Three-dimensional de Sitter horizon thermodynamics

TL;DR

This work investigates how three-dimensional de Sitter horizon thermodynamics is shaped by metric fluctuations and Chern-Simons gauge fields, using a Euclidean path integral about the round $S^3$ saddle to access subleading corrections to the gravitational partition function. It develops a Lorentzian edge-mode interpretation, drawing on topological entanglement entropy ideas, and analyzes complexified Chern-Simons theories as tractable toy models that capture essential features of gravitational edge dynamics. The authors compute and interpret sphere partition functions for Abelian and complexified CS theories, showing that edge modes encode universal finite contributions (e.g., $- frac{1}{2}\log k$) and connect these to the dS horizon entropy and its subleading structure, while outlining the non-Abelian extensions and potential holographic links to AdS$_4$/CFT$_3$. Overall, the paper provides a cohesive framework linking Euclidean CS calculations, Lorentzian edge modes, and holographic perspectives to deepen the microscopic understanding of horizon thermodynamics in 3D de Sitter space.

Abstract

We explore thermodynamic contributions to the three-dimensional de Sitter horizon originating from metric and Chern-Simons gauge field fluctuations. In Euclidean signature these are computed by the partition function of gravity coupled to matter semi-classically expanded about the round three-sphere saddle. We investigate a corresponding Lorentzian picture - drawing inspiration from the topological entanglement entropy literature - in the form of an edge-mode theory residing at the de Sitter horizon. We extend the discussion to three-dimensional gravity with positive cosmological constant, viewed (semi-classically) as a complexified Chern-Simons theory. The putative gravitational edge-mode theory is a complexified version of the chiral Wess-Zumino-Witten model associated to the edge-modes of ordinary Chern-Simons theory. We introduce and solve a family of complexified Abelian Chern-Simons theories as a way to elucidate some of the more salient features of the gravitational edge-mode theories. We comment on the relation to the AdS$_4$/CFT$_3$ correspondence.

Figures (5)

• Figure 1: The Penrose diagram of dS$_3$, with the static patch highlighted. Constant time slices are two-spheres, with the $\mathcal{T}=0$ shown explicitly, and the left and right-hand sides of the diagram are the poles of $S^2$.
• Figure 2: The $\mathcal{T}=0$ slice of global de Sitter split into two static patch hemispheres at the $S^1$ horizon. These hemispheres are topologically disks, and we can imagine 'tracing out' the degrees of freedom in one of the static patches that is behind the horizon of the other.
• Figure 3: The Penrose diagram on the left shows the Lorentzian picture of the edge modes, which live on the thick, blue line at a distance $\beta/2\pi \ll \ell$ from the horizon. On the right is the Euclidean picture, where the edge modes live on the surface of a solid torus with thermal cycle $\beta$. Shrinking the thermal cycle is the same as taking the high temperature limit, ot taking the edge modes to live very close to the horizon.
• Figure 4: A constant $t$ slice of the geometry (\ref{['LAdS']}). The space is foliated by static patch hemispheres with exponentially increasing radius in the $\vartheta$ direction and the horizon at $\rho = \pi/2$ is the surface of this shape.
• Figure 5: Plot showing the shaded regions $|x| > |y|$, where the Hamiltonian (\ref{['xgryH']}) is valid. The blue arrows indicate increasing $t = \tanh^{-1}(y/x)$. By gluing this solution along $|x| = |y|$ with the one in the $|y|>|x|$ region, we arrive at the solution (\ref{['Loreenergies']}).