An entropic puzzle in periodic dilaton gravity and DSSYK

TL;DR

We study 2d periodic dilaton gravity with a periodic potential, demonstrating that the momentum-shift symmetry conjugate to geodesic length must be gauged, which discretizes bulk lengths and renders negative-length states null, yielding a finite Hilbert space and an entropy S below the Bekenstein-Hawking value. By constructing ungauged and gauged quantizations, we connect the latter to the DSSYK chord Hilbert space and derive an exact density of states expressed through q-Pochhammer objects, resolving divergences and revealing a gauge-driven entropic paradox. The work identifies two universal limits: a flat-space JT gravity limit with a Gaussian (harmonic-oscillator) spectrum, and a highly quantum limit that reduces to topological (Gaussian) matrix-model physics, both tied to Gaussian matrix integrals and the q-Schwarzian. It also conjectures an exact disk partition function for periodic dilaton gravity and discusses potential path-integral realizations, cosmological interpretations, and extensions to dS JT gravity, positioning periodic dilaton gravity as a new solvable universality class for 2d quantum cosmology.

Abstract

We study 2d dilaton gravity theories with a periodic potential, with special emphasis on sine dilaton gravity, which is holographically dual to double-scaled SYK. The periodicity of the potentials implies a symmetry under (discrete) shifts in the momentum conjugate to the length of geodesic slices. This results in divergences. The correct definition is to gauge this symmetry. This discretizes the geodesic lengths. Lengths below a certain threshold are null states. Because of these null states, the entropy deviates drastically from Bekenstein-Hawking and the Hilbert space becomes finite dimensional. The spacetimes have a periodic radial coordinate. These are toy models of 2d quantum cosmology with a normalizable wavefunction. We study two limiting dualities: one between flat space quantum gravity and the Heisenberg algebra, and one between topological gravity and the Gaussian matrix integral. We propose an exact density of states for certain classes of periodic dilaton gravity models.

Figures (7)

• Figure 1: The left wavefunction \ref{['left1']} interpolates between the q-Hermite polynomials \ref{['left2']}, which are indicated by red dots at positive integer values of $n$.
• Figure 2: The right wave function $\psi^{\mathsf{ R}}_\theta(n+\mathrm{i} \epsilon)$ with $q=0.4$, $\theta=0.1$, and $\epsilon=0.1$ and $0.02$ in red and black respectively. Crucially, the wavefunction diverges only at positive integers as explained around \ref{['inf']}.
• Figure 3: Cartoon of the horizon function \ref{['4.7']}. Flat space gets regulated by a cosmological horizon, which is infinitely far away in the strict $\abs{\log q}\to 0$ limit of exactly flat space \ref{['4.4flat']} (blue). The light red region is the normal signature region outside of the black hole horizon.
• Figure 4: As $q\to 1$ the rescaled $\psi^{\mathsf{ L}}_\theta(n)$\ref{['eq:psilqto1']} (black) becomes a parabolic cylinder function (dotted gray). At integer $n$ these functions are $q$-Hermite (red dots) respectively ordinary Hermite polynomials (blue dots). To illustrate the limit we considered $q=0.8$ (left) and $q=0.99$ (right).
• Figure 5: On the left, we plot an example of a potential $V(\Phi)$ that obeys the assumptions we use. This potential takes the form $V(\Phi) = \sin(\Phi) + c_9 \sin(9 \Phi)$ for some small enough $c_9$. On the right, we demonstrate the qualitative branch cut structure for an example potential in the integral \ref{['eqn:momentumintegral']}. The two horizontal branch cuts connect the branch points at $V(\Phi) = \pm e^{-L/2}$. The space on the real $\Phi$ line between the branch cuts is where the integrand of \ref{['eqn:momentumintegral']} takes real values. Since the integrand is uniformized on the double cover of the complex plane, the contour $\mathcal{C}$ (in red) is a closed cycle which starts on the principal branch, goes to the second sheet and then comes back.