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]


1 Answer 1


Alright so first of all the quotient $//$ is something known as the symplectic quotient if I am not mistaken. I am however not very well acquainted with using such a quotient so I won't try to here but the name might give a pointer on what to search for.

As for why the dimension of the moduli space should be given by the number of entries in $Q, \tilde{Q}$ minus the number of broken generators:

First of all, why would we want to look at $Q, \tilde{Q}$ in general? We can interpret these components as coordinates of our moduli space. Now, however, these coordinates are restricted by the condition you mentioned above. This leads to the fact that on any generic point of the moduli space the gauge group is broken as you mentioned as well. Now each of those broken generators is eaten by one of the quark fields through the Higgs mechanism yielding a heavy field which does not contribute to the moduli space by definition.

Another way to find this dimension is by looking at the moduli space as an algebraic variety which I believe to be the more intuitive approach here. Here we look at the set of all gauge invariant monomials we can build from $Q, \tilde{Q}$ conforming to out symmetry. This leads to $M_j^i = \tilde{Q}^a_j Q^i_a$ which we call meson fields. Now we can just count the number of these fields (which here are all independent) giving us $F^2$ as a dimension. Now why is this the same? Well generally the problem is that even given our constraints the system still has a gauge freedom which should not lead to physically distinct systems. Before we removed this by actually looking at the gauge group an taking the corresponding quotient. Here however this is already included in the definition of $M$. Both a slightly different version of the above discussion as well as the algebraic variety approach can be found in David Tong's notes on Supersymmetry which goes into more detail than I have the time to here so I can only recommend reading the section on exactly this problem (its only a few pages and really well written).


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