Carroll-Weyl Gauge Symmetry

• Carroll-Weyl gauge symmetry is a local scale invariance in Carrollian geometry that introduces two distinct Weyl-type scalings, extending ordinary Weyl symmetry.
• In null-string formulations, this symmetry modifies the ILST action by incorporating a Carroll-Weyl gauge field, which recovers standard constraints via partial gauge fixing.
• Hamiltonian and path-integral analyses reveal that the symmetry leads to an extended BMS₃ algebra and a novel bcs ghost system, crucial for anomaly cancellation and the correct physical mode count.

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 (Sheikh-Jabbari et al., 25 May 2026). In related work, the same phrase is also used for a residual gauge ambiguity of compatible Carroll connections determined by conformal and projective data (Schwartz et al., 30 May 2026), and for a Carroll-Weyl covariant gauge choice at null infinity in asymptotically flat gravity (Mittal et al., 2022). 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 (Sheikh-Jabbari et al., 25 May 2026).

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 $(\ell_a,\ v^a,\ n_a),$0-scaling.

The second is the antisymmetric, “volume-preserving” Carroll-Weyl scaling,

$(\ell_a,\ v^a,\ n_a),$1

with

$(\ell_a,\ v^a,\ n_a),$2

so that

$(\ell_a,\ v^a,\ n_a),$3

This second scaling has no analogue in ordinary Lorentzian $(\ell_a,\ v^a,\ n_a),$4D geometry and is the genuinely new Carrollian possibility emphasized in the null-string construction (Sheikh-Jabbari et al., 25 May 2026).

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

The standard ILST null-string action is

$(\ell_a,\ v^a,\ n_a),$5

It is invariant under diffeomorphisms and the $(\ell_a,\ v^a,\ n_a),$6-scaling, but not under the $(\ell_a,\ v^a,\ n_a),$7-scaling. To make the $(\ell_a,\ v^a,\ n_a),$8-scaling local, the Carroll-Weyl gauge field $(\ell_a,\ v^a,\ n_a),$9 is introduced and

$$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$0

The resulting action is

$$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$1

In the normalization used in the worldsheet-gauging analysis, the infinitesimal transformations for $$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$2 are

$$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$3

$$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$4

$$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$5

Accordingly, the full gauge symmetry consists of $$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$6D diffeomorphisms, the $$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$7-scaling, and local $$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$8-scaling. In this formulation the $$h_{ab}=\ell_a\ell_b,\qquad v^a h_{ab}=0,\qquad v^a n_a=1.$$9-scaling is inert on the dynamical fields, but geometrically it remains part of the Carroll-Weyl structure (Sheikh-Jabbari et al., 25 May 2026).

The same construction is presented in later Hamiltonian and path-integral treatments with a different normalization of the scaling parameter, for example

$v^a$0

together with the gauged action

$v^a$1

This indicates a normalization choice rather than a change in the underlying gauge content (Duary et al., 3 Jun 2026).

A further point made in the worldsheet construction is that one cannot build a useful $v^a$2-invariant kinetic term for $v^a$3 from $v^a$4, $v^a$5, and

$v^a$6

For that reason, the action above is described as essentially the most general diffeomorphism- and $v^a$7-invariant null-string action of this type (Sheikh-Jabbari et al., 25 May 2026).

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

$v^a$8

Under a $v^a$9-transformation,

$\ell_a$0

after imposing the gauge condition itself. Hence the gauge fixes $\ell_a$1 only up to transformations satisfying

$\ell_a$2

In this gauge,

$\ell_a$3

so the gauged action reduces exactly to the ILST action (Sheikh-Jabbari et al., 25 May 2026).

The residual $\ell_a$4-transformations are therefore those for which $\ell_a$5 is constant along the integral curves of $\ell_a$6. Equivalently, the gauge parameter depends only on one worldsheet coordinate and is a codimension-$\ell_a$7 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 $\ell_a$8-gauge symmetry after a legitimate gauge choice (Sheikh-Jabbari et al., 25 May 2026).

