Longitudinal Carrollian Brane Limit

• The longitudinal Carrollian brane limit is a scaling procedure that singles out time and longitudinal spatial directions to produce a degenerate, Carrollian worldvolume geometry.
• It suppresses time-derivative terms and leads to ultralocal dynamics with a constrained Hamiltonian structure, while preserving nontrivial transverse momentum evolution.
• Multiple realizations in M-theory, Type II* strings, and non-relativistic settings illustrate its implications for de Sitter holography, tachyon condensation, and brane dualities.

Searching arXiv for recent and foundational papers on the longitudinal Carrollian brane limit and closely related Carroll brane constructions. The longitudinal Carrollian brane limit is an ultra-relativistic scaling procedure in which the longitudinal directions of a brane, including time, are singled out so that the resulting worldvolume or worldsheet geometry becomes Carrollian: the metric degenerates, time becomes absolute, and the dynamical system becomes ultra-local in time. In the literature this appears in several closely related forms: as the longitudinal or “stringy” Carroll limit of M2- and unstable M3-branes in eleven-dimensional supergravity, as a longitudinal Carrollian limit of non-relativistic strings with a Carrollian worldsheet, and as a decoupling limit of Type II$^*$ branes whose near-horizon geometries are conformal to de Sitter space. Across these realizations, the characteristic features are degenerate induced geometry, suppressed time-derivative terms, a constraint algebra with vanishing Hamiltonian-Hamiltonian bracket, and nontrivial transverse momentum dynamics despite frozen longitudinal embedding data (Roychowdhury, 2019, Bidussi et al., 2023, Argandoña et al., 8 Jul 2025).

1. Scaling prescriptions and limiting procedures

A standard realization is the longitudinal, or “stringy,” Carroll limit for relativistic branes in eleven dimensions. For M2- and unstable M3-branes one rescales only the first two target-space directions, treated as longitudinal, according to

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$

while the transverse sector is left unscaled,

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$

and then one sends $\omega\to\infty$. In Hamiltonian form the worldvolume multipliers and tensions must also scale. For the M2-brane,

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$

and for the unstable M3-brane,

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$

The background fields are unscaled, or else assigned only the minimal scaling required by pullback consistency (Roychowdhury, 2019).

A distinct, but closely related, longitudinal Carrollian brane limit appears in Type II$^*$ string theory. There the scaling is imposed directly on spacetime coordinates and parameters: $\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$

$$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$

with $\hat\ell_s=\ell_s$, $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$0, and $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$1. In static gauge this can be read as a limit of the Dirac–Born–Infeld action in which the worldvolume time scales as $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$2 and the brane tension scales so that time-derivative terms are suppressed. The net effect is a Carrollian, “ultra-local in time,” theory on the $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$3-brane (Argandoña et al., 8 Jul 2025).

For non-relativistic strings on a torsional string Newton–Cartan background, the longitudinal Carrollian limit is formulated by introducing an explicit longitudinal speed of light $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$4 through

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$5

defining the Carrollian tension

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$6

and sending $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$7 while keeping $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$8 and the background fields fixed. All $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$9 terms then drop out, producing a Carrollian worldsheet theory (Bidussi et al., 2023).

These examples suggest that the expression “longitudinal Carrollian brane limit” denotes a family of ultra-relativistic contractions in which longitudinal directions are scaled so that the induced metric loses its Lorentzian time component and the residual theory is governed by Carrollian rather than Galilean kinematics.

2. Effective worldvolume and worldsheet actions

For the M2-brane, the Hamiltonian form of the Carroll limit is

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$0

with primary constraints

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$1

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$2

and

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$3

The Nambu–Goto description leads, after the same scaling, to a first-order action of the form

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$4

and the Hamiltonian and Nambu–Goto Carrollian formulations are equivalent up to redefinitions of the lapse and shift variables (Roychowdhury, 2019).

For the unstable M3-brane, one starts from the tachyon–DBI-type action

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$5

and obtains the Carrollian first-order action

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$6

with

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$7

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$8

The tachyon therefore enters the Carrollian phase space on the same footing as the embedding coordinates, but subject to the degenerate limit and its associated constraints (Roychowdhury, 2019).

