Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)

TL;DR

This paper explores the mathematical framework of Q-systems within tensor categories and their applications to extensions of von Neumann algebras, with significant implications for local quantum field theory and conformal models.

Contribution

It provides a detailed review of Q-systems, their categorical structures, and operations, connecting these to von Neumann algebra homomorphisms and applications in quantum field theory.

Findings

Q-systems encode extensions of von Neumann algebras.

Operations like decompositions and centers are clarified in the algebraic context.

Applications include modeling boundary conditions in conformal quantum field theories.

Abstract

Q-systems describe "extensions" of an infinite von Neumann factor $N$, i.e., finite-index unital inclusions of $N$ into another von Neumann algebra $M$. They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of $N$. We review the relation between Q-systems, their modules and bimodules as structures in a category on one side, and homomorphisms between von Neumann algebras on the other side. We then elaborate basic operations with Q-systems (various decompositions in the general case, and the centre, the full centre, and the braided product in braided categories), and illuminate their meaning in the von Neumann algebra setting. The main applications are in local quantum field theory, where Q-systems in the subcategory of DHR endomorphisms of a local algebra encode extensions $A(O)\subset B(O)$ of local nets. These applications, notably in conformal quantum field theories with boundaries, are briefly exposed, and are discussed in more detail in two separate papers [arXiv:1405.7863, 1410.8848].