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

185 submissions , 145 unreviewed
4,725 questions , 1,928 unanswered
5,268 answers , 22,442 comments
1,470 users with positive rep
743 active unimported users
More ...

  What's the symmetry group $SU(N)/Z_N$?

+ 1 like - 0 dislike
86 views

I'm trying to understand David Tong's  http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html, specifically the discussion around page 92 where he's arguing that a different symmetry group may the group of QCD, namely $G'=SU(N)/Z_N$ instead of $G=SU(N)$. 

I understand that having $G'$ as a group leads to a different quantization of the $\theta$ term, but I want to know if this is really the crucial aspect? Namely, how does the cyclic group $Z_N$ act on the fundamental fields (quarks, gluons)? Can anybody show this invariance to me at the level of the Lagrangian? I'm a bit confused so even the most pedestrian computation would help :)


  

asked Feb 4 in Theoretical Physics by ucci (25 points) [ revision history ]

1 Answer

+ 1 like - 0 dislike

$\mathbb{Z}_N$ is realized in $SU(N)$ as the subgroup of scalar matrices which are diagonal matrices with a $N$-th root of the unity ($e^{2i\pi k/N}$) on the diagonal (exercise: check that these matrices are in $SU(N)$). These matrices are in the center of $SU(N)$ so they act trivially in the adjoint representation (where gluons are living). But they act non-trivially in the fundamental representation (where quarks are living).

answered Feb 26 by 40227 (5,140 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$\hbar$y$\varnothing$icsOverflow
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
...