The Sum over Topologies in Three-Dimensional Euclidean Quantum Gravity

TL;DR

This paper investigates the divergence of the sum over topologies in three-dimensional Euclidean quantum gravity, analyzing how different signs of the cosmological constant affect the behavior of the partition function.

Contribution

It provides a detailed analysis of the divergence mechanisms in the sum over topologies in 3D quantum gravity for both positive and negative cosmological constants.

Findings

Sum over topologies diverges for both signs of , but due to different reasons.

For >0, divergence from low-volume, complex manifolds.

For <0, divergence from infinite sequences of high-volume manifolds.

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 $\Lambda$, but for dramatically different reasons: for $\Lambda>0$, the divergent behavior comes from the contributions of very low volume, topologically complex manifolds, while for $\Lambda<0$ 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.