Sine Dilaton Gravity: 2D Quantum Dualities

• Sine Dilaton Gravity (sDG) is a 2D dilaton gravity theory with a periodic sine potential, serving as a UV-completion of JT gravity and the dual of DSSYK.
• Canonical quantization and discrete geodesic lengths rigorously connect sDG with matrix integrals and establish its precise holographic duality.
• The model’s quantum group symmetry and Poisson–sigma formulation enable exact matter correlator computations and reveal a rich integrable structure.

Sine Dilaton Gravity (sDG) is a two-dimensional dilaton gravity theory characterized by a periodic "sine" potential for the dilaton field. It occupies a unique position at the intersection of quantum gravity, low-dimensional holography, and quantum integrable systems, providing the precise bulk dual for the double-scaled Sachdev-Ye-Kitaev (DSSYK) model via canonical quantization. The theory is exactly solvable, admits a discrete Hilbert space structure, and encodes a concrete renormalization group (RG) flow from a pair of Liouville conformal field theories (CFTs) in the ultraviolet (UV) to Jackiw-Teitelboim (JT) gravity in the infrared (IR) (Mahapatra et al., 25 Jan 2026, Blommaert et al., 2024, &&&2&&&, Blommaert et al., 2024).

1. Formulation and Classical Structure

The bulk action for sDG is

$$S_{\rm sDG} = \frac{1}{4 |\log q|} \int d^2x\, \sqrt{-g} \left( \Phi R + 2\sin\Phi \right)$$

where $\Phi$ is the dilaton, $R$ is the Ricci scalar, and $|\log q|$ is a coupling inherited from the double-scaled SYK parameter $q$. The potential $V(\Phi) = 2 \sin\Phi$ renders the action $2\pi$-periodic in $\Phi$, leading to crucial consequences for the phase space and quantum spectrum (Mahapatra et al., 25 Jan 2026, Blommaert et al., 2024).

The classical equations of motion are

$$(\nabla_\mu\nabla_\nu - g_{\mu\nu} \Box)\Phi + g_{\mu\nu} \sin\Phi = 0,\qquad R + 2\cos\Phi = 0$$

Stationary black hole solutions in Schwarzschild gauge are given by $\Phi(r) = r$, $F(r) = 2\cos\theta - 2\cos r$, with event horizon at $r = \theta \in [0,\pi]$ and locally AdS$_2$ geometry realized via the Weyl-rescaled metric $ds^2_{AdS} = e^{-i\Phi} ds^2$ (Mahapatra et al., 25 Jan 2026, Blommaert et al., 2024).

The theory can be reformulated as two copies of Liouville theory coupled only through a global Wheeler–DeWitt (WdW) constraint. Under field redefinitions

$$A = \tfrac{Z_1 + iZ_2}{4},\qquad \Phi = \tfrac{Z_1 - iZ_2}{2i}$$

the action decomposes into $S = S_r(Z_1) + S_\ell(Z_2)$, each a standard Liouville action with central charge $1 + 6Q^2$ (Mahapatra et al., 25 Jan 2026).

2. Canonical Quantization and Discretization

Canonical analysis reveals that the renormalized geodesic length $L$ conjugate to momentum $P$ satisfies $[L, P] = 2i|\log q|$. The Hamiltonian in these variables,

$$H_{\rm grav}(L, P) = -\frac{\cos P}{2|\log q|} + \frac{1}{4|\log q|} e^{iP} e^{-L}$$

exactly matches the transfer matrix of DSSYK and the boundary $q$-Schwarzian quantum mechanics. Quantization enforces $L_n = 2 |\log q|\, n$, $n \in \mathbb{N}_0$, discretizing the geodesic length spectrum and, by extension, the entire gravitational Hilbert space. The Wheeler–DeWitt equation yields physical wavefunctions in terms of $q$-Hermite polynomials (Blommaert et al., 2024, Blommaert et al., 2024, Blommaert et al., 28 Jan 2025).

