Carroll-Weyl Scalings in Carrollian Geometry

• Carroll-Weyl scalings are conformal rescalings that adjust Carrollian geometry’s degenerate spatial metrics and distinguished temporal directions for ultra-relativistic regimes.
• They manifest in anisotropic field theories, null string gauge symmetries, and holographic models, organizing distinct scaling laws, symmetry algebras, and Ward identities.
• Their formalism unifies geometric, algebraic, and analytic frameworks, impacting correlators, gauge invariance, and gravitational applications in modern theoretical physics.

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 (Despontin et al., 29 May 2025, Sheikh-Jabbari et al., 25 May 2026, Schwartz et al., 30 May 2026). 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 $z\equiv 2/k$ (Despontin et al., 29 May 2025).

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 $\Phi\to \Omega(x)^{-\Delta}\Phi$, $h_{\mu\nu}$0, and $h_{\mu\nu}$1 for primary scalars and spatial delta functions (Afshar et al., 2024).

The case $h_{\mu\nu}$2 is structurally special. In the field-theoretic formulation it corresponds to $h_{\mu\nu}$3, so the Carrollian time direction is not rescaled while the spatial metric scales as $h_{\mu\nu}$4. In this limit the Weyl Ward identity reduces to the tracelessness of the spatial stress,

$h_{\mu\nu}$5

rather than a full spacetime trace condition (Despontin et al., 29 May 2025).

A more intrinsic geometric formulation replaces $h_{\mu\nu}$6 by a Carroll structure $h_{\mu\nu}$7 with $h_{\mu\nu}$8. A conformal Carroll structure is then the equivalence class

$h_{\mu\nu}$9

and a compatible connection satisfies

$\tau^\mu$0

with Weyl one-form transforming as $\tau^\mu$1 (Schwartz et al., 30 May 2026). 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 $\tau^\mu$2-conformal Carroll algebra is generated by modes $\tau^\mu$3 and $\tau^\mu$4 with

$\tau^\mu$5

In three spacetime dimensions one introduces $\tau^\mu$6, $\tau^\mu$7, and $\tau^\mu$8, obtaining two Witt algebras acting on supertranslations with $\tau^\mu$9-dependent weights (Despontin et al., 29 May 2025). The general infinite-dimensional structure was systematized as $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$0, with finite-dimensional truncations for integer or half-integer $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$1 (Afshar et al., 2024).

The $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$2 algebra is closely related to warped conformal symmetry, but the equivalence is only algebraic. In two dimensions, the $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$3 Carrollian algebra coincides as an infinite-dimensional Lie algebra with the Warped Conformal algebra, yet the global subgroups differ: the WCFT vacuum respects $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$4, whereas the Carrollian vacuum respects a four-generator subgroup $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$5 in $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$6 and a seven-generator type-D subgroup in $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$7. The distinction propagates to normal ordering, spectra, and correlators (Despontin et al., 29 May 2025).

The stress-tensor sector reflects the same anisotropic structure. On a Carroll manifold, local tangent symmetries imply

$$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$8

so there is no energy flux and the spatial stress is symmetric. For general $$\tau^\mu = (1,0^i), \qquad h_{\mu\nu}=\mathrm{diag}(0,\delta_{ij}).$$9, the Weyl Ward identity is

$$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$0

while for $$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$1 it becomes purely spatial tracelessness (Despontin et al., 29 May 2025). In $$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$2 and $$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$3, the local operators

$$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$4

transform as Virasoro primaries with weights $$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$5, $$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$6, and $$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$7, respectively; for general $$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$8, $$\mathcal{L}_\xi \tau^\mu = -\frac{\lambda(x)}{k}\tau^\mu, \qquad \mathcal{L}_\xi h_{\mu\nu}=\lambda(x) h_{\mu\nu},$$9 acquires weights $z\equiv 2/k$0 while $z\equiv 2/k$1 and $z\equiv 2/k$2 remain unchanged (Despontin et al., 29 May 2025).

At $z\equiv 2/k$3, conformal Carroll theories on curved backgrounds admit explicit Weyl rules for the Carrollian fields:

$z\equiv 2/k$4

with scalar weight

$z\equiv 2/k$5

These transformations support boost-invariant conformal Carroll scalar actions whose energy-momentum tensors are traceless on shell (Baiguera et al., 2022).

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,

$z\equiv 2/k$6

and for finite $z\equiv 2/k$7 dilatation covariance gives

$z\equiv 2/k$8

When spatial special conformal invariance is imposed and $z\equiv 2/k$9, the result is

$D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$0

whereas distinct dimensions are still allowed in the electric sector (Afshar et al., 2024).

