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Bipartite Graphs, Quivers and Gauge Theories

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We discuss how bipartite graphs on Riemann surfaces encapture a wealth of information about the physics of large classes of supersymmetric gauge theories, especially those with quiver structure and arising from the AdS/CFT context. The correspondence between the gauge theory, the underlying algebraic geometry of it space of vacua, the combinatorics of dimers and toric varieties, as well as the number theory of dessin d'enfants becomes particular intricate under this light.