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

TL;DR

This work establishes a precise isomorphism between the physical Hilbert spaces of the 3D Ponzano–Regge lattice gravity and ISO(3) Chern–Simons theory, linking lattice Wheeler–DeWitt states to gauge-invariant CS wavefunctions built from Wilson lines. By employing Hartle–Hawking-type states and Heegaard splittings, it shows that partition functions and a class of topology-changing amplitudes agree between the two frameworks, with a careful match of inner products and normalization factors dependent on genus. The results provide a concrete realization of the Ponzano–Regge model as a lattice manifestation of three-dimensional gravity and suggest connections to Turaev–Viro, with extensions to boundary processes and potential Lorentzian or cosmological-constant deformations. The analysis also highlights convergence caveats and points toward Lorentzian $SO(2,1)$ constructions and other topological field theory generalizations as promising avenues for future work.

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.