Sine-Dilaton Gravity: Periodic Dilaton Dynamics

• Sine-dilaton gravity is a two-dimensional dilaton gravity theory characterized by a periodic dilaton potential that underpins innovative holographic dual models.
• Its formulation bridges JT, sinh-dilaton, and Liouville models, offering analytic continuations and discrete quantization to address black hole thermodynamics.
• Explicit actions and boundary terms yield insights into gauge symmetries, entropy puzzles, and the emergence of a dual q-Schwarzian or DSSYK description.

Searching arXiv for recent and foundational papers on sine-dilaton gravity to ground the article in the current literature. Sine-dilaton gravity is a two-dimensional dilaton gravity theory whose defining feature is a periodic dilaton potential. In commonly used conventions the potential is written as $V(\Phi)=\sin(2|\ln q|\Phi)/|\ln q|$ or, after a rescaling of the dilaton, $V(\Phi)=2\sin\Phi$. In recent work it has been formulated as a holographic dual of the double-scaled SYK model through the $q$-Schwarzian boundary theory, while closely related analytic continuations connect it to sinh-dilaton and Liouville gravity (Blommaert et al., 2024, Blommaert et al., 2023, Fan et al., 2021).

1. Defining structure and action

A standard Euclidean presentation is

$$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$

while a closely related normalization used in the DSSYK literature is

$$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$

After rescaling the dilaton, another common form is

$$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$

These expressions differ by normalization and field conventions, but they all exhibit the same periodic potential structure (Blommaert et al., 2023, Blommaert et al., 2024, Mahapatra et al., 25 Jan 2026).

Boundary terms are essential in the holographic formulation. One explicit choice is

$$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$

and in the $\hbar\equiv 2|\ln q|$ convention the action on a manifold $\mathcal M$ with boundary $\partial\mathcal M$ is

$V(\Phi)=2\sin\Phi$0

Associated asymptotic conditions are written as

$V(\Phi)=2\sin\Phi$1

or equivalently, in the rescaled convention,

$V(\Phi)=2\sin\Phi$2

The periodicity $V(\Phi)=2\sin\Phi$3 is repeatedly identified as the defining feature of the theory (Cui et al., 1 Sep 2025, Blommaert et al., 28 Jan 2025).

In conformal gauge, the bulk theory is also written as

$V(\Phi)=2\sin\Phi$4

with $V(\Phi)=2\sin\Phi$5. The same literature notes the analytic continuation

$V(\Phi)=2\sin\Phi$6

which places sine-dilaton gravity adjacent to hyperbolic-sine dilaton models and Liouville gravity (Bossi et al., 2024).

2. Classical equations, black holes, and thermodynamics

Varying the sine-dilaton action yields a two-dimensional dilaton-gravity system with periodic curvature source. In the $V(\Phi)=2\sin\Phi$7 normalization the field equations are

$V(\Phi)=2\sin\Phi$8

and in the rescaled $V(\Phi)=2\sin\Phi$9 normalization they become

$q$0

These equations admit static black-hole-like solutions (Cui et al., 1 Sep 2025, Blommaert et al., 2024).

A commonly used family is

$q$1

with horizon at $q$2. In the $q$3 convention, the same geometry is written as

$q$4

where the horizon is at $q$5 (Blommaert et al., 2024, Blommaert et al., 2023).

Regularity at the Euclidean horizon fixes the Hawking temperature. In the $q$6 parametrization,

$q$7

with $q$8. In the $q$9 parametrization,

$$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$0

The thermodynamic saddle of the boundary theory is often expressed through $$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$1, for which

$$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$2

These formulas are the direct gravitational input for the comparison with the $$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$3-Schwarzian and DSSYK (Blommaert et al., 2023, Blommaert et al., 2024).

A useful bulk observable is the renormalized geodesic length of a spatial slice in a Weyl-rescaled auxiliary AdS$$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$4 geometry. In the periodic-dilaton literature this is written as

$$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$5

with physical ADM energy $$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$6 when $$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$7. This quantity becomes central in the canonical description and in the DSSYK dictionary (Blommaert et al., 2024).

