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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.

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    $\begingroup$ You can define it up to phase. The idea Is to see your setting as a fiber Bundle over Teichmuller space. There Is a projectively flat connection That relates state spaces over different Points. Complete with stable curves. Over A surface that has been pinched down to a collection of spheres with three singular points there is a canonical vector, drag it back. $\endgroup$ – Charlie Frohman Jul 30 '11 at 21:18
  • $\begingroup$ It is even a little more complicated than Charlie says. Not only does defining the invariant of a 3-fold require extra info (framing), but vector space associated with $\Sigma$ requires extra info to define (they are all isomorphic, but to find a natural basis in which to specify $Z(y^3)$ you will have to address this. I can speak about all of this precisely in surgery / 4 fold terms, but no idea how to relate it to $SU(2)$ character varieties. If you want my spiel let me know. $\endgroup$ – Steve Sawin Oct 8 '11 at 15:03
  • $\begingroup$ See this answer to the question on PhysicsOverflow. $\endgroup$ – Dilaton Sep 12 '14 at 20:13
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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 and part2 )

-In my thesis, 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.

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(Making this an answer, since it will be a little long, even though it doesn't actually answer the question.)

One thing to be aware of is that there are multiple ways to talk about this same TQFT. In particular, Reshetikhin and Touraev took a much more combinatorial point of view in defining the 3-manifold invariants. There's an entirely different way of getting at the vector space associated to a surface $\Sigma$ from the combinatorial point of view. You can take a handlebody $H$ with boundary $\Sigma$, pick a trivalent graph spine $\Gamma$ for $H$, and look at the vector space spanned by admissible colorings on $\Gamma$. You then have to work a little bit to see that different choices of $H$ and $\Gamma$ give canonically isomorphic vector spaces.

This is quite different from the description in terms of line bundles, as you described. It's not obvious that the two descriptions are isomorphic, and I'm not sure where to point you to get a proof that they are isomorphic. (Of course Witten gives a beautiful physics argument for isomorphism.)

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