All class \( \mathcal{S} \) theories of type-A originate from orbi-instantons

Abstract

A surprising relation between 4d \( \mathcal{N} \) = 2 class \( \mathcal{S} \) superconformal field theories of Type-A and 6d \( \mathcal{N} \) = (1, 0) orbi-instanton theories is investigated. We find that all of the theories in the former class can be obtained by a series of deformations of the 4d theories arising from compactifying the latter on a torus. This is demonstrated by examining Fayet-Iliopoulos (FI) deformations of the E₈-shaped magnetic quivers of the orbi-instanton theories whose body fits into the affine E₈ Dynkin diagram with a tail attached. Turning on FI parameters at the appropriate gauge groups leads, in stages, to E₇-shaped, E₆-shaped, and general star-shaped quivers, where the latter are magnetic quivers for the class \( \mathcal{S} \) theory of Type-A on a sphere with punctures. Deforming a suitable star-shaped quiver, one obtains a magnetic quiver of the Type-A class \( \mathcal{S} \) theory with general genus and an arbitrary number of punctures. Given such a theory, we also propose the inverse algorithm, thereby determining a parent orbi-instanton theory. This is achieved by uplifting the corresponding magnetic quiver step by step to the star-shaped, E₆-shaped, E₇-shaped, and E₈-shaped quivers, where at each step all of the underbalanced nodes, possessing non-zero FI parameters, are dualized. The latter E₈-shaped quiver then characterizes the 6d orbi-instanton theory from which the class \( \mathcal{S} \) theory in question originates.