A crucial symmetry arises from the periodicity of the potential: $P \to P + 2\pi$ is a gauge redundancy. Gauging this symmetry projects onto states with nonnegative integer $n$ and removes null Hilbert space states with $n < 0$. This structure leads to finite-dimensional Hilbert space for the associated cosmological models and topologically discrete spectra for closed-universe amplitudes (Blommaert et al., 2024, Blommaert et al., 28 Jan 2025).

3. Holographic Renormalization Group Flow and Central Charge

sDG provides an explicit bulk realization of a holographic RG flow. The holographic $c$-function, constructed via a superpotential formalism or the null energy condition, interpolates monotonically from the UV to the IR: $$\hat c(u) = c_r(u) + c_\ell(u), \qquad \hat c(u) \downarrow \text{ with } u$$ with $c_r(u)$/$c_\ell(u)$ derived from integrating a positive density $D(u)$ dependent on the solution profile.

In the UV limit ($u\to0$), sDG maps to two Liouville sectors with total central charge

$c_{\rm UV} = 26$

In the deep IR ($u\to\infty$), the potential and c-function flow to those of JT gravity, where the central charge strictly vanishes: $$S_{\rm JT} = \frac{1}{16\pi G_2} \int d^2x \sqrt{-g} \, \Phi(R+2),\qquad c_{\rm IR} = 0$$ This interpolation realizes a 2D $c$-theorem explicitly in a strongly coupled gravitational system (Mahapatra et al., 25 Jan 2026, Mahapatra et al., 19 Feb 2025).

The semiclassical central charge in sDG is parametrically large, $c_{sDG} \sim 1/|\log q|$, reflecting the exponentially large density of microstates in the dual DSSYK system. In the IR, $c_{sDG}$ decreases and matches the ordinary JT value, confirming the identification of sDG as a UV-completion of JT gravity with additional degrees of freedom decoupling for $|\log q| \to 0$ (Mahapatra et al., 19 Feb 2025).

4. Holography, Matrix Integrals, and Dualities

Canonical quantization establishes a precise holographic correspondence between sDG and DSSYK, with the $q$-Schwarzian theory as the boundary effective model. The chord number $n$ in the DSSYK combinatorics is matched by the discrete Weyl-rescaled length $L$ in sDG, and the partition functions and correlators coincide exactly (Blommaert et al., 2024). At the level of correlators, the bulk geodesic length, via operators $e^{-\Delta L}$, implements the insertion of bi-local or matter lines in the SYK model—the bulk-boundary dictionary is exact for all thresholds of $\Delta$.

The gravitational amplitudes decompose through factorization over a discrete set of trumpet and wormhole "length" eigenstates. The double trumpet amplitude precisely matches the spectral correlation function of a Hermitian matrix integral with finite support (the "finite-cut" $q$-deformed JT ensemble): $$Z_{\rm wormhole}(\beta_1, \beta_2) = \sum_{b=1}^\infty b\, I_b(\beta_1/\hbar) I_b(\beta_2/\hbar)$$ with $I_b$ a modified Bessel function, and higher-genus and higher-point amplitudes determined recursively by topological gluing rules involving these states (Blommaert et al., 28 Jan 2025, Blommaert et al., 2024).

Open and closed end-of-the-world (EOW) branes are realized as FZZT brane boundary conditions in the Liouville description, and their amplitudes yield a rich brane Hilbert space. The Lefschetz thimble structure in path integrals ensures normalizability of no-boundary wavefunctions in sDG, in stark contrast with JT where the Hartle–Hawking state is non-normalizable (Blommaert et al., 28 Jan 2025, Blommaert et al., 2024).

5. Quantum Group Structure and Integrability

sDG can be recast as a Poisson–sigma model (PSM) whose target space is equipped with a non-linear Poisson algebra. Upon quantization, the PSM structure generates the quantum group $\mathrm{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ for the bosonic case and $\mathrm{U}_q(\mathfrak{osp}(1|2,\mathbb{R}))$ for the $\mathcal{N}=1$ supersymmetric extension. The canonical quantization thus realizes sDG as an exactly solvable model governed by $q$-deformed representation theory (Fan et al., 2021, Blommaert et al., 2023).

