Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v\_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the Witten-Reshetikhin-Turaev TQFT. I'd like to understand what this vector is.


In short, $Z(\Sigma)$ is a space of sections of a line bundle over the $\mathrm{SU}(2)$ character variety of $\Sigma$. I am hoping that the section $v\_{WRT}(Y^3)$ achieves its maximum value (with respect to the canonical inner product on the line bundle) on the Lagrangian submanifold of the character variety consisting of those representations which extend to $Y^3$. [EDIT: there is a good reason to believe this holds, since then high powers of the section will concentrate on this Lagrangian, giving Volume Conjecture-like convergence to the classical Lagrangian intersection theory as the level of the TQFT goes to infinity]


In more detail, let's discuss an explicit description of $Z(\Sigma)$. There is a natural line bundle $\mathcal L$ over the character variety $X:=\operatorname{Hom}(\pi\_1(\Sigma),\mathrm{SU}(2))/\mathrm{SU}(2)$. There is a natural symplectic form on $X$, and choosing a complex structure on $\Sigma$ equips $X$ with a complex structure which together with the symplectic form makes $X$ a Kahler manifold. Then $Z(\Sigma)$ is the Hilbert space of square integrable holomorphic sections of $\mathcal L$ ($\mathcal L$ carries a natural inner product, and the curvature form of the induced connection coincides with the natural symplectic form on $X$).


My question is then: how can one describe $v(Y^3)\in Z(\Sigma)$? Does the corresponding section achieve its maximum value on the Lagrangian subvariety of $X$ comprised of those characters of $\pi\_1(\Sigma)$ extending to characters of $\pi\_1(Y)$?


Comment: answering this question for an arbitrary $3$-manifold $Y^3$ seems unlikely to yield a clean answer, since it includes as a special case calculating the value of the WRT TQFT applied to $Y$ (and the description of this requires the introduction of a whole bunch of extra stuff, e.g. surgery diagrams for $Y^3$, etc.). This is why I am restricting to the case that $Y^3$ is a handlebody, in hopes that in this special case, there is a clean answer to this question.



What is the state in the WRT TQFT associated to a handlebody?
I can think of two cases where the Witten-Reshetikhin-Turaev vector $Z\_k(Y^3)\in Z\_k(\Sigma)$ has been connected to a Lagrangian state as you described:


-Laurent Charles and Julien Marché showed that the WRT vector of the figure eight knot complement $Z\_k(E\_K)$ is a Lagrangian state concentrating on $\mathcal{M}(E\_K)=\mathrm{Hom}(\pi\_1 E\_K, \mathrm{SU}\_2)/\mathrm{SU}\_2$ and used this to prove Witten's asymptotic expansion conjecture for Dehn-fillings of the figure eight. 
(see here [part1](https://webusers.imj-prg.fr/~julien.marche/papiers/knot1.pdf) and [part2](https://webusers.imj-prg.fr/~julien.marche/papiers/knot2.pdf) )


-In my [thesis](https://arxiv.org/pdf/1412.4333.pdf), I considered the vectors $Z\_k(H,\Gamma)$ where $H$ is a handlebody and $\Gamma$ is a colored trivalent banded graph in $H$ that is a spine of $H.$ Such vectors form a basis of $Z\_k(\Sigma),$ and the vector $Z\_k(H)$ of the empty handlebody is obtained taking all colors=0.
I showed for generic colors, the vectors $Z\_k(H,\Gamma)$ are Lagrangian states concentrating on subsets of the form $\lbrace \mathrm{Tr}\rho(C\_i)=x\_i \rbrace$ where the curves $C\_i$ bound disks in $H$ that are dual to the edges of $\Gamma$.


Both results were obtained from the combinatorial definition of TQFT (Reshetikhin-Turaev, or rather Blanchet-Habegger-Masbaum-Vogel), after choosing an isomorphism $Z\_k(\Sigma) \rightarrow H^1(\mathcal{M}(\Sigma),\mathcal{L}^k)$ that turns curve operators in TQFT into Toeplitz operators whose principal symbols are trace functions on the moduli space $\mathcal{M}(\Sigma)$.


Although the problem of showing that the WRT vector $Z\_k(H)$ of a handlebody concentrates on the character variety of $H$ seem reasonable, one gets into trouble because of the singularities of $\mathcal{M}(\Sigma)=\mathrm{Hom}(\pi\_1\Sigma,\mathrm{SU}\_2)/\mathrm{SU}\_2,$ and also somewhat because one has to work at the critical levels of the principal symbols of the curve operators.


Laurent and Julien's work avoided this difficulty because $\mathcal{M}(T^2)$ is not really singular, and in my paper because I only considered the geometric quantization of dense open sets of $\mathcal{M}(\Sigma)$ and not the whole moduli space.


It is my feeling that if one was able to give a clean answer to your question, one would not be too far from proving Witten's asymptotic expansion conjecture, at least for $3$-manifolds with a "generic" Heegaard splitting. 


