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Testing our understanding of SCFTs: a catalogue of rank-2 $\mathcal{N}$=2 theories in four dimensions

Mario Martone

TL;DR

The work provides a thorough bottom-up catalog of rank-2 ${\mathcal{N}}=2$ SCFTs in four dimensions, building a lattice of theories organized by RG-flow connected mass deformations and isolating a set of truly isolated models. By analyzing moduli-space stratifications on both the Coulomb and Higgs branches, the authors derive detailed data for each theory, including Coulomb/Higgs/mixed branches, central charges, and flavor symmetries, and connect rank-2 data to rank-1 theories via the GHW framework and the doubling rule. The study highlights patterns and gaps, with many rank-2 theories descending from 6d constructions or class-S/S-fold frameworks, while several entries hint at undiscovered AD-type theories or new series. This bottom-up program provides a concrete test-bed for the completeness of the string-theoretic SCFT landscape and strengthens tools for translating geometric moduli-space information into quantum-field-theoretic data. Overall, the paper advances a systematic, data-rich foundation for understanding rank-2 ${\mathcal{N}}=2$ theories and their interconnections through RG flows and Higgsings, with implications for the broader classification of 4d SCFTs.

Abstract

In this paper we begin mapping out the space of rank-2 $\mathcal{N}=2$ 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.

Testing our understanding of SCFTs: a catalogue of rank-2 $\mathcal{N}$=2 theories in four dimensions

TL;DR

The work provides a thorough bottom-up catalog of rank-2 SCFTs in four dimensions, building a lattice of theories organized by RG-flow connected mass deformations and isolating a set of truly isolated models. By analyzing moduli-space stratifications on both the Coulomb and Higgs branches, the authors derive detailed data for each theory, including Coulomb/Higgs/mixed branches, central charges, and flavor symmetries, and connect rank-2 data to rank-1 theories via the GHW framework and the doubling rule. The study highlights patterns and gaps, with many rank-2 theories descending from 6d constructions or class-S/S-fold frameworks, while several entries hint at undiscovered AD-type theories or new series. This bottom-up program provides a concrete test-bed for the completeness of the string-theoretic SCFT landscape and strengthens tools for translating geometric moduli-space information into quantum-field-theoretic data. Overall, the paper advances a systematic, data-rich foundation for understanding rank-2 theories and their interconnections through RG flows and Higgsings, with implications for the broader classification of 4d SCFTs.

Abstract

In this paper we begin mapping out the space of rank-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.

Paper Structure

This paper contains 43 sections, 29 equations, 91 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Overview of the space of rank-2 theories
  3. Background on ${\mathcal{N}}=2$ moduli space
  4. Different branches of the moduli space
  5. Note on color coding
  6. Coulomb branch
  7. Doubling rule
  8. Higgs branch stratification
  9. GHW central charge formula
  10. Detailed description of rank-2 theories
  11. $\mathfrak{e}_8-\mathfrak{so}(20)$ series
  12. $\mathfrak{sp}(12)-\mathfrak{sp}(8)-\mathfrak{f}_4$ series
  13. $\mathfrak{su}(6)$ series
  14. $\mathfrak{sp}(14)$ series
  15. $\mathfrak{su}(5)$ series
  16. ...and 28 more sections

Figures (91)

  • Figure 1: The Coulomb and Higgs stratification of $\mathfrak{so}(20)_{16}$.
  • Figure 2: Central charges, parameters and dimension.
  • Figure 4: The Coulomb and Higgs stratification of $[\mathfrak{e}_8]_{20}$
  • Figure 5: Central charges, parameters and dimension.
  • Figure 7: The Coulomb and Higgs stratification of $\mathfrak{su}(2)_8{\times}\mathfrak{so}(16)_{12}$.
  • ...and 86 more figures