Network, Cluster coordinates and N=2 theory I

TL;DR

We address constructing cluster coordinates for the moduli space of flat connections on punctured Riemann surfaces for higher-rank class ${\mathcal S}$ theories. The approach triangulates the surface, tessellates each triangle, and builds a bipartite network whose read-out quiver undergoes mutations that realize cluster coordinates. The method covers irreducible and reducible punctures, three- and four-punctured cases, and extends to arbitrary punctures and higher genus, with square moves encoding flips and preserving the cluster structure. The resulting framework connects to BPS spectra, line/surface operators, Teichmüller theory, Chern-Simons theory, and integrable systems, thereby generalizing Fock–Goncharov coordinates to non-full punctures. It sets up a versatile, combinatorial toolkit for analyzing higher-rank ${\mathcal S}$ theories and their dimensional reductions.

Abstract

Combinatorial methods are developed to find the cluster coordinates for moduli space of flat connections which is describing the Coulomb branch of higher rank N=2 theories derived by compactifying six dimensional (2,0) theory on a punctured Riemann surface. The construction starts with a triangulation of the punctured Riemann surface and a further tessellation of all the triangles. The tessellation is used to construct a bipartite network from which a quiver can be read straightforwardly. We prove that the quivers for different triangulations are related by quiver mutations and justify that these are really the cluster coordinates. These coordinates are important in studying BPS wall crossing, line operators, and surface operators of these theories; and they are also useful in exploring three dimensional Chern-Simons theory and the corresponding N=2 gauge theory, two dimensional integrable system, etc.}

Figures (41)

• Figure 1: A four punctured sphere is decomposed into two three punctured spheres in the degeneration limit. A new puncture $e$ appears on each three punctured sphere.
• Figure 2: Three duality frames for a SCFT defined by compactifying six dimensional $(2,0)$ theory on a four punctured sphere.
• Figure 3: The quiver mutation.
• Figure 4: A triangulation of four punctured sphere.
• Figure 5: Two special pieces in an ideal triangulation.