Laughlin wave function and CFT

Question

I have a question regarding Eq. (3.5) in Moore & Read's paper. They said \begin{equation} \Psi_{\text{Laughlin}}=\left\langle\prod_{i=1}^{N}e^{i\sqrt{q}\phi(z_i)}\exp\left[-i\int \mathrm d^2z^{\prime}\sqrt{q}\rho_0\phi(z^{\prime})\right]\right\rangle \end{equation} is equivalent to Laughlin wave function described below \begin{equation} \Psi_{\text{Laughlin}}(z_1,\cdots,z_N)=\prod_{i<j}(z_i-z_j)^q\exp\left[-\frac{1}{4}\sum|z_i|^2\right] \end{equation} $\phi(z)$ is a free massless scalar filed satisfying $\langle\phi(z)\phi(w)\rangle=-\log(z-w)$. My question is how do I get the second equation from the first one. Especially, I don't know how do I obtain nonholomorphic part, that is, $\exp\left[-\frac{1}{4}\sum|z_i|^2\right]$ starting from the first equation. I don't understand at all the explanation below Eq. (3.7) in their paper.

Answer 1

Perhaps you could have a look at David Tong's lectures on QHE (http://www.damtp.cam.ac.uk/user/tong/qhe.html) you might find Section 6.2 helpful. The paper in question is available here (http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf)