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 ...

  Is the U(1) gauge theory in 2+1D dual to a U(1) or an integer XY model?

+ 6 like - 0 dislike
586 views

The compact U(1) lattice gauge theory is described by the action $$S_0=-\frac{1}{g^2}\sum_\square \cos\left(\sum_{l\in\partial \square}A_l\right),$$ where the gauge connection $A_l\in$U(1) is defined on the link $l$. I was told that this theory in 2+1D spacetime is dual to a U(1) XY model on the dual lattice, described by the following action $$S_1=-\chi\sum_l \cos\left(\sum_{i\in\partial l}\theta_i\right)-K\sum_i\cos(\theta_i),$$ where the XY variable $\theta_i\in$U(1) is defined on the dual site $i$. It was said that the K term in the action is to take into account the instanton effect in the compact U(1) lattice gauge theory (which I don't understand). However when I tried to derive the the dual theory, I arrived at the following integer XY model (or height model?) $$S_2=-\chi\sum_l \cos\left(\sum_{i\in\partial l}m_i\right),$$ with the integer variable $m_i\in\mathbb{Z}$ defined on the dual site $i$. Because the Pontryagin dual group of U(1) is simply $\mathbb{Z}$ but not U(1), so I believe that the U(1) gauge theory $S_0$ should dual to an integer XY model $S_2$, and this duality is exact. But every book or paper that I have encountered did not mention anything about $S_2$, instead they all point to the U(1) XY model $S_1$. Therefore I was forced to conjecture that the integer XY model is equivalent to the U(1) XY model with additional K term. Can anyone tell me if my conjecture is correct or not? How to go from $S_2$ to $S_1$ (or maybe directly from $S_0$ to $S_1$)? How is the K term being added? What is the physical meaning of the K term?

This post imported from StackExchange Physics at 2014-04-05 17:26 (UCT), posted by SE-user Everett You
asked Jun 17, 2012 in Theoretical Physics by Everett You (785 points) [ no revision ]
Very interesting question. Could you briefly comment on how you derived $S_2$ from $S_0$, and how $\chi$ and $m_i$ are related to $g$ and $A_l$?

This post imported from StackExchange Physics at 2014-04-05 17:26 (UCT), posted by SE-user Heidar

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$\hbar$ysicsOverfl$\varnothing$w
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
...