# Are Turaev--Viro invariants secretly a discretized path integral?

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?

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