  Representations of certain vertex algebras, here called of CohFT-type, can be
used to construct vector bundles of coinvariants and conformal blocks on moduli
spaces of stable curves [DGT2]. We show that such bundles define semisimple
cohomological field theories. As an application, we give an expression for
their total Chern character in terms of the fusion rules, following the
approach and computation in [MOPPZ] for bundles given by integrable modules
over affine Lie algebras. It follows that the Chern classes are tautological.
Examples and open problems are discussed.