The equations of motion and constraints in the gauged formulation sharpen this interpretation. Variation with respect to $\ell_a$9 gives

$\phi:=\chi_t=\chi_s,$0

Defining

$\phi:=\chi_t=\chi_s,$1

this becomes

$\phi:=\chi_t=\chi_s,$2

Variation with respect to $\phi:=\chi_t=\chi_s,$3 gives the constraints

$\phi:=\chi_t=\chi_s,$4

while variation with respect to $\phi:=\chi_t=\chi_s,$5 yields

$\phi:=\chi_t=\chi_s,$6

In the gauge $\phi:=\chi_t=\chi_s,$7, these reduce to the usual null-string constraints, including

$\phi:=\chi_t=\chi_s,$8

The gauge field $\phi:=\chi_t=\chi_s,$9 has no classical dynamics and is a pure gauge field (Sheikh-Jabbari et al., 25 May 2026).

4. Hamiltonian structure, extended BMS$$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$0, and classical degrees of freedom

The consistent Hamiltonian treatment shows that Carroll-Weyl symmetry is not optional. In the gauged system with

$$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$1

the momentum conjugate to $$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$2 is

$$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$3

while the auxiliary fields satisfy the primary constraints

$$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$4

The canonical Hamiltonian is

$$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$5

Stability of the primary constraints produces the secondary constraints

$$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$6

The third constraint is the Carroll-Weyl generator that earlier null-string analyses had overlooked (Sheikh-Jabbari et al., 26 May 2026).

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 $$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$7D worldsheet metric and the familiar Weyl symmetry acting on that metric, whereas the Carrollian null-string theory is built from the vector density $$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$8, and the relevant local symmetry rescales $$n_a\to e^\phi n_a,\qquad v^a\to e^{-\phi}v^a,\qquad \ell_a\to e^\phi \ell_a.$$9 and $$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$0 rather than a worldsheet metric (Sheikh-Jabbari et al., 26 May 2026).

The constraint algebra enlarges the usual centerless BMS$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$1 algebra. For smeared generators,

$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$2

the Poisson brackets are

$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$3

$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$4

$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$5

$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$6

$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$7

$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$8

At mode level,

$$h_{ab}\to e^{2\phi}h_{ab},\qquad \epsilon_{ab}\to e^{2\phi}\epsilon_{ab}.$$9

and

$2$0

$2$1

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$2$2 algebra rather than by the older two-constraint BMS$2$3 system (Sheikh-Jabbari et al., 26 May 2026).

The same analysis concludes that the classical null string in a $2$4-dimensional Minkowski target space has

$2$5

propagating modes, not $2$6. This one-mode reduction is the direct consequence of the extra first-class constraint generated by Carroll-Weyl symmetry (Sheikh-Jabbari et al., 26 May 2026).

5. Path-integral quantization and the $2$7 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

$2$8

Because the gauge group is $2$9, the Faddeev-Popov determinant must include three gauge directions rather than two (Duary et al., 3 Jun 2026).

A convenient set of gauge conditions is

$\mathcal V^a:=\sqrt{h}\,v^a,$0

On the gauge slice $\mathcal V^a:=\sqrt{h}\,v^a,$1, the infinitesimal variations are

$\mathcal V^a:=\sqrt{h}\,v^a,$2

$\mathcal V^a:=\sqrt{h}\,v^a,$3

$\mathcal V^a:=\sqrt{h}\,v^a,$4

The Faddeev-Popov operator is therefore

$\mathcal V^a:=\sqrt{h}\,v^a,$5

acting on $\mathcal V^a:=\sqrt{h}\,v^a,$6. The off-diagonal entry $\mathcal V^a:=\sqrt{h}\,v^a,$7 expresses the fact that the gauge condition $\mathcal V^a:=\sqrt{h}\,v^a,$8 is moved both by a diffeomorphism and by a Carroll-Weyl rescaling (Duary et al., 3 Jun 2026).

