# Relation between the momentum map and D-terms

Can someone explain to me the relation between the momentum map linked to symplectic quotients and the D-terms of a scalar potential for a $\mathcal{N}\geq 2$ supersymmetric gauge theory? I am interested in the definition of the momentum map when we talk about moduli space of vacua in supersymmetric theory, why and how they can be related to the D-terms equations for the minima.

Any reference and good review of these topics are really welcomed.

I give to you my definition of D-terms equation and moment map. Also in my other question

Why is important to know the geometry of the moduli space of vacua in a SUSY gauge theory?

I pointed out the Kahler geometry of the moduli space of vacua.

In general D-terms in a superspace integral are all those integrals that can be converted to an integral in full superspace. For $\mathcal{N}=1$ they are of the form $D(x)\theta\theta\bar\theta\bar\theta$

They become important when we consider the vector multiplet since in a SuperYM they gives another condition of minimum on the superpotential. There is also another my question

Can someone help me to expand the superpotential for $\mathcal{N}=2$ supersymmetry?

in which I write explicitly the scalar potential to minimize for a $\mathcal{N}=2$ supersymmetry and there is clear the distinction between F-terms and D-terms equation.

For the moment map, actually, I only know what Wikipedia

https://en.wikipedia.org/wiki/Moment_map

says about it. I know from literature that the moduli space of vacua can be seen as the space of the F-term equation and D-term equation modulo gauge transformation. However it is possible to complexify the gauge group with the D-terms and we can obtain the moduli space formed by the F-term equations modulo a complexified gauge group. Since the moduli space is a complex manifold and there is a quotient over a Lie group I think that the relation with the moment map is that concerning symplectic quotients.

However my question wants exactly to understand everything I asked on StackExchange and I'm fragmenting my doubts in many question hoping that some good man can give to me an exhaustive answer.