I am trying to learn supersymmetric localisation computation in which the path integral of a supersymmetric gauge theory (placed on a sphere) can be localised exactly to a BPS configuration parameterised by single parameter. For the case of superconformal CS theory (hep-th/0909.4559),

$Z = \int d\sigma_0 \exp\left[-4i\pi^2 Tr(\sigma^2_0)\right]\,Z^g_\text{1-loop}[\sigma_0]$

where $\sigma_0$ parameterises the moduli space and takes value in the adjoint representation of the Lie group $G$.

Restricting the value of the constant $\sigma_0$ to the Cartan subalgebra, Eq.(3.20) gives the result of the partition function

$Z = \frac{1}{|\mathcal W|}\int da\left(\prod_\alpha \alpha(a)\right) \exp\left[-4i\pi^2 Tr(a^2)\frac{}{}\right]\,Z^g_\text{1-loop}[a]$

where $\alpha$ runs over the roots and $a$ labels the Cartan sub algebra. $|\mathcal W|$ is the Weyl group of $G$. There is one sentence stating that "we can replace the integral over the entire Lie algebra with an integral over some chosen Cartan subalgebra. This introduces a Vandermonde determinant in the measure. "

I found that introductory notes on localisation computation often just mentions the Vandermonde determinant without going into details about what it is or how to compute it explicitly. My question is, what is a Vandermonde determinant and how does it arise in the measure? Any help whatsoever is appreciated. Thanks !

Also, is it always true that one should divide out the Weyl group of $G$?


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