2D Sinh Dilaton Gravity

• 2D sinh dilaton gravity is a class of two-dimensional dilaton-gravity theories featuring a hyperbolic sine potential that deforms JT gravity and connects to q-Schwarzian duality.
• The models include realizations via minimal string constructions and Liouville reductions, enabling exact solutions through Liouville equations and Schwarzian constraints.
• They also describe classical black hole solutions and holographic correspondences, linking boundary q-Schwarzian mechanics with bulk hyperbolic potential deformations.

2D sinh dilaton gravity denotes a family of two-dimensional dilaton-gravity theories in which the dilaton potential is of hyperbolic-sine type, or a closely related hyperbolic deformation of Jackiw–Teitelboim gravity. In the most direct contemporary usage, it is the model with Euclidean action

$$S_{\text{grav}}= -\frac{1}{2}\int d x \sqrt{g}\bigg(\Phi R + \frac{\sinh(2\pi b^2\Phi)}{\pi b^2}\bigg) -\int d\tau \sqrt{h}\,\Phi K ,$$

together with a boundary counterterm, and it is presented as a one-parameter deformation of JT gravity that is dual on the disk to the q-Schwarzian quantum mechanics (Blommaert et al., 2023). Closely related realizations include the finite-$p$ minimal-string model with

$$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$

the hyperbolic-potential deformation

$$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$

and Liouville dilaton gravity coupled to a sinh-Gordon matter sector (Turiaci et al., 2020, Kyono et al., 2017, Frolov et al., 2017). The expression therefore refers not to a single universally fixed action, but to a technically coherent cluster of $1+1$-dimensional models organized around hyperbolic dilaton interactions, Liouville reductions, and deformations of the JT/Schwarzian correspondence.

1. Core action principles and model realizations

A standard starting point is the generic first-order dilaton-gravity form

$$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$

Within this class, the finite-$p$ minimal-string construction yields the exact sinh potential

$U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$

so that the undeformed $(2,p)$ minimal string can be rewritten as a 2D dilaton-gravity theory with a sinh dilaton potential (Turiaci et al., 2020). In the q-Schwarzian construction, the potential is normalized instead as

$$V_{\text{qsch}}(\Phi)=\frac{\sinh(2\pi b^2\Phi)}{\pi b^2},$$

with $p$0, and the theory is explicitly called sinh dilaton gravity (Blommaert et al., 2023).

A second, distinct realization is the hyperbolic deformation of JT gravity,

$p$1

which is described as a $p$2-dimensional dilaton-gravity system with a hyperbolic dilaton potential (Kyono et al., 2017). Its potential is genuinely of sinh type, but it is a deformation in $p$3, not the linear-dilaton first-order form used in the q-Schwarzian and minimal-string literature.

A third usage appears in Liouville dilaton gravity with backreacting sinh-Gordon matter. There the full action is

$p$4

with

$p$5

$p$6

This model is not a pure sinh potential for the dilaton itself, but in the vacuum truncation it reduces to the hyperbolic deformation of JT gravity, and in the full theory the sinh-Gordon interaction is the structural reason exact solvability survives with backreaction (Frolov et al., 2017).

2. Relation to JT gravity, minimal strings, and Liouville gravity

The central organizing principle is that sinh dilaton gravity is a deformation of JT gravity. In the minimal-string formulation,

$p$7

so keeping

$p$8

fixed gives

$p$9

which is the JT limit (Turiaci et al., 2020). Likewise, in the q-Schwarzian formulation,

$$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$0

so the model is a one-parameter deformation of JT gravity (Blommaert et al., 2023).

Liouville gravity provides the main field-redefinition bridge. In the minimal-string derivation, one introduces

$$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$1

after which the theory becomes

$$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$2

(Turiaci et al., 2020). In the q-Schwarzian/Liouville construction, the corresponding change of variables is

$$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$3

and under this redefinition the Liouville-gravity action reduces to the sinh dilaton gravity action used in the disk duality (Blommaert et al., 2023).

The status of this identification is not uniform across the literature. One strand presents “preliminary evidence that the bulk theory can be interpreted as a 2d dilaton gravity model with a $$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$4 dilaton potential,” and reconstructs

$$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$5

from Liouville fixed-length disk thermodynamics (Mertens et al., 2020). A later strand presents the disk-level duality to q-Schwarzian quantum mechanics and the Poisson-sigma reformulation as an exact solution of sinh dilaton gravity on disk topology (Blommaert et al., 2023). This suggests a shift from motivated bulk interpretation to a more explicit disk-level equivalence, while leaving higher-topology completion outside the established claims.

3. Classical solutions and black-hole thermodynamics

For the q-Schwarzian normalization, the classical Euclidean black-hole solution is written as

$$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$6

with

$$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$7

The horizon sits at $$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$8, the asymptotic region is $$I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right],$$9, and the energy-temperature map is

$$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$0

At the semiclassical level, the entropy satisfies

$$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$1

which matches the gravity-side Bekenstein–Hawking term $$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$2 because $$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$3 (Blommaert et al., 2023).

In the hyperbolic-potential deformation of JT gravity, the vacuum and black-hole sectors are naturally described after the redefinition to two Liouville fields, but the paper also gives explicit thermodynamics for the black hole with conformal matter. The Hawking temperature is

$$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$4

and the energy obtained from the boundary stress tensor is

$$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$5

The entropy derived from integrating $$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$6 agrees with the Bekenstein–Hawking entropy obtained from the effective Newton constant, up to a temperature-independent integration constant (Kyono et al., 2017).

