$\mathcal{N}=1$ Curves on Generalized Coulomb Branches

We study the low energy effective dynamics of four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories of class $\mathcal{S}_k$ on the generalized Coulomb branch. The low energy effective gauge couplings are naturally encoded in algebraic curves $\mathcal{X}$, which we derive for general values of the couplings and mass deformations. We then recast these IR curves $\mathcal{X}$ to the UV or M-theory form $\mathcal{C}$: the punctured Riemann surfaces on which the six-dimensional $\mathcal{N}=(1,0)$ ${A_{k-1}}$ SCFTs are compactified giving the class $\mathcal{S}_k$ theories. We find that the UV curves $\mathcal{C}$ and their corresponding meromorphic differentials take the same form as those for their mother four-dimensional $\mathcal{N}=2$ theories of class $\mathcal{S}$. They have the same poles, and their residues are functions of all the exactly marginal couplings and the bare mass parameters which we can compute exactly.