The classification of 4d $$ \mathcal{N}=2 $$ N = 2 SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected ℚ-factorial log-Fano variety with Hodge numbers h p,q = δ p,q . With some plausible restrictions, this means that the Coulomb branch chiral ring "Image missing" is a graded polynomial ring generated by global holomorphic functions u i of dimension Δ i . The coarse-grained classification of the CSG consists in listing the (finitely many) dimension k-tuples {Δ1 , Δ2 , ⋯ , Δ k } which are realized as Coulomb branch dimensions of some rank-k CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible {Δ1 , ⋯ , Δ k }’s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number N (k) of dimensions allowed in rank k is given by a certain sum of the Erdös-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large k $$ \boldsymbol{N}(k)=\frac{2\zeta (2)\zeta (3)}{\zeta (6)}{k}^2+o\left({k}^2\right). $$ N k = 2 ζ 2 ζ 3 ζ 6 k 2 + o k 2 . In the special case k = 2 our dimension formula reproduces a recent result by Argyres et al. Class Field Theory implies a subtlety: certain dimension k-tuples {Δ1 , ⋯ , Δ k } are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted. Since the arguments tend to be abstract, we illustrate the various aspects with several concrete examples and perform a number of explicit checks. We include detailed tables of dimensions for the first few k’s.

Abstract:We revisit the classification of rank-1 4d N = 2 QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-Néron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs (E, F ∞ ) where E is a relatively minimal, rational elliptic surface with section, and F ∞ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on (E, F ∞ ) equivalent to the "safely irrelevant conjecture". The Mordell-Weil group of E (with the Néron-Tate pairing) contains a canonical root system arising from (−1)-curves in special position in the Néron-Severi group. This canonical system is identified with the roots of the flavor group F: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.

Argyres and co-workers started a program to classify all 4d N = 2 QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 N = 2 QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces.The classification of 4d N = 2 QFT is also conjectured to be equivalent to the representation theoretic (RT) classification of all 2-Calabi-Yau categories with suitable properties. Since the RT approach smells to be much simpler than the Special-Geometric one, it is worthwhile to check this expectation by reproducing the rank-1 result from the RT side. This is the main purpose of the present paper. Along the route we clarify several issues and learn new details about the rank-1 SCFT. In particular, we relate the rank-1 classification to mirror symmetry for Fano surfaces.In the follow-up paper we apply the RT methods to higher rank 4d N = 2 SCFT.

The S-duality group S(F ) of a 4d N = 2 supersymmetric theory F is identified with the group of triangle equivalences of its cluster category C (F ) modulo the subgroup acting trivially on the physical quantities. S(F ) is a discrete group commensurable to a subgroup of the Siegel modular group Sp(2g, Z) (g being the dimension of the Coulomb branch). This identification reduces the determination of the S-duality group of a given N = 2 theory to a problem in homological algebra. In this paper we describe the techniques which make the computation straightforward for a large class of N = 2 QFTs. The group S(F ) is naturally presented as a generalized braid group.The S-duality groups are often larger than expected. In some models the enhancement of S-duality is quite spectacular. For instance, a QFT with a huge S-duality group is the Lagrangian SCFT with gauge group SO(8) × SO(5) 3 × SO(3) 6 and half-hypermultiplets in the bi-and tri-spinor representations.

We review the categorical approach to the BPS sector of a 4d N = 2 QFT, clarifying many tricky issues and presenting a few novel results. To a given N = 2 QFT one associates several triangle categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. A basic theme of this review is the emphasis on the full web of categories, rather than on what we can learn from a single description. A second general theme is viewing the cluster category as a sort of 'categorification' of 't Hooft's theory of quantum phases for a 4d non-Abelian gauge theory. The S-duality group is best described as the auto-equivalences of the full web of categories. This viewpoint leads to a combinatorial algorithm to search for S-dualities of the given N = 2 theory. If the ranks of the gauge and flavor groups are not too big, the algorithm may be effectively run on a laptop. This viewpoint also leads to a clearer view of 3d mirror symmetry. For class S theories, all the relevant triangle categories may also be constructed in terms of geometric objects on the Gaiotto curve, and we present the dictionary between triangle categories and the WKB approach of GMN. We also review how the VEV's of UV line operators are related to cluster characters.