Abstract
We define a physical Hilbert space for the three-dimensional lattice gravity of Ponzano and Regge and establish its isomorphism to the one 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, the partition functions of each of these theories is expressed as an inner product of such wave functions. Since the isomorphism preserves the inner products, the partition functions of the two theories are the same for any closed orientable manifold. We also discuss a class of topology-changing amplitudes in the lattice gravity and their relation to the ones in the Chern-Simons theory.
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