---
title: 'Sine-Dilaton Gravity: Periodic Dilaton Dynamics'
url: https://www.emergentmind.com/topics/sine-dilaton-gravity
type: topic
---

# Sine-Dilaton Gravity: Periodic Dilaton Dynamics

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 [2404.03535][2312.00871][2109.07770].

## 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 [2312.00871][2404.03535][2601.17698].

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
\[
S_{\rm SD}
=
\frac{1}{2\hbar}
\Biggl[
\frac12\int_{\mathcal M} d^2x\sqrt g\,(\Phi R+2\sin\Phi)
+\int_{\partial\mathcal M}du\,\sqrt h\,\Phi K
-i\int_{\partial\mathcal M}du\,\sqrt h\,e^{-i\Phi/2}
\Biggr].
\]
Associated asymptotic conditions are written as
\[
\Phi\big|_{\rm bdy}=\frac{\pi}{2}+i\,u,
\qquad
\sqrt h\,e^{i\Phi/2}\big|_{\rm bdy}=\frac{i}{2|\ln q|},
\]
or equivalently, in the rescaled convention,
\[
2|\ln q|\,\Phi_{\rm bdy}=\frac\pi2+i\,\infty,
\qquad
\sqrt h\,e^{i|\ln q|\Phi_{\rm bdy}}=\frac{i}{2|\ln q|}.
\]
The periodicity \(\Phi\to \Phi+2\pi\) is repeatedly identified as the defining feature of the theory [2509.01680][2501.17091].

In conformal gauge, the bulk theory is also written as
\[
S_{\rm grav}
=
-\frac12\int d^2x\sqrt g\,\bigl[\phi R+U(\phi)\bigr],
\qquad
U(\phi)=\frac{1}{\epsilon}\sin(2\epsilon\phi),
\]
with \(\epsilon\equiv |\log q|\simeq \pi b^2\). The same literature notes the analytic continuation
\[
U(\phi)=2\mu\,\sinh(b\phi)\iff U(\phi)=2\mu\,\sin(i b\phi),
\]
which places sine-dilaton gravity adjacent to hyperbolic-sine dilaton models and Liouville gravity [2411.15957].

## 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 \(\sin\Phi\) normalization the field equations are
\[
R+2\cos\Phi=0,
\qquad
\nabla_\mu\nabla_\nu\Phi-g_{\mu\nu}\Box\Phi+g_{\mu\nu}\sin\Phi=0,
\]
and in the rescaled \(\sin(2|\ln q|\Phi)\) normalization they become
\[
R+2|\ln q|\,\cos\!\bigl(2|\ln q|\Phi\bigr)=0,
\qquad
\nabla_\mu\nabla_\nu\Phi-g_{\mu\nu}\nabla^2\Phi
+g_{\mu\nu}\,\frac{\sin(2|\ln q|\Phi)}{2|\ln q|}=0.
\]
These equations admit static black-hole-like solutions [2509.01680][2404.03535].

A commonly used family is
\[
\Phi(r)=r,
\qquad
ds^2=F(r)\,d\tau^2+\frac{dr^2}{F(r)},
\qquad
F(r)=2\cos\theta-2\cos r,
\qquad
\theta\in[0,\pi],
\]
with horizon at \(r=\theta\). In the \(\mu=-\ln q\) convention, the same geometry is written as
\[
ds^2=F(r)\,d\tau^2+\frac{dr^2}{F(r)},
\qquad
\Phi(r)=r,
\qquad
F(r)=\frac{\cos(2\mu r)-\cos(2\mu\Phi_h)}{2\mu^2},
\]
where the horizon is at \(r=\Phi_h\) [2404.03535][2312.00871].

Regularity at the Euclidean horizon fixes the Hawking temperature. In the \(\theta\) parametrization,
\[
\beta_{\rm BH}=\frac{2\pi}{\sin\theta},
\qquad
E=-\frac{\cos\theta}{2|\ln q|},
\qquad
S_{\rm BH}=\frac{\pi\theta}{|\ln q|},
\]
with \(8\pi G_N=1\). In the \(\mu\) parametrization,
\[
T=\frac{\sin(2\mu\Phi_h)}{4\pi\mu},
\qquad
S=2\pi\Phi_h,
\qquad
M=-\frac{\cos(2\mu\Phi_h)}{2\mu^2}.
\]
The thermodynamic saddle of the boundary theory is often expressed through \(\alpha=2\mu\Phi_h\), for which
\[
\beta=\frac{2\pi}{\sin\alpha},
\qquad
S=\frac{\alpha}{\mu},
\qquad
E=-\frac{\cos\alpha}{2\mu^2}.
\]
These formulas are the direct gravitational input for the comparison with the \(q\)-Schwarzian and DSSYK [2312.00871][2404.03535].

