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Abstract The Coulomb phase of a quantum field theory, when present, illuminates the analysis of its line operators and one-form symmetries. For 4d N $$ \mathcal{N} $$ = 2 field theories the low energy physics of this phase is encoded in the special Kähler geometry of the moduli space of Coulomb vacua. We clarify how the information on the allowed line operator charges and one-form symmetries is encoded in the special Kähler structure. We point out the important difference between the lattice of charged states and the homology lattice of the abelian variety fibered over the moduli space, which, when principally polarized, is naturally identified with a choice of the lattice of mutually local line operators. This observation illuminates how the distinct S-duality orbits of global forms of N $$ \mathcal{N} $$ = 4 theories are encoded geometrically.

Abstract It is expected on general grounds that the moduli space of 4d N $$ \mathcal{N} $$ = 3 theories is of the form ℂ3r /Γ, with r the rank and Γ a crystallographic complex reflection group (CCRG). As in the case of Lie algebras, the space of CCRGs consists of several infinite families, together with some exceptionals. To date, no 4d N $$ \mathcal{N} $$ = 3 theory with moduli space labelled by an exceptional CCRG (excluding Weyl groups) has been identified. In this work we show that the 4d N $$ \mathcal{N} $$ = 3 theories proposed in [1], constructed via non-geometric quotients of type- e $$ \mathfrak{e} $$ 6d (2,0) theories, realize nearly all such exceptional moduli spaces. In addition, we introduce an extension of this construction to allow for twists and quotients by outer automorphism symmetries. This gives new examples of 4d N $$ \mathcal{N} $$ = 3 theories going beyond simple S-folds.

Abstract In this paper we begin mapping out the space of rank-2 N $$ \mathcal{N} $$ = 2 superconformal field theories (SCFTs) in four dimensions. This represents an ideal set of theories which can be potentially classified using purely quantum field-theoretic tools, thus providing a precious case study to probe the completeness of the current understanding of SCFTs, primarily derived from string theory constructions. Here, we collect and systematize a large amount of field theoretic data characterizing each theory. We also provide a detailed description of each case and determine the theories’ Coulomb, Higgs and Mixed branch stratification. The theories naturally organize themselves into series connected by RG flows but which have gaps suggesting that our current understanding is not complete.

Abstract There is a well-known map from 4d N $$ \mathcal{N} $$ = 2 superconformal field theories (SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds to the VOA vacuum character, and must be a solution with integral coefficients of a modular differential equation. This suggests a classification program for 4d N $$ \mathcal{N} $$ = 2 SCFTs that starts with modular differential equations and proceeds by imposing all known constraints that follow from the 4d → 2d map. This program becomes fully algorithmic once one specifies the order of the modular differential equation and the rank (complex dimension of the Coulomb branch) of the N $$ \mathcal{N} $$ = 2 theory. As a proof of concept, we apply the algorithm to the study of rank-two N $$ \mathcal{N} $$ = 2 SCFTs whose Schur indices satisfy a fourth-order untwisted modular differential equation. Scanning over a large number of putative cases, only 15 satisfy all of the constraints imposed by our algorithm, six of which correspond to known 4d SCFTs. More sophisticated constraints can be used to argue against the existence of the remaining nine cases. Altogether, this indicates that our knowledge of such rank-two SCFTs is surprisingly complete.

Abstract In this note we construct magnetic quivers for the known rank-2 four dimensional N $$ \mathcal{N} $$ = 2 superconformal field theories. For every rank-1 theory one can find a unitary magnetic quiver; we observe that this is no longer possible at rank 2. Our list of magnetic quivers necessarily includes orthosymplectic quivers, in addition to unitary ones, of both the simply and non simply laced variety. Using quiver subtraction, one can compute Higgs branch Hasse diagrams and compare with the results obtained via other methods finding nearly perfect agreement.

Abstract We study general properties of the mapping between 5d and 4d superconformal field theories (SCFTs) under both twisted circle compactification and tuning of local relevant deformation and CB moduli. After elucidating in generality when a 5d SCFT reduces to a 4d one, we identify nearly all N $$ \mathcal{N} $$ = 1 5d SCFT parents of rank-2 4d N $$ \mathcal{N} $$ = 2 SCFTs. We then use this result to map out the mass deformation trajectories among the rank-2 theories in 4d. This can be done by first understanding the mass deformations of the 5d N $$ \mathcal{N} $$ = 1 SCFTs and then map them to 4d. The former task can be easily achieved by exploiting the fact that the 5d parent theories can be obtained as the strong coupling limit of Lagrangian theories, and the latter by understanding the behavior under compactification. Finally we identify a set of general criteria that 4d moduli spaces of vacua have to satisfy when the corresponding SCFTs are related by mass deformations and check that all our RG-flows satisfy them. Many of the mass deformations we find are not visible from the corresponding complex integrable systems.

