Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway.
Let $\Sigma$ be a two-manifold and $M$ a moduli space of flat connections on it with some gauge group $G$. $M$ will carry a determinant line bundle $L$. In a number of situations, if we pick a holomorphic structure on $\Sigma$, then we will get one on $M$ and $L$. Now, let's assume that $\Sigma$ is the boundary of a 3-manifold $B$. I would like to understand the process whereby Chern-Simons theory on $B$ gives rise to a section $v$ of $L$ over $M$. Now, I'm told this was first described in the paper of Witten on the Jones polynomial. There, by some process I don't really understand, there is a path integral formalism that fits together into a 3D TQFT so that $v$ is simply the image of the vacuum vector under the map induced by $B$. On the other hand, if you look at treatments like Lecture 4 in
https://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf
I get the impression that one can get such sections in a very elementary manner by using just the classical Chern-Simons functional. (Last displayed formula on page 41 of the notes.) I seem to find the same thing in more recent treatments, such as papers of Andersen (which I've hardly looked into at all). So I thought I would ask if this understanding is indeed correct and, if so, what the relation is between the classical and quantum constructions of sections.
Best Answer
This question has been answered at PhysicsOverflow.