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

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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,862 answers , 20,637 comments
1,470 users with positive rep
502 active unimported users
More ...

Why are boundary terms important in Chern-Simons theory?

+ 3 like - 0 dislike
159 views

I am learning about Chern-Simons theory. I work in Euclidean space. The action is given by
$$
I=\int_{\mathbb{R}^4}d\omega=\int_{\partial\mathbb{R}^4}\omega
$$
where $\omega$ is the usual Chern-Simons form, and I have used stokes' theorem. My first question is, what is the boundary of for dimensional Euclidean space? and why are boundary terms important in this theory? I mean, up till now I have shamelessly ignored boundary terms. Why are they important now?

asked Jul 4, 2016 in Theoretical Physics by Dmitry hand me the Kalashnikov (720 points) [ no revision ]

1 Answer

+ 3 like - 0 dislike

The action of Chern-Simons theory on a 3-manifold $M_3$ can be written as a boundary term of Yang-Mills theory on a 4-manifold $M_4$ of boundary $M_3$. If one is interested in $M_3=\mathbb{R}^3$ then a natural choice is to take $M_4$ to be a four dimensional half space $\mathbb{R}^+ \times \mathbb{R}^3$ of boundary $\{0\} \times \mathbb{R}^3$. Strictly speaking, $\mathbb{R}^4$ has no boundary (at least in the most naive sense) and so writing things like $\partial \mathbb{R}^4$ is not really correct if no additional precision is given.

To ignore boundary terms is only reasonable in some cases (when they are none or when their contributions obviously vanish). Chern-Simons theory is a natural theory living on some 3-dimensional boundary of 4-dimensional Yang-Mills theory. One can define and study Chern-Simons theory independently of this fact but to remember this often sheds light on important issues (example: quantization of the Chern-Simons level is related to the quantization of the instanton number).

answered Jul 6, 2016 by 40227 (4,660 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$\hbar\varnothing$sicsOverflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...