3d Quantum Gravity Partition Function at 3 Loops: a brute force computation

TL;DR

This paper explicitly computes the 3-loop partition function in 3D quantum gravity, confirming the 1-loop exactness by showing higher loops cancel out through intricate diagrammatic cancellations.

Contribution

It provides a detailed 3-loop calculation in 3D quantum gravity, demonstrating the vanishing of higher-order contributions and validating the 1-loop exactness conjecture.

Findings

All 2-loop diagrams vanish after regularization.

A series of cancellations at 3-loops lead to a zero result.

Methodology applicable to higher loops and dimensions.

Abstract

The partition function of 3-dimensional quantum gravity has been argued to be 1-loop exact. Here, we verify the vanishing of higher-orders in perturbation theory by explicit computation in the second-order, metric formulation at 3-loops. The number of 1-particle irreducible Feynman diagrams involving both gravitons and ghosts turns out to be 17. Using dimensional regularization, we solve all the diagrams. At 2-loops, we find that all such diagrams vanish separately after regularization. At 3-loops, in contrast, a series of remarkable cancellations between different diagrams takes place, with 9 diagrams beautifully conspiring to yield a vanishing result. Our techniques are suitable to be applied to higher loops as well as to similar computations in higher dimensions.