---
title: Carroll-Weyl Gauge Symmetry
url: https://www.emergentmind.com/topics/carroll-weyl-gauge-symmetry
type: topic
---

# Carroll-Weyl Gauge Symmetry

Carroll-Weyl gauge symmetry is a local scale symmetry associated with Carrollian, rather than ordinary Lorentzian, geometry. In the null-string literature, it denotes the local scaling symmetry of a Carrollian worldsheet, where two inequivalent Weyl-type scalings exist and the standard Isberg-Lindström-Sundborg-Theodoridis (ILST) action is recovered only after gauge fixing a more complete Carroll-gauged theory [2605.25817]. In related work, the same phrase is also used for a residual gauge ambiguity of compatible Carroll connections determined by conformal and projective data [2606.00799], and for a Carroll-Weyl covariant gauge choice at null infinity in asymptotically flat gravity [2212.14062]. The common structural feature is that degenerate Carrollian data admit Weyl-type rescalings with no ordinary Lorentzian analogue.

## 1. Carrollian worldsheet geometry and the two Weyl-type scalings

The null-string worldsheet is treated as a \(2\)D Carrollian manifold with basic data
\[
(\ell_a,\ v^a,\ n_a),
\]
where
\[
h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.
\]
Because Carrollian geometry has both a distinguished temporal direction \(v^a\) and spatial form \(\ell_a\), it admits two independent scaling options rather than the single Weyl scaling familiar from an ordinary tensile-string worldsheet [2605.25817].

The first is the symmetric, “volume-modulating” Carroll-Weyl scaling,
\[
\phi:=\chi_t=\chi_s,
\]
under which
\[
n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.
\]
This implies
\[
h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.
\]
It is the analogue of ordinary Weyl scaling in \(2\)D. A key invariant is
\[
\mathcal V^a:=\sqrt{h}\,v^a,
\]
which remains invariant under \(\phi\)-scaling.

The second is the antisymmetric, “volume-preserving” Carroll-Weyl scaling,
\[
\chi:=\chi_t=-\chi_s,
\]
with
\[
n_a\to e^\chi n_a,\qquad v^a\to e^{-\chi}v^a,\qquad \ell_a\to e^{-\chi}\ell_a,
\]
so that
\[
h_{ab}\to e^{-2\chi}h_{ab},\qquad \epsilon_{ab}\to \epsilon_{ab}.
\]
This second scaling has no analogue in ordinary Lorentzian \(2\)D geometry and is the genuinely new Carrollian possibility emphasized in the null-string construction [2605.25817].

## 2. Gauging Carroll-Weyl symmetry in the null-string action

The standard ILST null-string action is
\[
S_{\text{ILST}}=\frac{\kappa}{2}\int d^2\sigma\;\mathcal V^a\mathcal V^b\,\partial_aX^\mu\partial_bX^\nu\,\eta_{\mu\nu}.
\]
It is invariant under diffeomorphisms and the \(\phi\)-scaling, but not under the \(\chi\)-scaling. To make the \(\chi\)-scaling local, the Carroll-Weyl gauge field \(\mathcal W_a\) is introduced and
\[
\partial_a X^\mu\quad\to\quad D_a X^\mu:=\partial_a X^\mu+\mathcal W_a X^\mu.
\]
The resulting action is
\[
S=\frac{\kappa}{2}\int d^2\sigma\;\mathcal V^a\mathcal V^b\,D_aX^\mu D_bX^\nu\,\eta_{\mu\nu}.
\]
In the normalization used in the worldsheet-gauging analysis, the infinitesimal transformations for \(\eta=(\xi^a,\chi,\phi)\) are
\[
\delta_\eta X^\mu=\xi^a\partial_aX^\mu+2\chi X^\mu,
\]
\[
\delta_\eta \mathcal V^a=\xi^b\partial_b\mathcal V^a-\mathcal V^b\partial_b\xi^a+\frac12(\partial_b\xi^b)\mathcal V^a-2\chi\mathcal V^a,
\]
\[
\delta_\eta \mathcal W_a=\xi^b\partial_b\mathcal W_a+\mathcal W_b\partial_a\xi^b-2\partial_a\chi.
\]
Accordingly, the full gauge symmetry consists of \(2\)D diffeomorphisms, the \(\phi\)-scaling, and local \(\chi\)-scaling. In this formulation the \(\phi\)-scaling is inert on the dynamical fields, but geometrically it remains part of the Carroll-Weyl structure [2605.25817].

