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First, maybe some quick exposition would be helpful.

In the original work on 3d mirror symmetry, by Intriligator and Seiberg, they first define a $3d$ $\mathcal{N}=4$ gauge theory, whose moduli space flows to two branches in the infra-red. Note that the global symmetry here is $SU(2)_N\times SU(2)_R$.

On one side, the Higgs mechanism completely breaks the gauge symmetry to give us the Higgs branch. Here, only one hypermultiplet is left massless, which corresponds to Goldstone particles. Furthermore, the Fayet-Iliopolous (FI) parameters here transforms trivially under the abovementioned global symmetry, and they parameterise the Higgs branch -- in other words, FI parameters control the geometry of the Higgs branch.

On the other side, the Higgs mechanism only breaks the gauge symmetry to its maximal torus, which leads to the more well-studied Coulomb branch. Here, mass parameters control the geometry of the Coulomb branch.

In summary, mirror symmetry exchanges the roles of FI and mass terms, and also the Higgs and Coulomb branches. These moduli spaces are also necessarily hyper-Kahler.

I really have two questions.

What happens to the moduli space when supersymmetry is broken? There has been some work done on this aspect, with folks like Kachru, Karch and Tong, who have found non-supersymmetric analogues of mirror symmetry. From what I remember, there was no mention of how the moduli spaces were altered.

The second, perhaps a far more fundamental (and stupid) one, is this: What kind of gauge theories lie on the Higgs and Coulomb branches? Are these sigma-models, like their 2d analogues? Or are they perhaps Yang-Mills theories (or SQED as with Borokhov and Kapustin's work)?

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