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
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Path Integral of Chern-Simons Theory

+ 2 like - 0 dislike
1195 views

Can the path-integral of Abelian Chern-Simons theory be valuated exactly? 

$$\int \mathcal{D}[A] \exp\left\{{\frac{i}{2\pi}\int A\wedge dA}\right\}$$

I found Witten's paper "Quantum Field Theory and Jones Polynomial" very hard to understand. Is there any pedagogical way to find out the gauge fixing of the above action? Is it possible to perform the path-integral non-perturbatively? 

asked Feb 28, 2018 in Theoretical Physics by Libertarian Feudalist Bot (270 points) [ no revision ]
recategorized Mar 2, 2018 by Dilaton

1 Answer

+ 1 like - 0 dislike

It's not an easy calculation (like anything nonperturbative and interesting). Witten evaluates it using cut-and-paste techniques which first he has to devise a generally covariant regularization scheme to use.

The first step is to quantize the theory on an arbitrary closed oriented surface. The Hilbert space can be identified with the space of conformal blocks in an associated WZW CFT. Then to a filling of this surface one needs to associate a state in this Hilbert space. Finally, to compute the partition function on a 3-manifold, one decomposes it into two such filled surfaces (a "Heegaard splitting") and computes the inner product between the associated states.

One nice thing about all this is that every one of these partition functions are full of deep meaning! For example, the partition function on a sphere is the exponentiated entanglement entropy across a disk, and encodes the quasiparticle content in a nontrivial way.

Wilson loops can be evaluated similarly by studying surfaces with punctures.

I return to that paper at least every few months and always find new interesting things in it I didn't see before. :)

In other news, there is some perturbative->nonperturbative progress recently by Gukov, Marino, Putrov, and others which use this Lefschetz thimble technique to better control the expansion around saddle points $dA = 0$. This approach is more direct in some ways from the path integral. Of course, you should expect significant reorganization of degrees of freedom to occur for us to go from a propagating gauge field to a nontrivial TQFT!

answered Feb 28, 2018 by Ryan Thorngren (1,925 points) [ revision history ]

Thank you very much. But I don't understand anything.

It's just really hard to evaluate anything exactly...

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$\varnothing$ysicsOverflow
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
...