Geometric classification of 4d $\mathcal{N}=2$ SCFTs

Abstract

The classification of 4d $\mathcal{N}=2$ 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 $\mathbb{Q}$-factorial log-Fano variety with Hodge numbers $h^{p,q}={\delta }_{p,q}$. With some plausible restrictions, this means that the Coulomb branch chiral ring $\mathcal{R}$ is a graded polynomial ring generated by global holomorphic functions $u_i$ of dimension ${\Delta }_i$. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension $k$-tuples $\{{\Delta }_1,{\Delta }_2,\cdots ,{\Delta }_k\}$ 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 $\{{\Delta }_1,\cdots ,{\Delta }_k\}$'s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number $\mathbf{N}(k)$ 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 $k$ $$\mathbf{N}(k)=\frac{2\zeta (2)\zeta (3)}{\zeta (6)}{k}^2+o(k^2).$$ 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 $\{{\Delta }_1,\cdots ,{\Delta }_k\}$ 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. We illustrate the various aspects with several examples and perform a number of explicit checks. We include tables of dimensions for the first few $k$'s.

arXiv comment119 pages, 11 tables (3 of them multi-page), 52 footnotes. ADDED: a clarification in section 6.2 and example 20. ADDED v3: better presentation of the tables

References in corpus (41)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• N=2 dualities
• CFT's From Calabi-Yau Four-folds
• Symmetries and Strings in Field Theory and Gravity
• Supersymmetric Yang-Mills Systems And Integrable Systems
• Geometric Engineering of Quantum Field Theories
• The Exact Superconformal R-Symmetry Extremizes Z
• Four-dimensional wall-crossing via three-dimensional field theory
• On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version)
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• Webs of five-branes and N=2 superconformal field theories
• Superconformal Fixed Points with E_n Global Symmetry
• R-Twisting and 4d/2d Correspondences
• Special Kahler Manifolds
• Spectral networks
• Branes, Calabi-Yau Spaces, and Toroidal Compactification of the N=1 Six-Dimensional E_8 Theory
• Introduction to Arithmetic Groups
• Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2
• More on the N=2 superconformal systems of type $D_p(G)$
• Classification of Argyres-Douglas theories from M5 branes
• Infinite coupling duals of N=2 gauge theories and new rank 1 superconformal field theories
• Infinitely many N=2 SCFT with ADE flavor symmetry
• The Power of Holomorphy -- Exact Results in 4D SUSY Field Theories
• Geometric constraints on the space of N=2 SCFTs I: physical constraints on relevant deformations
• On Arnold's 14 `exceptional' N=2 superconformal gauge theories
• Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated
• Geometric constraints on the space of N=2 SCFTs III: enhanced Coulomb branches and central charges
• The Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches
• Fake weighted projective spaces
• Geometric constraints on the space of N=2 SCFTs II: Construction of special Kähler geometries and RG flows
• 4d N=2 SCFT and singularity theory Part I: Classification
• Superconformal Sigma Models in Three Dimensions
• Classification of N=2 Superconformal Field Theories with Two-Dimensional Coulomb Branches, II
• Lectures on Exotic Algebraic Structures on Affine Spaces
• 4d N=2 SCFT and singularity theory Part II: Complete intersection
• 4d N=2 SCFT from Complete Intersection Singularity
• Classification of N=2 Superconformal Field Theories with Two-Dimensional Coulomb Branches
• Expanding the landscape of $\mathcal{N}$=2 rank 1 SCFTs
• 4d N=2 SCFT and singularity theory Part III: Rigid singularity
• $Y$ systems, $Q$ systems, and 4D $\mathcal{N}=2$ supersymmetric QFT
• Algebraic Stein Varieties

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