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Coulomb branches with complex singularities

Philip C. Argyres, Mario Martone

TL;DR

The paper constructs 4d ${ m N}=2$ SCFTs with Coulomb branches possessing complex singularities by gauging discrete symmetries of ${ m N}=4$ sYM, yielding CB coordinate rings that are not freely generated and enabling distinct SCFTs to share identical moduli spaces. The authors develop a systematic framework using Weyl groups, S-duality, and discrete outer automorphisms to build ${ m N}=3$-preserving and ${ m N}=4$-preserving symmetries, then analyze the resulting CB geometries via Hilbert/Molien series and plethystic techniques. They provide explicit examples, including rank-2 and rank-4 cases, where CBs remain regular (Freely generated) and where CBs acquire complex singularities (non-free invariants), some realized as hypersurfaces and others as non-complete intersections with higher-rank syzygies. The work shows that CB geometry does not uniquely determine SCFT data and that moduli spaces can be identical across different theories, offering a testing ground for the link between geometric moduli space data and conformal data, and raises open questions about extension to ${ m N}=2$ theories and the role of discretely gauged symmetries in line operators and RG flows.

Abstract

We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of distinct SCFTs which have identical moduli space (Coulomb, Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting arena in which to test the relationship between moduli space geometries and conformal field theory data. We construct these SCFTs by gauging certain discrete global symmetries of $\mathcal N=4$ superYang-Mills (sYM) theories. In the simplest cases, these discrete symmetries are outer automorphisms of the sYM gauge group, and so these theories have lagrangian descriptions as $\mathcal N=4$ sYM theories with disconnected gauge groups.

Coulomb branches with complex singularities

TL;DR

The paper constructs 4d SCFTs with Coulomb branches possessing complex singularities by gauging discrete symmetries of sYM, yielding CB coordinate rings that are not freely generated and enabling distinct SCFTs to share identical moduli spaces. The authors develop a systematic framework using Weyl groups, S-duality, and discrete outer automorphisms to build -preserving and -preserving symmetries, then analyze the resulting CB geometries via Hilbert/Molien series and plethystic techniques. They provide explicit examples, including rank-2 and rank-4 cases, where CBs remain regular (Freely generated) and where CBs acquire complex singularities (non-free invariants), some realized as hypersurfaces and others as non-complete intersections with higher-rank syzygies. The work shows that CB geometry does not uniquely determine SCFT data and that moduli spaces can be identical across different theories, offering a testing ground for the link between geometric moduli space data and conformal data, and raises open questions about extension to theories and the role of discretely gauged symmetries in line operators and RG flows.

Abstract

We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of distinct SCFTs which have identical moduli space (Coulomb, Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting arena in which to test the relationship between moduli space geometries and conformal field theory data. We construct these SCFTs by gauging certain discrete global symmetries of superYang-Mills (sYM) theories. In the simplest cases, these discrete symmetries are outer automorphisms of the sYM gauge group, and so these theories have lagrangian descriptions as sYM theories with disconnected gauge groups.

Paper Structure

This paper contains 28 sections, 104 equations.

Table of Contents

  1. Introduction and summary
  2. ${\mathcal{N}}{=}3$-preserving discrete symmetries with CB action
  3. Constructing the symmetries
  4. ${\mathcal{N}}{=}4$-preserving symmetries
  5. ${\mathcal{N}}{=}3$-preserving symmetries
  6. Gauging the symmetries
  7. Geometry of ${\mathcal{N}}=4$ sYM moduli space
  8. Moduli space geometry of the gauged theories
  9. Complex structure of the CB
  10. General considerations
  11. Hilbert series of rings of polynomial invariants
  12. Counting generators and relations
  13. Comments on Higgs branch complex geometry
  14. Examples
  15. ${\mathcal{N}}=4$ theories with regular CBs
  16. ...and 13 more sections