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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 $SU(2)$ 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 $X:=\operatorname{Hom}(\pi_1(\Sigma),SU(2))/\\!/SU(2)$. 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 $\pi_1(Y)$?

A 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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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. –  Charlie Frohman Jul 30 '11 at 21:18
    
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. –  Steve Sawin Oct 8 '11 at 15:03
    
See this answer to the question on PhysicsOverflow. –  Dilaton Sep 12 at 20:13

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