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Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated

Philip C. Argyres, Yongchao Lü, Mario Martone

TL;DR

This work probes whether four-dimensional ${\mathcal N}=2$ SCFTs necessarily have freely generated Coulomb branch (CB) chiral rings by focusing on rank-1 examples with non-freely generated CB rings. It develops a Seiberg–Witten (SW) framework for non-planar, scale-invariant CB geometries, showing the CB is a bouquet of cones connected at a common tip and that IR deformations generically induce irregular SK singularities ($m>0$). The authors prove a central result: rank-1 SCFTs with non-freely generated CB rings flow under relevant deformations to IR fixed points that also have non-freely generated CB rings, indicating these theories form a distinct RG-connected subset from the planar, freely-generated sector. They also construct non-planar SK geometries via multi-sheeted covers of planar SK CB solutions and discuss the physical interpretation of irregular singularities, suggesting exotic IR physics and potential quantum corrections to CB chiral ring relations; together, these results refine the landscape of possible rank-1 SCFTs and provide concrete criteria to test the existence of non-freely generated CB rings.

Abstract

Coulomb branch chiral rings of $\mathcal N=2$ SCFTs are conjectured to be freely generated. While no counter-example is known, no direct evidence for the conjecture is known either. We initiate a systematic study of SCFTs with Coulomb branch chiral rings satisfying non-trivial relations, restricting our analysis to rank 1. The main result of our study is that (rank-1) SCFTs with non-freely generated CB chiral rings when deformed by relevant deformations, always flow to theories with non-freely generated CB rings. This implies that if they exist, they must thus form a distinct subset under RG flows. We also find many interesting characteristic properties that these putative theories satisfy which may behelpful in proving or disproving their existence using other methods.

Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated

TL;DR

This work probes whether four-dimensional SCFTs necessarily have freely generated Coulomb branch (CB) chiral rings by focusing on rank-1 examples with non-freely generated CB rings. It develops a Seiberg–Witten (SW) framework for non-planar, scale-invariant CB geometries, showing the CB is a bouquet of cones connected at a common tip and that IR deformations generically induce irregular SK singularities (). The authors prove a central result: rank-1 SCFTs with non-freely generated CB rings flow under relevant deformations to IR fixed points that also have non-freely generated CB rings, indicating these theories form a distinct RG-connected subset from the planar, freely-generated sector. They also construct non-planar SK geometries via multi-sheeted covers of planar SK CB solutions and discuss the physical interpretation of irregular singularities, suggesting exotic IR physics and potential quantum corrections to CB chiral ring relations; together, these results refine the landscape of possible rank-1 SCFTs and provide concrete criteria to test the existence of non-freely generated CB rings.

Abstract

Coulomb branch chiral rings of SCFTs are conjectured to be freely generated. While no counter-example is known, no direct evidence for the conjecture is known either. We initiate a systematic study of SCFTs with Coulomb branch chiral rings satisfying non-trivial relations, restricting our analysis to rank 1. The main result of our study is that (rank-1) SCFTs with non-freely generated CB chiral rings when deformed by relevant deformations, always flow to theories with non-freely generated CB rings. This implies that if they exist, they must thus form a distinct subset under RG flows. We also find many interesting characteristic properties that these putative theories satisfy which may behelpful in proving or disproving their existence using other methods.

Paper Structure

This paper contains 36 sections, 73 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction and summary of results
  2. Rank-1 CBs of SCFTs
  3. Global structure of the CB
  4. Local CB geometry
  5. Irregular geometries.
  6. Complex singularities.
  7. Comparison to constraints from the superconformal bootstrap
  8. Constraints from deformations of rank-1 CBs
  9. Seiberg Witten curve and form
  10. Scaling behaviors at finite points and at infinities
  11. Scaling behaviors at finite points.
  12. Scaling behaviors at infinity.
  13. Global analysis of SW curve and form
  14. Patching SW data over smooth CBs.
  15. Closedness of the set of theories with non-freely-generated CB chiral rings under RG flows.
  16. ...and 21 more sections

Figures (1)

  • Figure 1: (a) A scale-invariant CB which is a bouquet of 3 cones: the conformal vacuum is the common tip of the cones, while at all other points conformality is spontaneously broken. (b) A "near" deformation of the scale-invariant geometry, in which the conformal singularity splits into several conical or cusp-like metric singularities (the red dots) without affecting the asymptotically far geometry, while the complex geometry desingularizes into that of a 3-punctured genus-1 Riemann surface.