---
title: Sine Dilaton Gravity in 2D Holography
url: https://www.emergentmind.com/topics/sine-dilaton-gravity-37520b53-cf95-47d1-b37f-942ee449a4d5
type: topic
---

# Sine Dilaton Gravity in 2D Holography

Searching arXiv for recent papers on sine dilaton gravity and closely related holography.
Sine dilaton gravity is a two-dimensional dilaton-gravity theory with a periodic dilaton potential, commonly written in rescaled variables as \(V(\Phi)=2\sin\Phi\). In current lower-dimensional holography it is proposed as the bulk dual of double-scaled SYK (DSSYK), with canonical quantization reproducing the \(q\)-Schwarzian auxiliary system, while at the same time it admits an interpretation as \(2\)d quantum cosmology in which the periodic potential gives classical solutions with a Big-Bang and Big-Crunch, and a finite maximal size [2404.03535] [2505.24633].

## 1. Action and field content

In the cosmological normalization, and in units where the dS length \(L=1\) with an overall \(1/\hbar\) suppressed, the action is
\[
S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.
\]
Equivalently,
\[
S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,
\]
with boundary term \(\oint \phi K\). The field content is the metric \(g_{\mu\nu}\) and the dilaton \(\Phi\) or \(\phi\). The periodic dilaton potential \(V(\Phi)=2\sin\Phi\) is the defining structural ingredient: it is this periodicity that underlies the finite maximal size of the closed universe solutions and the discrete features of the quantum theory [2505.24633].

In the DSSYK normalization one starts from
\[
S_{\rm bulk}=\frac12\int d^2x\sqrt g\,[\,\Phi R+U(\Phi)\,]+S_{\rm bdy},\qquad
U(\Phi)=\frac{\sin(2|\log q|\,\Phi)}{|\log q|},
\]
with
\[
S_{\rm bdy}=\int d\tau \sqrt h\left[\Phi K-\frac{i\,e^{-i|\log q|\Phi}}{2|\log q|}\right].
\]
After the rescaling \(2|\ln q|\,\Phi\to\Phi\), this becomes
\[
S_{sDG}=\frac{1}{4|\ln q|}\int d^2x\,\sqrt{-g}\,\bigl(\Phi R+2\sin\Phi\bigr),
\]
up to the corresponding boundary terms. In this formulation the parameter \(|\ln q|\ll1\) controls the flow to JT gravity in the IR, and the non-minimally coupled probe sector is naturally described using the Weyl-rescaled metric
\[
g_{\rm eff}=e^{-2i|\log q|\Phi}\,g,
\]
with renormalized length
\[
L=\oint ds\,e^{-i|\log q|\Phi(x(s))}.
\]
This \(L\) is the bulk observable that is identified with chord number in DSSYK [2404.03535].

A distinct but related canonical presentation emphasizes the phase-space variables \((L,P)\) and the symplectic form \(\omega=dL\wedge dP\). For periodic dilaton potentials, the asymptotic Hamiltonian obeys \(H=W(P)\), and since \(V(\Phi)\) is periodic, \(W(P)\) is invariant under a discrete shift \(P\to P+2\pi\Phi_0\). Quantum mechanically this discrete shift symmetry must be gauged; otherwise the density of states diverges. This gauging is one of the main structural differences between periodic dilaton gravity and linear-dilaton JT gravity [2411.16922].

## 2. Classical geometry, horizons, and cosmological interpretation

A standard classical solution is obtained in Schwarzschild gauge by setting the dilaton equal to the radial coordinate,
\[
\Phi(r)=r,\qquad
ds^2=F(r)\,d\tau^2+\frac{dr^2}{F(r)},\qquad
F(r)=-2\cos r+2\cos\theta,
\]
with \(\theta\in[0,\pi]\) an integration constant. In this geometry there is a black-hole horizon at \(r=\theta\), and a cosmological horizon at \(r=2\pi-\theta\). The ADM energy is
\[
E_{\rm ADM}=-\frac{\cos\theta}{2|\log q|}.
\]
In the limit \(|\log q|\to0\), the potential expands as \(\sin(2|\log q|\Phi)/|\log q|\to2\Phi\), and one recovers JT gravity with linear dilaton [2411.15957].

