Topological recursion, topological quantum field theory and Gromov-Witten invariants of BG

The purpose of this paper is to give a decorated version of the Eynard-Orantin topological recursion using a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic reformulation of a topological recursion to define how to decorate a standard topological recursion by a 2D TQFT. The A-model side enumerative problem consists of counting cell graphs where in addition vertices are decorated by elements in a Frobenius algebra, and which are a decorated version of the generalized Catalan numbers. We show that the function that counts these decorated graphs, which is a decoration of the counting function of the generalized Catalan numbers by a Frobenius algebra, satisfies a topological recursion with respect to the edge-contraction axioms. The path we follow to pass from the A-model side to the remodeled B-model side is to use a discrete Laplace transform as a mirror symmetry map. We show that a decorated version by a 2D TQFT of the Eynard-Orantin differentials satisfies a decorated version of the Eynard-Orantin recursion formula. We illustrate these results using a toy model for the theory arising from the orbifold cohomology of the classifying space of a finite group. In this example, the graphs are orbifold cell graphs (graphs drawn on an orbifold punctured Riemann surface) defined out of the moduli space (M) over bar (g,n)(BG) of stable morphisms from twisted curves to the classifying space of a finite group G. In particular we show that the cotangent class intersection numbers on the moduli space (M) over bar (g,n)(BG) satisfy a decorated Eynard-Orantin topological recursion and we derive an orbifold DVV equation as a consequence of it. This proves from a different perspective the known result which states that the psi-class intersection numbers on (M) over bar (g,n)(BG) satisfy the Virasoro constraint condition.