Spinfoam 2d quantum gravity and discrete bundles

TL;DR

This paper develops a spinfoam quantization of 2D Riemannian general relativity as a constrained BF theory, incorporating bundle topology, and discusses implications for quantum gravity.

Contribution

It extends the spinfoam approach to 2D gravity, demonstrating the viability of the Barrett-Crane technique in this setting and analyzing bundle topology effects.

Findings

Partition function matches expectations, supporting the technique.

Highlights the role of bundle topology in quantum gravity.

Discusses potential finiteness of the quantum gravity partition function.

Abstract

In 4 dimensions, general relativity can be formulated as a constrained $BF$ theory; we show that the same is true in 2 dimensions. We describe a spinfoam quantization of this constrained BF-formulation of 2d riemannian general relativity, obtained using the Barrett-Crane technique of imposing the constraint as a restriction on the representations summed over. We obtain the expected partition function, thus providing support for the viability of the technique. The result requires the nontrivial topology of the bundle where the gravitational connection is defined, to be taken into account. For this purpose, we study the definition of a principal bundle over a simplicial base space. The model sheds light also on several other features of spinfoam quantum gravity: the reality of the partition function; the geometrical interpretation of the Newton constant, and the issue of possible finiteness of the partition function of quantum general relativity.