In the anisotropic conformal Carroll theories analyzed holographically, the $D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$1 case is vacuum-sensitive. In $D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$2, primaries are labeled by eigenvalues of $D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$3, and time evolution takes the form

$D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$4

For the $D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$5 vacuum $D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$6, the two-point function is the standard two-dimensional power law on the spatial slice multiplied by $D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$7. For the type-D Carroll vacuum $D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$8, the $D^{(z)} = z\, t\,\partial_t + x^i \partial_i,$9 branch has only the spatial power law, while the $t\to \lambda^z t,\qquad x^i\to \lambda x^i.$0 branch is ultra-local in space,

$t\to \lambda^z t,\qquad x^i\to \lambda x^i.$1

together with dimension and charge constraints. For generic $t\to \lambda^z t,\qquad x^i\to \lambda x^i.$2, the quoted two-point function is spatially ultra-local and scales as a power of $t\to \lambda^z t,\qquad x^i\to \lambda x^i.$3 (Despontin et al., 29 May 2025).

Momentum space exhibits the same structure in a different analytic form. The momentum-space dilatation Ward identity is

$t\to \lambda^z t,\qquad x^i\to \lambda x^i.$4

and for scalar two-point functions the general solution is

$t\to \lambda^z t,\qquad x^i\to \lambda x^i.$5

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 $t\to \lambda^z t,\qquad x^i\to \lambda x^i.$6 the mixed branches develop logarithmic terms such as

$t\to \lambda^z t,\qquad x^i\to \lambda x^i.$7

which originate from singularities in the Fourier transform rather than from new position-space branches (Marotta et al., 7 Dec 2025).

4. Carroll-Weyl gauge symmetry on null strings

On a two-dimensional Carrollian worldsheet, the basic data are a rank-one degenerate spatial metric $t\to \lambda^z t,\qquad x^i\to \lambda x^i.$8, its kernel vector field $t\to \lambda^z t,\qquad x^i\to \lambda x^i.$9, and a clock one-form $dx'^2=\Omega^2(x)\,dx^2$0 normalized by $dx'^2=\Omega^2(x)\,dx^2$1. Writing $dx'^2=\Omega^2(x)\,dx^2$2, the structure admits two independent local scalings:

$dx'^2=\Omega^2(x)\,dx^2$3

Two linear combinations are singled out. The $dx'^2=\Omega^2(x)\,dx^2$4-scaling has $dx'^2=\Omega^2(x)\,dx^2$5 and leaves $dx'^2=\Omega^2(x)\,dx^2$6 invariant. The $dx'^2=\Omega^2(x)\,dx^2$7-scaling has $dx'^2=\Omega^2(x)\,dx^2$8, preserves the volume form $dx'^2=\Omega^2(x)\,dx^2$9, and rescales $\Phi\to \Omega(x)^{-\Delta}\Phi$0 as $\Phi\to \Omega(x)^{-\Delta}\Phi$1 (Sheikh-Jabbari et al., 25 May 2026).

Gauging the volume-preserving scaling requires a Carroll-Weyl connection $\Phi\to \Omega(x)^{-\Delta}\Phi$2 and covariant derivative

$\Phi\to \Omega(x)^{-\Delta}\Phi$3

The fully gauged null-string action is

$\Phi\to \Omega(x)^{-\Delta}\Phi$4

which is invariant under worldsheet diffeomorphisms, the $\Phi\to \Omega(x)^{-\Delta}\Phi$5-scaling, and the $\Phi\to \Omega(x)^{-\Delta}\Phi$6-scaling (Sheikh-Jabbari et al., 25 May 2026). Fixing the gauge $\Phi\to \Omega(x)^{-\Delta}\Phi$7 reduces the theory to the ILST action, and the residual symmetry satisfies

$\Phi\to \Omega(x)^{-\Delta}\Phi$8

This residual codimension-one symmetry was identified as the overlooked partial gauge symmetry of the ILST formulation (Sheikh-Jabbari et al., 25 May 2026).

The Hamiltonian analysis adds a genuinely new first-class constraint,

$\Phi\to \Omega(x)^{-\Delta}\Phi$9

alongside

$h_{\mu\nu}$00

In mode language, $h_{\mu\nu}$01 generates a weight-one current extending the standard centerless $h_{\mu\nu}$02 algebra of null strings. The extended algebra contains modes $h_{\mu\nu}$03, $h_{\mu\nu}$04, and $h_{\mu\nu}$05 with

$h_{\mu\nu}$06

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 (Sheikh-Jabbari et al., 26 May 2026).

Quantization reflects the same enlargement. In the path-integral treatment, complete gauge fixing of $h_{\mu\nu}$07 produces not the old BMS $h_{\mu\nu}$08 system alone but a $h_{\mu\nu}$09 system, with a scalar ghost $h_{\mu\nu}$10 and scalar antighost $h_{\mu\nu}$11 for Carroll-Weyl scaling. The ghost action contains the characteristic mixing term $h_{\mu\nu}$12, and the BRST complex must therefore include the $h_{\mu\nu}$13-sector generated by $h_{\mu\nu}$14 (Duary et al., 3 Jun 2026).

