New Modular Hopf Algebras related to rational $k$ $\widehat{sl(2)}$

Abstract

We show that the Hopf link invariants for an appropriate set of finite dimensional representations of $U_{q}SL(2)$ are identical, up to overall normalisation, to the modular S matrix of Kac and Wakimoto for rational $k$ $\widehat{sl(2)}$ representations. We use this observation to construct new modular Hopf algebras, for any root of unity $q=e^{-i\pi m/r}$, obtained by taking appropriate quotients of $U_{q}SL(2)$, that give rise to 3-manifold invariants according to the approach of Reshetikin and Turaev. The phase factor correcting for the `framing anomaly' in these invariants is equal to $e^{-\frac{i\pi }{4}(\frac{3k}{k+2})}$, an analytic continuation of the anomaly at integer $k$. As expected, the Verlinde formula gives fusion rule multiplicities in agreement with the modular Hopf algebras. This leads to a proposal, for $(k+2)=r/m$ rational with an odd denominator, for a set of $\widehat{sl(2)}$ representations obtained by dropping some of the highest weight representations in the Kac-Wakimoto set and replacing them with lowest weight representations. For this set of representations the Verlinde formula gives non-negative integer fusion rule multiplicities. We discuss the consistency of the truncation to highest and lowest weight representations in conformal field theory.

arXiv comment30 pages (minor typos corrected, refs added)

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