The Liouville–sinh-Gordon model also contains black-hole geometries. In the vacuum sector $$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$7, the action reduces in the $$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$8 frame to

$$S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K ,$$9

which the authors state reproduces, up to normalization, the hyperbolic deformation of the Jackiw–Teitelboim model (Frolov et al., 2017).

4. Exact solvability, Liouville reduction, and Schwarzian constraints

A defining feature of 2D sinh dilaton gravity is that several of its realizations are exactly tractable because they reduce to Liouville equations plus constraints. In the Liouville dilaton gravity with sinh-Gordon matter, one performs the Weyl transformation

$1+1$0

then introduces

$1+1$1

At the special coupling

$1+1$2

the conformal factor $1+1$3 satisfies a Liouville equation and the combinations

$1+1$4

also satisfy Liouville equations. The full system therefore reduces to three Liouville equations plus Schwarzian constraints,

$1+1$5

$1+1$6

This is the core integrability mechanism behind the exact solution families, including vacuum black holes and non-vacuum geometries with nontrivial sinh-Gordon matter (Frolov et al., 2017).

The hyperbolic JT deformation exhibits a closely parallel structure. In conformal gauge,

$1+1$7

one introduces

$1+1$8

The action becomes the difference of two Liouville-type systems, and the equations reduce to

$1+1$9

The constraints are expressible as Schwarzian equalities,

$$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$0

or with matter as a Schwarzian equation sourced by $$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$1 (Kyono et al., 2017).

At the action-theory level, broader embedding results also exist. Generalized 2D dilaton gravity with arbitrary $$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$2, $$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$3, and $$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$4 can be rewritten in kinetic gravity braiding form, so $$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$5- and $$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$6-type potentials are naturally included as special choices of $$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$7 even though the exact solutions derived there are restricted to the shift-symmetric sector (Takahashi et al., 2018). This suggests that exact Liouville reduction is special, whereas action-level accommodation of hyperbolic potentials is generic.

5. Holography, q-Schwarzian duality, and matrix-model formulations

The most explicit holographic statement is

$$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$8

with

$$I= -\frac{1}{2} \int \sqrt{g} \left[ \Phi R + 2 U(\Phi) \right].$$9

The exact disk partition function is

$p$0

and the Poisson-sigma-model formulation has Poisson tensor

$p$1

After reduction to the boundary phase space, the Hamiltonian is

$p$2

and canonical quantization leads to the exact difference equation

$p$3

with

$p$4

This is the basis for the claim that the disk sector of sinh dilaton gravity is exactly solvable (Blommaert et al., 2023).

A complementary matrix-model description arises from minimal strings. A large class of asymptotically AdS$p$5 dilaton gravities are dual to a matrix integral, and the undeformed finite-$p$6 theory is precisely the sinh dilaton gravity model

$p$7

When tachyon deformations are included, the potential becomes

$p$8

In the large-$p$9 limit these deformations become JT gravity plus exponentially decaying defect terms, not an exact sinh potential. This is why the clean exact sinh structure belongs to the finite-$U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$0 minimal-string rewriting, whereas the large-$U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$1 regime is JT-like (Turiaci et al., 2020).

6. Scope, related models, and major distinctions

A persistent misconception is that every hyperbolic or semiclassical 2D dilaton-gravity paper belongs to sinh dilaton gravity. This is not correct. The 2024 study of self-consistent backreaction in a two-horizon dilaton black hole uses the action

$U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$2

with exponential potential $U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$3, and it explicitly states that the model is not a sinh-dilaton model (Akhmedov et al., 2024). Its relevance is methodological: it shows how exact 2D anomalous stress tensors can reorganize multi-horizon geometry under self-consistent semiclassical backreaction.

A second distinction concerns periodic versus hyperbolic deformations. Sine dilaton gravity, with potential $U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$4 or $U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$5, is presented as an analytic continuation or cousin of the sinh theory, but periodicity changes the canonical structure drastically. In the periodic case one finds compact momentum, discrete geodesic lengths, null states below a threshold, and an effectively finite Hilbert space; these are stated as consequences of periodicity and are not presented as generic properties of non-periodic sinh models (Blommaert et al., 2024, Blommaert et al., 2024). This suggests that exact q-Schwarzian solvability has two sharply different branches: hyperbolic $U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$6 sinh dilaton gravity and periodic $U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$7 sine dilaton gravity.

A third distinction is between exact model-specific results and general embedding frameworks. Generalized 2D dilaton gravity in KGB/Horndeski form and the most general local-Lorentz-invariant consistent deformation of JT gravity both accommodate $U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$8- or $U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right),$9-type choices of potential, but they do not by themselves define the canonical sinh dilaton model or solve it in the same sense as the q-Schwarzian or Liouville constructions (Takahashi et al., 2018, Grumiller et al., 2021). A plausible implication is that “2D sinh dilaton gravity” names both a concrete solved theory in the disk/q-Schwarzian literature and a wider design space of generalized 2D dilaton gravities with hyperbolic interactions.

In that wider sense, the subject is best understood as the intersection of three developments: hyperbolic deformations of JT gravity, Liouville and minimal-string rewritings that produce exact $(2,p)$0 potentials, and boundary dual descriptions in terms of q-deformed Schwarzian mechanics and matrix models. The narrowest, most established formulation is the disk-level theory with

$(2,p)$1

while the broader literature shows that hyperbolic dilaton interactions also arise in Yang–Baxter-deformed JT gravity, Liouville dilaton gravity with sinh-Gordon matter, and matrix-model deformations of minimal strings (Blommaert et al., 2023, Kyono et al., 2017, Frolov et al., 2017, Turiaci et al., 2020).