• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius

+ 3 like - 0 dislike

Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained that the symmetry U(1)$_L$ x U(1)$_R$ is enhanced to SU(2)$_L$ x SU(2)$_R$ for a string theory at the self dual radius.

The text further explains that to complete the U(1) symmetries to SU(2), new generators J$^{\pm}$ of the SU(2) group with corresponding charge densities

$\rho^{\pm} = :exp(iX^5):$

are needed. To see that the old U(1) generators ($\partial_{\tau}X^5$ and $\partial_{\sigma}X^5$) together with the generators corresponding to the new charge densities combine to SU(2) one would do an operator product expansion (OPE) to retrieve the commutator of the SU(2) algebra.

I'd like to see some details (or at least how to get started) of this calculation.

asked Aug 5, 2012 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
I think this is in Polchinsky, I'll check.

This post imported from StackExchange Physics at 2014-03-17 03:20 (UCT), posted by SE-user Ron Maimon
Aah ok thanks @RonMaimon for checking. I dont have that in my bookshelf ... :-/

This post imported from StackExchange Physics at 2014-03-17 03:20 (UCT), posted by SE-user Dilaton
Yes, Polchinski section 8.3 (volume 1).

This post imported from StackExchange Physics at 2014-03-17 03:20 (UCT), posted by SE-user Guy Gur-Ari
Dear downvoter, what is wrong with this question ...?

This post imported from StackExchange Physics at 2014-03-17 03:20 (UCT), posted by SE-user Dilaton

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:
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