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

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!

• 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? – Meng Cheng Nov 21 '15 at 18:53
• Hi @MengCheng, thanks very much for your comments! For the reference, I was simply reading page 4 of this paper: arxiv.org/abs/hep-th/0105038. Let me double check if it is about terminology or not later. – IsingX Nov 22 '15 at 16:28
• A more conceptual reason may be the following: the level of the edge CFT is tied to the level of the bulk Chern-Simons theory. In this case, if $m$ is odd, then the bulk Chern-Simons theory is a spin TQFT, namely it requires that the system is made of physical fermions (not emergent fermionic quasiparticles). At the level of CFT, you can see that there is a primary $e^{im\phi}$ that has trivial OPE with all other primaries, but still has a nontrivial monodromy. I can see this is a reason that people use $U(1)_m$ to actually refer to level "$2m$". – Meng Cheng Nov 22 '15 at 16:59
• Hi @MengCheng, thanks for the comments and sorry for the late feedback. I checked some literature on fractional quantum hall states, and it seems this is indeed a terminology issue. As an example, in the paper by Moore and Read: physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf, or in the recent paper by Kareljan Schoutens and Xiao-Gang Wen: arxiv.org/abs/1508.01111, which I think should be reliable, they use U(1)_{m/2} – IsingX Dec 3 '15 at 18:10
• they use $U(1)_{m/2}$ current algebra to study the fermionic $1/m$ Laughlin state, where $m$ is odd. In $U(1)_{m/2}$ current algebra, there are in total $\frac{m}{2}\times 2=m$ primary fields, corresponding to the $m$ quasiparticles/anyons. On the other hand, In Eduardo Fradkin's textbook, he uses $U(1)_m$ for this. – IsingX Dec 3 '15 at 18:23