Exponentiating the determinant yields a $\mathcal V^a:=\sqrt{h}\,v^a,$9 ghost system rather than only the old BMS $(\ell_a,\ v^a,\ n_a),$00 system: $(\ell_a,\ v^a,\ n_a),$01 Here $(\ell_a,\ v^a,\ n_a),$02 is the fermionic scalar ghost for Carroll-Weyl transformations and $(\ell_a,\ v^a,\ n_a),$03 is its scalar antighost. The gauge-fixed action is

$(\ell_a,\ v^a,\ n_a),$04

The term $(\ell_a,\ v^a,\ n_a),$05 is the kinetic term for the new ghost sector, while $(\ell_a,\ v^a,\ n_a),$06 is the mixing term that encodes the nontrivial coupling between Carroll-Weyl scaling and the temporal BMS ghost sector (Duary et al., 3 Jun 2026).

The residual gauge-preserving conditions are

$(\ell_a,\ v^a,\ n_a),$07

so that

$(\ell_a,\ v^a,\ n_a),$08

The extra arbitrary function $(\ell_a,\ v^a,\ n_a),$09 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 $(\ell_a,\ v^a,\ n_a),$10, and the BRST operator must now include the $(\ell_a,\ v^a,\ n_a),$11 sector of the extended BMS algebra (Duary et al., 3 Jun 2026).

A central consequence is that the familiar $(\ell_a,\ v^a,\ n_a),$12 consistency check based only on the older BMS $(\ell_a,\ v^a,\ n_a),$13 ghosts is a partially gauge-fixed calculation. In the Carroll-Weyl covariant quantum theory, the anomaly problem must be recomputed for the full matter-plus-$(\ell_a,\ v^a,\ n_a),$14 system, and the physical-state conditions must include

$(\ell_a,\ v^a,\ n_a),$15

alongside the usual $(\ell_a,\ v^a,\ n_a),$16 and $(\ell_a,\ v^a,\ n_a),$17 constraints (Duary et al., 3 Jun 2026).

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

$(\ell_a,\ v^a,\ n_a),$18

under

$(\ell_a,\ v^a,\ n_a),$19

and a compatible connection $(\ell_a,\ v^a,\ n_a),$20 satisfies

$(\ell_a,\ v^a,\ n_a),$21

Here the scale connection $(\ell_a,\ v^a,\ n_a),$22 transforms as

$(\ell_a,\ v^a,\ n_a),$23

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 $(\ell_a,\ v^a,\ n_a),$24, are determined only up to

$(\ell_a,\ v^a,\ n_a),$25

Under this shift,

$(\ell_a,\ v^a,\ n_a),$26

This residual $(\ell_a,\ v^a,\ n_a),$27-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 (Schwartz et al., 30 May 2026).

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

$(\ell_a,\ v^a,\ n_a),$28

with Weyl transformations

$(\ell_a,\ v^a,\ n_a),$29

Within this framework, a Weyl-Carroll derivative is constructed, the modified Newman-Unti gauge makes null infinity manifestly Carrollian, and the hidden bulk Ehlers $(\ell_a,\ v^a,\ n_a),$30 action becomes local on boundary Carrollian data (Mittal et al., 2022).

Several 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

$(\ell_a,\ v^a,\ n_a),$31

and states that it provides no direct support for “Carroll-Weyl Gauge Symmetry” because Carroll symmetry is absent (Ghilencea, 8 Apr 2026). 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 (Mohammedi, 2024, Attard et al., 2015, Sauro et al., 2022, Quiros, 2023).

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 (Sheikh-Jabbari et al., 25 May 2026, Sheikh-Jabbari et al., 26 May 2026, Duary et al., 3 Jun 2026). In differential geometry it names the residual temporal Weyl-like freedom left after fixing conformal, projective, and torsional Carroll data (Schwartz et al., 30 May 2026). In asymptotically flat gravity it denotes a Carrollian and Weyl-covariant gauge language for null infinity rather than an independent gauge group (Mittal et al., 2022).