Network, Cluster coordinates and N=2 theory I

Details Network, Cluster coordinates and N=2 theory I Network, Cluster coordinates and N=2 theory I

2012-03-20

arxiv.org/abs/1203.4573v1

Physics

High Energy Physics

High Energy Physics - Theory

40 figures, 43 pages

Scientific paper

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.}

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