Partition Functions and Topology-Changing Amplitudes in the 3D Lattice Gravity of Ponzano and Regge

TL;DR

This paper establishes an isomorphism between the Hilbert space of 3D lattice gravity and $ISO(3)$ Chern-Simons theory, showing their partition functions and topology-changing amplitudes are equivalent for any closed orientable manifold.

Contribution

It constructs a Hilbert space for Ponzano-Regge lattice gravity, demonstrating its equivalence to $ISO(3)$ Chern-Simons theory and relating their partition functions and topology-changing amplitudes.

Findings

Hilbert space of lattice gravity is isomorphic to Chern-Simons theory.

Partition functions of both theories are identical for any closed orientable manifold.

Topology-changing amplitudes in lattice gravity relate directly to those in Chern-Simons theory.

Abstract

We define a physical Hilbert space for the three-dimensional lattice gravity of Ponzano and Regge and establish its isomorphism to the ones in the $ISO(3)$ Chern-Simons theory. It is shown that, for a handlebody of any genus, a Hartle-Hawking-type wave-function of the lattice gravity transforms into the corresponding state in the Chern-Simons theory under this isomorphism. Using the Heegaard splitting of a three-dimensional manifold, a partition function of each of these theories is expressed as an inner product of such wave-functions. Since the isomorphism preserves the inner products, the partition function of the two theories are the same for any closed orientable manifold. We also discuss on a class of topology-changing amplitudes in the lattice gravity and their relation to the ones in the Chern-Simons theory.