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

Question

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.

Comments

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

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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

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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

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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