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

A bstract A surprising relation between 4d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 class $$ \mathcal{S} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> superconformal field theories of Type- A and 6d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = (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 $$ \mathcal{S} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> 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 $$ \mathcal{S} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> 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 $$ \mathcal{S} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> theory in question originates.