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On Arnold's 14 `exceptional' N=2 superconformal gauge theories

Sergio Cecotti, Michele Del Zotto

TL;DR

<3-5 sentence high-level summary>This work extends the CNV framework to Arnold's 14 exceptional unimodal singularities, constructing the corresponding ${\cal N}=2$ SCFTs and computing their BPS spectra in strongly coupled chambers. It develops a cluster-algebraic CNV strategy to derive finite BPS chambers and uses Weyl-/sink-factorized subquiver sequences to obtain direct-sum spectra, while connecting these results to a family of periodic ${Y}$-systems predicted by a general holographic/TBA correspondence. The paper also identifies IR gauge-theory interpretations of these exceptional models, demonstrates decoupling limits to Argyres–Douglas theories, and confirms the periodicity of the associated ${Y}$-systems (with explicit checks for several models). These findings illuminate deep links between singularity theory, quiver representations, and integrable structures in four-dimensional ${\cal N}=2$ theories, expanding the landscape of exactly solvable BPS spectra and their thermodynamic descriptions.

Abstract

We study the four-dimensional superconformal N=2 gauge theories engineered by the Type IIB superstring on Arnold's 14 exceptional unimodal singularities (a.k.a. Arnold's strange duality list), thus extending the methods of 1006.3435 to singularities which are not the direct sum of minimal ones. In particular, we compute their BPS spectra in several `strongly coupled' chambers. From the TBA side, we construct ten new periodic Y-systems, providing additional evidence for the existence of a periodic Y-system for each isolated quasi-homogeneous singularity with $\hat c<2$ (more generally, for each N=2 superconformal theory with a finite BPS chamber whose chiral primaries have dimensions of the form N/l).

On Arnold's 14 `exceptional' N=2 superconformal gauge theories

TL;DR

<3-5 sentence high-level summary>This work extends the CNV framework to Arnold's 14 exceptional unimodal singularities, constructing the corresponding SCFTs and computing their BPS spectra in strongly coupled chambers. It develops a cluster-algebraic CNV strategy to derive finite BPS chambers and uses Weyl-/sink-factorized subquiver sequences to obtain direct-sum spectra, while connecting these results to a family of periodic -systems predicted by a general holographic/TBA correspondence. The paper also identifies IR gauge-theory interpretations of these exceptional models, demonstrates decoupling limits to Argyres–Douglas theories, and confirms the periodicity of the associated -systems (with explicit checks for several models). These findings illuminate deep links between singularity theory, quiver representations, and integrable structures in four-dimensional theories, expanding the landscape of exactly solvable BPS spectra and their thermodynamic descriptions.

Abstract

We study the four-dimensional superconformal N=2 gauge theories engineered by the Type IIB superstring on Arnold's 14 exceptional unimodal singularities (a.k.a. Arnold's strange duality list), thus extending the methods of 1006.3435 to singularities which are not the direct sum of minimal ones. In particular, we compute their BPS spectra in several `strongly coupled' chambers. From the TBA side, we construct ten new periodic Y-systems, providing additional evidence for the existence of a periodic Y-system for each isolated quasi-homogeneous singularity with (more generally, for each N=2 superconformal theory with a finite BPS chamber whose chiral primaries have dimensions of the form N/l).

Paper Structure

This paper contains 34 sections, 138 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Arnold's $14$ exceptional unimodal singularities
  3. Coxeter--Dynkin graphs, Coxeter transformations
  4. Arnold's ${\cal N}=2$ superconformal theories
  5. Quivers and superpotentials
  6. Set up
  7. $2d/4d$ correspondence revisited
  8. The square and the Coxeter--Dynkin forms of the quiver
  9. Minimal non--complete models
  10. Flavor symmetries
  11. Order of the quantum monodromy
  12. Arnold's exceptional ${\cal N}=2$ models as gauge theories
  13. The cluster strategy of CNV
  14. Quantum cluster mutations
  15. Quantum mutations
  16. ...and 19 more sections

Figures (7)

  • Figure 1: The quiver corresponding to the $3$--Calabi--Yau completion of the Coxeter--Dynkin algebra of extended canonical type $\widehat{D}(p,q,r)$; the table gives the correspondence (singularity type) $\longleftrightarrow (p,q,r)$.
  • Figure 2: The $A_4\,\square\,A_2$ and the $A_3\,\square\,A_2$ quivers.
  • Figure 3: Other quivers in the $E_7$ class with a complete family of Dynkin subquivers
  • Figure 4: Quivers for the Z familiy.
  • Figure 5: Quivers for the Q familiy.
  • ...and 2 more figures