3. Boundary description and the DSSYK duality

A principal result of the recent literature is that canonical quantization of sine-dilaton gravity reproduces the $$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$8-Schwarzian quantum mechanics that appears as the auxiliary transfer-matrix system of double-scaled SYK. In a convenient canonical pair $$S[g_{\mu\nu},\Phi] = -\frac12\int d^2x\,\sqrt{g}\,\bigl[\Phi R+V(\Phi)\bigr] -\int_{\partial} d\tau\,\sqrt{h}\,\Phi K, \qquad V(\Phi)=\frac{\sin(2\mu\Phi)}{\mu}, \quad \mu=-\ln q>0,$$9,

$$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$0

The bulk interpretation is that the chord number of DSSYK is played by the Weyl-rescaled geodesic length,

$$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$1

The positivity of $$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$2 is therefore the bulk positivity constraint $$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$3 (Blommaert et al., 2024).

At the level of the boundary path integral, the same dynamics is encoded in a one-dimensional canonical action. In the $$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$4 variables used in the DSSYK/chord description,

$$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$5

A closely related formulation uses $$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$6, canonical conjugate $$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$7, and

$$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$8

The canonical map

$$S[g,\Phi] = \frac1{2G_N}\int d^2x\sqrt g\; \Bigl[\Phi R+\frac{\sin\!\bigl(2|\ln q|\,\Phi\bigr)}{|\ln q|}\Bigr] +S_{\rm bdy}.$$9

reproduces the DSSYK transfer-matrix dynamics (Blommaert et al., 2024, Bossi et al., 2024).

A central subtlety is the distinction between the physical inverse temperature and the “fake” inverse temperature. In DSSYK the physical inverse temperature is

$$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$0

whereas the smooth Lorentzian black hole has

$$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$1

Without the positivity constraint one would obtain $$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$2. Imposing $$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$3 changes the allowed classical orbits of the $$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$4-Schwarzian and shifts $$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$5 away from $$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$6 by $$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$7. Semiclassically this is described by inserting a curvature defect with opening angle

$$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$8

at the horizon. The Euclidean periodicity is then $$S_{\rm sDG} = \frac{1}{4\alpha} \int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr), \qquad \alpha\equiv |\ln q|>0.$$9, while the Lorentzian two-sided geometry remains a smooth black hole of temperature $$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$0 (Blommaert et al., 2024).

4. Gauging, discreteness, and the entropy problem

Because the dilaton potential is periodic, the Hamiltonian is periodic in the momentum conjugate to the length variable. In the Schwarzschild-gauge analysis of periodic dilaton gravity one obtains for sine-dilaton gravity

$$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$1

Naive ungauged quantization on $$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$2 produces continuous wavefunctions with exact density of states

$$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$3

and the corresponding partition function diverges because the energy is $$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$4-periodic in $$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$5. The proposed resolution is to gauge the redundant shift symmetry $$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$6 (Blommaert et al., 2024).

Gauging forces the conjugate variable $$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$7 to be discrete,

$$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$8

with $$S_{\rm bdy} = \frac1{2G_N}\int d\tau\sqrt h\; \Bigl[\Phi K-i\,\frac{e^{-i|\ln q|\Phi}}{2|\ln q|}\Bigr],$$9 acting as $\hbar\equiv 2|\ln q|$0. The same analysis states that the projected wavefunctions vanish for $\hbar\equiv 2|\ln q|$1, so those levels are null and must be removed, leaving the physical Hilbert space spanned by

$\hbar\equiv 2|\ln q|$2

The exact gauged density of states is written as

$\hbar\equiv 2|\ln q|$3

The same work emphasizes that the entropy then departs strongly from the Bekenstein-Hawking form. In the semiclassical limit,

$\hbar\equiv 2|\ln q|$4

This is the “entropic puzzle” of periodic dilaton gravity (Blommaert et al., 2024).

A Wheeler-DeWitt treatment leads to an equivalent discretization. In minisuperspace,

$\hbar\equiv 2|\ln q|$5

and gauge invariance under $\hbar\equiv 2|\ln q|$6 quantizes the conjugate momentum as $\hbar\equiv 2|\ln q|$7, $\hbar\equiv 2|\ln q|$8. For trumpet boundaries this gives a discrete AdS length

$\hbar\equiv 2|\ln q|$9

and the same analysis states that the discreteness of $\mathcal M$0 implies a maximal energy and makes the spectrum UV-complete. It also replaces the ungauged delta-function no-boundary state by a normalizable Gaussian disk wavefunction in the discrete label $\mathcal M$1 (Blommaert et al., 28 Jan 2025).

