and the second the magnetic branch. Three-point functions decompose into five branches—one purely electric, one purely magnetic, and three mixed branches—and when $\Delta_i=(d-1)/2$ the mixed branches develop logarithmic terms such as
$$
\omega^{\Delta_t-2d+1}\ln\!\left(\frac{k}{\omega}\right),
$$
which originate from singularities in the Fourier transform rather than from new position-space branches [2512.06881].

## 4. Carroll-Weyl gauge symmetry on null strings

On a two-dimensional Carrollian worldsheet, the basic data are a rank-one degenerate spatial metric $h_{ab}$, its kernel vector field $v^a$, and a clock one-form $n_a$ normalized by $v^a n_a=1$. Writing $h_{ab}=\ell_a\ell_b$, the structure admits two independent local scalings:
$$
n_a\to e^{\chi_t} n_a,\qquad
v^a\to e^{-\chi_t} v^a,\qquad
\ell_a\to e^{\chi_s}\ell_a.
$$
Two linear combinations are singled out. The $\phi$-scaling has $\chi_t=\chi_s=\phi$ and leaves $\mathcal V^a=\sqrt h\,v^a$ invariant. The $\chi$-scaling has $\chi_t=-\chi_s=\chi$, preserves the volume form $\epsilon_{ab}$, and rescales $\mathcal V^a$ as $\mathcal V^a\to e^{-2\chi}\mathcal V^a$ [2605.25817].

Gauging the volume-preserving scaling requires a Carroll-Weyl connection $\mathcal W_a$ and covariant derivative
$$
D_a X^\mu=\partial_a X^\mu+\mathcal W_a X^\mu.
$$
The fully gauged null-string action is
$$
S_{\mathrm{Carroll}}=\frac{\kappa}{2}\int d^2\sigma\,
\mathcal V^a\mathcal V^b D_aX^\mu D_bX^\nu \eta_{\mu\nu},
$$
which is invariant under worldsheet diffeomorphisms, the $\phi$-scaling, and the $\chi$-scaling [2605.25817]. Fixing the gauge $\mathcal V^a\mathcal W_a=0$ reduces the theory to the ILST action, and the residual symmetry satisfies
$$
\mathcal V^a\partial_a \chi=0.
$$
This residual codimension-one symmetry was identified as the overlooked partial gauge symmetry of the ILST formulation [2605.25817].

The Hamiltonian analysis adds a genuinely new first-class constraint,
$$
\mathcal C_3=P\cdot X,
$$
alongside
$$
\mathcal C_1=P^2,\qquad \mathcal C_2=P\cdot X'.
$$
In mode language, $\mathcal C_3$ generates a weight-one current extending the standard centerless $\mathrm{BMS}_3$ algebra of null strings. The extended algebra contains modes $L_n$, $M_n$, and $N_n$ with
$$
\{L_m,N_n\}=i n N_{m+n},\qquad
\{M_m,N_n\}=-2 M_{m+n},\qquad
\{N_m,N_n\}=0.
$$
A central point of the recent null-string literature is that this Carroll-Weyl gauge symmetry is intrinsic to Carrollian worldsheets and is not obtained from the ultra-relativistic limit of the tensile string gauge algebra [2605.26822].

Quantization reflects the same enlargement. In the path-integral treatment, complete gauge fixing of $\mathrm{Diff}\ltimes \mathrm{CW}$ produces not the old BMS $bc$ system alone but a $bcs$ system, with a scalar ghost $s$ and scalar antighost $b^s$ for Carroll-Weyl scaling. The ghost action contains the characteristic mixing term $-2 s b^0$, and the BRST complex must therefore include the $S$-sector generated by $C_3=P\cdot X$ [2606.04999].

## 5. Compatible connections and Weyl-type uniqueness

In differential-geometric terms, Carroll-Weyl scaling is encoded by the conformal class
$$
(v,\gamma)\sim \bigl(e^{-\frac12\lambda}v,\ e^\lambda \gamma\bigr),
$$
together with a compatible connection satisfying
$$
\nabla v=-\tfrac12\,\varphi\otimes v,\qquad
\nabla\gamma=\varphi\otimes\gamma,\qquad
\varphi\mapsto \varphi+d\lambda.
$$
This is the Carrollian analogue of a Weyl connection, but the torsion constraints differ sharply from the pseudo-Riemannian case [2606.00799].

If $T^\rho{}_{\mu\nu}$ denotes torsion, compatibility implies
$$
T_{(\mu\nu)\rho}v^\rho
=
\frac12\,\varphi(v)\,\gamma_{\mu\nu}
-\frac12\,(\mathcal L_v \gamma)_{\mu\nu}.
$$
Hence torsion-free Carroll-Weyl connections exist only when
$$
\mathcal L_v\gamma=\varphi(v)\gamma.
$$
This condition has no exact analogue in ordinary Weyl geometry and expresses the fact that conformal rescaling and the degenerate temporal direction are intertwined [2606.00799].

Projective structure enters through unparametrized autoparallels. Two linear connections are projectively equivalent when their difference satisfies
$$
D^\rho_{(\mu\nu)}=\delta^\rho_{(\mu}\eta_{\nu)},
$$
which preserves the set of unparametrized geodesics. The Carroll Weyl-type theorem states that if two compatible Carroll-Weyl connections are projectively equivalent and have the same free torsion components with respect to a timelike co-direction $[\tau]$, then their difference is
$$
D^\rho_{\mu\nu}=f\,\tau_\mu \delta^\rho_\nu
$$
for some function $f$ [2606.00799].