A useful bulk observable is the renormalized geodesic length of a spatial slice in a Weyl-rescaled auxiliary AdS\(_2\) geometry. In the periodic-dilaton literature this is written as
\[
e^{-L}
=
\frac{V(\Phi_h)^2}{\cosh\!\bigl(V(\Phi_h)\,T/2\bigr)^2},
\]
with physical ADM energy \(H=W(\Phi_h)\) when \(W'(\Phi)=V(\Phi)\). This quantity becomes central in the canonical description and in the DSSYK dictionary [2411.16922].

## 3. Boundary description and the DSSYK duality

A principal result of the recent literature is that canonical quantization of sine-dilaton gravity reproduces the \(q\)-Schwarzian quantum mechanics that appears as the auxiliary transfer-matrix system of double-scaled SYK. In a convenient canonical pair \((L,P)\),
\[
[\mathbf L,\mathbf P]=2i|\ln q|,
\qquad
\mathbf H
=
-\frac{\cos(\mathbf P)}{2|\ln q|}
-\frac{1}{4|\ln q|}\,e^{i\mathbf P}e^{-\mathbf L}.
\]
The bulk interpretation is that the chord number of DSSYK is played by the Weyl-rescaled geodesic length,
\[
\mathbf L=2|\ln q|\,\mathbf n,
\qquad
\mathbf n\ge 0.
\]
The positivity of \(\mathbf n\) is therefore the bulk positivity constraint \(L\ge 0\) [2404.03535].

At the level of the boundary path integral, the same dynamics is encoded in a one-dimensional canonical action. In the \((\varphi,p)\) variables used in the DSSYK/chord description,
\[
S_{\rm qSch}
=
\int_0^\beta du\,
\Bigl[
i\,p\,\dot\varphi
+\tfrac12\cos p
-\tfrac14 e^{ip}e^{-2\varphi}
\Bigr].
\]
A closely related formulation uses \(L\equiv 2\phi\), canonical conjugate \(p\), and
\[
S_{q{\rm Sch}}[\phi,p]
=
\frac{1}{\epsilon}\int_0^\beta d\tau
\Bigl[
i\,p\,\phi'
+\tfrac12\cos p
-\tfrac14 e^{ip}e^{-2\phi}
\Bigr].
\]
The canonical map
\[
q=e^{i\pi b^2},
\qquad
\hat L\leftrightarrow 2\epsilon\,\hat n,
\qquad
e^{i\hat p}\leftrightarrow -\hat\alpha
\]
reproduces the DSSYK transfer-matrix dynamics [2404.03535][2411.15957].

A central subtlety is the distinction between the physical inverse temperature and the “fake” inverse temperature. In DSSYK the physical inverse temperature is
\[
\beta=\frac{2\pi-4\theta}{\sin\theta},
\]
whereas the smooth Lorentzian black hole has
\[
\beta_{\rm BH}=\frac{2\pi}{\sin\theta}.
\]
Without the positivity constraint one would obtain \(\beta=\beta_{\rm BH}\). Imposing \(L\ge 0\) changes the allowed classical orbits of the \(q\)-Schwarzian and shifts \(\beta\) away from \(\beta_{\rm BH}\) by \(4\theta/\sin\theta\). Semiclassically this is described by inserting a curvature defect with opening angle
\[
\gamma=2\pi-4\theta
\]
at the horizon. The Euclidean periodicity is then \(\beta\), while the Lorentzian two-sided geometry remains a smooth black hole of temperature \(1/\beta_{\rm BH}\) [2404.03535].

## 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
\[
H(P,L)=-\cos(P)+\tfrac12 e^{iP}e^{-L},
\qquad
P\sim P+2\pi.
\]
Naive ungauged quantization on \(L\in\mathbb R\) produces continuous wavefunctions with exact density of states
\[
\rho_{\rm ungauged}(\theta)
=
\sin(\theta)\,
\sinh\!\Bigl(\frac{\pi\theta}{|\log q|}\Bigr),
\qquad
E=-\cos(\theta),
\]
and the corresponding partition function diverges because the energy is \(2\pi\)-periodic in \(\theta\). The proposed resolution is to gauge the redundant shift symmetry \(P\to P+2\pi\) [2411.16922].