5. Liouville reformulations, central charge flow, and one-loop structure

Several papers reformulate sine-dilaton gravity in Liouville language. One result states that the $\mathcal M$2-Schwarzian is holographically dual to Liouville gravity and that, for real $\mathcal M$3, this same $\mathcal M$4-Schwarzian corresponds to double-scaled SYK and is dual to a sine-dilaton gravity (Blommaert et al., 2023). A more explicit conformal-gauge rewriting introduces fields $\mathcal M$5 by

$\mathcal M$6

after which the sine-dilaton action decomposes as

$\mathcal M$7

with each sector an ordinary Liouville CFT of central charge

$\mathcal M$8

In this representation the total UV central charge is

$\mathcal M$9

(Mahapatra et al., 25 Jan 2026).

The same work constructs a holographic $\partial\mathcal M$0-function in domain-wall gauge,

$\partial\mathcal M$1

by defining

$\partial\mathcal M$2

Because $\partial\mathcal M$3, the flow is monotonic,

$\partial\mathcal M$4

In the deep infrared, $\partial\mathcal M$5 implies $\partial\mathcal M$6, and the action reduces to pure Jackiw-Teitelboim gravity,

$\partial\mathcal M$7

with vanishing boundary central charge in two dimensions (Mahapatra et al., 25 Jan 2026).

The quantum comparison with DSSYK has been pushed to one loop. A careful path-integral analysis selects the Hartle-Hawking state with $\partial\mathcal M$8 as the empty state and imposes Dirichlet boundary conditions

$\partial\mathcal M$9

Expanding around the classical saddle,

$V(\Phi)=2\sin\Phi$00

one computes the one-loop determinant using Forman’s extension of the Gel'fand-Yaglom theorem. The one-loop partition function is

$V(\Phi)=2\sin\Phi$01

and the logarithmic correction to the free energy matches the DSSYK result up to an ordering ambiguity of the gravity Hamiltonian. The same work also computes the one-loop correction to the boundary-to-boundary propagator of a non-minimally coupled bulk matter field and finds agreement with the corresponding DSSYK matter correlator (Bossi et al., 2024).

A structurally related strand of the literature studies the quantization of hyperbolic-sine dilaton gravity through the Poisson-sigma model and the quantum group $V(\Phi)=2\sin\Phi$02. In that framework the Whittaker functions, $V(\Phi)=2\sin\Phi$03-symbols, and Plancherel measure organize the dilaton-gravity Hilbert space, and the classical limit $V(\Phi)=2\sin\Phi$04 recovers JT gravity (Fan et al., 2021). This suggests a precise analytic-continuation bridge between periodic and hyperbolic-sine potentials.

6. Matter, end-of-the-world branes, and wormhole observables

Matter observables are naturally expressed through geodesic lengths in the Weyl-rescaled AdS$V(\Phi)=2\sin\Phi$05 geometry. For a massive scalar or worldline of mass $V(\Phi)=2\sin\Phi$06, the boundary-to-boundary propagator in a fixed background is written as

$V(\Phi)=2\sin\Phi$07

A key geometric idea is to regard the matter geodesic as a thin shell cutting the disk into left and right regions, then split it into two end-of-the-world branes with $V(\Phi)=2\sin\Phi$08 and glue the two bulk regions back together (Cui et al., 1 Sep 2025).

This leads to two complementary quantization channels. In the fully-open channel the brane Hilbert space is the empty sine-dilaton Hilbert space $V(\Phi)=2\sin\Phi$09, with EOW-state wavefunctions in the length basis given by continuous $V(\Phi)=2\sin\Phi$10-Hermite polynomials. In the semi-open channel one obtains a twisted Hilbert space $V(\Phi)=2\sin\Phi$11 with the same discrete length basis but an energy basis diagonalized by Al-Salam-Chihara polynomials. The resulting one-particle wormhole Hilbert space has a factorized length basis,

$V(\Phi)=2\sin\Phi$12

whereas the energy basis is explicitly nonlocal and state dependent. Different choices of splitting, including $V(\Phi)=2\sin\Phi$13-split, $V(\Phi)=2\sin\Phi$14-split, and more general $V(\Phi)=2\sin\Phi$15-split representations, are stated to be equivalent for physical observables. Within this framework disk correlators, crossed four-point functions, and the OTOC reproduce the DSSYK chord-diagram expressions, and comparing different split representations yields a new identity for the $V(\Phi)=2\sin\Phi$16-symbol of $V(\Phi)=2\sin\Phi$17 (Cui et al., 1 Sep 2025).

