A freely generated ring for N = 1 models in class S_k

We study 4d N = 1 supersymmetric theories of class S_k, obtained from flux compactifications on a Riemann surface of 6d (1, 0) conformal theories describing the low energy physics on a stack of M5 branes probing a @@ℤk singularity. We conjecture that the protected spectrum of class S_k theories contains a freely generated ring, generalizing the Coulomb branch of the N = 2 theories. We derive this by examining a limit of the supersymmetric index of 4d N = 1 class S_k theories. The limit generalizes the Coulomb limit of N = 2 theories, which coincides with the case of k = 1 for a particular choice of flux. We conjecture a general simple formula for the index in the aforementioned limit.