We propose a generalization of S-folds to 4d N = 2 theories. This construction is motivated by the classification of rank one 4d N = 2 super-conformal field theories (SCFTs), which we reproduce from D3-branes probing a configuration of N = 2 S-folds combined with 7-branes. The main advantage of this point of view is that realizes both Coulomb and Higgs branch flows and allows for a straight forward generalization to higher rank theories.

A large family of 4d N = 2 SCFT's was introduced in 1210.2886. Its elements D p (G) are labelled by a positive integer p ∈ N and a simply-laced Lie group G; their flavor symmetry is at least G. In the present paper we study their physics in detail. We also analyze the properties of the theories obtained by gauging the diagonal symmetry of a collection of D p i (G) models. In all cases the computation of the physical quantities reduces to simple Lie-theoretical questions.To make the analysis more functorial, we replace the notion of the BPS-quiver of the N = 2 QFT by the more intrinsic concept of its META-quiver.In particular: 1) We compute the SCFT central charges a, c, k, and flavor group F for all D p (G) models. 2) We identify the subclass of D p (G) theories which correspond to previously known SCFT's (linear SU and SO-U Sp quiver theories, Argyres-Douglas models, superconformal gaugings of Minahan-Nemeshanski E r models, etc.), as well as to non-trivial IR fixed points of known theories. The D p (E r ) SCFT's with p ≥ 3 cannot be constructed by any traditional method. 3) We investigate the finite BPS chambers of some of the models. 4) As a by product, we prove three conjectures by Xie and Zhao, and provide new checks of the Argyres-Seiberg duality.

Canonical threefold singularities in M-theory and Type IIB string theory give rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. In this paper, we study canonical hypersurface singularities whose resolutions contain residual terminal singularities and/or 3-cycles. We focus on a certain class of ‘trinion’ singularities which exhibit these properties. In Type IIB, they give rise to 4d $$ \mathcal{N} $$ N = 2 SCFTs that we call $$ {D}_p^b $$ D p b (G)-trinions, which are marginal gaugings of three SCFTs with G flavor symmetry. In order to understand the 5d physics of these trinion singularities in M-theory, we reduce these 4d and 5d SCFTs to 3d $$ \mathcal{N} $$ N = 4 theories, thus determining the electric and magnetic quivers (or, more generally, quiverines). In M-theory, residual terminal singularities give rise to free sectors of massless hypermultiplets, which often are discretely gauged. These free sectors appear as ‘ugly’ components of the magnetic quiver of the 5d SCFT. The 3-cycles in the crepant resolution also give rise to free hypermultiplets, but their physics is more subtle, and their presence renders the magnetic quiver ‘bad’. We propose a way to redeem the badness of these quivers using a class $$ \mathcal{S} $$ S realization. We also discover new S-dualities between different $$ {D}_p^b $$ D p b (G)-trinions. For instance, a certain E8 gauging of the E8 Minahan-Nemeschansky theory is S-dual to an E8-shaped Lagrangian quiver SCFT.

We propose a general way to complete supersymmetric theories with operators below the unitarity bound, adding gauge-singlet fields which enforce the decoupling of such operators. This makes it possible to perform all usual computations, and to compactify on a circle. We concentrate on a duality between an N = 1 SU (2) gauge theory and the N = 2 A3 Argyres-Douglas [1,2], mapping the moduli space and chiral ring of the completed N = 1 theory to those of the A3 model. We reduce the completed gauge theory to 3d, finding a 3d duality with N = 4 SQED with two flavors. The naive dimensional reduction is instead N = 2 SQED. Crucial is a concept of chiral ring stability, which modifies the superpotential and allows for a 3d emergent global symmetry.In gauge theories with four supercharges, many nonperturbative properties of the infrared strongly coupled fixed points are known, for instance the scaling dimensions of Sometimes a BPS operator violates the bound imposed by conformal invariance and unitarity, which in 4d (3d) is ∆ > 1 (∆ > 1 2 ). The standard lore is that the operator decouples and becomes free [6]: the infrared fixed point is described by some interacting superconformal theory (SCFT) plus a free chiral field. How to perform computations in such theories is however an open problem: it is known how to perform a/Z-extremizations [3-5] or compute supersymmetric indices/partition functions, but it is not known how to compute, for instance, the chiral ring or the moduli space of vacua.In this note we propose a prescription to re-formulate theories with decoupled operators: introduce a gaugesinglet chiral multiplet β O for each operator O violating the bound, and add the superpotential term β O O. Gauge singlet fields entering the superpotential in this way are usually said to "flip the operator O". The F-term of β O sets O = 0 in the chiral ring, there are no unitarity violations and all usual computations can be performed.This "completion" isolates the interacting sector and also allows to compactify dualities where at least one side has decoupled operators. Unitarity bounds change as we change the dimension of spacetime and what decouples in higher dimension may not decouple in lower dimension, so a compactification of dual theories without introducing the β O fields generically fails to produce a dual pair.We check the validity of our proposal focusing on a class of theories in four dimensions recently discovered in [1,2,7]: certain N = 1 gauge theories exhibit unitarity bound violations, the interacting sector is proposed to be equivalent to a well-known class of N = 2 SCFT's called Argyres-Douglas (AD) theories [8][9][10][11], which cannot have a manifestly N = 2 lagrangian description.We focus on a simple case, the A 3 AD theory, which admits an N = 1 lagrangian description in terms of an SU (2) gauge theory with an adjoint and two doublets [2].First we point out that the superpotential as written in [2,7] are inconsistent: a superpotential term must be discarded, in order to satisfy a chiral ring stability ...