Summing over homology groups of 3-manifolds

TL;DR

This paper models a 3D topological quantum gravity scenario using abelian TQFTs, expressing the sum over 3-manifolds as a sum over abelian groups, and explores conditions for convergence and ensemble interpretations.

Contribution

It introduces a framework to rewrite the sum over 3-manifolds as a sum over finitely generated abelian groups with bounds on weights, and discusses ensemble averages of 2D TQFTs.

Findings

Bounds on weights ensure sum convergence

Sum can be factorized into p-group sums for certain measures

Ensemble interpretation of the sum as average over 2D TQFTs

Abstract

We consider a toy model of a 3-dimensional topological quantum gravity. In this model, a contribution of a given 3-manifold is given by the partition function of an abelian Topological Quantum Field Theory (TQFT), with a topological boundary condition at the boundary. Using the fact that the TQFT partition function depends only on the first homology group of the 3-manifold with some additional structure, the sum over all 3-manifolds with fixed boundary can be rewritten as a sum over finitely generated abelian groups (and the extra structure). We present bounds on the universal weights in the sum, that is, the measure on the set of isomorphism classes of finitely generated abelian groups (with the extra structure) sufficient for the sum to be convergent. Moreover, with further assumptions on the measure, we argue the existence of a distribution of 2d TQFTs, such that the sum is equal to the ensemble average of their partition functions evaluated on the boundary. For a certain kind of measure, the sum over the abelian groups can be factorized into sums over abelian $p$-groups, and the analysis can be performed independently for each prime $p$.