Geometric realization of algebraic conformal field theories

We explore new connections between the fields and local observables in two dimensional chiral conformal field theory. We show that in a broad class of examples, the von Neumann algebras of local observables (a conformal net) can be obtained from the fields (a unitary vertex operator algebra) via a continuous geometric interpolation procedure involving Graeme Segal's functorial definition of conformal field theory, and that the conformal net may be thought of as a boundary value of the Segal CFT. In particular, we construct conformal nets from these unitary vertex operator algebras by showing that ‘geometrically mollified’ versions of the fields yield bounded, local observables on the Hilbert space completion of the vertex algebra. These are the first results which unite the three major definitions of chiral conformal field theory. This work is inspired by Henriques' picture of conformal nets arising from degenerate Riemann surfaces.