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Network, cluster coordinates and N = 2 theory II: Irregular singularity

Dan Xie

TL;DR

This work develops a geometric framework to construct cluster coordinates and BPS quivers for a broad class of Argyres-Douglas and asymptotically free N=2 theories by using Stokes data from irregular singularities on bordered Riemann surfaces. By translating irregular singularity data into dot diagrams, bipartite networks, and triangulation-based quivers, the authors connect the Coulomb-branch geometry to Grassmannian cells and propose that the network quiver is the BPS quiver. The method covers Type I–IV AD theories and linear/quiver gauge theories with AD matter, providing explicit examples where quiver mutations and Grassmannian cell structures align with known spectra and dualities. The approach promises practical tools for computing BPS spectra, wall-crossing, and line-operator data, and suggests deep links to spectral networks and Y-systems across N=2 theories.

Abstract

Cluster coordinates for a large class of Argyres-Douglas and asymptotical free theories are constructed using network on bordered Riemann surface. Such N = 2 theories are engineered using six dimensional (2, 0) theory on Riemann surface with irregular and regular singularities. The Stokes phenomenon plays an important role in our construction. Our results are expected to be very useful in studying BPS spectrum, wall crossing, and line operators of these theories, etc. In particular, we conjecture that the quiver from the network is the BPS quiver. Moreover, our construction provides a simple way to build the minimal network for cells of positive Grassmannia .

Network, cluster coordinates and N = 2 theory II: Irregular singularity

TL;DR

This work develops a geometric framework to construct cluster coordinates and BPS quivers for a broad class of Argyres-Douglas and asymptotically free N=2 theories by using Stokes data from irregular singularities on bordered Riemann surfaces. By translating irregular singularity data into dot diagrams, bipartite networks, and triangulation-based quivers, the authors connect the Coulomb-branch geometry to Grassmannian cells and propose that the network quiver is the BPS quiver. The method covers Type I–IV AD theories and linear/quiver gauge theories with AD matter, providing explicit examples where quiver mutations and Grassmannian cell structures align with known spectra and dualities. The approach promises practical tools for computing BPS spectra, wall-crossing, and line-operator data, and suggests deep links to spectral networks and Y-systems across N=2 theories.

Abstract

Cluster coordinates for a large class of Argyres-Douglas and asymptotical free theories are constructed using network on bordered Riemann surface. Such N = 2 theories are engineered using six dimensional (2, 0) theory on Riemann surface with irregular and regular singularities. The Stokes phenomenon plays an important role in our construction. Our results are expected to be very useful in studying BPS spectrum, wall crossing, and line operators of these theories, etc. In particular, we conjecture that the quiver from the network is the BPS quiver. Moreover, our construction provides a simple way to build the minimal network for cells of positive Grassmannia .

Paper Structure

This paper contains 23 sections, 47 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Review
  3. UV and IR data of AD theory
  4. Dot diagram, network and quiver
  5. Quiver and mutation
  6. Stokes phenomenon
  7. Integer pole
  8. Type I irregular singularity
  9. Type II irregular singularity
  10. Cluster coordinates for Argyres-Douglas theory
  11. Type I AD theory
  12. $(A_1,A_{N-1})$ theory
  13. $(A_{k-1},A_{nk-1})$ theory
  14. Generic case
  15. Type II AD theory
  16. ...and 8 more sections

Figures (30)

  • Figure 1: The Newton polygon for type I and type II irregular singularity.
  • Figure 2: Left: The orientation of the big triangle. Middle: Put a white vertex to each polygon whose triangle completion has the same orientation as the big triangle. Right: Put a black vertex to each polygon whose triangle completion has opposite orientation.
  • Figure 3: The network would be non-minimal if the boundary of the dot diagram has this form at any vertex.
  • Figure 4: The quiver mutation.
  • Figure 5: The square move which represents the quiver mutation.
  • ...and 25 more figures