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The chiral ring of the Coulomb branch of a 4D ${\cal N}=2$ supersymmetric gauge theory is given by the Casimirs of the vector multiplet scalars, and they don't have non-trivial relations; the Casimirs are always independent.

Also in Gaiotto's class of ${\cal N}=2$ non-Lagrangian theories, the chiral ring of the Coulomb branch doesn't (seem to) have relations.

Is it a general fact? If so, how can we deduce it from the $N=2$ supersymmetric algebras?


I was asked to clarify the definition of the Coulomb branch in non-Lagrangian theories; let's define them for $N=2$ SCFT by the fact that $SU(2)_R$ symmetry acts on the Coulomb branch operators trivially.

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As far as I remember it goes something like this:

If ($g$) is a finite dimensional and a semi-simple Lie algebra with a "rank" ($n$) then the center $[Z(U(g))]$ should be isomorphic to the polynomial algebra ($K[C^i]$) over the base field of ($n$) variables ($C^i$) where ($i=1, 2, 3, ... , n$). The number of algebraically independent Casimirs is equal to the rank.

So if your question is why are Casimirs independent, that's the best answer I can think of.

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