I'm interested in the Witten-Reshetikhin-Turaev invariants for 3-manifolds, and in particular, how to calculate them from a triangulation of the 3-manifold (recall that as they were first introduced, their calculation is based on a surgery diagram for the manifold). Is there any some sort of "state sum" model giving the WRT invariant for a $3$-manifold, which is based on the triangulation of the $3$-manifold?
I am well aware of the Turaev-Viro construction, and it is exactly the type of definition I'm hoping for, but of course the TV invariant is $\left|\cdot\right|^2$ of the WRT invariant, not the WRT invariant itself.
The simpler the construction, the better. Certainly one can go algorithmically from a triangulation to a surgery diagram, but I want something which is conceptually based on the triangulation.
I guess any answer I get will also include the case for a pair $(Y^3,L)$, and even perhaps allow for $M^3$ to have nonempty boundary (i.e. the full tqft). However, I will be satisfied even if I just get a construction for a closed $3$-manifold $M$.
Best Answer
This can be done if you extend the triangulation of your 3-manifold $M$ to a triangulation of a 4-manifold $W$ whose boundary is $M$. You can find the basic idea in Chapter 9 of these draft notes on my web page, and there are very explicit details for the associated Hilbert space in this arXiv preprint (with Z. Wang). (In the latter paper we use cubes instead of simplices to make computer implementation easier.) There's also a summary in the notes from a talk.
The main idea is that the Witten-Reshetikhin-Turaev TQFT can be reinterpreted as a 3+1-dimensional TQFT, and the 3+1-dimensional version is "fully extended" (goes all the way down to points). Once you have a fully extended TQFT, standard techniques allow you to construct a state sum model. In this case, the state sum you get is a modified version of the Crane-Yetter state sum. (You would get a Turaev-Viro state sum for a fully extended 2+1 dimensional TQFT.)
EDIT:
If you don't want to wade through the above sources, here's the explicit formula.
Let $C$ be a modular ribbon category. Let $D= \sqrt{\sum_i d_i^2}$, the square root of the sum of the squares of the quantum dimensions. Choose a handle decomposition of $W^4$. Keep in mind is the case where the $i$-handles are thickenings of the $i$-cells of some cell decomposition of $W$, such as the dual cell decomposition to a triangulation. Let ${\cal H}_i$ be the set of $i$-handles. Let ${\cal L}_2$ be the set of labelings of the 2-handles by simple objects of $C$. For fixed $\alpha\in{\cal L}_2$, let ${\cal L}_1(\alpha)$ denote the set of labelings of the 1-handles by orthogonal basis elements of the associated vertex spaces. (To each 1-handle is associated a $C$-picture on the linking 2-sphere, that is, a collection $c$ of ribbon endpoints on the 2-sphere labeled by simple objects $a_1\otimes \cdots\otimes a_m$ according to $\alpha$. The "vertex space" is $\hom_C(1, a_1\otimes\cdots\otimes a_m)$ or, more canonically, the vector space associated to $(B^3; c)$, where we think of $B^3$ as the normal fiber to the core of the 1-handle.) Then $$Z(W^4) = \sum_{\alpha\in{\cal L}_2} \sum_{\beta\in{\cal L}_1(\alpha)} \prod_{h_4\in {\cal H}_4} D \prod_{h_3\in {\cal H}_3} D^{-1} \prod_{h_2\in {\cal H}_2} D^{-1} \,\text{Loop}(h_2, \alpha)\quad\quad\quad\quad$$ $$\quad\quad\quad\quad\prod_{h_1\in {\cal H}_1} D \,\text{Th}(h_1, \beta)^{-1} \prod_{h_0\in {\cal H}_0} D^{-1} \, \text{Link}(h_0, \alpha, \beta) , $$
where
If $W$ has a boundary $M$ then we can place a labeled link $L$ in the boundary, and the labels of the link play a role similar to the labels of the 2-handles above. In this case the state sum computes $WRT(M, L)$.
For a generic cell decomposition of $W$ (i.e. dual to a triangulation), the $\text{Link}(h_0, \alpha, \beta)$ factor is an evaluation of a labeled ribbon graph which looks like the 1-skeleton of a 4-simplex. If we resolve the five 4-valent vertices oin this graph into pairs of 3-valent vertices, then this becomes the "15-j symbol" used by Crane and Yetter. (Keep in mind that Crane and Yetter use a different normalization which obscures the relation to the WRT invariant.)
On the other hand, if we choose a handle decomposition of $W$ with a single 0-handle and several 2-handles (no 1- 3- or 4-handles), then it is easy to see that the above state sum reduces to the Reshetikhin-Turaev surgery formula. The factor $\text{Link}(h_0, \alpha, \beta)$ is the generalized Jones polynomial (evaluated at a root of unity) in this case.
A similar variant yields the Turaev "shadow" state sum.