Calabi-Yau Varieties: from Quiver Representations to Dessins d'Enfants
  The connections amongst (1) quivers whose representation varieties are Calabi-Yau, (2) the combinatorics of bipartite graphs on Riemann surfaces, and (3) the geometry of mirror symmetry have engendered a rich subject at whose heart is the physics of gauge/string theories.   We review the various parts of this intricate story in some depth, for a mathematical audience without assumption of any knowledge of physics, emphasizing a plethora of results residing at the intersection between algebraic geometry, representation theory and number theory. 
Yang-Hui He
math.AG hep-th math-ph math.MP math.NT