# Classical and Quantum Chern-Simons Theory

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

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.

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Minhyong: instructions for reconnecting with your old account are at meta.stackoverflow.com/help/merging-accounts, if you care. – Neil Strickland Mar 7 '14 at 18:33
This has received an answer on PhysicsOverflow if you're still interested: Classical and Quantum Chern-Simons Theory – Dimensio1n0 Aug 14 '14 at 13:14