A Weyl rescaling maps the solution to an AdS\(_2\)-type form. In particular,
\[
ds_{\rm eff}^2=e^{-i\Phi}ds^2,
\]
and in the Euclidean black-hole language one may write
\[
e^{-L}=\frac{\sin^2\theta}{\cosh^2\!\bigl(\tfrac12\,\sin\theta\,T\bigr)},
\]
where \(L\) is the renormalized geodesic length and \(T\) is the two-boundary time separation. The same classical family therefore admits both a black-hole interpretation and a closed-universe interpretation. This suggests that the model interpolates between a holographic boundary description and a minisuperspace description of quantum cosmology [2411.15957].

In the cosmological reading, \(\Phi\) plays the role of a time coordinate or “clock,” while \(\ell=\oint ds\) is the size of the spatial circle. The periodic potential \(\sin\Phi\) acts like a closed-universe cosmological constant that enforces a maximal universe size and a Big-Bang/Crunch. In that sense the same periodic structure responsible for replicated horizons in the static description becomes the mechanism that bounds the universe size in the cosmological description [2505.24633].

## 3. Canonical quantization, \(q\)-Schwarzian dynamics, and the DSSYK duality

In minisuperspace, the canonical variables are \((\ell,\Phi)\), and the Wheeler–DeWitt constraint is
\[
H_{\rm WDW}=\hbar^2\,\partial_\Phi\partial_\ell-\ell\sin\Phi,
\]
acting on wavefunctions \(\Psi(\ell,\Phi)\). A complete set of energy-labeled solutions is
\[
\psi_E(\ell,\Phi)=\sqrt{\pi/2}\;H_0^{(1)}\!\bigl(\ell\sqrt{2\cos\Phi-E}/\hbar\bigr),\qquad E\in[-2,2].
\]
These oscillate in the Lorentzian region \(|\Phi|<\arccos(E/2)\) and decay elsewhere. The Hartle–Hawking or no-boundary state is
\[
\psi_{\rm NB}(\ell,\Phi)=D\int_{-2}^{2} dE\,\rho(E)\,\psi_E(\ell,\Phi),
\]
with coefficients \(\rho(E)\) fixed by matching the disk path integral to DSSYK [2505.24633].

The holographic matching is explicit. The path integral on a disk with asymptotic thermal boundary conditions reproduces the DSSYK thermal partition function \(Z_{\rm DSSYK}(\beta)\). More precisely, the exact no-boundary wavefunction \(\psi_{\rm NB}(\ell,\Phi)\) obeys
\[
\Phi\to\frac{\pi}{2}+i\infty,\qquad \ell\,e^{-i\Phi/2}=i\beta,
\]
under which it matches \(Z_{\rm DSSYK}(\beta)\). The spectral density \(\rho(E)\) appearing in the expansion of \(\psi_{\rm NB}\) is exactly the DSSYK density of states, supported on the finite interval \(E\in[-2,2]\) [2505.24633].

The same duality appears in canonical phase space. One introduces Darboux variables \((L,P)\) obeying \(\{L,P\}=2|\log q|\), so that after quantization \([\,\hat L,\hat P\,]=2i|\log q|\). The Hamiltonian becomes
\[
H_{\rm grav}(L,P)=-\frac{\cos P}{2|\log q|}+\frac{1}{4|\log q|}e^{iP}e^{-L},
\]
and in first-order form the boundary action is the \(q\)-Schwarzian,
\[
\int_0^\beta du\left[i\,p_\phi\,\partial_u\phi+\frac{1}{2|\log q|}\cos(|\log q|\,p_\phi)-\frac{1}{4|\log q|}e^{i|\log q|p_\phi}e^{-2\phi}\right].
\]
This is exactly the transfer-matrix Hamiltonian of DSSYK once one identifies
\[
\hat L\equiv 2|\log q|\,\hat n,
\]
with \(\hat n\) the DSSYK chord number operator [2404.03535].

