Solving the $CP^N$ model in large $N$ limit

Question

I have trouble filling in the essential step of solving the $CP^N$ model in large $N$ limit, described on Page 84 to 86 of Michael Dine's Supersymmetry and String Theory.

The Lagrangian is given by
$$ mathcal{L} = frac{1}{g^2} [|D_mu z_i|^2 -lambda(x)(|z_i|^2-1)] $$

Without explanation, Michael claimed that

the effective action after you integrate out $z_i$ is
$$ Gamma_{eff} = -N operatorname{tr} log (-D^2-lambda) - frac{1}{g^2}int d^2x lambda $$
What is $D$ here? The propagator?
After this, then you take the large $N$ limit, fixing $g^2N$. It is claimed that the path integral is dominated by a single field configuration, which solves
$$ frac{delta Gamma_{eff}}{delta lambda} =0. $$
Alternately, you can set the gauge field to zero, and get
$$ N int frac{d^2k}{(2pi)^2} frac{1}{k^2+lambda^2} = frac{1}{g^2}.$$
I'm totally confused here. How did Michael derive this integral equation of $lambda$?

Kangle · Answer

I have some understanding which may be not so accurate.

(1) Here the $D^2$ is just the inverse propagator, which represents the eigenvalue matrices $p^2$. The effective action $Gamma_{eff}$ is integrated as $z_i$ should be rescaled by factor $g$. Would that be any question about the integration process?

(2) Once you get the effective action, then you make variation about $lambda$ and set it to be zero:
begin{equation}
begin{split}
frac{delta Gamma_{eff}}{deltalambda(x)} &=-frac{1}{g^2}-Nfrac{delta(logdet(-D^2-lambda))}{deltalambda} 
&= -frac{1}{g^2}-Ntr((-D^2-lambda)^{-1}) 
&=-frac{1}{g^2}+Nintfrac{dk^2}{(2pi)^2}frac{1}{k^2+lambda}
end{split}
end{equation}
So set it equal to 0 and you get the result. Here the variation of logdet is from the $Gravitation and cosmology$ of Weinberg Page 107, but actually the sign before $N$ has some issue. I should concern it later. The last step follows from the propagator definition, but I am not sure it's accurate.

Solving the $CP^N$ model in large $N$ limit

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