Conformal nets IV: The 3-category

TL;DR

This paper constructs a symmetric monoidal 3-category structure for conformal nets, defects, sectors, and intertwiners, capturing the algebraic interactions among conformal field theories.

Contribution

It establishes the 3-category framework for conformal nets, including defects and sectors, advancing the mathematical understanding of conformal field theory interactions.

Findings

The collection forms a symmetric monoidal 3-category.

Fusion operations define composition of defects and sectors.

The 3-category encodes interactions among conformal field theories.

Abstract

Conformal nets are a mathematical model for conformal field theory, and defects between conformal nets are a model for an interaction or phase transition between two conformal field theories. In the preceding paper of this series, we introduced a notion of composition, called fusion, between defects. We also described a notion of sectors between defects, modeling an interaction among or transformation between phase transitions, and defined fusion composition operations for sectors. In this paper we prove that altogether the collection of conformal nets, defects, sectors, and intertwiners, equipped with the fusion of defects and fusion of sectors, forms a symmetric monoidal 3-category. This 3-category encodes the algebraic structure of the possible interactions among conformal field theories.