Because the periodic dilaton implies a discrete shift symmetry in the conjugate momentum, one must gauge \(P\sim P+2\pi\). The projected theory retains only discrete lengths
\[
L_n=n\,\hbar,\qquad n\in\mathbb Z,
\]
with wavefunctions at \(n<0\) becoming null and the physical sector spanned by \(n=0,1,2,\ldots\). In this basis the wavefunctions are \(q\)-Hermite polynomials \(H_n(\cos\theta|q^2)\), and the physical density of states becomes finite. This discretization is the gravitational counterpart of chord-number positivity in DSSYK [2411.16922].

## 4. No-boundary states, sphere amplitude, and universe-size predictions

The no-boundary state is central in the cosmological use of the model. Its norm squared, computed with the Klein–Gordon inner product on minisuperspace, is the sphere amplitude:
\[
Z_{\rm sphere}=\langle\psi_{\rm NB}\mid\psi_{\rm NB}\rangle
=-\int_{-2}^{2} dE_1\,\rho(E_1)\int_{-2}^{2} dE_2\,\rho(E_2)\,\log|E_1-E_2|.
\]
The same expression follows by expanding in the Chebyshev basis. This quantity is finite, and via canonical quantization it matches the on-shell action of a dual matrix integral whose leading sphere free energy is
\[
F_0=-D^2\iint dE_1\,dE_2\,\rho(E_1)\rho(E_2)\,\ln|E_1-E_2|.
\]
The match provides direct evidence that the exact sphere amplitude of sine dilaton gravity is captured by a finite-cut matrix integral [2505.24633].

The cosmological interpretation focuses on the size observable. In pure dS JT gravity, written as the linear-dilaton limit of \(\sin\Phi\), the no-boundary distribution at fixed “time” \(\Phi\) for the asymptotic length \(L=\ell/\Phi\) diverges as \(L\to0\) and gives
\[
P(L)\sim \frac{1}{L^4},
\]
so the state is non-normalizable and heavily weighted to small universes. In sine-dilaton gravity, by contrast, the spectral density has compact support, the sphere amplitude converges, and one finds a finite probability density \(P_{\rm sphere}(\ell\mid\Phi)\) that vanishes as \(\ell\to0\). At large \(\ell\), the sphere contribution decays only as
\[
P_{\rm sphere}\sim \frac{1}{\ell^2}.
\]
The short-distance singularity associated with the Big-Bang is therefore smoothed out quantum-mechanically, although the sphere contribution still favors smaller universes over very large ones [2505.24633].

The observer’s no-boundary state changes this conclusion. When one includes a pointlike observer whose worldline must end smoothly, the usual Hartle–Hawking sphere does not contribute to the density matrix. The leading contribution is instead a bra–ket wormhole, or cylinder geometry connecting bra and ket boundaries. Canonically, this cylinder path integral imposes the projector \(\delta(H_{\rm WDW})\) onto the physical Hilbert space, equivalently the identity operator on the space of solutions \(L^2(\ell,\Phi)\). In any basis that diagonalizes \(\ell\) at fixed \(\Phi\), the identity gives
\[
P_{\rm obs}(\ell,\Phi)=1.
\]
Hence an observer sees no preference toward small or large universes; the resulting distribution is flat [2505.24633].

