Thanks, I am wondering how to modify the normal ordering definition. Because when we work in orbifold CFT, we will come across non-integer cases. For example, (3.4) and (3.9) in this paper inspirehep.net/literature/558659.

It seems that $h_A,h_B$ are assumed to be integers. How about when this is not the case?

How to distinguish this symmetry with electric-magnetic U(1) symmetry? It seems that these two symmetry has the same U(1) transformation on quark field.

Yes, you are right! But this is in contracdict with $\pi_0|0\rangle=0=\langle 0|\pi_0$ which seems also be right.

Sorry, I made a typo in my last comments. Indeed, $a_0$ is identified with $\pi_0$, then the term like this $\langle 0|\phi_0 \pi_0|0\rangle=0= \langle 0| \pi_0\phi_0|0\rangle$ will not contribute, since $\pi_0|0\rangle=0=\langle0|\pi_0$.

Thanks for you comments. I think phi0 is just a0, the latter acting on $|0\rangle$ gives 0, so phi0 will not contribute.

@Heidar Very nice answer! Could you give more details on how the path integral measures are equal to each other?