Gauging forces the conjugate variable \(L\) to be discrete,
\[
L_n=n\,\Delta L,
\qquad
n\in\mathbb Z,
\qquad
\Delta L=2|\log q|,
\]
with \(e^{iP}\) acting as \(n\to n+1\). The same analysis states that the projected wavefunctions vanish for \(n<0\), so those levels are null and must be removed, leaving the physical Hilbert space spanned by
\[
\{\ket n\mid n=0,1,2,\dots\}.
\]
The exact gauged density of states is written as
\[
\rho_{\rm gauged}(E)
=
(e^{+2i\theta};q^2)_\infty
+
(e^{-2i\theta};q^2)_\infty,
\qquad
E=-\cos\theta,
\qquad
\theta\in[0,\pi].
\]
The same work emphasizes that the entropy then departs strongly from the Bekenstein-Hawking form. In the semiclassical limit,
\[
S(\theta)=S_{\rm BH}-2\theta^2+\mathcal O(\log|\log q|),
\qquad
S_{\rm BH}=2\pi\Phi_h.
\]
This is the “entropic puzzle” of periodic dilaton gravity [2411.16922].

A Wheeler-DeWitt treatment leads to an equivalent discretization. In minisuperspace,
\[
H_{\rm WdW}\,\psi(\ell,\Phi)=0,
\qquad
H_{\rm WdW}
=
\hbar^2\frac{\partial}{\partial\Phi}\frac{\partial}{\partial \ell}
-\ell\sin\Phi,
\]
and gauge invariance under \(\Phi\to \Phi+2\pi\) quantizes the conjugate momentum as \(\hbar b\), \(b\in\mathbb Z\). For trumpet boundaries this gives a discrete AdS length
\[
L_{\rm AdS}=\hbar\,b,
\qquad
b\in\mathbb N,
\]
and the same analysis states that the discreteness of \(b\) 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 \(b\) [2501.17091].

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

Several papers reformulate sine-dilaton gravity in Liouville language. One result states that the \(q\)-Schwarzian is holographically dual to Liouville gravity and that, for real \(q\), this same \(q\)-Schwarzian corresponds to double-scaled SYK and is dual to a sine-dilaton gravity [2312.00871]. A more explicit conformal-gauge rewriting introduces fields \(Z_1,Z_2\) by
\[
A=\frac{Z_1+iZ_2}{4},
\qquad
\Phi=\frac{Z_1-iZ_2}{2i},
\]
after which the sine-dilaton action decomposes as
\[
S_{\rm sDG}=S_r(Z_1)+S_\ell(Z_2),
\]
with each sector an ordinary Liouville CFT of central charge
\[
c_i=1+6Q_i^2,
\qquad
Q_i=b_i+b_i^{-1}.
\]
In this representation the total UV central charge is
\[
c_{\rm UV}=c_1+c_2=26
\]
[2601.17698].

The same work constructs a holographic \(c\)-function in domain-wall gauge,
\[
ds^2=e^{2A(u)}(-dt^2+du^2),
\qquad
\Phi=\Phi(u),
\]
by defining
\[
\hat c(u)=\frac{c_{\rm UV}}{N}\int_u^\infty dv\,D(v),
\qquad
N=\int_0^\infty dv\,D(v).
\]
Because \(D(u)>0\), the flow is monotonic,
\[
\frac{d\hat c}{du}<0,
\qquad
\hat c(0)=26,
\qquad
\hat c(\infty)=0.
\]
In the deep infrared, \(\Phi\ll 1\) implies \(\sin\Phi\approx\Phi\), and the action reduces to pure Jackiw-Teitelboim gravity,
\[
S_{\rm IR}
=
\frac{1}{4\alpha}\int d^2x\sqrt{-g}\,(\Phi R+2\Phi)
\equiv
\frac{1}{16\pi G_2}\int d^2x\sqrt{-g}\,\Phi(R+2),
\]
with vanishing boundary central charge in two dimensions [2601.17698].

The quantum comparison with DSSYK has been pushed to one loop. A careful path-integral analysis selects the Hartle-Hawking state with \(L=0\) as the empty state and imposes Dirichlet boundary conditions
\[
\phi(0)=\phi(\beta)=0.
\]
Expanding around the classical saddle,
\[
\phi=\phi_{\rm cl}+\sqrt\epsilon\,\delta\phi,
\qquad
p=p_{\rm cl}+\sqrt\epsilon\,\delta p,
\]
one computes the one-loop determinant using Forman’s extension of the Gel'fand-Yaglom theorem. The one-loop partition function is
\[
Z_{\rm one-loop}=Z_{\rm cl}\cdot \mathrm{Det}^{-1/2}L,
\]
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 [2411.15957].