The same construction is presented in later Hamiltonian and path-integral treatments with a different normalization of the scaling parameter, for example
\[
\delta_\chi X^\mu=\chi X^\mu,\qquad \delta_\chi V^a=-\chi V^a,\qquad \delta_\chi W_a=-\partial_a\chi,
\]
together with the gauged action
\[
S_{CW}[X,V,W] = \frac{1}{2\pi}\int d^2\sigma\, V^aV^b D_aX\cdot D_bX.
\]
This indicates a normalization choice rather than a change in the underlying gauge content [2606.04999].

A further point made in the worldsheet construction is that one cannot build a useful \(\chi\)-invariant kinetic term for \(\mathcal W_a\) from \(\mathcal V^a\), \(h^{ab}\), and
\[
F_{ab}=\partial_{[a}\mathcal W_{b]}.
\]
For that reason, the action above is described as essentially the most general diffeomorphism- and \(\chi\)-invariant null-string action of this type [2605.25817].

## 3. Gauge fixing, recovery of the ILST action, and the overlooked partial-gauge symmetry

The gauge choice used to recover the standard ILST formulation is
\[
\mathcal V^a\mathcal W_a=0.
\]
Under a \(\chi\)-transformation,
\[
\delta_\chi(\mathcal V^a\mathcal W_a)=-2\mathcal V^a\partial_a\chi
\]
after imposing the gauge condition itself. Hence the gauge fixes \(\chi\) only up to transformations satisfying
\[
\mathcal V^a\partial_a\chi=0.
\]
In this gauge,
\[
\mathcal V^a D_a X^\mu=\mathcal V^a\partial_a X^\mu,
\]
so the gauged action reduces exactly to the ILST action [2605.25817].

The residual \(\chi\)-transformations are therefore those for which \(\chi\) is constant along the integral curves of \(\mathcal V^a\). Equivalently, the gauge parameter depends only on one worldsheet coordinate and is a codimension-\(1\) gauge parameter. This residual transformation is precisely the “partial-gauge symmetry” that had previously appeared as an overlooked feature of the ILST action. In the gauge-fixed ILST description it looked anomalous or ad hoc; in the Carroll-gauged formulation it is simply the leftover of an ordinary local \(\chi\)-gauge symmetry after a legitimate gauge choice [2605.25817].

The equations of motion and constraints in the gauged formulation sharpen this interpretation. Variation with respect to \(X^\mu\) gives
\[
D_a\!\left(\mathcal V^a\mathcal V^b D_bX_\mu\right)=0.
\]
Defining
\[
P_\mu:=\kappa \mathcal V^a D_aX_\mu,
\]
this becomes
\[
D_a(\mathcal V^a P^\mu)=0.
\]
Variation with respect to \(\mathcal V^a\) gives the constraints
\[
\mathcal C_a=\kappa\,\mathcal V^b D_aX^\mu D_bX_\mu=D_aX^\mu P_\mu=0,
\]
while variation with respect to \(\mathcal W_a\) yields
\[
\frac{\delta S}{\delta \mathcal W_a}=\kappa\,\mathcal V^a\mathcal V^b X^\mu D_bX_\mu=\mathcal V^a\,\mathcal C_3=0,
\qquad
\mathcal C_3:=P\cdot X.
\]
In the gauge \(\mathcal V^a\mathcal W_a=0\), these reduce to the usual null-string constraints, including
\[
P\cdot P=0,\qquad P\cdot(\ell^a\partial_aX)=0,\qquad P\cdot X=0.
\]
The gauge field \(\mathcal W_a\) has no classical dynamics and is a pure gauge field [2605.25817].

