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I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the quantum group $U_{q}\mathfrak{sl}_{2}$ when $q$ is not a root of unity.

In particular if convergence of expressions for the invariants of all manifolds isn't guaranteed this wouldn't be so much of an issue for my interests. For example something like $S^{1}\times\Sigma$ having infinite invariant isn't so bad for me (i.e having infinite dimensional vector spaces on boundaries).

Thanks in advance!

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  • $\begingroup$ I'm not sure that it is a bad thing at all. Maybe it has more to do with the physical applications? The Witten-Reshetikhin-Turaev invariant has close ties to Chern-Simons theory where they are interested in finite sets of modules (i.e. q is root of unity). After Turaev wrote his book the volume conjecture was introduced and if you are interested in hyperbolic knot volume q being a root of unity may not be what you are looking for. $\endgroup$ – Bob Dec 12 '18 at 18:09
  • $\begingroup$ Hyperbolic knots and their volumes are what I'm trying to understand. A TQFT can't have infinite dimensional vector spaces with the usual definition of a TQFT. Is this the only thing (and its consequences) that goes wrong in their construction of a TQFT? Seemingly they can define invariants for some links in $\mathbb{R}^{3}$ at arbitrary q is it just that occasionally when they glue they will get infinite invariants? I'm not sure if the anomalies they fix will be a bigger problem for arbitrary q for example. $\endgroup$ – Campbell Dec 14 '18 at 20:48
  • $\begingroup$ arxiv.org/pdf/1503.02547.pdf Maybe that article will be useful. $\endgroup$ – Bob Dec 21 '18 at 20:03

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