Conformal nets I: coordinate-free nets

Understand

We describe a coordinate-free perspective on conformal nets, as functors from intervals to von Neumann algebras.

• We discuss an operation of fusion of intervals and observe that a conformal net takes a fused interval to the fiber product of von Neumann algebras.
• Though coordinate-free nets do not a priori have vacuum sectors, we show that there is a vacuum sector canonically associated to any circle equipped with a conformal structure.
• This is the first in a series of papers constructing a 3-category of conformal nets, defects, sectors, and intertwiners.