# What is the state in the WRT TQFT associated to a handlebody?

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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.

This post imported from StackExchange MathOverflow at 2014-09-04 08:37 (UCT), posted by SE-user John Pardon

edited Sep 4, 2014
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.

This post imported from StackExchange MathOverflow at 2014-09-04 08:37 (UCT), posted by SE-user Charlie Frohman
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.

This post imported from StackExchange MathOverflow at 2014-09-04 08:37 (UCT), posted by SE-user Steve Sawin

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One way to figure this out is to perform the path integral

$\int DA e^{iS}$

over the handlebody with boundary conditions specified by fixing the (flat) connection along the boundary Riemann surface. This number is the value of the wavefunction you want on the boundary configuration specified. In the abelian case this should actually only depend on $g$ numbers describing the holonomies of the boundary gauge field around the cycles which bound discs in your handle-body. In general it is a section of the line bundle you mention on the character variety.

This path integral can be (presumably) performed using the state-sum representation of the RT invariant.

answered Sep 12, 2014 by (1,895 points)

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