## 4. Hamiltonian structure, extended BMS\(_3\), and classical degrees of freedom

The consistent Hamiltonian treatment shows that Carroll-Weyl symmetry is not optional. In the gauged system with
\[
q^A(\tau,\sigma)=\{X^\mu,\mathcal V^a,\mathcal W_a\},
\]
the momentum conjugate to \(X^\mu\) is
\[
P_\mu=\kappa\,\mathcal V^0\mathcal V^a D_aX_\mu,
\]
while the auxiliary fields satisfy the primary constraints
\[
\Pi_a=0,\qquad \mathcal P^a=0.
\]
The canonical Hamiltonian is
\[
H_c=\int d\sigma\left[ \frac{P_\mu P^\mu}{2\kappa(\mathcal V^0)^2} -\frac{\mathcal V^1}{\mathcal V^0}P_\mu D_1X^\mu -\mathcal W_0 (P_\mu X^\mu) \right].
\]
Stability of the primary constraints produces the secondary constraints
\[
\mathcal C_1:=P_\mu P^\mu\approx 0,\qquad 
\mathcal C_2:=P_\mu D_1X^\mu\approx 0,\qquad 
\mathcal C_3:=P_\mu X^\mu\approx 0.
\]
The third constraint is the Carroll-Weyl generator that earlier null-string analyses had overlooked [2605.26822].

The paper states explicitly that this extra symmetry cannot be obtained from the ultra-relativistic Carrollian limit of tensile strings. The ordinary tensile string has a \(2\)D worldsheet metric and the familiar Weyl symmetry acting on that metric, whereas the Carrollian null-string theory is built from the vector density \(\mathcal V^a\), and the relevant local symmetry rescales \(X^\mu\) and \(\mathcal V^a\) rather than a worldsheet metric [2605.26822].

The constraint algebra enlarges the usual centerless BMS\(_3\) algebra. For smeared generators,
\[
\mathcal C_1[f]=\int d\sigma\, f(\sigma)\mathcal C_1(\sigma),\quad 
\mathcal C_2[g]=\int d\sigma\, g(\sigma)\mathcal C_2(\sigma),\quad 
\mathcal C_3[k]=\int d\sigma\, k(\sigma)\mathcal C_3(\sigma),
\]
the Poisson brackets are
\[
\{\mathcal C_1[f],\mathcal C_1[g]\}=0,
\]
\[
\{\mathcal C_1[f],\mathcal C_2[g]\}=\mathcal C_1[fg'-f'g],
\]
\[
\{\mathcal C_1[f],\mathcal C_3[g]\}=-2\,\mathcal C_1[fg],
\]
\[
\{\mathcal C_2[f],\mathcal C_2[g]\}=\mathcal C_2[fg'-f'g],
\]
\[
\{\mathcal C_2[f],\mathcal C_3[g]\}=\mathcal C_3[fg'],
\]
\[
\{\mathcal C_3[f],\mathcal C_3[g]\}=0.
\]
At mode level,
\[
L_n=\int_0^{2\pi}d\sigma\,e^{-in\sigma}\mathcal C_2(\sigma),\qquad 
M_n=\int_0^{2\pi}d\sigma\,e^{-in\sigma}\mathcal C_1(\sigma),\qquad 
N_n=\int_0^{2\pi}d\sigma\,e^{-in\sigma}\mathcal C_3(\sigma),
\]
and
\[
\{L_m,L_n\}=i(m-n)L_{m+n},\qquad
\{L_m,M_n\}=i(m-n)M_{m+n},\qquad
\{M_m,M_n\}=0,
\]
\[
\{L_m,N_n\}=in\,N_{m+n},\qquad
\{M_m,N_n\}=-2M_{m+n},\qquad
\{N_m,N_n\}=0.
\]
The new generator transforms as a weight-one operator under the Witt algebra, and the classical null string is therefore described by an extended BMS\(_3\) algebra rather than by the older two-constraint BMS\(_3\) system [2605.26822].

