Discrete and Continuum Approaches to Three-Dimensional Quantum Gravity

TL;DR

This paper explores the relationship between discrete lattice gravity and continuum gauge theories in three dimensions, revealing an isomorphism between physical states in Ponzano-Regge gravity and gauge-invariant functions on moduli spaces.

Contribution

It establishes a precise correspondence between three-dimensional lattice gravity and continuum Chern-Simons gauge theory, linking discrete and continuum approaches.

Findings

Physical state space in lattice gravity is isomorphic to gauge-invariant functions on moduli space.

Connects Ponzano-Regge model with $ISO(3)$ Chern-Simons theory.

Provides a framework for understanding quantum states in 3D gravity.

Abstract

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat $SU(2)$ connections over a two-dimensional surface, which gives physical states in the $ISO(3)$ Chern-Simons gauge theory.