Large-N of non-linear sigma model

Question

Consider the partition function of non-linear sigma model is: $$Z=\int\mathcal{D}[\textbf n]\mathcal{D}[\lambda]\exp(-S)\\S=\frac{1}{f} \int d^dx \int d \tau [(\partial_\tau \textbf{n})^2+c^2(\nabla_x \textbf n)^2+i \lambda (\textbf n^2-1)]$$

I have two questions when I want to generalize it to large $N$:

1. I know that the coefficient $f$ need to be scaling as $1/N$, thus, the total pre-factor $1/f$ is $N/f_0$, where $f_0$ is fixed. But, I cannot understand the motivation and reason of it because it seems like we just put it for our goal, i.e. the whole pre-factor $N$.

2. After integral out the $\textbf{n}$ field, we can obtain one term of the effective action: $$\frac{N}{2} Tr \ln(-\partial_\tau^2-c^2\nabla_x+i\lambda)$$

I can understand that the $N$ comes from the $N$ independent components, but why $\frac{1}{2}$? Also, why does the coupling $f$ not enter in the logarithm？

Answer 1

The half comes from the fract that the gaussian integral over the bosonic fields gives you the reciprocal of the square root of the Fredholm determinant: In $A$ is an $n$-by-$n$ matrix
$$ \int d^nx\, e^{ -x^TAx}= \pi^{n/2} [{\rm det}(A)]^{-1/2}. $$