The same analysis concludes that the classical null string in a \(D\)-dimensional Minkowski target space has
\[
D-3
\]
propagating modes, not \(D-2\). This one-mode reduction is the direct consequence of the extra first-class constraint generated by Carroll-Weyl symmetry [2605.26822].

## 5. Path-integral quantization and the \(bcs\) ghost system

The path-integral quantization of the tensionless bosonic string changes once all local gauge symmetries of the Carrollian worldsheet are gauge fixed. The unfixed functional integral is written as
\[
Z_{CW} = \int \frac{DX\,DV\,DW}{\mathrm{Vol}(\mathrm{Diff}\ltimes \mathrm{CW})} \,e^{iS_{CW}[X,V,W]}.
\]
Because the gauge group is \(\mathrm{Diff}\ltimes \mathrm{CW}\), the Faddeev-Popov determinant must include three gauge directions rather than two [2606.04999].

A convenient set of gauge conditions is
\[
G^0=V^0-1=0,\qquad G^1=V^1=0,\qquad G^s=V^aW_a=0.
\]
On the gauge slice \(V^a=(1,0)\), the infinitesimal variations are
\[
\delta G^0 = -\frac12\partial_0\epsilon^0+\frac12\partial_1\epsilon^1-\lambda,
\]
\[
\delta G^1 = -\partial_0\epsilon^1,
\]
\[
\delta G^s = -\partial_0\lambda.
\]
The Faddeev-Popov operator is therefore
\[
\Delta_{CW} = \begin{pmatrix}
-\frac12\partial_0 & \frac12\partial_1 & -1 \\
0 & -\partial_0 & 0 \\
0 & 0 & -\partial_0
\end{pmatrix},
\]
acting on \((\epsilon^0,\epsilon^1,\lambda)\). The off-diagonal entry \(-1\) expresses the fact that the gauge condition \(V^0=1\) is moved both by a diffeomorphism and by a Carroll-Weyl rescaling [2606.04999].

Exponentiating the determinant yields a \(bcs\) ghost system rather than only the old BMS \(bc\) system:
\[
c^A=(c^0,c^1,s),\qquad b_A=(2b^0,2b^1,b^s).
\]
Here \(s\) is the fermionic scalar ghost for Carroll-Weyl transformations and \(b^s\) is its scalar antighost. The gauge-fixed action is
\[
S_{\mathrm{gf}}=\frac{1}{2\pi}\int d^2\sigma\, \left[ \dot X^2 +i\left( c^0\partial_0b^0 -c^1\partial_1b^0 +2c^1\partial_0b^1 +s\partial_0b^s -2sb^0 \right) \right].
\]
The term \(s\partial_0 b^s\) is the kinetic term for the new ghost sector, while \(-2sb^0\) is the mixing term that encodes the nontrivial coupling between Carroll-Weyl scaling and the temporal BMS ghost sector [2606.04999].

The residual gauge-preserving conditions are
\[
\partial_0\epsilon^1=0,\qquad \partial_0\lambda=0,\qquad \partial_0\epsilon^0=\partial_1\epsilon^1-2\lambda,
\]
so that
\[
\epsilon^1=f(\sigma),\qquad \lambda=\chi(\sigma),\qquad \epsilon^0=\tau\big(f'(\sigma)-2\chi(\sigma)\big)+g(\sigma).
\]
The extra arbitrary function \(\chi(\sigma)\) is the residual Carroll-Weyl scaling. The corresponding ghost equations of motion, mode expansions, and oscillator algebra enlarge the BRST complex by a new fermionic scalar pair \((s,r)\), and the BRST operator must now include the \(S_n\) sector of the extended BMS algebra [2606.04999].

A central consequence is that the familiar \(D=26\) consistency check based only on the older BMS \(bc\) ghosts is a partially gauge-fixed calculation. In the Carroll-Weyl covariant quantum theory, the anomaly problem must be recomputed for the full matter-plus-\(bcs\) system, and the physical-state conditions must include
\[
S_n|\Psi\rangle=0
\]
alongside the usual \(L_n|\Psi\rangle=0\) and \(M_n|\Psi\rangle=0\) constraints [2606.04999].