This result is weaker than the corresponding pseudo-Riemannian and Galilei statements. A Carroll Weyl metric can be represented by the data
$$
[v,\gamma,\varphi,\hat\chi],\qquad [\tau],\qquad T_{\mathrm{free}},
$$
with scaling
$$
(v,\gamma,\varphi,\hat\chi)\sim
\bigl(e^{-\frac12\lambda}v,\ e^\lambda\gamma,\ \varphi+d\lambda,\ e^{\frac12\lambda}\hat\chi\bigr),
$$
but compatibility plus projective structure determines $\varphi$ only up to
$$
\varphi\mapsto \varphi+2 f\tau.
$$
A plausible implication is that Carroll-Weyl geometry retains an irreducible ambiguity along the temporal co-direction that has no direct pseudo-Riemannian counterpart.

## 6. Gravity, holography, and broader applications

In anisotropic Carrollian holography, Carroll-Weyl scaling appears both on the boundary and in the bulk. For $z=0$, a canonical four-dimensional plane wave metric is
$$
ds^2=2\,du\,dv+e^u(dx^2+dy^2),
$$
with Brinkmann form
$$
ds^2=2\,dx^+dx^-+\frac{x^2+y^2}{4}(dx^+)^2+dx^2+dy^2.
$$
More generally,
$$
ds^2_{(k)}=2\,du\,dv+u^k(dx^2+dy^2),
$$
with $z=2/k$, realizes anisotropic scale transformations in the bulk. Residual diffeomorphisms of suitable asymptotically plane-wave phase spaces reproduce the expected infinite-dimensional conformal Carroll algebra at the null boundary, thereby furnishing a framework for anisotropic Carrollian holography [2505.23755].

In scaling-Carroll gravity, the local dilatation generator $D$ acts on the gravity multiplet by
$$
\delta_D \tau_\mu = z\lambda_D \tau_\mu,\qquad
\delta_D e_\mu{}^a=\lambda_D e_\mu{}^a,\qquad
\delta_D b_\mu=\partial_\mu \lambda_D,
$$
so that
$$
\delta_D h_{\mu\nu}=2\lambda_D h_{\mu\nu},\qquad
\delta_D h^{\mu\nu}=-2\lambda_D h^{\mu\nu}.
$$
The extrinsic curvature
$$
K_{\mu\nu}=-\tfrac12 \mathcal L_\tau h_{\mu\nu}
$$
transforms as
$$
\delta_D K_{\mu\nu}=(2-z)\lambda_D K_{\mu\nu}-h_{\mu\nu}\partial_0\lambda_D.
$$
After scale fixing, one has $b_0=-K/d$ and $\delta_G b_a=(1/d)K\lambda_a$, and the theory admits three regimes: dynamical Carroll gravity, Aristotelian gravity, and a fracton gauge theory coupled to Aristotelian geometry [2512.20736]. Earlier ultra-relativistic gravity work already identified anisotropic Weyl invariance in $2+1$ dimensions for $z=0$, where $\tau_\mu$ and $v^\mu$ are inert while spatial frames scale [1505.05011].

Matter models realize the same symmetry in explicit Lagrangians. Two conformal Carroll scalar actions with local boost symmetry are known on curved Carrollian backgrounds. The timelike theory is built from $(v^\mu\partial_\mu\phi)^2$ plus improvement terms involving $K_{\mu\nu}$ and $K$, while the spacelike theory combines $h^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi$ with the Weyl-covariant curvature combination $h^{\mu\nu}\tilde R_{\mu\nu}-\tilde\nabla_\mu a^\mu$ and Lagrange-multiplier constraints. In both cases the on-shell Ward identity gives $T^\mu{}_\mu=0$, and in the spacelike branch the theory reduces to a lower-dimensional Euclidean CFT after solving the constraints and using Weyl invariance [2207.03468].

At null infinity, Carroll-Weyl rescaling is also part of the natural boundary calculus. For Carrollian boundary data $(a_{ij},b_i,\Omega)$ one has
$$
a_{ij}\to e^{2\phi} a_{ij},\qquad
b_i\to e^\phi b_i,\qquad
\Omega\to e^\phi \Omega,
$$
so that the clock one-form $u$ scales with weight $+1$ and the observer field $v$ with weight $-1$. The acceleration $\varphi_i$ and expansion $\Theta$ serve as the spatial and temporal components of the Weyl connection, and the resulting Weyl-Carroll covariant tensors organize electric and magnetic towers of $\mathrm{BMS}_4$ charges [2212.14062].

Carroll-Weyl weights also control ultra-relativistic limits of ordinary conformal field theory. In the Carrollian limit of CFT$_3$ OPE blocks, a scalar primary is normalized with weight $w=\Delta-1$, and spinning components with $w=\Delta+\ell-1$. These weights select finite Carrollian operators and lead, after Mellin-Fourier projection, to towers of $\mathfrak{sl}(2,\mathbb C)$ primaries of negative integer dimensions. The resulting blocks generate conformally soft photon and graviton towers and reproduce the celestial $S$ and $w_{1+\infty}$ algebras [2508.19981].

Taken together, these developments show that Carroll-Weyl scaling is not a marginal refinement of Carrollian kinematics. It is a unifying organizing principle for conformal algebras, Ward identities, ultra-local correlation structures, worldsheet gauge symmetries, Weyl-compatible connections, and holographic or gravitational realizations in the ultra-relativistic regime.

Source: https://www.emergentmind.com/topics/carroll-weyl-scalings