Gauge Theories: Quivers, Dessins and Calabi-Yau

We discuss how bipartite graphs on Riemann surfaces capture a wealth of information about the physics and the mathematics of large classes of gauge theories, especially those arising from string theory in the context of string theory.
The dialogue between the physics, the underlying algebraic geometry of Calabi-Yau varieties, the combinatorics of dimers and toric varieties, as well as the number theory of dessin d'enfants becomes particularly intricate and fruitful under this light.