  Conformal nets provide a mathematical formalism for conformal field theory.
Associated to a conformal net with finite index, we give a construction of the
`bundle of conformal blocks', a representation of the mapping class groupoid of
closed topological surfaces into the category of finite-dimensional projective
Hilbert spaces. We also construct infinite-dimensional spaces of conformal
blocks for topological surfaces with smooth boundary. We prove that the
conformal blocks satisfy a factorization formula for gluing surfaces along
circles, and an analogous formula for gluing surfaces along intervals. We use
this interval factorization property to give a new proof of the modularity of
the category of representations of a conformal net.
