Modular functors, cohomological field theories and topological recursion

Given a topological modular functor V in the sense of Walker, we construct vector bundles [Z _{⃗ λ} ] _g,n over M _g,n, whose Chern characters 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 indices of the Chern character with the ψ-classes in M _g,n is computed by the topological recursion of Eynard and Orantin, for a local spectral curve that we describe. In particular, we show how the Verlinde formula for the ranks [D _{⃗ λ} ] _g,n = rank [V _{⃗ λ} ] _g,n is recovered 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 _{⃗ λ} ] _g,n enumerates certain G-principle bundles over a genus g surface with n boundary conditions specified by ^⃗λ), and the modular functor obtained from Wess-Zumino-Witten conformal field theory associated to a simple, simply-connected Lie group G (for which [Z _{⃗ λ} ] _g,n is the Verlinde bundle).