Carroll Dilaton Gravity Overview

• Carroll dilaton gravity is a two-dimensional ultra-relativistic analogue of dilaton gravity that replaces Lorentzian metrics with degenerate Carroll geometry, with the dilaton controlling curvature and vacuum structure.
• It admits first-order Cartan, BF, and Poisson-sigma-model formulations, enabling analytical treatment of black-hole-like solutions and conserved mass (Casimir) dynamics.
• The theory supports innovative matter couplings (yielding propagating swiftons) and extends to supersymmetric, postcarrollian, and boundary formulations, enriching its gravitational dynamics.

Searching arXiv for recent and foundational papers on Carroll dilaton gravity to ground the article in the literature. Carroll dilaton gravity is the two-dimensional ultra-relativistic counterpart of dilaton gravity in which Lorentzian metric geometry is replaced by Carroll geometry, the fundamental fields are organized in first-order Cartan or BF/Poisson-sigma-model variables, and the dilaton remains the field that controls the curvature sector and the vacuum structure. In the contemporary literature it appears in several closely related forms: as a Carrollian contraction of Jackiw–Teitelboim gravity, as a generic $1+1$-dimensional Carroll dilaton theory with black-hole-like solutions and thermodynamics, as a graded Poisson-sigma model admitting $\mathcal N=1$ and $\mathcal N=2$ supersymmetric extensions, and as a matter-coupled framework in which genuinely propagating Carroll “swiftons” can exist in two dimensions (Grumiller et al., 2020, Ecker et al., 2023, Grumiller et al., 2024, Ecker et al., 2024).

1. Ultra-relativistic origin and the special role of two dimensions

Carroll dilaton gravity is defined by an ultra-relativistic $c\to 0$ limit. In the two-dimensional JT context, the Carrollian limit is the contraction $\hat C\to 0$, performed directly at the level of algebra, action, gauge transformations, and equations of motion, and it yields a theory supported on a $2d$ Carroll geometry rather than a Lorentzian metric geometry (Grumiller et al., 2020). In the black-hole literature, the same class of models is described as the magnetic ultra-relativistic limit of ordinary Lorentzian $2d$ dilaton gravity, with the additional statement that spherical reduction and the magnetic Carroll limit commute (Ecker et al., 2023). The broader thesis literature presents the same sector as the classical approximation to the “tantum gravity limit,” a triple scaling in which $c\to 0$, $\hbar\to \infty$, and $G_N\to 0$ while keeping $\mathcal N=1$0 and $\mathcal N=1$1 fixed (Ecker, 13 Mar 2026).

Two dimensions are singled out repeatedly. One reason is technical: in $\mathcal N=1$2 there is only one spatial direction, so the Cartan data reduce to $\mathcal N=1$3, $\mathcal N=1$4, and $\mathcal N=1$5, and the resulting first-order theory is tractable in BF and Poisson-sigma-model language (Ecker et al., 2024, Ecker et al., 2023). A second reason is physical: all known Carroll black hole solutions are described by $\mathcal N=1$6 models, either intrinsically or by dimensional reduction (Ecker et al., 2024). A third reason is structural: the scalar swifton coupling introduced in the matter-coupled theory uses a special $\mathcal N=1$7 Stückelberg-like completion involving one of the Carroll gravity multipliers, and the authors explicitly state that the crucial term “does not generalize to higher dimensions” (Ecker et al., 2024).

The relation to relativistic $\mathcal N=1$8 dilaton gravity is therefore not merely heuristic. The literature consistently treats Carroll dilaton gravity as a genuine non-Lorentzian analogue of generic $\mathcal N=1$9 dilaton gravity: it inherits constant- and linear-dilaton sectors, conserved Casimirs, and BF/PSM solvability, but it replaces the Lorentzian causal structure by a degenerate Carrollian one (Grumiller et al., 2020, Ecker et al., 2023).

2. Geometric variables, gauge structure, and first-order actions

The underlying geometry is Carrollian. In metric language one uses a degenerate Carroll metric $\mathcal N=2$0 together with a kernel vector $\mathcal N=2$1; in Cartan language one uses the temporal einbein $\mathcal N=2$2, the spatial vielbein $\mathcal N=2$3, and the Carroll boost connection $\mathcal N=2$4 (Ecker et al., 2024). In the higher-dimensional Carroll literature these variables arise from the ultra-relativistic contraction of the Poincaré algebra, while in $\mathcal N=2$5 only one spatial vielbein survives, so the spatial tangent index is absent (Bergshoeff et al., 2017, Ecker et al., 2024).

