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Geometric constraints on the space of N=2 SCFTs III: enhanced Coulomb branches and central charges

Philp Argyres, Matteo Lotito, Yongchao Lü, Mario Martone

TL;DR

The paper advances the classification of rank-1 ${ m N}=2$ SCFTs with planar CBs by detailing Higgs and mixed (ECB) branches, elucidating their geometric and chiral-ring structures, and developing a nonperturbative framework to extract conformal and flavor central charges from the twisted ECB partition function. By leveraging S-duality, class ${ m S}$ constructions, and Hall-Littlewood indices, it determines ECB/HB data for both Lagrangian and non-lagrangian rank-1 theories and derives consistency constraints (bounds and integrality) that shape the landscape of possible theories. The work provides explicit central-charge calculations for multiple series (I$_1$, I$_2$, I$_4$, I$^*_1$, IV$^*_{Q=1}$, etc.), analyzes discrete obstructions to gauging certain flavor factors, and proposes a conservative conjecture for the complete set of planar rank-1 ${ m N}=2$ SCFTs, including their discrete gaugings. Overall, the results solidify the link between CB geometry, ECB/HB structures, and CFT data, with implications for broader nonperturbative SCFT classifications and potential higher-rank generalizations. The methods have broad relevance for understanding moduli spaces, central charges, and flavor symmetries in strongly coupled ${ m N}=2$ theories.

Abstract

This is the third in a series of three papers on the systematic analysis of rank 1 four dimensional $\mathcal{N}=2$ SCFTs. In the first two papers we developed and carried out a strategy for classifying and constructing physical planar rank-1 Coulomb branch geometries of $\mathcal{N}=2$ 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 $\mathcal{N}=2$ SCFTs.

Geometric constraints on the space of N=2 SCFTs III: enhanced Coulomb branches and central charges

TL;DR

The paper advances the classification of rank-1 SCFTs with planar CBs by detailing Higgs and mixed (ECB) branches, elucidating their geometric and chiral-ring structures, and developing a nonperturbative framework to extract conformal and flavor central charges from the twisted ECB partition function. By leveraging S-duality, class constructions, and Hall-Littlewood indices, it determines ECB/HB data for both Lagrangian and non-lagrangian rank-1 theories and derives consistency constraints (bounds and integrality) that shape the landscape of possible theories. The work provides explicit central-charge calculations for multiple series (I, I, I, I, IV, etc.), analyzes discrete obstructions to gauging certain flavor factors, and proposes a conservative conjecture for the complete set of planar rank-1 SCFTs, including their discrete gaugings. Overall, the results solidify the link between CB geometry, ECB/HB structures, and CFT data, with implications for broader nonperturbative SCFT classifications and potential higher-rank generalizations. The methods have broad relevance for understanding moduli spaces, central charges, and flavor symmetries in strongly coupled theories.

Abstract

This is the third in a series of three papers on the systematic analysis of rank 1 four dimensional SCFTs. In the first two papers we developed and carried out a strategy for classifying and constructing physical planar rank-1 Coulomb branch geometries of 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 SCFTs.

Paper Structure

This paper contains 32 sections, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Moduli space of rank-1 SCFTs
  3. Higgs branches (HBs) and mixed branches
  4. Enhanced Coulomb branches (ECBs)
  5. ECBs in ${\mathcal{N}}=2$ gauge theories
  6. Some examples.
  7. Moduli space for generic, non-lagrangian, rank-1 SCFTs
  8. I$^*_{\bf 1}$ series.
  9. IV$^*_{\bf Q=1}$ series.
  10. I$_{\bf 1}$ series.
  11. I$_{\bf 4}$ series.
  12. Chiral rings of SCFTs and coordinate ring of HBs and ECBs
  13. ${\mathcal{N}}$=2 SCFT chiral ring
  14. Moduli space coordinate ring for rank-1 SCFTs
  15. Coordinate ring of $V_d:=\mathbb{C}^d/\mathbb{Z}_2$.
  16. ...and 17 more sections

Figures (2)

  • Figure 1: Visualization of an ${\mathcal{N}}=2$ moduli space. The hyperkähler directions are vertical and the special Kähler directions horizontal, with the different types of branches labelled. Mixed branches are metrically a cartesian product of hyperkähler and special Kähler directions except perhaps over complex codimension one subvarieties of their special Kähler base.
  • Figure 2: Moduli space of a planar rank-1 N=2 SCFT.