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

July 17, 2014

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].

Keywords:

*none*