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The Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches

Philip C. Argyres, Cody Long, Mario Martone

TL;DR

This work extends the rank-1 Coulomb branch classification to rank-2 ${\mathcal N}=2$ SCFTs by linking the SK geometry, EM duality monodromies, and the topology of the singular locus ${\mathcal{V}}$ on the CB ${\mathcal{C}}$. It shows that the CB operator dimensions $\{\Delta_u,\Delta_v\}$ must be rational and belong to a finite set of 24 possibilities, with $\Delta_u$ and $\Delta_v$ commensurate; this finiteness is a consequence of unit-norm eigenvalues in the ${\rm Sp}_{\Delta}(4,\mathbb{Z})$ monodromy and the knot/link structure of ${\mathcal{V}}$. The authors develop a toolkit—describing the algebraic form of ${\mathcal{V}}$, the topology of ${\mathcal{V}}\subset{\mathcal{C}}$, factorized monodromies, and U(1)$_R$ eigenspaces—that will be essential for constructing all scale-invariant rank-2 CB geometries and, potentially, higher-rank generalizations. The results unify with known rank-1 CB dimensions and connect to concrete rank-2 Lagrangian theories via explicit knot/link descriptions, while outlining future work on deformations and complete classifications across ranks.

Abstract

We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 $\mathcal{N}{=}2$ superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special Kähler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the $\rm\, U(1)_R$ symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.

The Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches

TL;DR

This work extends the rank-1 Coulomb branch classification to rank-2 SCFTs by linking the SK geometry, EM duality monodromies, and the topology of the singular locus on the CB . It shows that the CB operator dimensions must be rational and belong to a finite set of 24 possibilities, with and commensurate; this finiteness is a consequence of unit-norm eigenvalues in the monodromy and the knot/link structure of . The authors develop a toolkit—describing the algebraic form of , the topology of , factorized monodromies, and U(1) eigenspaces—that will be essential for constructing all scale-invariant rank-2 CB geometries and, potentially, higher-rank generalizations. The results unify with known rank-1 CB dimensions and connect to concrete rank-2 Lagrangian theories via explicit knot/link descriptions, while outlining future work on deformations and complete classifications across ranks.

Abstract

We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special Kähler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.

Paper Structure

This paper contains 39 sections, 93 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction and summary of results
  2. Topology of Coulomb branch singularities for rank-2 SCFTs
  3. Basic ingredients of SK geometry
  4. Some regularity assumptions
  5. Complex scaling action and orbits in rank-2
  6. Complex scaling orbits and singularities.
  7. Topology of ${\mathcal{V}}\subset{\mathcal{C}}$
  8. The fundamental group of ${\mathcal{C}}\setminus{\mathcal{V}}$.
  9. A few concrete examples
  10. ${\rm\, SU}(3)$ with 1 adjoint hypermultiplet
  11. ${\rm\, SU}(3)$ with 6 fundamental hypermultiplets
  12. Other rank-2 lagrangian SCFTs
  13. SK geometry of the Coulomb branch in rank-2
  14. SK geometry of ${\mathcal{M}}$
  15. SK geometry near ${\mathcal{V}}$
  16. ...and 24 more sections

Figures (4)

  • Figure 1: Depiction of an $L_{(1,6)}(1,1,1)$ link consisting of the blue ($K_0$), red ($K(1,6)$), and green ($K_\infty$) circles. The solid gray torus is there for visualization purposes.
  • Figure 2: Depiction of an $L_{(1,6)}(0,1,0)$ link consisting of the red circle. The ${\gamma}_0$ cycle threads the interior of the donut, while ${\gamma}_\infty$ threads the hole of the donut.
  • Figure 3: Depiction of an $L_{(1,2)}(0,3,0)$ link consisting of the red, orange, and yellow circles. The $f_1$ cycle links the first strand in the direction of the ${\gamma}_0$ cycle, while $f_2$ links the first two strands. The ${\gamma}_0$ and ${\gamma}_\infty$ cycles, as in Figure \ref{['knot2']}, are not shown.
  • Figure 4: Depiction of an $L_{(1,6)}(1,1,1)$ link consisting of the blue ($K_0$), red ($K(1,6)$), and green ($K_\infty$) circles. The ${\delta}_0$ cycle links only the $K_0$ unknot, while ${\delta}_\infty$ links only the $K_\infty$ unknot. The ${\gamma}_0$ and ${\gamma}_\infty$ cycles, as in Figure \ref{['knot2']}, are not shown.