Network, Cluster coordinates and N=2 theory I

TL;DR

This paper develops combinatorial methods to determine cluster coordinates for the moduli space of flat connections, illuminating aspects of N=2 theories derived from six-dimensional compactifications and their applications.

Contribution

It introduces a triangulation-based construction of cluster coordinates for moduli spaces, proving their invariance under quiver mutations and linking to various physical theories.

Findings

Constructed bipartite networks from triangulations to derive quivers.

Proved quivers for different triangulations are related by mutations.

Established the relevance of these coordinates in BPS wall crossing and other theories.

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.}