--- title: 2D Sinh Dilaton Gravity url: https://www.emergentmind.com/topics/2d-sinh-dilaton-gravity type: topic --- # 2D Sinh Dilaton Gravity 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 [2312.00871]. 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 [2011.06038] [1704.07410] [1712.07700]. 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 [2011.06038]. 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 \(q=e^{\pi i b^2}\), and the theory is explicitly called sinh dilaton gravity [2312.00871]. A second, distinct realization is the hyperbolic deformation of JT gravity, $$ S_\Phi = \frac{1}{16\pi G}\int d^2x\,\sqrt{-g} \left[ \Phi^2 R +\frac{1}{\eta}\sinh\!\bigl(2\eta\Phi^2\bigr) \right] +\frac{1}{8\pi G}\int dt\,\sqrt{-\gamma_{tt}}\,K , $$ which is described as a \(1+1\)-dimensional dilaton-gravity system with a hyperbolic dilaton potential [1704.07410]. Its potential is genuinely of sinh type, but it is a deformation in \(\Phi^2\), 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 $$ S[g_{\mu\nu},\phi,\chi,f]=S_\phi+S_\chi+S_f, $$ with $$ S_\phi = -\frac{1}{4\pi}\int d^2x\,\sqrt{-g}\, \left[ Q\,\phi R+(1-2bQ)(\nabla\phi)^2-\lambda e^{-2b\phi} \right], $$ $$ S_\chi = -\frac{1}{4\pi}\int d^2x\,\sqrt{-g}\, \left[ (\nabla\chi)^2+m\cosh(2b\chi) \right]. $$ 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 [1712.07700]. ## 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, $$ \sinh(2\pi b^2\Phi)\sim 2\pi b^2 \Phi \qquad (b\to 0), $$ so keeping $$ \Lambda \equiv 4\pi b^2 \mu $$ fixed gives $$ I= - \frac{1}{2} \int \sqrt{g}\, \Phi (R +2 \Lambda), $$ which is the JT limit [2011.06038]. Likewise, in the q-Schwarzian formulation, $$ \frac{\sinh(2\pi b^2\Phi)}{\pi b^2} = 2\Phi + O(b^4\Phi^3), $$ so the model is a one-parameter deformation of JT gravity [2312.00871]. Liouville gravity provides the main field-redefinition bridge. In the minimal-string derivation, one introduces $$ b\phi = \rho - \pi b^2 \Phi,\qquad b \chi = \rho + \pi b^2 \Phi,\qquad g = e^{2\rho}\hat g, $$ after which the theory becomes $$ I= - \frac{1}{2} \int \sqrt{g} \left[ \Phi R + 4\mu \sinh \left( 2\pi b^2 \Phi \right) \right] $$ [2011.06038]. In the q-Schwarzian/Liouville construction, the corresponding change of variables is $$ \varphi=\frac{\rho}{b}-b\pi\Phi, \qquad \chi=\frac{\rho}{b}+b\pi\Phi , $$ and under this redefinition the Liouville-gravity action reduces to the sinh dilaton gravity action used in the disk duality [2312.00871]. 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 \(\sinh \Phi\) dilaton potential,” and reconstructs $$ V(\Phi)\propto \sinh(2\pi b^2\Phi) $$ from Liouville fixed-length disk thermodynamics [2006.07072]. 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 [2312.00871]. 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 $$ ds^2=F(r)d\tau^2+\frac{dr^2}{F(r)}, \qquad \Phi=r, $$ with $$ F(r)=\frac{\cosh(2\pi b^2 r)}{2\pi^2 b^4} -\frac{\cosh(2\pi b^2 \Phi_h)}{2\pi^2 b^4}. $$ The horizon sits at \(r=\Phi_h\), the asymptotic region is \(r\to\infty\), and the energy-temperature map is $$ \alpha = 2\pi b^2\Phi_h,\qquad \frac{\beta}{\pi b^2}=\frac{2\pi}{\sinh(\alpha)},\qquad E(\alpha)=\frac{\cosh(\alpha)}{2\pi^2 b^4}. $$ At the semiclassical level, the entropy satisfies $$ S \sim \frac{\alpha}{b^2}, $$ which matches the gravity-side Bekenstein–Hawking term \(2\pi\Phi_h\) because \(\alpha=2\pi b^2\Phi_h\) [2312.00871]. 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 $$ T_H=\frac{\sqrt{\mu}}{\pi}, $$ and the energy obtained from the boundary stress tensor is $$ E = -\frac{\left(1+\frac{4GN\eta}{3}\right)\log(1-\pi^2T_H^2\eta^2)}{8\pi G\eta^2} +\frac{N}{6}T_H. $$ The entropy derived from integrating \(dE=T_H\,dS\) agrees with the Bekenstein–Hawking entropy obtained from the effective Newton constant, up to a temperature-independent integration constant [1704.07410]. The Liouville–sinh-Gordon model also contains black-hole geometries. In the vacuum sector \(\chi=0\), the action reduces in the \(\hat g_{\mu\nu}\) frame to $$ S = -\frac{1}{4\pi}\int d^2x\,\sqrt{-\hat g}\, \left[ \phi \hat R +2m\sinh(b\phi) +\frac12(\hat\nabla f)^2 \right], $$ which the authors state reproduces, up to normalization, the hyperbolic deformation of the Jackiw–Teitelboim model [1712.07700]. ## 