de Sitter Microstates from $T\bar T+Λ_2$ and the Hawking-Page Transition

TL;DR

This work constructs a concrete microstate count for the de Sitter horizon in $dS_3$ by applying a solvable $T\bar{T}+\Lambda_2$ deformation to a seed CFT with sparse spectrum, yielding microstates from the $\Delta\simeq c/6$ band associated with the Hawking-Page transition. A new Cosmic Horizon patch is introduced, with a careful joining point $y_0=\tfrac{3}{c\pi^2}$ that dresses BTZ microstates into the de Sitter static patch, reproducing the Gibbons-Hawking entropy and its logarithmic correction in the pure-gravity sector. The analysis ties the microscopic counting to modular structure and Hawking-Page physics, and discusses subleading, model-dependent bulk physics requiring further generalizations to fully capture higher-dimensional cases and bulk matter. Overall, the results reinforce the utility of finite-cutoff holography and $T\bar{T}$-type deformations as a framework for understanding de Sitter microstates and horizon entropy, while outlining clear directions for extending to more general spacetimes and late-time cosmology.

Abstract

We obtain microstates accounting for the Gibbons-Hawking entropy in $dS_3$, along with a subleading logarithmic correction, from the solvable $T\bar T+Λ_2$ deformation of a seed CFT with sparse light spectrum. The microstates arise as the dressed CFT states near dimension $Δ=c/6$, associated with the Hawking-Page transition; they dominate the real spectrum of the deformed theory. We exhibit an analogue of the Hawking-Page transition in de Sitter. Appropriate generalizations of the $T\bar T+Λ_2$ deformation are required to treat model-dependent local bulk physics (subleading at large central charge) and higher dimensions. These results add considerably to the already strong motivation for the continued pursuit of such generalizations along with a more complete characterization of $T\bar T$ type theories, building from existing results in these directions.

Figures (3)

• Figure 1: A schematic of our prescription for holographic reconstruction of the static patch, including a microstate count, for a given deformation parameter $y=\lambda/L^2$. This may be summarized as follows, with details in the bulk of the paper. On the left, we obtain the indicated patch containing the cosmic horizon by dressing the $\Delta\simeq c/6$ microstates comprising a BTZ black hole at the Hawking-Page transition and switching between the $T\bar{T}$ and $T\bar{T}+\Lambda_2$ trajectories at a value $y_0$ of the deformation parameter such that the BTZ and Cosmic Horizons degenerate to indistinguishable near-horizon regions. On the right we depict the trajectory described in GSTLLST which formulates the complementary patch for the same boundary geometry. The extrinsic curvature of the boundary, and hence the square root term in the dressed energy formula, are equal and opposite for the two patches. As contributions to the thermal partition function at fixed boundary geometry, these dominate for complementary ratios of $(\beta/L)^2=y/y_0-1$, providing a de Sitter analogue of the Hawking-Page transition. In both cases, we can capture the bulk of the static patch by continuing the trajectory in the manner indicated by the arrow. The Pole patch with a Dirichlet boundary condition always excludes the cosmic horizon and does not directly account for the microstates, in keeping with the sparse set of real dressed energies in that case. However, for the cosmic horizon patch -- the main subject of the current paper -- proceeding with the trajectory to $y_{final} \gg 1$ formulates the static patch with boundary at the observer position at the North Pole.
• Figure 2: A schematic of a time slice of the bulk dual for dressed $\Delta < c/6$ states. The spatial geometry piecewise forms a cone. A domain wall sits between bulk regions with different signs of $\Lambda_3$. Deforming with $T\bar{T}+\Lambda_2$ to $y>y_0$ has the effect of increasing the volume of the positive-$\Lambda_3$ domain. The radius of the internal circle decreases away from the wall at $y=y_{0}$ on both sides, unlike in the diagram.
• Figure 3: The two phases of the doubled system, including the entangled cosmic horizon patches in the right panel. In that case, the upper and lower triangles should be reconstructable by bulk evolution. One may treat these patches as conditioned on an observer at the Dirichlet wall boundary, or alternatively use them as building blocks, joining these patches together at their common boundary, integrating over its geometry, to obtain global $dS_3$ with its symmetries. It would be interesting to connect the full nonlinearity of the deformed theory as a function of the fundamental seed CFT variables to the notions of hyper-scrambling and complexity suggested recently in Susskind:2021esxChapman:2021eyy.