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

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.