The sum over topologies in three-dimensional Euclidean quantum gravity

Abstract

In the Euclidean path-integral approach to quantum gravity, the partition function for Hawking's 'volume canonical ensemble' is computed by summing contributions from all possible topologies. The behaviour such a sum can be estimated in three spacetime dimensions in the limit of small cosmological constant. The sum over topologies diverges for any sign of Lambda , but for dramatically different reasons: for Lambda )0, the divergent behaviour 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.