An entropic puzzle in periodic dilaton gravity and DSSYK

TL;DR

This paper explores 2D periodic dilaton gravity theories, revealing how gauge fixing of symmetries leads to discretized geodesic lengths, finite entropy, and connections to quantum cosmology and matrix models.

Contribution

It introduces a gauge-invariant framework for periodic dilaton gravity, deriving an exact density of states and establishing dualities with quantum algebra and matrix integrals.

Findings

Discretization of geodesic lengths due to gauging symmetry

Finite-dimensional Hilbert space with entropy deviating from Bekenstein-Hawking

Dualities between flat space quantum gravity, topological gravity, and matrix models

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.