Classical and Quantum Chern-Simons Theory

Question

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.

This post imported from StackExchange MathOverflow at 2014-08-14 08:34 (UCT), posted by SE-user Minhyong Kim

Comments

Minhyong: instructions for reconnecting with your old account are at meta.stackoverflow.com/help/merging-accounts, if you care.

This post imported from StackExchange MathOverflow at 2014-08-14 08:34 (UCT), posted by SE-user Neil Strickland

Answer 1

The last displayed formula on page 41 of the lecture notes does not defined a section of the complex line bundle $L$. It is rather a definition of $L$. It states that if $C$ is a point of $M$, i.e. a flat connection on $\Sigma$, then for every 3-manifold $B$ such that $\Sigma = \partial B$ and for every $A$ connection on $B$ whose restriction to the boundary is $C$, the quantity $e^{2i \pi S(A)}$ is an element of the fiber of $L$ at $C$ (where $S$ denotes the classical Chern-Simons functional). This statement does not define a section because it does not say what to associate to a point $C$ of the basis but what do associate to a point $C$ and to a connection $A$ of boundary  $C$.

The way to define a section is quantum and is recalled in the question. To define a section from the procedure of the preceding paragraph, one has to eliminate the dependence on the choice of $A$. But quantum field theory does that for us because it tell us to integrate over the all possible $A$. In other words, if $B$ is fixed, a section $v$ of $L$ is defined by assigning to a point $C$ the quantity defined formally by the path integral (over the space of all gauge equivalence classes of connections $A$ on $B$ of boundary $C$):

\[ \int_{A, \partial A =C} DA e^{2i \pi S(A)}, \]

which is indeed an element of the fiber of $L$ at $C$ because it is a "sum" of such elements by the definition of $L$.