In a physics paper I found a very complicated Gaussian matrix model:
$$ Z = \int \frac{d\mu}{(2\pi)^n} \frac{d\nu}{(2\pi)^n} \frac{ \prod_{i < j}\left[2 \sinh \frac{\mu_i - \mu_j}{2} \right]^2 \prod_{i < j}\left[2 \sinh \frac{\nu_i - \nu_j}{2} \right]^2 }{\prod_{i,j}\left[ 2 \cosh \frac{\mu_i - \nu_j}{2} \right]^2} e^{\frac{ik}{4\pi}\sum (\mu_i^2 - \nu_i^2)} $$
Such a formula looks vaguely like the formula for Haar measure on the grou of Unitary matrices - whose exact formula I forget. Let's derive several simplified versions of this measure:
$$ Z = \int \frac{d\mu}{(2\pi)^n} \frac{d\nu}{(2\pi)^n} e^{\frac{ik}{4\pi}\sum (\mu_i^2 - \nu_i^2)} $$ One question is how to meaningfully ascribe a value to a Gaussian integral where the variance is a complex number. And actually $\mu^2 - \nu^2$ could be negative... so this could be like integrating over the maximual torus of a super-matrix model.
$$ Z = \int \frac{d\mu}{(2\pi)^n} \frac{d\nu}{(2\pi)^n} \frac{ \prod_{i < j}\left[\mu_i - \mu_j \right]^2 \prod_{i < j}\left[\nu_i - \nu_j \right]^2 }{\prod_{i,j}\left[\mu_i - \nu_j \right]^2} e^{\frac{ik}{4\pi}\sum (\mu_i^2 - \nu_i^2)} $$ This could be like Haar measure on the unitary group $SU(N|N)$ except for the Gaussian. One more variant is if we use $\sin$ instaed of $\sinh$:
$$ Z = \int \frac{d\mu}{(2\pi)^n} \frac{d\nu}{(2\pi)^n} \frac{ \prod_{i < j}\left[2 \sin \frac{\mu_i - \mu_j}{2} \right]^2 \prod_{i < j}\left[2 \sin \frac{\nu_i - \nu_j}{2} \right]^2 }{\prod_{i,j}\left[ 2 \cos \frac{\mu_i - \nu_j}{2} \right]^2} $$ This could be an integral over some matrix-like objects or over a Lie super algebra. Also there is something like the Cauchy-Binet formula.
What are the appropriate Lie Groups for these setups? In physics these go under various names like Lens Space model or Chern-Simons theory
