All Class $\mathcal{S}$ Theories of Type-$A$ Originate from Orbi-instantons

TL;DR

This paper reveals that all Type-A class S 4d $ ext{N}=2$ superconformal theories can be derived from 6d $ ext{N}=(1,0)$ orbi-instanton theories through a series of deformations involving magnetic quivers and FI parameters.

Contribution

It establishes a systematic method to obtain all Type-A class S theories from 6d orbi-instanton theories using magnetic quivers and FI deformations, and proposes an inverse algorithm for theory reconstruction.

Findings

All class $ ext{S}$ theories of Type-$A$ originate from 6d orbi-instanton theories.

Deformations of magnetic quivers connect 6d theories to 4d class $ ext{S}$ theories.

An inverse algorithm reconstructs parent 6d theories from 4d magnetic quivers.

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 $E_8$-shaped magnetic quivers of the orbi-instanton theories whose body fits into the affine $E_8$ Dynkin diagram with a tail attached. Turning on FI parameters at the appropriate gauge groups leads, in stages, to $E_7$-shaped, $E_6$-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, $E_6$-shaped, $E_7$-shaped, and $E_8$-shaped quivers, where at each step all of the underbalanced nodes, possessing non-zero FI parameters, are dualized. The latter $E_8$-shaped quiver then characterizes the 6d orbi-instanton theory from which the class $\mathcal{S}$ theory in question originates.