Abstract We carry out a systematic study of 4d N $$ \mathcal{N} $$ = 2 preserving S-folds of F-theory 7-branes and the worldvolume theories on D3-branes probing them. They consist of two infinite series of theories, which we denote following [1, 2] by S G , ℓ r $$ {\mathcal{S}}_{G,\mathrm{\ell}}^{(r)} $$ for ℓ = 2, 3, 4 and T G , ℓ r $$ {\mathcal{T}}_{G,\mathrm{\ell}}^{(r)} $$ for ℓ = 2, 3, 4, 5, 6. Their distinction lies in the discrete torsion carried by the S-fold and in the difference in the asymptotic holonomy of the gauge bundle on the 7-brane. We study various properties of these theories, using diverse field theoretical and string theoretical methods.

Abstract We derive explicit formulae to compute the a and c central charges of four dimensional N $$ \mathcal{N} $$ = 2 superconformal field theories (SCFTs) directly from Coulomb branch related quantities. The formulae apply at arbitrary rank. We also discover general properties of the low-energy limit behavior of the flavor symmetry of N $$ \mathcal{N} $$ = 2 SCFTs which culminate with our N $$ \mathcal{N} $$ = 2 UV-IR simple flavor condition. This is done by determining precisely the relation between the integrand of the partition function of the topologically twisted version of the 4d N $$ \mathcal{N} $$ = 2 SCFTs and the singular locus of their Coulomb branches. The techniques developed here are extensively applied to many rank-2 SCFTs, including new ones, in a companion paper. This manuscript is dedicated to the memory of Rayshard Brooks, George Floyd, Breonna Taylor and the countless black lives taken by US police forces and still awaiting justice. Our hearts are with our colleagues of color who suffer daily the consequences of this racist world.

Abstract We study the stratification of the singular locus of four dimensional N $$ \mathcal{N} $$ = 2 Coulomb branches. We present a set of self-consistency conditions on this stratification which can be used to extend the classification of scale-invariant rank 1 Coulomb branch geometries to two complex dimensions, and beyond. The calculational simplicity of the arguments presented here stems from the fact that the main ingredients needed — the rank 1 deformation patterns and the pattern of inclusions of rank 2 strata — are discrete topological data which satisfy strong self-consistency conditions through their relationship to the central charges of the SCFT. This relationship of the stratification data to the central charges is used here, but is derived and explained in a companion paper [1] by one of the authors. We illustrate the use of these conditions by re-analyzing many previously-known examples of rank 2 SCFTs, and also by finding examples of new theories. The power of these conditions stems from the fact that for Coulomb branch stratifications a conjecturally complete list of physically allowed “elementary slices” is known. By contrast, constraining the possible elementary slices of symplectic singularities relevant for Higgs branch stratifications remains an open problem.

Abstract The F-theory realization of the rank-1 Minahan Nemeschansky (MN) E 6, E 7 and E 8 theories leads to a description of the BPS states on the Coulomb branch in terms of Type IIB (p, q)-string networks. Subject to a simple ansatz for the types of networks which can occur, we study the representations of the flavor symmetry group which occur in the BPS spectrum. The results we find for the E 6 and E 7 theories are in perfect agreement with previous calculations by other methods (in particular, we find that arbitrarily large representations occur), but our scheme is easier to implement and more computationally efficient. The string network picture also gives a possible explanation of the experimental observation that in rank-1 MN theories, BPS states whose charge is n times a primitive charge occur with BPS index divisible by (−1)n+1 n.

Abstract We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of distinct SCFTs which have identical moduli space (Coulomb, Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting arena in which to test the relationship between moduli space geometries and conformal field theory data. We construct these SCFTs by gauging certain discrete global symmetries of N $$ \mathcal{N} $$ = 4 superYang-Mills (sYM) theories. In the simplest cases, these discrete symmetries are outer automorphisms of the sYM gauge group, and so these theories have lagrangian descriptions as N $$ \mathcal{N} $$ = 4 sYM theories with disconnected gauge groups.

Abstract According to the standard lore only single monopoles contribute to the superpotential on the Coulomb branch of 3D N=2 $$ \mathcal{N}=2 $$ SUSY gauge theories. However we argue that multi-monopole configurations can also generate superpotential terms in the presence of squark VEVs on the mixed Higgs-Coulomb branch. The new ingredient is the confinement of monopoles via Nielsen-Olesen flux tubes. Such confined multi-monopoles will yield a pre-ADS superpotential which depends both on the local Coulomb moduli and matter superfields but has no fractional powers. Once the lifted moduli are integrated out the familiar ADS superpotential is obtained. Our results demonstrate the important role multi-monopoles can play in generating non-perturbative effects and also sheds light on the still somewhat mysterious dynamical origin of the general 4D ADS superpotential.

