Sine Dilaton Gravity in 2D Holography

• Sine dilaton gravity is a 2D dilaton-gravity theory defined by a periodic potential (V(Φ)=2 sinΦ) that leads to discrete quantum signatures and a maximal closed universe size.
• Its duality with double-scaled SYK is established through canonical quantization and q-Schwarzian dynamics, linking black-hole horizons with cosmological Big-Bang/Crunch models.
• Quantization methods and matrix-integral techniques reveal finite spectral densities, discretized geodesic lengths, and one-loop corrections that connect to Liouville reformulations and holographic flows.

Searching arXiv for 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 (Blommaert et al., 2024, Blommaert et al., 30 May 2025).

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 $q$0. The periodic dilaton potential $q$1 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 (Blommaert et al., 30 May 2025).

In the DSSYK normalization one starts from

$q$2

with

$q$3

After the rescaling $q$4, this becomes

$q$5

up to the corresponding boundary terms. In this formulation the parameter $q$6 controls the flow to JT gravity in the IR, and the non-minimally coupled probe sector is naturally described using the Weyl-rescaled metric

$q$7

with renormalized length

$q$8

This $q$9 is the bulk observable that is identified with chord number in DSSYK (Blommaert et al., 2024).

A distinct but related canonical presentation emphasizes the phase-space variables $2$0 and the symplectic form $2$1. For periodic dilaton potentials, the asymptotic Hamiltonian obeys $2$2, and since $2$3 is periodic, $2$4 is invariant under a discrete shift $2$5. 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 (Blommaert et al., 2024).

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,

$2$6

with $2$7 an integration constant. In this geometry there is a black-hole horizon at $2$8, and a cosmological horizon at $2$9. The ADM energy is

$L=1$0

In the limit $L=1$1, the potential expands as $L=1$2, and one recovers JT gravity with linear dilaton (Bossi et al., 2024).

A Weyl rescaling maps the solution to an AdS$L=1$3-type form. In particular,

$L=1$4

and in the Euclidean black-hole language one may write

$L=1$5

where $L=1$6 is the renormalized geodesic length and $L=1$7 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 (Bossi et al., 2024).

In the cosmological reading, $L=1$8 plays the role of a time coordinate or “clock,” while $L=1$9 is the size of the spatial circle. The periodic potential $1/\hbar$0 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 (Blommaert et al., 30 May 2025).

3. Canonical quantization, $1/\hbar$1-Schwarzian dynamics, and the DSSYK duality

In minisuperspace, the canonical variables are $1/\hbar$2, and the Wheeler–DeWitt constraint is

$1/\hbar$3

acting on wavefunctions $1/\hbar$4. A complete set of energy-labeled solutions is

$1/\hbar$5

These oscillate in the Lorentzian region $1/\hbar$6 and decay elsewhere. The Hartle–Hawking or no-boundary state is

$1/\hbar$7

with coefficients $1/\hbar$8 fixed by matching the disk path integral to DSSYK (Blommaert et al., 30 May 2025).

The holographic matching is explicit. The path integral on a disk with asymptotic thermal boundary conditions reproduces the DSSYK thermal partition function $1/\hbar$9. More precisely, the exact no-boundary wavefunction $$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$0 obeys

$$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$1

under which it matches $$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$2. The spectral density $$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$3 appearing in the expansion of $$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$4 is exactly the DSSYK density of states, supported on the finite interval $$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$5 (Blommaert et al., 30 May 2025).

The same duality appears in canonical phase space. One introduces Darboux variables $$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$6 obeying $$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$7, so that after quantization $$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$8. The Hamiltonian becomes

$$S=\frac12\int d^2x\,\sqrt g\,[\,\Phi R+2\sin\Phi\,]+\oint du\,\sqrt h\,\Phi K.$$9

and in first-order form the boundary action is the $$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$0-Schwarzian,

$$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$1

This is exactly the transfer-matrix Hamiltonian of DSSYK once one identifies

$$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$2

with $$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$3 the DSSYK chord number operator (Blommaert et al., 2024).

Because the periodic dilaton implies a discrete shift symmetry in the conjugate momentum, one must gauge $$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$4. The projected theory retains only discrete lengths

$$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$5

with wavefunctions at $$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$6 becoming null and the physical sector spanned by $$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$7. In this basis the wavefunctions are $$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$8-Hermite polynomials $$S=\frac12\int d^2x\,\sqrt g\,[\,\phi R+V(\phi)\,],\qquad V(\phi)=2\sin\phi,$$9, and the physical density of states becomes finite. This discretization is the gravitational counterpart of chord-number positivity in DSSYK (Blommaert et al., 2024).

