Rational Conformal Field Theory and Multi-Wormhole Partition Function in 3-dimensional Gravity

TL;DR

This paper links the Turaev-Viro invariant to 3D gravity partition functions, showing how rational conformal field theories inform the invariant's initial data and analyzing the behavior of the partition function near positive cosmological constant.

Contribution

It demonstrates the construction of Turaev-Viro invariant data from rational conformal field theories and explores the partition function's bounds and divergence behavior in multi-wormhole configurations.

Findings

Partition function bounded by handlebody genus

Partition function diverges near positive cosmological constant

Multi-wormhole configurations dominate near term

Abstract

We study the Turaev-Viro invariant as the Euclidean Chern-Simons-Witten gravity partition function with positive cosmological constant. After explaining why it can be identified as the partition function of 3-dimensional gravity, we show that the initial data of the TV invariant can be constructed from the duality data of a certain class of rational conformal field theories, and that, in particular, the original Turaev-Viro's initial data is associated with the $A_{k+1}$ modular invariant WZW model. As a corollary we then show that the partition function $Z(M)$ is bounded from above by $Z((S^2\times S^1)^{\sharp g}) =(S_{00})^{-2g+2}\sim \Lambda^{-\frac{3g-3}{2}}$, where $g$ is the smallest genus of handlebodies with which $M$ can be presented by Hegaard splitting. $Z(M)$ is generically very large near $\Lambda\sim +0$if $M$ is neither $S^3$ nor a lens space, and many-wormholeconfigurations dominate near $\Lambda\sim +0$ in the sense that $Z(M)$ generically tends to diverge faster as the ``number of wormholes'' $g$ becomes larger.