A structurally related strand of the literature studies the quantization of hyperbolic-sine dilaton gravity through the Poisson-sigma model and the quantum group \(\mathrm U_q(\mathfrak{sl}(2,\mathbb R))\). In that framework the Whittaker functions, \(3j\)-symbols, and Plancherel measure organize the dilaton-gravity Hilbert space, and the classical limit \(b\to 0\) recovers JT gravity [2109.07770]. 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\(_2\) geometry. For a massive scalar or worldline of mass \(m\), the boundary-to-boundary propagator in a fixed background is written as
\[
\langle \mathcal O(x_1)\mathcal O(x_2)\rangle
=
\exp\bigl(-\Delta\,L(x_1,x_2)\bigr),
\qquad
m=\cosh(\hbar\Delta).
\]
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 \(\Delta_L+\Delta_R=\Delta\) and glue the two bulk regions back together [2509.01680].

This leads to two complementary quantization channels. In the fully-open channel the brane Hilbert space is the empty sine-dilaton Hilbert space \(\mathcal H_{\rm SD}\), with EOW-state wavefunctions in the length basis given by continuous \(q\)-Hermite polynomials. In the semi-open channel one obtains a twisted Hilbert space \(\mathcal H_{\Delta,\bar\alpha}\) 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,
\[
\bigl|\Delta;n_L,n_R\bigr\rangle_1
=
|n_L\rangle\otimes|n_M=0\rangle\otimes|n_R\rangle,
\]
whereas the energy basis is explicitly nonlocal and state dependent. Different choices of splitting, including \(0\)-split, \(1\)-split, and more general \(n\)-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 \(6j\)-symbol of \(\mathcal U_q(\mathfrak{su}(1,1))\) [2509.01680].

The same discrete structure appears in closed-universe and wormhole amplitudes. WdW quantization gives trumpet wavefunctions \(Z_{\rm trumpet}(b;\beta)=I_b(\beta/\hbar)\) and the connected two-boundary amplitude
\[
Z_{\rm wormhole}(\beta_1,\beta_2)
=
\sum_{b=1}^\infty
b\,
I_b\!\bigl(\tfrac{\beta_1}{\hbar}\bigr)
I_b\!\bigl(\tfrac{\beta_2}{\hbar}\bigr).
\]
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 [2501.17091].

Another wormhole observable is the renormalized quantum length operator \(\hat L\). In the infinite-temperature limit of DSSYK, the equality
\[
\langle \hat L(\tau)\rangle_{\rm SDG}=C_{\rm Krylov}(\tau)
\]
is used to identify the quantum wormhole length with Krylov spread complexity. At \(\beta=0\), one finds
\[
C(t)=\langle \hat L(t)\rangle
=
2\log\cosh(Jt)+\mathcal O(\ell^0),
\]
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 \(e^{-\pi^2/\ell^2}\) indicated by numerical data [2606.20220].

## 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 \(U(X)=\kappa\,\sinh(\lambda X)\) leads after quantization to the quantum group \(\mathrm U_q(\mathfrak{sl}(2,\mathbb R))\). The associated Whittaker functions, \(3j\)-symbols, and Plancherel measure reproduce structural features of Liouville gravity and its supersymmetric extensions, and the classical \(q\to 1\) limit recovers JT gravity [2109.07770]. 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
\[
S=\frac1\kappa\int d^2x\sqrt{-g}\,
\bigl[\varphi R+\mathcal F(\mathcal X)+\kappa\mathcal L_m\bigr],
\qquad
\mathcal X=-\tfrac12(\nabla\varphi)^2,
\]
and, for a specific choice
\[
\mathcal F(\mathcal X)=-\frac{2}{l}\sqrt{-2\mathcal X},
\qquad
W(\phi)=k v^2[\sin(\phi/v)-c],
\]
obtains a sine-Gordon-type kink with pure AdS\(_2\) warp factor \(A(x)=x/l\). For the special case \(v=2\), \(c=0\), the scalar profile reduces to the familiar sine-Gordon kink \(\phi(x)=4\arctan e^{kx}\), and linear perturbations are governed by a factorizable Schrödinger operator [2308.13786]. 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 \(r_H=\beta_s\). In that setup the Hawking temperature in the soliton comoving frame is
\[
T_H=\frac{\beta_s}{2\pi},
\]
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 [1808.03125].

Source: https://www.emergentmind.com/topics/sine-dilaton-gravity