3D Quantum Gravity Partition Function at Three Loops

The partition function of three-dimensional quantum gravity has been argued to be one-loop exact. Here, we verify the vanishing of higher orders in perturbation theory by explicit computation in the second-order metric formulation at three loops. The number of one-particle irreducible Feynman diagrams involving both gravitons and ghosts turns out to be 17. Using dimensional regularization, we solve all the diagrams. At two loops, we find that all such diagrams vanish separately after regularization. At three loops, in contrast, a series of remarkable cancellations between different diagrams takes place, with nine 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.