This is the ABJM partition function on the 3-sphere,

$$ Z(2) = \int \frac{d^2\mu}{(2\pi)^2} \frac{d^2\nu}{(2\pi)^2} \frac{\left[ 2 \sinh \frac{\mu_1 - \mu_2}{2}\right]^2\left[ 2 \sinh \frac{\mu_1 - \mu_2}{2}\right]^2}{\left[ 2 \cosh \frac{\mu_1 - \nu_1}{2}\right]^2 \left[ 2 \cosh \frac{\mu_2 - \nu_2}{2}\right]^2} \exp \left[ \frac{ik}{4\pi} \left( (\mu_1^2 + \mu_2^2) - (\nu_1^2 + \nu_2^2) \right) \right]$$

The strategies I've seen in the paper involve merging all the possible $Z(N)$ into the grand canonial partition fuction, as in [1, 2] but I'd prefer not to do that. Also they only solve $k = 1$ and possibly $k = 2$, for $N \leq 9$.

The physics paper has written these without domains of integration. Our choices are $[0, 2\pi]^2 \times [0, 2\pi]^2$ and $\mathbb{R}^2 \times \mathbb{R}^2$.

I'm sure someone sufficiently determined could find the answer. Both sources agree that ($k=1$):

$$ Z(2) = \frac{1}{16\pi} $$

For the Gaussian we have a choice of derivations what connect this measure to probability and Lie grou theory. At this moment I'd like to have more understanding of that the $\sinh$ and $\cosh$ term come from and a more conceptual derivation of what is offerend in the paper.

I'd imagine $k > 2$ is still open...