The gauge structure is fixed by local Carroll boosts. In the $\mathcal N=2$6-dimensional black-hole formulation, the action is invariant under

$\mathcal N=2$7

$\mathcal N=2$8

and there are two further gauge symmetries associated with time and space translations, which reproduce diffeomorphisms on shell after adding a compensating boost (Ecker et al., 2023). In the swifton-coupled model, the transformation law $\mathcal N=2$9 is precisely the ingredient that makes a boost-invariant spatial derivative possible (Ecker et al., 2024).

The literature presents several closely related first-order actions. In the generic Carrollian analogue of $c\to 0$0 dilaton gravity obtained from JT, the bulk action is

$c\to 0$1

The $c\to 0$2-dependence of $c\to 0$3 determines the curvature, while nontrivial $c\to 0$4-dependence produces torsion (Grumiller et al., 2020). In the generic $c\to 0$5-dimensional Carroll black-hole analysis the first-order action is

$c\to 0$6

with the standard $c\to 0$7-$c\to 0$8 family given by

$c\to 0$9

Here $\hat C\to 0$0 is the curvature, $\hat C\to 0$1 is the torsion, and $\hat C\to 0$2 is the intrinsic torsion (Ecker et al., 2023). In the thesis formulation, the same structural pattern is written as

$\hat C\to 0$3

with $\hat C\to 0$4 for the standard dilaton family (Ecker, 13 Mar 2026).

What is common to these formulations is that $\hat C\to 0$5 is always the dilaton, $\hat C\to 0$6 encode the Carrollian geometry, and the remaining scalar multipliers enforce the Carrollian structure equations. The theory is topological in the same broad sense as ordinary $\hat C\to 0$7 dilaton gravity: there are no local bulk graviton degrees of freedom, and the nontrivial content is carried by the dilaton sector, boundary data, and matter couplings (Grumiller et al., 2020, Ecker et al., 2023).

3. BF, Poisson-sigma, and second-order formulations

A central structural feature of Carroll dilaton gravity is that it admits BF and Poisson-sigma-model descriptions. In the BF formulation of the JT limit, the universal bulk equations are the flatness conditions

$\hat C\to 0$8

together with a flat $\hat C\to 0$9 sector after eliminating a trivial $2d$0 field by redefinition (Grumiller et al., 2020). In the black-hole formulation, the same theory is written as a Poisson sigma model,

$2d$1

with

$2d$2

and

$2d$3

The degenerate kernel of $2d$4 yields a conserved Casimir interpreted as mass (Ecker et al., 2023). The thesis uses the same PSM structure with $2d$5 and

$2d$6

again emphasizing the conserved Casimir and exact solvability (Ecker, 13 Mar 2026).

A second-order description also exists, but it retains a characteristically Carrollian ambiguity. In the black-hole analysis, the boost connection decomposes as

$2d$7

with $2d$8 torsionless and $2d$9 undetermined. The second-order action becomes

$2d$0

and the Carroll metric is the degenerate spatial metric

$2d$1

The thesis writes the same second-order structure as

$2d$2

with the undetermined connection component acting as a Lagrange multiplier enforcing $2d$3 (Ecker et al., 2023, Ecker, 13 Mar 2026).

This incompletely determined connection is a recurrent Carrollian feature. Already in the JT-limit analysis it is stated that, unlike the Lorentzian case, one cannot solve for $2d$4 entirely in terms of the vielbein; one component remains undetermined and acts as a Lagrange multiplier (Grumiller et al., 2020). The higher-dimensional Carroll gravity analysis reaches the same conclusion in general dimensions: the equations of motion do not determine all spin-connection components, and in the second-order formulation the independent components enforce Carrollian geometric constraints (Bergshoeff et al., 2017).

4. Classical solution space, conserved Casimir, and Carroll black holes

The classical solution space is organized into constant- and linear-dilaton sectors. In the generic Carrollian analogue of dilaton gravity obtained from JT, the constant dilaton sector is

$2d$5

while the linear dilaton sector has nonconstant $2d$6, $2d$7, and $2d$8 (Grumiller et al., 2020). In the black-hole analysis, constant dilaton vacua are defined by $2d$9, which forces $c\to 0$0 to be constant and requires $c\to 0$1; the generic sector has $c\to 0$2 and is analytically controlled by one conserved mass $c\to 0$3 together with the functions $c\to 0$4 and $c\to 0$5 (Ecker et al., 2023).

