# About deriving the multi-trace index in terms of the single-trace index

This question is in reference to this paper

Combining their equations 5.2, 5.3, 5.6 and 5.7 one seems to be looking at the integral/partition function,

$Z(x) = \prod_{n=1}^{n =\infty}\left [ \int d\rho_n e^{-N^2\frac{\rho_n^2}{n}(1 - I_L(x,n))} \right ]$

where $I_L(x,n) = z_B(x^n) + (-1)^{n+1}z_F(x^n)$

("x" is the fugacity, $z_B$ and $z_F$ are single particle Bosonic and Fermionic partition functions respectively and $\rho_n$ are Fourier components of the eigenvalue density on the circle of the gauge group $U(N \rightarrow \infty)$)

-I would like to know as to how in the large N limit does this become their claimed equation 5.10,

$Z(x) = \prod_{n=1}^{\infty} \frac{1}{1 - I_L(x,n)}$

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At least provide a link if you're going to ask people to read a paper for you... – user1504 May 14 '13 at 0:41
Its this paper jp.arxiv.org/abs/hep-th/0310285 – user6818 May 14 '13 at 0:51
But I guess the issue is of some mathematical trick here and there probably isn't any physics here - the paper just brushes over this calculation in a line or so. – user6818 May 14 '13 at 0:52
First of all, they renormalize the partition function en route (just below Eq. 5.10), so that takes care of any (divergent) prefactors $\sim \prod n \pi /N$. Are you sure that $\rho_n$ is a single real variable? If it's a complex variable (seems more logical) then you're out of the woods. – Vibert May 14 '13 at 21:50
Also, please refrain from saying things like "this clearly doesn't give the right answer" if you don't understand what is going on. Other people are trying to help you in their spare time by looking at 90-page papers, so when you don't understand their comments it's nicer to ask about any misunderstanding than to criticise their suggestions straight off the bat. – Vibert May 14 '13 at 21:59