Title: Sphere partition functions and quantum de Sitter thermodynamics
Description: Driven by a tiny positive cosmological constant, our observable universe asymptotes into a casual patch in de Sitter space in the distant future. Due to the exponential cosmic expansion, a static observer in a de Sitter space is surrounded by a horizon. A semi-classical gravity analysis by Gibbons and Hawking implies that the de Sitter horizon has a temperature and entropy, obeying laws of thermodynamics. Understanding the statistical origin of these thermodynamic quantities requires a precise microscopic model for the de Sitter horizon. With the vision of narrowing the search of such a model with quantum-corrected macroscopic data, we aim to exactly compute the leading quantum (1-loop) corrections to the Gibbons-Hawking entropy, mathematically defined as the logarithm of the effective field theory path integral expanded around the round sphere saddle, i.e. sphere partition functions. This thesis discusses sphere partition functions and their relations to de Sitter (dS) thermodynamics.

It consists of three main parts: The first part addresses the subtleties of 1-loop partition functions for totally symmetric tensor fields on 𝑆^{d⁺¹, and generalizes all known results to arbitrary spin 𝑠 ≥ 0 in arbitrary dimensions 𝑑 ≥ 1. Starting from a manifestly covariant and local path integral on the sphere, we carry out a detailed analysis for any massive, shift-symmetric, massless, and partially massless fields. For any field with spin 𝑠 ≥ 1, we find a finite contribution from longitudinal modes; for any massless and partially massless fields, there is a residual group volume factor due to modes generating constant gauge transformations; for any massless and partially massless fields with spin 𝑠 ≥ 2, we derive the phase factor resulted from Wick-rotating negative conformal modes, generalizing the phase factor first obtained by Polchinski for the case of massless spin 2 to arbitrary spins. The second part presents a novel formalism for studying 1-loop quantum de Sitter thermodynamics.

We first argue that the Harish-Chandra character for the de Sitter group 𝑆𝑂(1,𝑑+1) provides a manifestly de Sitter-invariant regularization for normal mode density of states in the static patch, without introducing boundary ambiguities as in the traditional brick wall approach. These characters encode quasinormal mode spectrums in the static patch. With these, we write down a simple integral formula for the thermal (quasi)canonical partition function, which straightforwardly generalizes to arbitrary spin representations. Then, we derive a universal formula for 1-loop sphere partition functions in terms of the 𝑆𝑂(1,𝑑+1)$ characters. We find a precise relation between these and the (quasi)canonical partition function mentioned earlier: they are equal for scalars and spinors; for any fields with spin 𝑠 ≥ 1, they differ by ``edge'' degrees of freedom living on the de Sitter horizon. This formalism allows us to efficiently compute the exact 1-loop corrected de Sitter horizon entropy, which as we argue provides non-trivial constraints on microscopic models for the de Sitter horizon.

In three dimensions, higher-spin gravity can be alternatively formulated as an sl(𝑛) Chern-Simons theory, which as we show possesses an exponentially large landscape of de Sitter vacua. For each vacuum, we obtain the all-loop exact sphere partition function, given by the absolute value squared of a topological string partition function. Finally, our formalism elegantly proves the relations between generic dS, AdS, and conformal higher-spin partition functions. The last part extends our studies in the previous part to grand (quasi)canonical partition functions on the dS static patch, where we generalize the (quasi)canonical partition functions by allowing non-zero chemical potentials in some of the angular directions. For these, we derive a generalized character integral formula in terms of the full 𝑆𝑂(1,𝑑+1) characters. In three dimensions, we relate them to path integrals on Lens spaces. Similar to its sphere counterpart, the Lens space path integral exhibits a ``bulk-edge'' structure.