Is there any discussion of topology of space of matrices $ ABA^{-1}B^{-1} = E $ with $E$ diagonal matrix, $A,B \in \mathrm{GL}(n,\mathbb{C})$?

E.g. is this a variety of just a scheme? How many components? I'm trying to understand this space in relation to integrable systems

I know it's related to flat connections on the once-punctured torus.

**REMARK**: The question is really this abstract. I am looking at representations of the fundamental group by $n \times n$ matrices: $\pi_1(\mathbb{T}^2 - \{\cdot\})\to\operatorname{GL}(n, \mathbb{C})$. This is a system of $n^2$ polynomial equations in $2n^2$ unknowns. There should be some relation to Fenchel-Nielsen coordinates for the once-puctured torus.