  We use string-net models to accomplish a direct, purely two-dimensional,
approach to correlators of two-dimensional rational conformal field theories.
We obtain concise geometric expressions for the objects describing bulk and
boundary fields in terms of idempotents in the cylinder category of the
underlying modular fusion category, comprising more general classes of fields
than is standard in the literature. Combining these idempotents with Frobenius
graphs on the world sheet yields string nets that form a consistent system of
correlators, i.e. a system of invariants under appropriate mapping class groups
that are compatible with factorization.
  Using markings, we extract operator products of field objects from specific
correlators; the resulting operator products are natural algebraic expressions
that make sense beyond semisimplicity. We also derive an Eckmann-Hilton
relation internal to a braided category, thereby demonstrating the utility of
string nets for understanding algebra in braided tensor categories. Finally we
introduce the notion of a universal correlator. This systematizes the treatment
of situations in which different world sheets have the same correlator and
allows for the definition of a more comprehensive mapping class group.
