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

Abstract: The purpose of this paper is to give a twisted version of the Eynard-Orantin topological recursion by a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic definition for a topological recursion to define how to twist 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 twisted version of the generalized Catalan numbers of Dumitrescu-Mulase-Safnuk-Sorkin. We show that the function which counts these decorated graphs satisfies a twisted version of the same type of recursion of Catalan numbers with respect to the edge-contraction axioms of Dumitrescu-Mulase. The path we follow to pass from the A-model side to the remodelled B-model side is to use a discrete Laplace transform based on the ideas of the group of Mulase. We show that a twisted version by a 2D TQFT of the Eynard-Orantin differentials satisfies a twisted generalization of the topological recursion formula. We shall illustrate these results with 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 drawn on an orbifold punctured Riemann surface and defined out of the moduli space of stable morphisms from twisted curves to the classifying space of a finite group. In particular we show that the cotangent class intersection numbers on this moduli space satisfy a twisted Eynard-Orantin topological recursion and we derive an orbifold DVV equation as a consequence of it. This proves from a different perspective a result of Jarvis-Kimura, which states that the cotangent class intersection numbers on that moduli space satisfy the Virasoro constraint condition.