Conformal nets I: coordinate-free nets

TL;DR

This paper introduces a coordinate-free framework for conformal nets using functors from intervals to von Neumann algebras, establishing foundational structures for a 3-category of conformal nets and related concepts.

Contribution

It presents a novel coordinate-free perspective on conformal nets and constructs the first step towards a 3-category framework including defects and sectors.

Findings

Fusion of intervals corresponds to fiber products of von Neumann algebras.

A canonical vacuum sector exists for any circle with a conformal structure.

Lays groundwork for a 3-category of conformal nets, defects, sectors, and intertwiners.

Abstract

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.