Edge-contraction on Dual Ribbon Graphs, 2d Tqft, and the Mirror of Orbifold Hurwitz Numbers

We present a new set of axioms for a 2D TQFT formulated on the category of the dual of ribbon graphs with edge-contraction operations as morphisms. Every Frobenius algebra A determines a contravariant functor from this category to the category of elements of the tensor category of Frobenius algebras over A∗. The functor evaluated at connected graphs is the TQFT corresponding to A. The edge-contraction operations also form an effective tool in various graph enumeration problems, such as counting Grothendieck’s dessins d’enfants, simple and double Hurwitz numbers. These counting problems can be solved by the topological recursion, which is the universal mirror B-model corresponding to these counting problems. We show that for the case of orbifold Hurwitz numbers, the mirror objects, the spectral curves and the differential forms on it, are constructed purely from the edge-contraction operations of the counting problem.