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The two-sphere partition function from timelike Liouville theory at three-loop order

While the Euclidean two-dimensional gravitational path integral is in general highly fluctuating, it admits a semiclassical two-sphere saddle if coupled to a matter CFT with large and positive central charge. In Weyl gauge this gravity theory is known as timelike Liouville theory, and is conjectured to be a non-unitary two-dimensional CFT. We explore the semiclassical limit of timelike Liouville theory by calculating the two-sphere partition function from the perspective of the path integral to three-loop order, extending the work in 2106.01665. We also compare our result to the conjectured all-loop sphere partition function obtained from the DOZZ formula. Since the two-sphere is the geometry of Euclidean two-dimensional de Sitter space our discussion is tied to the conjecture of Gibbons-Hawking, according to which the dS entropy is encoded in the Euclidean gravitational path integral over compact manifolds.