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 ...

  Why are topological properties described by surface terms?

+ 4 like - 0 dislike
608 views

An example are the anomalies in abelian and non-abelian gauge quantum field theories. 

For example, the abelian anomaly is $\tilde {F}_{\mu\nu}F^{\mu\nu}$ and the integral over this quantity is a topological invariant which measures a topological characteristic of the gauge field $A_\mu$.

All such quantities can be rewritten as total derivatives and then, using Gauss' law transformed into a surface integral. 

**What's the intuitive reason that quantities which describe topological properties can always be written as surface integrals?**

 Formulated a bit differently: Why are topological properties always completely encoded in the boundary of the system? 

asked Nov 20, 2017 in Theoretical Physics by JakobS (110 points) [ no revision ]

Locally (i.e., in the bulk), topology cannot enter as all manifolds are locally diffeomorphic to the unit ball. 

1 Answer

+ 4 like - 0 dislike

Effectively this is just Stokes' theorem: The integral over spacetime of an exact differential form is the integral of its potential over the boundary. The only subtlety is that the interesting topological invariants are only locally exact, not globally. But the analogous statement remains true with differential forms enhanced to higher gerbes with connection. Instead of Chern-Simons forms integrated over the boundary one then gets the volume holonomy of higher Chern-Simons gerbes over the boundary. That the analogue of Stokes's theorem still applies in this relevant generality is proposition 4.4.130 (p. 532) in arxiv:1310.7930 .

answered Nov 20, 2017 by Urs Schreiber (6,095 points) [ no revision ]

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$\hbar$ysicsOverf$\varnothing$ow
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
...