A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ is a root of unity. Components of $L$ are coloured by finite dimensional $G$-modules.

Since Murakami in 1995, people have been proving integrality results for increasingly general classed of simple Lie groups $G$ (first $SO(3)$ then $SU(n)$, then any compact simple Lie group), roots of unity $\xi$ (first prime, then non-prime), and $3$--manifolds $M$ (first integral homology $3$-sphere, then rational homology $3$-sphere, then the general case). These results usually state that the WRT invariant is an algebraic integer- an element of $\mathbb{Z}[\xi]$- or the stronger result that it's the evaluation at $\xi$ of an element in the Habiro ring.

Papers on the integrality of WRT invariants usually list wonderful things that can be done once integrality properties are established (e.g. integral TQFT or categorification or representations over $\mathbb{Z}$ of the mapping class group). But why would we expect $\tau_{M,L}^G(\xi)$ to be an element of $\mathbb{Z}[\xi]$? As far as I know that's always been the case. Is there some perhaps some not-quite-rigourous construction of WRT invariants which takes place entirely over $\mathbb{Z}[\xi]$, or maybe over the Habiro ring?

**Question**: Why are WRT invariants expected to be algebraic integers? Is there a conceptual explanation for their integrality? Are all WRT invariants in fact expected to come from analytic functions over roots of unity (i.e. elements of the Habiro ring), and if so, why?

This post imported from StackExchange MathOverflow at 2015-01-02 16:47 (UTC), posted by SE-user Daniel Moskovich