5. Compatible connections and Weyl-type uniqueness

In differential-geometric terms, Carroll-Weyl scaling is encoded by the conformal class

$h_{\mu\nu}$15

together with a compatible connection satisfying

$h_{\mu\nu}$16

This is the Carrollian analogue of a Weyl connection, but the torsion constraints differ sharply from the pseudo-Riemannian case (Schwartz et al., 30 May 2026).

If $h_{\mu\nu}$17 denotes torsion, compatibility implies

$h_{\mu\nu}$18

Hence torsion-free Carroll-Weyl connections exist only when

$h_{\mu\nu}$19

This condition has no exact analogue in ordinary Weyl geometry and expresses the fact that conformal rescaling and the degenerate temporal direction are intertwined (Schwartz et al., 30 May 2026).

Projective structure enters through unparametrized autoparallels. Two linear connections are projectively equivalent when their difference satisfies

$h_{\mu\nu}$20

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 $h_{\mu\nu}$21, then their difference is

$h_{\mu\nu}$22

for some function $h_{\mu\nu}$23 (Schwartz et al., 30 May 2026).

This result is weaker than the corresponding pseudo-Riemannian and Galilei statements. A Carroll Weyl metric can be represented by the data

$h_{\mu\nu}$24

with scaling

$h_{\mu\nu}$25

but compatibility plus projective structure determines $h_{\mu\nu}$26 only up to

$h_{\mu\nu}$27

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 $h_{\mu\nu}$28, a canonical four-dimensional plane wave metric is

$h_{\mu\nu}$29

with Brinkmann form

$h_{\mu\nu}$30

More generally,

$h_{\mu\nu}$31

with $h_{\mu\nu}$32, 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 (Despontin et al., 29 May 2025).

In scaling-Carroll gravity, the local dilatation generator $h_{\mu\nu}$33 acts on the gravity multiplet by

$h_{\mu\nu}$34

so that

$h_{\mu\nu}$35

The extrinsic curvature

$h_{\mu\nu}$36

transforms as

$h_{\mu\nu}$37

After scale fixing, one has $h_{\mu\nu}$38 and $h_{\mu\nu}$39, and the theory admits three regimes: dynamical Carroll gravity, Aristotelian gravity, and a fracton gauge theory coupled to Aristotelian geometry (Afshar et al., 23 Dec 2025). Earlier ultra-relativistic gravity work already identified anisotropic Weyl invariance in $h_{\mu\nu}$40 dimensions for $h_{\mu\nu}$41, where $h_{\mu\nu}$42 and $h_{\mu\nu}$43 are inert while spatial frames scale (Hartong, 2015).

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 $h_{\mu\nu}$44 plus improvement terms involving $h_{\mu\nu}$45 and $h_{\mu\nu}$46, while the spacelike theory combines $h_{\mu\nu}$47 with the Weyl-covariant curvature combination $h_{\mu\nu}$48 and Lagrange-multiplier constraints. In both cases the on-shell Ward identity gives $h_{\mu\nu}$49, and in the spacelike branch the theory reduces to a lower-dimensional Euclidean CFT after solving the constraints and using Weyl invariance (Baiguera et al., 2022).

At null infinity, Carroll-Weyl rescaling is also part of the natural boundary calculus. For Carrollian boundary data $h_{\mu\nu}$50 one has

$h_{\mu\nu}$51

so that the clock one-form $h_{\mu\nu}$52 scales with weight $h_{\mu\nu}$53 and the observer field $h_{\mu\nu}$54 with weight $h_{\mu\nu}$55. The acceleration $h_{\mu\nu}$56 and expansion $h_{\mu\nu}$57 serve as the spatial and temporal components of the Weyl connection, and the resulting Weyl-Carroll covariant tensors organize electric and magnetic towers of $h_{\mu\nu}$58 charges (Mittal et al., 2022).

Carroll-Weyl weights also control ultra-relativistic limits of ordinary conformal field theory. In the Carrollian limit of CFT$h_{\mu\nu}$59 OPE blocks, a scalar primary is normalized with weight $h_{\mu\nu}$60, and spinning components with $h_{\mu\nu}$61. These weights select finite Carrollian operators and lead, after Mellin-Fourier projection, to towers of $h_{\mu\nu}$62 primaries of negative integer dimensions. The resulting blocks generate conformally soft photon and graviton towers and reproduce the celestial $h_{\mu\nu}$63 and $h_{\mu\nu}$64 algebras (Gioia et al., 27 Aug 2025).

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.