The Sum over Topologies in Three-Dimensional Euclidean Quantum Gravity

TL;DR

This work analyzes Hawking's Euclidean path integral for 3D gravity by summing over topologies in the small-Λ limit. Using a semiclassical, one-loop evaluation around constant-curvature Einstein spaces and expressing manifolds as quotients M̃/Γ with Ray-Singer torsion as the key weighting, the paper distinguishes the Λ>0 and Λ<0 cases. It finds that Λ>0 is driven to divergence by large numbers of small-volume, highly connected lens spaces, while Λ<0 is driven to divergence by infinite families of high-volume hyperbolic manifolds connected through cusp-rich Dehn surgeries. The results highlight a strong, sign-dependent interplay between action and entropy and suggest caution when extrapolating to four dimensions, while noting limitations such as the single-loop approximation and incomplete summation over saddles.

Abstract

In Hawking's Euclidean path integral approach to quantum gravity, the partition function is computed by summing contributions from all possible topologies. The behavior such a sum can be estimated in three spacetime dimensions in the limit of small cosmological constant. The sum over topologies diverges for either sign of $Λ$, but for dramatically different reasons: for $Λ>0$, the divergent behavior comes from the contributions of very low volume, topologically complex manifolds, while for $Λ<0$ it is a consequence of the existence of infinite sequences of relatively high volume manifolds with converging geometries. Possible implications for four-dimensional quantum gravity are discussed.