Network, Cluster coordinates and N=2 theory I

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.}

The view on non-minimal network is changed, and we clarify the dependence on the cyclic choices of the path in the triangulation

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• N=2 dualities
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• Four-dimensional wall-crossing via three-dimensional field theory
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• Supersymmetric vacua and Bethe ansatz
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• S-duality in N=2 supersymmetric gauge theories
• Quantum integrability and supersymmetric vacua
• Thermodynamic Bubble Ansatz
• Mirrors of 3d Sicilian theories
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• Gauge Theories Labelled by Three-Manifolds
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• Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories
• Quantum Riemann Surfaces in Chern-Simons Theory
• Chern-Simons Theory and S-duality
• Hyperbolic 3-manifolds and Cluster Algebras

Cited by in corpus (11)

• Classification of Argyres-Douglas theories from M5 branes
• Central charges and RG flow of strongly-coupled N=2 theory
• Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions
• New Directions in Bipartite Field Theories
• Higher laminations, webs and N=2 line operators
• Aspects of line operators of class S theories
• Network, cluster coordinates and N = 2 theory II: Irregular singularity
• BPS spectrum, wall crossing and quantum dilogarithm identity
• Confinement and duality in supersymmetric gauge theories
• Hitchin systems in N=2 field theory
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I