## 5. Trumpets, wormholes, branes, and matter correlators

The simplest connected wormhole observable is the double trumpet. In the trumpet minisuperspace problem one again uses the Wheeler–DeWitt operator
\[
H_{\rm WdW}=\hbar^2\,\partial_{(\Phi)}\partial_{(\ell)}-\ell\sin\Phi,
\]
supplemented by the periodicity condition \(\psi(\ell,\Phi+2\pi)=\psi(\ell,\Phi)\). Defining
\[
L=e^{i\Phi/2}\ell^{1/2},\qquad \bar L=e^{-i\Phi/2}\ell^{1/2},
\]
the equation factorizes into two identical eigenproblems labeled by \(b\in\mathbb Z\), and the AdS geodesic length is quantized as
\[
\oint_{\rm geo} ds\,e^{-i\Phi/2}=\hbar\,b,\qquad b\in\mathbb N.
\]
Imposing regularity as \(\ell\to0\) and the holographic boundary condition gives the exact trumpet wavefunction
\[
Z_{\rm trumpet}(b;\ell,\Phi)=\frac{\sqrt\pi}{2\sqrt2}\bigl[J_b(L/\hbar)\,Y_b(\bar L/\hbar)+Y_b(L/\hbar)\,J_b(\bar L/\hbar)\bigr],
\]
which approaches \(I_b(\beta/\hbar)\) in the holographic limit [2501.17091].

The physical Hilbert space on the trumpet is discrete. The Klein–Gordon norm at \(\ell\to0\) is
\[
\langle b_1|b_2\rangle=\frac{1}{b_1}\,\delta_{b_1b_2},
\]
so the physical identity operator is
\[
1_{\rm phys}=\sum_{b=1}^{\infty} b\,|b\rangle\langle b|.
\]
The connected double-trumpet amplitude therefore factorizes as
\[
Z_{\rm wormhole}(\beta_1,\beta_2)=\sum_{b=1}^\infty b\,I_b(\beta_1/\hbar)\,I_b(\beta_2/\hbar).
\]
In the energy basis this reproduces the universal finite-cut random-matrix sine-kernel correlation [2501.17091].

Open and closed end-of-the-world branes admit an exact quantization as well. The Wheeler–DeWitt constraint leaves a one-parameter family of gauge-invariant branes labeled by
\[
\Phi_h=-\,i\,\sinh^{-1}(\mu)-\,i\,\sinh^{-1}(\bar\mu),\qquad [\,b,\Phi_h\,]=i.
\]
The closed-channel cylinder with brane parameter \(\Phi_h\) is
\[
Z_{\rm EOW}(\Phi_h,\beta)
=\sum_{b=0}^{\infty}\frac{e^{-ib\Phi_h}}{1-q^{2b}}\,I_b(\beta/\hbar),
\]
and Fourier-transforming in \(\Phi_h\) localizes onto \(|b\rangle\), reproducing the trumpet. The same work gives the operator dictionary
\[
\mathcal O_G(b)=\frac{2}{b}\,\mathrm{Tr}\cos\bigl[b\,\arccos(-H)\bigr],\qquad
\mathcal O_{\rm FZZT}(\Phi_h)=\mathrm{Tr}\,\ln\bigl[H+\cos\Phi_h\bigr],
\]
which ties geodesic boundaries and FZZT branes to finite-cut matrix-integral observables [2501.17091].

Matter correlators can be derived directly from the bulk by treating matter lines as EOW branes and implementing a splitting-and-gluing procedure. In this construction the length basis factorizes across subregions, whereas the energy basis acquires a non-local, state-dependent structure determined by the EOW brane quantization in each subregion. General correlators, including the OTOC, exactly reproduce the DSSYK chord-diagram results, and the crossed four-point function exposes a new identity for the \(6j\)-symbol of the quantum group \(\mathcal U_q(su(1,1))\). The resulting wormhole Hilbert space also allows matter correlators on the double trumpet and the inclusion of bulk matter loops [2509.01680].

