Turaev-Viro Invariant, Rational Conformal Field Theory and 3-dimensional Gravity

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We study the Turaev-Viro (TV) topological invariant in relation to 3-dimensional gravity. We show that it may be viewed as a regularized Ponzano-Regge model, and effectively includes the cosmological term with positive cosmological constant Λ = (8π^2)/k^2 at Ο(k^<-2>) from q-deformation, where q is a 2(k + 2)th root of unity. We argue that the subtleties in the Ponzano-Regge model may be understood by identifying the TV invariant with the Euclidean Chern-Simons-Witten gravity partition function. 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 SU(2) WZW model. As a corollary we then show that the partition function Z(M) is bounded from above by Z((S^2 × S^1)^<#g>)=(S_<00>)^<-2g+2>〜Λ-(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 Λ 〜 +0 if M is neither S^3 nor a lens space, and many-wormhole configurations dominate near Λ 〜 +0 in the sense that Z(M) generically tends to diverge faster as the "number of wormholes" g becomes larger.
素粒子論グループ素粒子研究編集部の論文
1993-04-20