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: $\oint \phi K$0 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

$\oint \phi K$1

The match provides direct evidence that the exact sphere amplitude of sine dilaton gravity is captured by a finite-cut matrix integral (Blommaert et al., 30 May 2025).

The cosmological interpretation focuses on the size observable. In pure dS JT gravity, written as the linear-dilaton limit of $\oint \phi K$2, the no-boundary distribution at fixed “time” $\oint \phi K$3 for the asymptotic length $\oint \phi K$4 diverges as $\oint \phi K$5 and gives

$\oint \phi K$6

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 $\oint \phi K$7 that vanishes as $\oint \phi K$8. At large $\oint \phi K$9, the sphere contribution decays only as

$g_{\mu\nu}$0

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 (Blommaert et al., 30 May 2025).

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 $g_{\mu\nu}$1 onto the physical Hilbert space, equivalently the identity operator on the space of solutions $g_{\mu\nu}$2. In any basis that diagonalizes $g_{\mu\nu}$3 at fixed $g_{\mu\nu}$4, the identity gives

$g_{\mu\nu}$5

Hence an observer sees no preference toward small or large universes; the resulting distribution is flat (Blommaert et al., 30 May 2025).

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

$g_{\mu\nu}$6

supplemented by the periodicity condition $g_{\mu\nu}$7. Defining

$g_{\mu\nu}$8

the equation factorizes into two identical eigenproblems labeled by $g_{\mu\nu}$9, and the AdS geodesic length is quantized as

$\Phi$0

Imposing regularity as $\Phi$1 and the holographic boundary condition gives the exact trumpet wavefunction

$\Phi$2

which approaches $\Phi$3 in the holographic limit (Blommaert et al., 28 Jan 2025).

The physical Hilbert space on the trumpet is discrete. The Klein–Gordon norm at $\Phi$4 is

$\Phi$5

so the physical identity operator is

$\Phi$6

The connected double-trumpet amplitude therefore factorizes as

$\Phi$7

In the energy basis this reproduces the universal finite-cut random-matrix sine-kernel correlation (Blommaert et al., 28 Jan 2025).

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$8

The closed-channel cylinder with brane parameter $\Phi$9 is

$q$00

and Fourier-transforming in $q$01 localizes onto $q$02, reproducing the trumpet. The same work gives the operator dictionary

$q$03

which ties geodesic boundaries and FZZT branes to finite-cut matrix-integral observables (Blommaert et al., 28 Jan 2025).

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 $q$04-symbol of the quantum group $q$05. The resulting wormhole Hilbert space also allows matter correlators on the double trumpet and the inclusion of bulk matter loops (Cui et al., 1 Sep 2025).

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 $q$06 (Alfinito et al., 18 Jun 2026).

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 $q$07, the full action is

$q$08

Using the $q$09-Schwarzian description, the Hartle–Hawking boundary condition $q$10, 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 $q$11-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 (Bossi et al., 2024).

The model also admits a Liouville reformulation. A $q$12-Schwarzian/Poisson-sigma-model construction shows that the trigonometric potential arises by analytic continuation of the $q$13-Schwarzian/Liouville duality to real $q$14, 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 (Blommaert et al., 2023).

A more recent reformulation in domain-wall gauge splits sine-dilaton gravity into two Liouville sectors. With

$q$15

one has

$q$16

and the associated Liouville central charges satisfy

$q$17

From this decomposition one constructs a monotonic holographic $q$18-function

$q$19

which interpolates from $q$20 in the UV to $q$21 in the deep IR. In parallel, the $q$22 limit reduces the theory to JT gravity, whose dual low-energy SYK description has vanishing two-dimensional central charge (Mahapatra et al., 25 Jan 2026).

Several conceptual issues are clarified by the periodic structure. First, the model distinguishes the “fake” Hawking inverse temperature

$q$23

from the “true” inverse temperature

$q$24

The difference is not a contradiction but the consequence of the positivity constraint on chord number, or equivalently the restriction $q$25 with $q$26 in the $q$27-Schwarzian path integral. Second, the entropy is modified from the Bekenstein–Hawking law to

$q$28

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 (Blommaert et al., 2024, Blommaert et al., 2024).