In the string case, the Carrollian Nambu–Goto action is

$X^I=y^I,\qquad P_I=p_I,\qquad I=2,\dots,10,$9

while the Carrollian Polyakov action is

$\omega\to\infty$0

The Polyakov multipliers impose Carrollian constraints and the zweibein equations reproduce the Nambu–Goto description (Bidussi et al., 2023).

A broader sigma-model formulation for Carroll $\omega\to\infty$1-branes is

$\omega\to\infty$2

where the Lagrange multiplier imposes the longitudinal-null constraint

$\omega\to\infty$3

In this formulation the kinetic term retains only the transverse sector, while the longitudinal pullbacks are projected out by constraint rather than by a nondegenerate worldvolume metric (Bergshoeff et al., 2020).

3. Constraint algebra, gauge fixing, and dynamical content

In the Hamiltonian description of Carroll branes, the smeared Hamiltonian and diffeomorphism constraints satisfy

$\omega\to\infty$4

with Poisson brackets

$\omega\to\infty$5

$\omega\to\infty$6

$\omega\to\infty$7

This is the ultra-local, Abelian limit of the relativistic $\omega\to\infty$8-brane Dirac algebra: the Hamiltonian-Hamiltonian bracket vanishes instead of generating a diffeomorphism constraint, which is one of the clearest canonical signatures of the Carroll limit (Roychowdhury, 2019).

Varying the canonical action

$\omega\to\infty$9

gives

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$0

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$1

supplemented by $$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$2 and $$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$3. In the static Carroll gauge

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$4

these reduce to

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$5

The longitudinal directions are thereby fixed by gauge choice, whereas the transverse sector still exhibits nontrivial momentum evolution (Roychowdhury, 2019).

For longitudinal Carrollian strings, static gauge is imposed as

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$6

with $$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$7 and nonzero winding $$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$8. In this gauge,

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2),\qquad T_2=\omega\,\tilde T_2,$$9

so that

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$0

The gauge-fixed kinetic term is only linear in time derivatives, and the Polyakov constraints are solved by

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$1

up to local Carroll–Weyl transformations. Residual symmetries include foliation-preserving diffeomorphisms, local Carroll boosts, and Carroll–Weyl rescalings (Bidussi et al., 2023).

A recurrent misconception is that Carrollian branes are dynamically trivial because longitudinal motion is frozen. The explicit equations instead show a more specific statement: longitudinal embedding data become nondynamical in appropriate gauge choices, while transverse momenta, constraint propagation, and spatial profile data remain nontrivial.

4. Geometric and algebraic structures

A $$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$2-brane Carrollian geometry is specified by a clock one-form of rank $$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$3,

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$4

and a spatial cometric of rank $$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$5,

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$6

subject to the orthogonality condition

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$7

and the completeness relation

$$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$8

Geometrically, the $$\lambda_0=\omega^2\tilde\lambda_0,\qquad \lambda_i=\tilde\lambda_i\;(i=1,2,3),\qquad T_3=\omega\,\tilde T_3.$$9 select the longitudinal directions, while the transverse sector carries a positive-definite metric structure. In the ultra-relativistic scaling

$^*$0

the spacetime metric becomes

$^*$1

and the limit $^*$2 leaves only the degenerate spatial piece. The worldvolume therefore degenerates along the null directions defined by the clock forms (Bergshoeff et al., 2023).

The associated adapted connection satisfies the vielbein postulates

$^*$3

$^*$4

and its intrinsic torsion is the class of the actual torsion in the cokernel of the Spencer differential. One finds

$^*$5

with

$^*$6

For generic $^*$7 the vanishing patterns of the boost-invariant intrinsic torsions

$^*$8

define five inequivalent Carrollian-brane geometries, ranging from generic torsion to a torsion-free Carrollian analogue of Newton–Cartan geometry (Bergshoeff et al., 2023).

At the algebraic level, the Carroll $^*$9-brane algebra is obtained from the Poincaré algebra by splitting generators into longitudinal and transverse sectors, taking

$\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$0

rescaling the $\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$1 generators by $\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$2, and sending $\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$3. This yields the Carroll contraction along longitudinal directions. A formal Galilei$\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$4Carroll map then exchanges longitudinal and transverse indices,

$\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$5

mapping the $\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$6-brane Galilei algebra onto the $\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$7-brane Carroll algebra. The sigma-model formulation carries $\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$8-dimensional diffeomorphisms, local Carroll boosts, local longitudinal Lorentz rotations, and local transverse rotations (Bergshoeff et al., 2020).

