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


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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.