Topology change in ISO(2,1) Chern-Simons gravity

TL;DR

This paper investigates topology change in 2+1 dimensional ISO(2,1) Chern-Simons gravity, revealing conditions under which topology-changing amplitudes are non-zero and constructing transition paths for different space topologies.

Contribution

It establishes a bundle condition for topology change in 2+1 gravity and constructs explicit transition paths for various space topologies.

Findings

Topology change amplitude is non-zero only under specific bundle conditions.

Constructed transition paths for all genus g ≥ 2 spaces.

Identified a selection rule for allowed space topologies.

Abstract

In 2+1 dimensional gravity, a dreibein and the compatible spin connection can represent a space-time containing a closed spacelike surface $\Sigma$ only if the associated SO(2,1) bundle restricted to $\Sigma$ has the same non-triviality (Euler class) as that of the tangent bundle of $\Sigma.$ We impose this bundle condition on each external state of Witten's topology-changing amplitude. The amplitude is non-vanishing only if the combination of the space topologies satisfies a certain selection rule. We construct a family of transition paths which reproduce all the allowed combinations of genus $g \ge 2$ spaces.