TITLE: Are Turaev--Viro invariants secretly a discretized path integral?
QUESTION [12 upvotes]: Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with $\left|WRT(M)\right|^2$ where $WRT(M)$ is the Witten--Reshetikhin--Turaev invariant of $M$.  Both invariants depend on a choice of quantum group $U_q(\mathfrak g)$ and a root of unity $q$.
Witten's "physics definition" of $WRT(M)$ http://www.ams.org/mathscinet-getitem?mr=990772 is via an "integral over all $\mathfrak g$-connections on $M$".  I have read in many places that the Turaev--Viro model is essentially a discretization of the path integral (which only converges if we use a quantum group; the nonconvergent analogue for a classical group is the "Ponzano--Regge model").

How can one see that the Turaev--Viro model as a discretization of the path integral?

The definition of $TV(M)$ is as a "state sum" over all assignments of representations of $U_q(\mathfrak g)$ to each of the edges of (a fixed triangulation of) $M$.  On the other hand, one would naively expect that a discretization of the path integral would be a "state sum" over all assignments of "elements" of $U_q(\mathfrak g)$ to each of the edges of $M$ (this being the most obvious way to think about a discretized connection on $M$).
How does the connection between these two state sums go?

REPLY [6 votes]: Perhaps I asked this question a bit too quickly.  An explanation exactly as I was looking for appears at the beginning of Section III of this paper: http://arxiv.org/abs/hep-th/0401076