Let Y3 be a handlebody with boundary Σ. By definition, there is some associated vector vWRT(Y3)∈Z(Σ), the (finite dimensional) Hilbert space associated to Σ by the Witten-Reshetikhin-Turaev TQFT. I'd like to understand what this vector is.
In short, Z(Σ) is a space of sections of a line bundle over the SU(2) character variety of Σ. I am hoping that the section vWRT(Y3) 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 Y3. [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(Σ). There is a natural line bundle L over the character variety X:=Hom(π1(Σ),SU(2))/!/SU(2). There is a natural symplectic form on X, and choosing a complex structure on Σ equips X with a complex structure which together with the symplectic form makes X a Kahler manifold. Then Z(Σ) is the Hilbert space of square integrable holomorphic sections of L (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(Y3)∈Z(Σ)? Does the corresponding section achieve its maximum value on the Lagrangian subvariety of X comprised of those characters of π1(Σ) extending to characters of π1(Y)?
A comment: answering this question for an arbitrary 3-manifold Y3 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 Y3, etc.). This is why I am restricting to the case that Y3 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