For a c=1 Boson on a circle at the self-dual rdius, we get an enhanced gauge symmetry ^SU(2)1. It is said that we can orbifold this model by any finite subgroup of SU(2) since SU(2) is a symmetry of the model. But the Lagrangian of a c=1 Boson does not have an SU(2) symmetry even at the self-dual radius which is
L=√2∫d2z ∂ϕˉ∂ϕ
right?
I know that at the self-dual radius we get extra marginal operators which close among each other to form the OPE of ^SU(2)1 but is there another form of the above Lagrangian which makes the SU(2) symmetry manifest?
Is the c=1 Boson at the self-dual radius equivalent to a level 1 WZW model? Thanks.
Answer
Yes, a c=1 boson at the self-dual radius is exactly equivalent to the SU(2) WZW model at k=1. The latter makes the SU(2) symmetry manifest. When one gets used to this and similar equivalences and to the extra marginal momentum/winding operators in the boson description, the SU(2) symmetry becomes "manifest" in both formalisms. There is always some degree of psychology or subjective judgement in what is "manifest".
If you want to be sure about all the c=1 CFTs, pages 261-262 of Joe Polchinski's book, volume I, may be helpful. They approximately look like this:
Special message for Joe: if you wonder why I use an electronic version of the book, it's because Nima Arkani-Hamed borrowed and lost my paper edition. Well, it was actually Volume II that disappeared and I still own this Volume I, but let's ignore those details. ;-)
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