The chiral ring of the Coulomb branch of a 4D $N=2$ 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 $N=2$ 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.

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.

The chiral ring of the Coulomb branch of a 4D $\mathcal N=2$ 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 $\mathcal N=2$ 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 $\mathcal N=2$ supersymmetric algebras?

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

The chiral ring of the Coulomb branch of a 4D $N=2$ 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 $N=2$ 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.

The chiral ring of the Coulomb branch of a 4d 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 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 susy 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.

The chiral ring of the Coulomb branch of a 4D $\mathcal N=2$ 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 $\mathcal N=2$ 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 $\mathcal N=2$ supersymmetric algebras?

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