Representation theory controls the spectrum and correlation functions: Whittaker functions (matrix elements in the mixed parabolic basis) solve the difference equations from the boundary Hamiltonian and encode the density of states and disk amplitudes. $3j$-symbols and $6j$-symbols, along with Faddeev's double-sine functions, appear as structure constants and crossing kernels—solutions yield all gravitational observables, including out-of-time-ordered correlators (OTOCs). These features connect sDG directly to integrable structures and quantum groups, in parallel with developments in Liouville gravity and topological field theory (Fan et al., 2021, Cui et al., 1 Sep 2025).

6. Matter Correlators, Wormhole Hilbert Space, and Gluing

sDG with bulk matter fields admits an exact computation of arbitrary boundary correlators through a splitting and gluing procedure. Matter geodesics are treated as EOW branes, and correlators are computed by constructing a wormhole Hilbert space factorized in the discrete length basis (labelled by $n$) but non-locally entangled in the energy basis. Distinct choices of splitting correspond to inequivalent but physically isomorphic Hilbert space representations. This geometric approach enables direct computation of all correlators—including higher-point and OTOCs—by explicit gluing rules derived from the $q$-group crossing kernels (Cui et al., 1 Sep 2025).

General results include:

• Exact matching with DSSYK matter correlators, including normalization and OTOC structure.
• The emergence of new integral identities for the $6j$-symbols of $\mathcal{U}_q(\mathrm{su}(1,1))$.
• Gluing on higher-genus manifolds (e.g., the double trumpet) produces finite, well-defined amplitudes, matching the regularized matrix integral and DSSYK predictions.

7. Thermodynamics, One-Loop Corrections, and Entropy

The thermodynamics of sDG can be analyzed semiclassically and at the quantum one-loop level. The entropy deviates from the naive Bekenstein–Hawking area law due to the projection enforcing the periodic shift symmetry for the conjugate momentum, leading to

$$S = \log \rho_{\rm phys}(\Phi_h) \approx 2\pi \Phi_h - 2 \Phi_h^2 < S_{\rm BH}$$

with $\rho_{\rm phys}$ the exact, periodic, $q$-deformed density of states. This nonmonotonic entropy profile is a signature of the finite-dimensional Hilbert space and the underlying "finite-cut" matrix integral universality (Blommaert et al., 2024, Blommaert et al., 28 Jan 2025).

One-loop corrections to the free energy and matter correlators in sDG exactly match the corresponding corrections in DSSYK. The quadratic fluctuation determinant is computable via Forman's extension of the Gelfand–Yaglom theorem adapted to matrix-valued operators with mixed boundary conditions. The boundary conditions selected in the Hartle–Hawking vacuum project out negative-length states, enforcing the discrete, positive-length nature of the bulk Hilbert space and fixing the effective physical temperature (Bossi et al., 2024).

• (Mahapatra et al., 25 Jan 2026) — Probing sine dilaton gravity with flow central charge
• (Fan et al., 2021) — From Quantum Groups to Liouville and Dilaton Quantum Gravity
• (Blommaert et al., 2024) — An entropic puzzle in periodic dilaton gravity and DSSYK
• (Blommaert et al., 28 Jan 2025) — Wormholes, branes and finite matrices in sine dilaton gravity
• (Bossi et al., 2024) — Sine-dilaton gravity vs double-scaled SYK: exploring one-loop quantum corrections
• (Blommaert et al., 2023) — The q-Schwarzian and Liouville gravity
• (Mahapatra et al., 19 Feb 2025) — Holographic central charge for double scaled SYK
• (Tian et al., 2018) — Analogue Hawking Radiation and Sine-Gordon Soliton in a Superconducting Circuit
• (Cui et al., 1 Sep 2025) — Splitting and gluing in sine-dilaton gravity: matter correlators and the wormhole Hilbert space
• (Blommaert et al., 2024) — The dilaton gravity hologram of double-scaled SYK