1/m Laughlin state and $U(1)_M$ chiral CFT

Question

I am a little confused that people claim that the edge theory of a 1/m Laughlin state corresponds to a $U(1)_m$ chiral CFT. This indicates there should be $m$ primary field operators in $U(1)_m$ chiral CFT. However, there are actually $2m$ primary field operators in $U(1)_m$ chiral CFT.

What is wrong here? Or did I make some naive mistakes above? Thanks!

This post imported from StackExchange Physics at 2015-11-21 21:50 (UTC), posted by SE-user IsingX

Comments

It is probably just an issue of terminology. There are $m$ primaries in the edge CFT of $1/m$ Laughlin, that's for sure, and the Lagrangian for the edge theory is simply $\mathcal{L}=\frac{m}{4\pi}\partial_t\phi\partial_x\phi - \dots$ with $\phi$ a compactified boson. And this is usually called $U(1)_m$. Maybe you can give a reference where "$U(1)_m$" CFT has $2m$ primaries?

This post imported from StackExchange Physics at 2015-11-21 21:50 (UTC), posted by SE-user Meng Cheng