5. Curved-background realizations: AdS–Carroll branes

A curved-background realization is provided by codimension-one branes embedded in AdS space and then contracted to the Carroll limit by coset methods. The relevant spontaneous breaking pattern is

$\hat x^A=\omega^{+\tfrac12}x^A,\qquad A=1,\dots,p,$9

with coset representative

$$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$0

The Maurer–Cartan one-forms reduce in the Carroll limit to the Carrollian forms $$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$1, $$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$2, $$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$3, $$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$4, and $$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$5. The inverse-Higgs condition $$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$6 is equivalent to the $$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$7 equation of motion and gives

$$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$8

so the Nambu–Goldstone field $$\hat x^0\equiv \hat t=\omega^{-\tfrac12}t,\qquad \hat x^{A'}=\omega^{-\tfrac12}x^{A'},\qquad A'=p+1,\dots,9,$$9 has a frozen spatial profile. The second inverse-Higgs relation,

$\hat\ell_s=\ell_s$0

eliminates the boost Goldstones in terms of spatial gradients of $\hat\ell_s=\ell_s$1 (Clark et al., 2016).

The induced vielbein is extracted from

$\hat\ell_s=\ell_s$2

and the unique lowest-derivative invariant action is

$\hat\ell_s=\ell_s$3

Using the explicit AdS–Carroll vielbein, the determinant becomes

$\hat\ell_s=\ell_s$4

Although $\hat\ell_s=\ell_s$5 is static, the canonical momentum density is not. The broken translation current conservation law

$\hat\ell_s=\ell_s$6

implies

$\hat\ell_s=\ell_s$7

The time variation of momentum is thus controlled by the spatial brane profile and by the AdS–Carroll geometry (Clark et al., 2016).

The same system admits a dual vector formulation. Introducing

$\hat\ell_s=\ell_s$8

one obtains, after a partial Legendre transform,

$\hat\ell_s=\ell_s$9

Eliminating the Lagrange multiplier reproduces the original Nambu–Goto–Carroll description, so the scalar and vector pictures are on-shell equivalent (Clark et al., 2016).

6. de Sitter holography, tachyon vacua, and interpretive scope

In the Type II$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$00 realization, the longitudinal Carrollian limit produces a degenerate worldvolume metric directly. The induced metric obeys

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$01

so that in the limit

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$02

with no $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$03 term. Time is absolute and does not appear in the metric. Applying the same scalings to the extremal SD$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$04 black-brane solution yields, in the near-horizon region, a geometry conformal in the dual frame to

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$05

with de Sitter radius

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$06

On the open-string side, the light degrees of freedom are those of Euclidean maximally-supersymmetric $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$07 Yang–Mills in $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$08 dimensions, with

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$09

while on the closed-string side one has Type II$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$10 string theory on $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$11. The proposed open/closed duality is therefore

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$12

which extends Hull’s proposal for $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$13 and is related to temporal T-duality (Argandoña et al., 8 Jul 2025).

The same work relates the construction to Buscher duality. Starting from an extremal D$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$14-brane in Type II, smearing along one transverse spatial direction, performing a spatial Buscher duality, and then a timelike Buscher duality, one recovers the SD$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$15-brane of Type II$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$16. In Buscher form,

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$17

together with the characteristic sign flip of RR kinetic terms in Type II$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$18 (Argandoña et al., 8 Jul 2025).

For unstable M3-branes, the Carroll limit has a different endpoint. Around the tachyon vacuum

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$19

the worldvolume action degenerates and the tachyon momentum must vanish: $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$20 The remaining theory is

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$21

with

$$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$22

This is identical to the action of a Carroll M3-brane with no tachyon, and can be reinterpreted via brane descent as a Carroll M2 or string, depending on the embedding (Roychowdhury, 2019).

Taken together, these constructions show that the longitudinal Carrollian brane limit is not merely a kinematical $$X^\mu=\omega^{-1}x^\mu,\qquad P_\mu=\omega\,\pi_\mu,\qquad \mu=0,1,$$23 contraction. It is a framework in which degenerate worldvolume geometry, ultra-local canonical structure, curved-background embeddings, tachyon condensation, and de Sitter holography are organized by a common longitudinal ultra-relativistic scaling, while still admitting multiple realizations whose precise field content and interpretation depend on the underlying brane system.