Abstract We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 N=2 $$ \mathcal{N}=2 $$ superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special Kähler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the U(1) R symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.

Abstract This is the third in a series of three papers on the systematic analysis of rank 1 four dimensional N $$ \mathcal{N} $$ = 2 SCFTs. In the first two papers [1, 2] we developed and carried out a strategy for classifying and constructing physical planar rank-1 Coulomb branch geometries of N $$ \mathcal{N} $$ = 2 SCFTs. Here we describe general features of the Higgs and mixed branch geometries of the moduli space of these SCFTs, and use this, along with their Coulomb branch geometry, to compute their conformal and flavor central charges. We conclude with a summary of the state of the art for rank-1 N $$ \mathcal{N} $$ = 2 SCFTs.

Abstract We initiate a systematic study of four dimensional N $$ \mathcal{N} $$ = 2 superconformal field theories (SCFTs) based on the analysis of their Coulomb branch geometries. Because these SCFTs are not uniquely characterized by their scale-invariant Coulomb branch geometries we also need information on their deformations. We construct all inequivalent such deformations preserving N $$ \mathcal{N} $$ = 2 supersymmetry and additional physical consistency conditions in the rank 1 case. These not only include all the ones previously predicted by S-duality, but also 16 additional deformations satisfying all the known N $$ \mathcal{N} $$ = 2 low energy consistency conditions. All but two of these additonal deformations have recently been identified with new rank 1 SCFTs; these identifications are briefly reviewed. Some novel ingredients which are important for this study include: a discussion of RG-flows in the presence of a moduli space of vacua; a classification of local N $$ \mathcal{N} $$ = 2 supersymmetry-preserving deformations of unitary N $$ \mathcal{N} $$ = 2 SCFTs; and an analysis of charge normalizations and the Dirac quantization condition on Coulomb branches. This paper is the first in a series of three. The second paper [1] gives the details of the explicit construction of the Coulomb branch geometries discussed here, while the third [2] discusses the computation of central charges of the associated SCFTs.

Abstract This is the second in a series of three papers on systematic analysis of rank 1 Coulomb branch geometries of four dimensional N $$ \mathcal{N} $$ = 2 SCFTs. In [1] we developed a strategy for classifying physical rank-1 CB geometries of N $$ \mathcal{N} $$ = 2 SCFTs. Here we show how to carry out this strategy computationally to construct the Seiberg-Witten curves and one-forms for all the rank-1 SCFTs. Explicit expressions are given for all 28 cases, with the exception of the N f =4 su(2) gauge theory and the E n SCFTs which were constructed in [2, 3] and [4, 5].

Abstract Coulomb branch chiral rings of N = 2 $$ \mathcal{N}=2 $$ SCFTs are conjectured to be freely generated. While no counter-example is known, no direct evidence for the conjecture is known either. We initiate a systematic study of SCFTs with Coulomb branch chiral rings satisfying non-trivial relations, restricting our analysis to rank 1. The main result of our study is that (rank-1) SCFTs with non-freely generated CB chiral rings when deformed by relevant deformations, always flow to theories with non-freely generated CB rings. This implies that if they exist, they must thus form a distinct subset under RG flows. We also find many interesting characteristic properties that these putative theories satisfy which may behelpful in proving or disproving their existence using other methods.

Abstract In this paper we present a beautifully consistent web of evidence for the existence of interacting 4d rank-1 N $$ \mathcal{N} $$ = 2 SCFTs obtained from gauging discrete subgroups of global symmetries of other existing 4d rank-1 N $$ \mathcal{N} $$ = 2 SCFTs. The global symmetries that can be gauged involve a non-trivial combination of discrete subgroups of the U(1) R , low-energy EM duality group S L 2 , ℤ $$ \mathrm{S}\mathrm{L}\left(2,\kern0.5em \mathbb{Z}\right) $$ , and the outer automorphism group of the flavor symmetry algebra, Out(F ). The theories that we construct are remarkable in many ways: (i) two of them have exceptional F 4 and G 2 flavor groups; (ii) they substantially complete the picture of the landscape of rank-1 N $$ \mathcal{N} $$ = 2 SCFTs as they realize all but one of the remaining consistent rank-1 Seiberg-Witten geometries that we previously constructed but were not associated to known SCFTs; and (iii) some of them have enlarged N $$ \mathcal{N} $$ = 3 SUSY, and have not been previously constructed. They are also examples of SCFTs which violate the ShapereTachikawa relation between the conformal central charges and the scaling dimension of the Coulomb branch vev. We propose a modification of the formulas computing these central charges from the topologically twisted Coulomb branch partition function which correctly compute them for discretely gauged theories.