For the $c\to 0$6-$c\to 0$7 family one introduces

$c\to 0$8

and the conserved Casimir becomes

$c\to 0$9

Equivalently, in the thesis notation,

$\hbar\to \infty$0

The local linear-dilaton solution can be written in gauge $\hbar\to \infty$1 as

$\hbar\to \infty$2

$\hbar\to \infty$3

In second-order form one has $\hbar\to \infty$4, so the nontrivial physics is encoded not by a Lorentzian black-hole metric but by the dilaton profile and the Carroll vector field (Ecker et al., 2023).

The notion of a black hole is consequently reformulated. Carroll black holes are defined as solutions with a Carroll extremal surface and Carroll thermal properties: $\hbar\to \infty$5 From the Poisson-sigma-model viewpoint, the Carroll extremal surface is the fixed-point locus of Carroll boosts, namely

$\hbar\to \infty$6

Using the on-shell relation $\hbar\to \infty$7, the second-order criterion is

$\hbar\to \infty$8

The literature emphasizes that this is not a Killing horizon or event horizon; it is a Carrollian structure singularity, where the Carroll vector field diverges or the clock shrinks (Ecker et al., 2023, Ecker, 13 Mar 2026).

Thermodynamics nevertheless follows in close formal analogy with relativistic dilaton gravity. The energy is

$\hbar\to \infty$9

the temperature is fixed by Carrollian holonomy or Gauss–Bonnet smoothness,

$G_N\to 0$0

and the entropy is

$G_N\to 0$1

The first law takes the standard form

$G_N\to 0$2

and the specific heat is

$G_N\to 0$3

These formulas are exhibited for generic $G_N\to 0$4-dimensional Carroll dilaton gravity and specialized to Carroll JT, Carroll CGHS, Carroll Witten, and Carroll–Schwarzschild families (Ecker et al., 2023, Ecker, 13 Mar 2026).

The explicit Carroll JT model is defined by

$G_N\to 0$5

so that $G_N\to 0$6. Its black holes exist for $G_N\to 0$7, with

$G_N\to 0$8

The Carroll Witten black hole uses

$G_N\to 0$9

and has

$\mathcal N=1$00

These examples underpin the claim that Carroll black holes are black-hole-like states without horizons, defined instead by thermal smoothness and a Carroll extremal surface (Ecker et al., 2023).

5. Matter couplings, swiftons, and dynamical torsion

A major development is the coupling of propagating matter to Carroll dilaton gravity. Ordinary Carroll scalar theories are usually either electric or magnetic: the electric theory is ultra-local in space, while the magnetic theory enforces time-independence. The swifton construction shows that in $\mathcal N=1$01 Carroll dilaton gravity one can write a scalar action with both time and space derivatives while preserving Carroll invariance (Ecker et al., 2024).

The matter action is

$\mathcal N=1$02

with $\mathcal N=1$03, $\mathcal N=1$04, $\mathcal N=1$05, and

$\mathcal N=1$06

Because $\mathcal N=1$07 transforms as

$\mathcal N=1$08

the ratio $\mathcal N=1$09 shifts in exactly the way needed to compensate the boost variation of $\mathcal N=1$10, so $\mathcal N=1$11 is boost invariant. The term $\mathcal N=1$12 is explicitly described as Stückelberg-like, requires no extra field beyond the gravity multiplet, and requires

$\mathcal N=1$13

The authors further state that one is “not allowed to sit on a Carroll extremal surface” $\mathcal N=1$14 unless additional care is taken, for example by choosing $\mathcal N=1$15 (Ecker et al., 2024).

This coupling produces genuinely propagating Carroll “swiftons,” meaning fields moving with speed strictly greater than the Carroll speed $\mathcal N=1$16. On a Carroll-Schwarzschild background with

$\mathcal N=1$17

and the field redefinition

$\mathcal N=1$18

the scalar equation becomes

$\mathcal N=1$19

The paper states that this equation is hyperbolic for $\mathcal N=1$20, elliptic for $\mathcal N=1$21, and for $\mathcal N=1$22 coincides with the $\mathcal N=1$23-wave Regge–Wheeler equation (Ecker et al., 2024).

