Modular functors, cohomological field theories, and topological recursion

Given a topological modular functor V in the sense of Walker, we construct vector bundles Z (lambda) over bar, over (M) over bar (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 1P-classes in (M) over bar (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 (lambda) over bar= rank nu(lambda) over bar 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 (lambda) over bar enumerates certain G-principle bundles over a genus g surface with n boundary conditions specified by and the modular functor obtained from Wess-Zumino-Witten conformal field theory associated to a simple, simply connected Lie group G (for which Z (lambda) over bar is the Verlinde bundle).