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

Abstract

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 ${\overline{\mathcal{M}}}_{g,n}(BG)$ 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 ${\overline{\mathcal{M}}}_{g,n}(BG)$ 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 ${\overline{\mathcal{M}}}_{g,n}(BG)$ satisfy the Virasoro constraint condition.