Recent higher-loop work has sharpened the dynamical interpretation of the bulk length observable: the quantum wormhole length in sine-dilaton gravity equals the Krylov spread complexity in DSSYK. At infinite temperature, a five-loop semiclassical expansion has been obtained for the complexity, the Krylov variance, and the third cumulant, together with large-time linear growth and non-perturbative corrections of the form \(\delta A_1(\lambda)\sim e^{-\pi^2/\lambda}\) [2606.20220].

## 6. One-loop structure, Liouville reformulations, RG flow, and conceptual issues

One-loop calculations provide a nontrivial check of the DSSYK duality. In Euclidean signature, with \(G_N\equiv|\log q|\), the full action is
\[
S_{\rm tot}
=\frac12\int d^2x\,\sqrt g\left[\Phi R+\frac{\sin(2|\log q|\Phi)}{|\log q|}\right]
+\int_{\partial} d\tau\,\sqrt h\left[\Phi K-\frac{i}{2|\log q|}e^{-i|\log q|\Phi}\right].
\]
Using the \(q\)-Schwarzian description, the Hartle–Hawking boundary condition \(\varphi(0)=\varphi(\beta)=0\), and a generalized Gel'fand–Yaglom computation of the fluctuation determinant, the logarithmic correction to the free energy matches the corresponding DSSYK quantity up to a \(1/2\)-ambiguity traceable to operator ordering choices. The gravitational one-loop correction to the boundary-to-boundary propagator of a non-minimally coupled matter field matches exactly the DSSYK one-loop matter correlator [2411.15957].

The model also admits a Liouville reformulation. A \(q\)-Schwarzian/Poisson-sigma-model construction shows that the trigonometric potential arises by analytic continuation of the \(q\)-Schwarzian/Liouville duality to real \(q\), yielding sine dilaton gravity as the bulk theory appropriate to DSSYK. In this language the exact disk problem reduces to special-function technology associated with quantum-group representation theory, and the quantum solutions are expressed in terms of non-compact double-sine functions [2312.00871].

A more recent reformulation in domain-wall gauge splits sine-dilaton gravity into two Liouville sectors. With
\[
A=\frac{Z_1+iZ_2}{4},\qquad \Phi=\frac{Z_1-iZ_2}{2i},
\]
one has
\[
S_{sDG}=S_r(Z_1)+S_\ell(Z_2),
\]
and the associated Liouville central charges satisfy
\[
c^{(r)}+\bar c^{(\ell)}=26.
\]
From this decomposition one constructs a monotonic holographic \(c\)-function
\[
\hat c(u)=\frac{26}{N}\int_u^\infty D(v)\,dv,\qquad \frac{d\hat c}{du}<0,
\]
which interpolates from \(\hat c(0)=26\) in the UV to \(0\) in the deep IR. In parallel, the \(|\ln q|\to0\) limit reduces the theory to JT gravity, whose dual low-energy SYK description has vanishing two-dimensional central charge [2601.17698].

Several conceptual issues are clarified by the periodic structure. First, the model distinguishes the “fake” Hawking inverse temperature
\[
\beta_{\rm BH}=\frac{2\pi}{\sin\theta}
\]
from the “true” inverse temperature
\[
\beta=\frac{2\pi-4\theta}{\sin\theta}.
\]
The difference is not a contradiction but the consequence of the positivity constraint on chord number, or equivalently the restriction \(\phi(u)\ge0\) with \(\phi(0)=\phi(\beta)=0\) in the \(q\)-Schwarzian path integral. Second, the entropy is modified from the Bekenstein–Hawking law to
\[
S(\theta)=2\pi\theta-2\theta^2,
\]
reflecting the existence of null states after gauging the discrete momentum-shift symmetry. A common misconception is that the smooth Euclidean black-hole saddle alone determines the thermodynamics; in the periodic theory the physical answer depends essentially on the projection to the gauge-invariant Hilbert space and on the normalizable Hartle–Hawking vacuum [2404.03535] [2411.16922].

Source: https://www.emergentmind.com/topics/sine-dilaton-gravity-37520b53-cf95-47d1-b37f-942ee449a4d5