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 $$ g_{\mu\nu}=e^{2b\phi}\,\tilde g_{\mu\nu}, $$ then introduces $$ \omega_1=\phi+\chi,\qquad \omega_2=\phi-\chi. $$ At the special coupling $$ Q=\frac1b, $$ the conformal factor \(\rho\) satisfies a Liouville equation and the combinations $$ \sigma_k=b\omega_k+\rho,\qquad k=1,2, $$ also satisfy Liouville equations. The full system therefore reduces to three Liouville equations plus Schwarzian constraints, $$ \{X_1^+,x^+\}+\{X_2^+,x^+\}=2\{Y^+,x^+\}+2b^2(f_+)^2, $$ $$ \{X_1^-,x^-\}+\{X_2^-,x^-\}=2\{Y^-,x^-\}+2b^2(f_-)^2. $$ This is the core integrability mechanism behind the exact solution families, including vacuum black holes and non-vacuum geometries with nontrivial sinh-Gordon matter [1712.07700]. The hyperbolic JT deformation exhibits a closely parallel structure. In conformal gauge, $$ ds^2=-e^{2\omega}dx^+dx^-, $$ one introduces $$ \omega_1\equiv \omega+\eta\Phi^2, \qquad \omega_2\equiv \omega-\eta\Phi^2 . $$ The action becomes the difference of two Liouville-type systems, and the equations reduce to $$ 4\partial_+\partial_-\omega_1+e^{2\omega_1}=0, \qquad 4\partial_+\partial_-\omega_2+e^{2\omega_2}=0. $$ The constraints are expressible as Schwarzian equalities, $$ \operatorname{Sch}\{X_1^+,x^+\}-\operatorname{Sch}\{X_2^+,x^+\}=0,\qquad \operatorname{Sch}\{X_1^-,x^-\}-\operatorname{Sch}\{X_2^-,x^-\}=0, $$ or with matter as a Schwarzian equation sourced by \(T_{\pm\pm}\) [1704.07410]. At the action-theory level, broader embedding results also exist. Generalized 2D dilaton gravity with arbitrary \(\xi(\phi)R\), \(k(\phi,X)\), and \(C(\phi,X)\nabla^\mu\phi\nabla_\mu X\) can be rewritten in kinetic gravity braiding form, so \(\sinh\)- and \(\cosh\)-type potentials are naturally included as special choices of \(K(\phi,X)\) even though the exact solutions derived there are restricted to the shift-symmetric sector [1812.08847]. 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 $$ \text{q-Schwarzian QM} \;\longleftrightarrow\; \text{2D sinh dilaton gravity} \;\equiv\; \text{Liouville gravity on the disk}, $$ with $$ q=e^{\pi i b^2}. $$ The exact disk partition function is $$ Z(\beta)=\int_0^\infty d s\,\frac{1}{S_b(\pm 2i s)}\, \exp\bigg(-\beta \frac{\cosh(2\pi b s)}{2\pi^2 b^4}\bigg), $$ and the Poisson-sigma-model formulation has Poisson tensor $$ \alpha_{H0}=-J_1,\qquad \alpha_{H1}=-J_0,\qquad \alpha_{01}=\frac{\sinh(2\pi b^2 J_H)}{2\pi b^2}. $$ After reduction to the boundary phase space, the Hamiltonian is $$ \mathbf{H}(j_A)=-j_1^2+j_0^2+\frac{\cosh(2\pi b^2 h)}{2\pi^2 b^4}, $$ and canonical quantization leads to the exact difference equation $$ \psi_\alpha(\varphi+i\pi b^2) + (1-e^{-2\varphi})\,\psi_\alpha(\varphi-i\pi b^2) = 2\cosh(\alpha)\,\psi_\alpha(\varphi), $$ with $$ E(\alpha)=\frac{\cosh(\alpha)}{2\pi^2 b^4}. $$ This is the basis for the claim that the disk sector of sinh dilaton gravity is exactly solvable [2312.00871]. A complementary matrix-model description arises from minimal strings. A large class of asymptotically AdS\(_2\) dilaton gravities are dual to a matrix integral, and the undeformed finite-\(p\) theory is precisely the sinh dilaton gravity model $$ U(\Phi)=2\mu \sinh \left( 2 \pi b^2 \Phi \right). $$ When tachyon deformations are included, the potential becomes $$ U(\Phi) = 2\mu \sinh \left( 2\pi b^2 \Phi \right) + \sum_{n=1}^{m-1} \tau_n \, e^{-2\pi b^2 n \Phi}. $$ In the large-\(p\) 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-\(p\) minimal-string rewriting, whereas the large-\(p\) regime is JT-like [2011.06038]. ## 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 $$ S=\frac{1}{16\pi G}\int d^2x\sqrt{-g}\,e^{-2\phi} \Big[R-4\omega(\partial\phi)^2+4\lambda^2\Big] -\frac12\int d^2x\,\partial_\mu\varphi\,\partial^\mu\varphi, $$ with exponential potential \(V(\phi)\sim e^{-2\phi}\), and it explicitly states that the model is not a sinh-dilaton model [2401.07645]. 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 \(V(\Phi)=2\sin\Phi\) or \(V(\Phi)=\sin(2|\log q|\Phi)/|\log q|\), 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 [2404.03535] [2411.16922]. This suggests that exact q-Schwarzian solvability has two sharply different branches: hyperbolic \(|q|=1\) sinh dilaton gravity and periodic \(0