The same discrete structure appears in closed-universe and wormhole amplitudes. WdW quantization gives trumpet wavefunctions $V(\Phi)=2\sin\Phi$18 and the connected two-boundary amplitude

$V(\Phi)=2\sin\Phi$19

This is described as the sine-dilaton double trumpet and is matched to the leading connected correlator of a finite-cut Hermitian matrix integral. The same work further identifies open and closed EOW branes with FZZT branes for the two Liouville theories that make up sine-dilaton gravity, and shows that a Legendre transform of the EOW-brane amplitude reproduces the trumpet (Blommaert et al., 28 Jan 2025).

Another wormhole observable is the renormalized quantum length operator $V(\Phi)=2\sin\Phi$20. In the infinite-temperature limit of DSSYK, the equality

$V(\Phi)=2\sin\Phi$21

is used to identify the quantum wormhole length with Krylov spread complexity. At $V(\Phi)=2\sin\Phi$22, one finds

$V(\Phi)=2\sin\Phi$23

which reproduces the classical JT wormhole length. The same analysis develops a semiclassical expansion up to five loops, as well as expansions for the variance and third cumulant of the length operator, and proposes an all-orders resummation for the large-time slope together with non-perturbative corrections of order $V(\Phi)=2\sin\Phi$24 indicated by numerical data (Alfinito et al., 18 Jun 2026).

7. Adjacent constructions and terminology

The terminology surrounding sine-dilaton gravity overlaps with two neighboring subjects. One is the analytic continuation to hyperbolic-sine dilaton gravity. In the Poisson-sigma model description, choosing a prepotential $V(\Phi)=2\sin\Phi$25 leads after quantization to the quantum group $V(\Phi)=2\sin\Phi$26. The associated Whittaker functions, $V(\Phi)=2\sin\Phi$27-symbols, and Plancherel measure reproduce structural features of Liouville gravity and its supersymmetric extensions, and the classical $V(\Phi)=2\sin\Phi$28 limit recovers JT gravity (Fan et al., 2021). This is not the same as the periodic theory, but the recent sine-dilaton literature repeatedly uses analytic continuation between sine and sinh potentials.

A second neighboring subject is the construction of sine-Gordon-type kinks in generalized two-dimensional dilaton gravity. In that setting one studies an action

$V(\Phi)=2\sin\Phi$29

and, for a specific choice

$V(\Phi)=2\sin\Phi$30

obtains a sine-Gordon-type kink with pure AdS$V(\Phi)=2\sin\Phi$31 warp factor $V(\Phi)=2\sin\Phi$32. For the special case $V(\Phi)=2\sin\Phi$33, $V(\Phi)=2\sin\Phi$34, the scalar profile reduces to the familiar sine-Gordon kink $V(\Phi)=2\sin\Phi$35, and linear perturbations are governed by a factorizable Schrödinger operator (Zhong et al., 2023). This construction is a different model, but it clarifies how sine-Gordon structures can emerge inside broader dilaton-gravity systems.

A more direct analogue-gravity line of work uses the duality between black holes in Jackiw-Teitelboim dilaton gravity and solitons in sine-Gordon field theory. In a dc-SQUID transmission line the superconducting phase obeys the sine-Gordon equation, and the induced metric of a one-soliton solution can be rewritten in the standard JT black-hole form with horizon at $V(\Phi)=2\sin\Phi$36. In that setup the Hawking temperature in the soliton comoving frame is

$V(\Phi)=2\sin\Phi$37

with a Doppler-shifted laboratory temperature and an emitted thermal spectrum interpreted as quantum soliton evaporation. This is conceptually adjacent to sine-dilaton gravity rather than an instance of it, but it underscores the broader role of periodic scalar structures in two-dimensional black-hole analogues (Tian et al., 2018).