The same coupling also changes the geometry. After matter is added to Carroll dilaton gravity, the $\mathcal N=1$24 and $\mathcal N=1$25 equations become

$\mathcal N=1$26

The left-hand sides are, respectively, intrinsic torsion and standard torsion. In the pure gravity sector the corresponding sources vanish; with swifton matter they are turned on by the mixed time-space derivative structure. This is the explicit sense in which scalar backreaction generates dynamical torsion in $\mathcal N=1$27 Carroll dilaton gravity (Ecker et al., 2024).

This matter-coupling result should be distinguished from the higher-dimensional electric/magnetic scalar couplings obtained by Carroll limits of relativistic matter. In the general Carroll gravity literature, the simplest scalar limit keeps only time derivatives,

$\mathcal N=1$28

and thus does not by itself produce finite-speed propagation (Bergshoeff et al., 2017). The $\mathcal N=1$29 swifton construction is special precisely because the Carroll dilaton multiplet supplies the compensating structure absent in ordinary Carroll field theory (Ecker et al., 2024).

6. Supersymmetric, postcarrollian, boundary, and discrete extensions

The generic bosonic theory admits a systematic supersymmetric extension. Two-dimensional $\mathcal N=1$30 and $\mathcal N=1$31 Carroll dilaton supergravity are constructed in a graded Poisson-sigma-model framework, with the bosonic fields $\mathcal N=1$32, dilaton $\mathcal N=1$33, multipliers $\mathcal N=1$34 and $\mathcal N=1$35, and fermionic partners $\mathcal N=1$36 and $\mathcal N=1$37. For generic $\mathcal N=1$38 models the nonvanishing Poisson tensor components are

$\mathcal N=1$39

$\mathcal N=1$40

with

$\mathcal N=1$41

The Carroll–Jackiw–Teitelboim model is recovered for $\mathcal N=1$42 and $\mathcal N=1$43, while $\mathcal N=1$44 admits distinct “democratic” and “despotic” versions (Grumiller et al., 2024).

A different extension keeps the Carroll framework but adds the first subleading sector in the small-$\mathcal N=1$45 expansion. The resulting postcarrollian $\mathcal N=1$46 dilaton gravity introduces additional one-forms $\mathcal N=1$47 and $\mathcal N=1$48, scalar multipliers $\mathcal N=1$49 and $\mathcal N=1$50, and the general first-order action

$\mathcal N=1$51

For the $\mathcal N=1$52-family,

$\mathcal N=1$53

and the solution space splits into an $\mathcal N=1$54 branch, which resembles Carroll black holes, and an $\mathcal N=1$55 branch, in which the dilaton becomes time-like and the solutions resemble cosmologies (Ecker et al., 22 Apr 2025).

Boundary dynamics is another major theme. In the AdS–Carroll limit of JT gravity, imposing metric BF boundary conditions leads to the universal particle-on-group boundary action

$\mathcal N=1$56

and in the AdS–Carroll$\mathcal N=1$57 case the reduced boundary theory is a twisted warped action,

$\mathcal N=1$58

which is identified as the AdS–Carroll analogue of the Schwarzian (Grumiller et al., 2020). In the postcarrollian JT theory the reduced boundary action becomes

$\mathcal N=1$59

a Schwarzian-type mechanics involving both $\mathcal N=1$60 and $\mathcal N=1$61 (Ecker et al., 22 Apr 2025).

The theory also admits a discrete BF realization. In the lattice formulation, one assigns holonomies

$\mathcal N=1$62

to links and face variables $\mathcal N=1$63 to plaquettes, with discrete action

$\mathcal N=1$64

Bulk flatness $\mathcal N=1$65 ensures that the lattice preserves the topological character of the continuum theory, while the boundary phase space carries a discrete affine Carroll algebra and, after reduction, a discrete Virasoro-type algebra (Özer et al., 24 Jun 2026).

Finally, the thesis literature extends the subject to quantum matter on fixed Carroll black-hole backgrounds and states that for the Carroll–Schwarzschild black hole there is a non-vanishing asymptotic energy density, called the Carroll–Hawking effect (Ecker, 13 Mar 2026). Within the present literature, this places Carroll dilaton gravity at the intersection of ultra-relativistic gravity, $\mathcal N=1$66 dilaton models, Carroll black-hole thermodynamics, boundary mechanics, and matter-coupled non-Lorentzian field theory.