Modular functors, cohomological field theories, and topological recursion

Abstract

Given a topological modular functor $\mathcal{V}$ in the sense of Walker \cite{Walker}, we construct vector bundles over $\bar{\mathcal{M}}_{g,n}$, whose Chern classes define semi-simple cohomological field theories. This construction depends on a determination of the logarithm of the eigenvalues of the Dehn twist and central element actions. We show that the intersection of the Chern class with the $\psi$-classes in $\bar{\mathcal{M}}_{g,n}$ is computed by the topological recursion of \cite{EOFg}, for a local spectral curve that we describe. In particular, we show how the Verlinde formula for the dimensions $D_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n}) = \dim \mathcal{V}_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ is retrieved from the topological recursion. We analyze the consequences of our result on two examples: modular functors associated to a finite group $G$ (for which $D_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ enumerates certain $G$-principle bundles over a genus $g$ surface with $n$ boundary conditions specified by $\vec{\lambda}$), and the modular functor obtained from Wess-Zumino-Witten conformal field theory associated to a simple, simply-connected Lie group $G$ (for which $\mathcal{V}_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ is the Verlinde bundle).