Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,786 comments
1,470 users with positive rep
820 active unimported users
More ...

  Classical and Quantum Chern-Simons Theory

+ 10 like - 0 dislike
1677 views

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
asked Mar 7, 2014 in Theoretical Physics by Minhyong Kim (50 points) [ no revision ]
retagged Aug 14, 2014
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

1 Answer

+ 3 like - 0 dislike

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 $L$.

answered Aug 14, 2014 by 40227 (5,140 points) [ revision history ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysic$\varnothing$Overflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...