Dominant Topologies in Euclidean Quantum Gravity

TL;DR

This paper investigates how the dominant topologies in Euclidean quantum gravity vary with the sign of the cosmological constant, revealing distinct topological phases and implications for the cosmological constant problem.

Contribution

It identifies the different topological phases in Euclidean quantum gravity depending on the sign of the cosmological constant, highlighting a boundary at zero.

Findings

For $ ext{ extLambda}>0$, dominant topologies have vanishing first Betti number.

For $ ext{ extLambda}<0$, the sum over topologies is dominated by complex fundamental groups.

The case $ ext{ extLambda}=0$ acts as a boundary between these phases.

Abstract

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according on the sign of the cosmological constant. For $\Lambda>0$, saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For $\Lambda<0$, on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the ``density of topologies'' grows fast enough to overwhelm this suppression. The value $\Lambda=0$ is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle-Hawking wave function.