Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

TL;DR

This paper develops a perturbation theory for three-dimensional quantum gravity using spin foam models, analyzing topological classes, and calculating partition functions with corrections related to the cosmological constant.

Contribution

It introduces a spin foam perturbation framework for 3D quantum gravity with a cosmological constant and challenges existing conjectures on topological classes of configurations.

Findings

Baez conjecture is false for general triangulations

Partition function computed for special triangulations shows vanishing corrections

Modified dilute-gas limit yields nonvanishing second-order corrections

Abstract

We formulate the spin foam perturbation theory for three-dimensional Euclidean Quantum Gravity with a cosmological constant. We analyse the perturbative expansion of the partition function in the dilute-gas limit and we argue that the Baez conjecture stating that the number of possible distinct topological classes of perturbative configurations is finite for the set of all triangulations of a manifold, is not true. However, the conjecture is true for a special class of triangulations which are based on subdivisions of certain 3-manifold cubulations. In this case we calculate the partition function and show that the dilute-gas correction vanishes for the simplest choice of the volume operator. By slightly modifying the dilute-gas limit, we obtain a nonvanishing correction which is related to the second order perturbative correction. By assuming that the dilute-gas limit coupling constant is a function of the cosmological constant, we obtain a value for the partition function which is independent of the choice of the volume operator.