Two 6d origins of 4d SCFTs: class S and 6d (1,0) on a torus

Abstract

We consider all 4d N = 2 theories of class S arising from the compactification of exceptional 6d (2, 0) SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4d theories has another origin as a 6d (1, 0) SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6d theory is identified with a complex structure deformation of (e_n, e_n) minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class S construction. We explain how the plurality of origins makes manifest some aspects of 4d SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4d theories from different origins via mirror symmetry.