Topology Change in (2+1)-Dimensional Gravity

TL;DR

This paper evaluates topology-changing amplitudes in (2+1)-dimensional gravity, showing they are nonzero but suppressed compared to topology-preserving processes, with some amplitudes being infrared divergent.

Contribution

It provides an explicit computation of topology-changing transition amplitudes in (2+1)-dimensional gravity using Ray-Singer torsion, highlighting their suppression and divergence properties.

Findings

Certain topology-changing amplitudes are nonvanishing.

These amplitudes can be infrared divergent.

Topology-preserving amplitudes dominate over topology-changing ones.

Abstract

In (2+1)-dimensional general relativity, the path integral for a manifold $M$ can be expressed in terms of a topological invariant, the Ray-Singer torsion of a flat bundle over $M$. For some manifolds, this makes an explicit computation of transition amplitudes possible. In this paper, we evaluate the amplitude for a simple topology-changing process. We show that certain amplitudes for spatial topology change are nonvanishing---in fact, they can be infrared divergent---but that they are infinitely suppressed relative to similar topology-preserving amplitudes.