Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions

TL;DR

The paper provides a universal, regulator-controlled framework to compute exact one-loop corrected de Sitter entropy via the sphere partition function, reframing it as a bulk quasicanonical piece minus an edge contribution encoded by Harish-Chandra characters. By deriving a bulk–edge decomposition that universally applies to scalars, fermions, and higher-spin fields (massive and massless), it yields closed-form expressions for the entropy and related thermodynamics, including all-loop results in 3D higher-spin gravity and a topological-string dual at large spin. The formalism reveals holographic-like bulk–edge cancelations and exposes the crucial role of edge degrees of freedom in determining the net quantum gravitational entropy, with broad implications for dS, AdS, and CHS theories. It also provides a covariant route to Euclidean thermodynamics on the sphere and clarifies how UV divergences are absorbed into renormalized couplings, while highlighting challenges in theories with infinite spin range. The results offer concrete tests for microscopic models and connect de Sitter entropy to topological-string partition functions in a novel duality framework.

Abstract

Motivated by the prospect of constraining microscopic models, we calculate the exact one-loop corrected de Sitter entropy (the logarithm of the sphere partition function) for every effective field theory of quantum gravity, with particles in arbitrary spin representations. In doing so, we universally relate the sphere partition function to the quotient of a quasi-canonical bulk and a Euclidean edge partition function, given by integrals of characters encoding the bulk and edge spectrum of the observable universe. Expanding the bulk character splits the bulk (entanglement) entropy into quasinormal mode (quasiqubit) contributions. For 3D higher-spin gravity formulated as an sl($n$) Chern-Simons theory, we obtain all-loop exact results. Further to this, we show that the theory has an exponentially large landscape of de Sitter vacua with quantum entropy given by the absolute value squared of a topological string partition function. For generic higher-spin gravity, the formalism succinctly relates dS, AdS$^\pm$ and conformal results. Holography is exhibited in quasi-exact bulk-edge cancelation.

Figures (23)

• Figure 1: a: Cartoon of observable universe evolving to its maximal-entropy equilibrium state. The horizon consumes everything once seen, growing until it reaches its de Sitter equilibrium area $A$. (The spiky dot is a reference point for ${\tt b,c}$; it will ultimately be gone, too.) b: Penrose diagram of dS static patch. c: Wick-rotated $(\tt b)$ = sphere. Metric details are given in appendix \ref{['app:dSWick']} + fig. \ref{['fig:penrose-app']} c,d.
• Figure 2: Contributions to dS$_3$ one-loop entropy from gravity and massive $s=0,1,2$.
• Figure 3: Regularized dS$_2$ scalar mode density with $\nu=2$, $\Lambda_{\rm uv} \ell \approx 4000$. Blue line = Fourier transform of $\chi_{\rm bulk}$: $\rho(\omega)/\ell = \frac{2}{\pi} \log(\Lambda_{\rm uv}\ell) -\frac{1}{2\pi} \sum \psi\bigl(\tfrac{1}{2} \pm i \nu \pm i\omega \ell)$. Red dots = inverse eigenvalue spacing of numerically diagonalized $4000 \times 4000$ matrix $H$ in globally truncated model (appendix \ref{['sec:DOS']}). Rightmost panel = $|\rho(\omega)|$ on complex $\omega$-plane, with quasinormal mode poles at $\omega \ell=\pm i (\frac{1}{2} \pm i\nu + n)$.
• Figure 4: One- and all-loop entropy corrections, and dual topological string $t$, $g_s$, for 3D HS$_n$ theory in its maximal-entropy de Sitter vacuum, for different values of $n$ at fixed ${\cal S}^{(0)}=10^8$, $l=0$.
• Figure 5: a: Penrose diagram of global dS, showing flows of $SO(1,1)$ generator $H=M_{0,d+1}$, S = southern static patch. b: Wick-rotated $S$ = sphere; Euclidean time = angle. c:Pelagibacter ubique inertial observer in dS with $\ell = 1.2 \, \mu{\rm m}$ finds itself immersed in gas of photons, gravitons and higher-spin particles at a pleasant $30 {}^\circ {\rm C}$. More details are provided in fig. \ref{['fig:penrose-app']} and appendix \ref{['app:dSWick']}.