Algebraic conformal nets, investigation of the locally constant case -- M1 internship supervised by Domenico Fiorenza

TL;DR

This paper develops a categorical framework for algebraic conformal nets, especially in the locally constant case, by defining a tricategory structure and relating it to commutative algebraic structures.

Contribution

It introduces a new categorical formulation of algebraic conformal nets and extends definitions for defects, including solitons, with a focus on the locally constant case.

Findings

Defined the tricategory of algebraic conformal nets, defects, sectors, and intertwiners.

Identified the locally constant case with the tricategory of commutative algebras and bimodules.

Proposed more general definitions for defects and solitons.

Abstract

We define the tricategory of algebraic conformal nets, defects, sectors and intertwiners where algebraic refers to the absence of a topology on the relevant algebras and modules. We aim at making these definitions satisfying from a categorical point of view, and in a few cases, e.g., for defects, we propose more general definitions than what can be found in the literature. This allows us in particular to propose an implementation of solitons as defects. Then we focus on the locally constant case where we provide an identification of the tricategory of locally constant algebraic conformal nets, defects, sectors and intertwiners with the tricategory of commutative algebras, algebras, bimodules and bimodule homomorphisms.