# Laughlin wave function and CFT

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 homorphic 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.

• Especially considering the paper you link is not freely available, please include all relevant context into the question. What does the notation means, what are you trying to compute, and what specifically about the given explanation is unclear to you? – ACuriousMind Apr 7 '16 at 10:44