Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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.

share|improve this question
I think this is in Polchinsky, I'll check. –  Ron Maimon Aug 5 '12 at 19:56
Aah ok thanks @RonMaimon for checking. I dont have that in my bookshelf ... :-/ –  Dilaton Aug 5 '12 at 20:49
Yes, Polchinski section 8.3 (volume 1). –  Guy Gur-Ari Aug 6 '12 at 1:09
Dear downvoter, what is wrong with this question ...? –  Dilaton Aug 6 '12 at 11:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.