3-dimensional Gravity from the Turaev-Viro Invariant

TL;DR

This paper demonstrates that the Turaev-Viro invariant provides a natural regularization of 3D quantum gravity path integrals, effectively incorporating the cosmological constant, and relates to Euclidean Chern-Simons-Witten gravity.

Contribution

It shows that the Turaev-Viro invariant defines a regularized 3D quantum gravity model including a cosmological term, connecting quantum invariants with gravitational path integrals.

Findings

The Turaev-Viro invariant acts as a regularized path integral for 3D quantum gravity.

The cosmological constant is effectively included and depends on the regularization parameter.

The model relates to Euclidean Chern-Simons-Witten gravity in three dimensions.

Abstract

We study the $q$-deformed su(2) spin network as a 3-dimensional quantum gravity model. We show that in the semiclassical continuum limit the Turaev-Viro invariant obtained recently defines naturally regularized path-integral $\grave{\rm a}$ la Ponzano-Regge, In which a contribution from the cosmological term is effectively included. The regularization dependent cosmological constant is found to be ${4\pi^2\over k^2} +O(k^{-4})$, where $q^{2k}=1$. We also discuss the relation to the Euclidean Chern-Simons-Witten gravity in 3-dimension.