## 6. Other Carroll-Weyl constructions and the broader Weyl context

The phrase “Carroll-Weyl gauge symmetry” is not confined to null strings. In the theoremic study of Galilei and Carroll geometry, a conformal Carroll structure is an equivalence class
\[
[v,\gamma]
\]
under
\[
(v,\gamma)\sim \big(e^{-\frac12\lambda}v,\; e^\lambda \gamma\big),
\]
and a compatible connection \(\nabla\) satisfies
\[
\nabla v = -\frac12\,\varphi\otimes v, \qquad \nabla \gamma = \varphi\otimes \gamma.
\]
Here the scale connection \(\varphi\) transforms as
\[
\varphi\mapsto \varphi + d\lambda.
\]
The Carroll analogue of Weyl’s theorem then shows that compatible connections with the same conformal and projective structures, and the same free torsion components relative to a timelike co-direction \([\tau]\), are determined only up to
\[
D^\rho_{\mu\nu}=f\,\tau_\mu \delta^\rho_\nu.
\]
Under this shift,
\[
\tilde\varphi=\varphi+2f\,\tau.
\]
This residual \(1\)-parameter freedom is identified as the hallmark of Carroll-Weyl gauge symmetry. Unlike the Lorentzian and Galilei cases, the compatible Carroll connection is therefore not uniquely fixed by conformal and projective data [2606.00799].

In asymptotically flat gravity, the expression has a different meaning. The paper on Ehlers symmetry and dual charges states explicitly that its “Carroll-Weyl gauge” is not a separate symmetry group in the group-theoretic sense, but rather the boundary-covariant gauge choice in which the null conformal boundary is manifestly a three-dimensional Carrollian geometry and residual rescalings are handled with Weyl covariance. The boundary data are
\[
a_{ij},\qquad b_i,\qquad \Omega,
\]
with Weyl transformations
\[
a_{ij} \to \mathcal{B}^2 a_{ij},\qquad b_i \to \mathcal{B}\, b_i,\qquad \Omega \to \mathcal{B}\, \Omega.
\]
Within this framework, a Weyl-Carroll derivative is constructed, the modified Newman-Unti gauge makes null infinity manifestly Carrollian, and the hidden bulk Ehlers \(SL(2,\mathbb{R})\) action becomes local on boundary Carrollian data [2212.14062].

Several recent papers provide context for the Weyl side of the phrase while explicitly not addressing Carroll symmetry. The review of Weyl gauge theory of gravity discusses local dilatations with
\[
g_{\mu\nu}^\prime=\Sigma^2 g_{\mu\nu},\qquad \omega_\mu'=\omega_\mu-\partial_\mu\ln\Sigma,
\]
and states that it provides no direct support for “Carroll-Weyl Gauge Symmetry” because Carroll symmetry is absent [2604.07508]. Related works on generalized Weyl affine connections, conformal Cartan geometry, metric-affine Weyl gauging, and the physical interpretation of Weyl gauge symmetry likewise do not discuss Carroll geometry, although they sharpen the distinction between a genuine Weyl gauge field, a Stueckelberg-like compensator, and the postulates used to interpret local scale symmetry [2402.04712, 1512.06907, 2203.08692, 2310.18854].

Taken together, these works delimit the modern meaning of Carroll-Weyl gauge symmetry. In null strings it is a genuine local gauge symmetry of the Carrollian worldsheet whose residual form explains the overlooked ILST partial-gauge symmetry and whose full treatment modifies both the constraint algebra and the BRST complex [2605.25817, 2605.26822, 2606.04999]. In differential geometry it names the residual temporal Weyl-like freedom left after fixing conformal, projective, and torsional Carroll data [2606.00799]. In asymptotically flat gravity it denotes a Carrollian and Weyl-covariant gauge language for null infinity rather than an independent gauge group [2212.14062].

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