We are in $\mathcal{N}=1$ SUSY. Consider massless SQCD with gauge group $SU(N)$ and $F$ flavours. The quarks superfields $Q$ and $\tilde{Q}$ are $F\times N$ and $N\times F$ matrices respectively and the superpotential is $W=0$. The moduli space is therefore given by the D-flatness condition only, which you can write as $$ \mathrm{Tr}\Big[t^A \Big(Q^\dagger Q-\tilde{Q}\tilde{Q}^\dagger\Big)\Big]=0 \quad \forall A=1,2,\dots,N^2-1 \ , $$ where $t^A$ for $A=1,2,\dots, N^2-1$ are the $SU(N)$ generators in the fundamental representation. Suppose that $F<N$, then it is possible to show, using gauge and flavour transformations (see for example [1]), that on the moduli space the matrices $Q$ and $\tilde{Q}$ can be put, at most, in the following form $$ Q=\begin{pmatrix} v_1& & &0 &\cdots&0\\ &\ddots& &\vdots&\ddots&\vdots\\ & &v_F&0 &\cdots& 0 \end{pmatrix}=\tilde{Q}^T \ . $$ This means that at a generic point of the moduli space the gauge group is broken to $SU(N-F)$. Now, in section 5.3.1 (specifically pag. 99-100) of [2], it is claimed that, given the previous result, the complex dimension of the classical moduli space is $$ \dim_{\mathbb{C}}\mathcal{M}_{cl}=2FN-\{N^2-1-[(F-N)^2-1]\}=F^2 \ . $$
I can't understand why it is so. I see that $\{N^2-1-[(F-N)^2-1]\}$ is the number of broken $SU(N)$ generators, but why the dimension of the moduli space should be exactly the number of entries of $Q$ and $\tilde{Q}$ matrices minus the number of broken gauge symmetry generators?
In another reference, [3], it is said that
[...] a supersymmetric gauge theory with gauge group $G$ is invariant under the complexified gauge group $G^c$. From this point of view, the usual D-flatness conditions can be viewed as a $G^c$ gauge artifact. By using a gauge in which $G^c$ invariance is preserved, we show that in the absence of a superpotential every constant value of the matter fields is $G^c$ gauge-equivalent (in an extended sense that we make precise) to a solution of the D-flatness conditions. This gives the result that the space of classical vacua is $$ \mathcal{M}_{cl} = \mathcal{F}//G^c $$ where $\mathcal{F}$ is the space of all constant matter field configurations and the quotient denoted by $//$ identifies any $G^c$ orbits that have common limit points.
Now, I am confused by the meaning of the quotient $//$; it can't be the usual quotient between a manifold and a Lie group, because in that case, I think, the dimension of $\mathcal{M}_{cl}$ should be $$ \dim_{\mathbb{C}}\mathcal{M}_{cl}=\dim_{\mathbb{C}}\mathcal{F}-\dim_{\mathbb{C}}G^c \ , $$ which in our case should read $$ \dim_{\mathbb{C}}\mathcal{M}_{cl}=2NF-\dim_{\mathbb{C}} SU(N)^c=2NF-N^2+1 \ , $$ which doesn't give the expected result. I haven't fully studied [3] because (1) it's above my actual knowledge to fully understand it and (2) it's time-consuming and I can't afford it right now. Anyway, I took a look at it but I didn't clarify anything, in fact I'm more confused.
Confusion is the best word to describe the situation I'm in now; I already spent a lot of time trying to understand the problem but I haven't made any progress. At this point, I don't want a complete answer, but at least an intuition that helps me justify the result.
[1] Terning, John. Modern Supersymmetry: Dynamics and Duality. Chapter 3.4, Oxford: Oxford University Press, 2005. Oxford Scholarship Online, 2007. doi:10.1093/acprof:oso/9780198567639.001.0001.
[2] M. Bertolini, Lectures on Supersymmetry, 2022, https://people.sissa.it/~bertmat/susycourse.pdf
[3] M.A. Luty and W. Taylor, Varieties of vacua in classical supersymmetric gauge theories, Phys. Rev. D 53 (1996) 3399 [hep-th/9506098]