Marcel Bischoff

Tensor categories and endomorphisms of von Neumann algebras

Introduction.- Homomorphisms of von Neumann algebras.- Endomorphisms of infinite factors.- Homomorphisms and subfactors.- Non-factorial extensions.- Frobenius algebras, Q-systems and modules.- C* Frobenius algebras.- Q-systems and extensions.- The canonical Q-system.- Modules of Q-systems.- Induced Q-systems and Morita equivalence.- Bimodules.- Tensor product of bimodules.- Q-system calculus.- Reduced Q-systems.- Central decomposition of Q-systems.- Irreducible decomposition of Q-systems.- Intermediate Q-systems.- Q-systems in braided tensor categories.- a-induction.- Mirror Q-systems.- Centre of Q-systems.- Braided product of Q-systems.- The full centre.- Modular tensor categories.- The braided product of two full centres.- Applications in QFT.- Basics of algebraic quantum field theory.- Hard boundaries.- Transparent boundaries.- Further directions.- Conclusions.

Construction of Wedge-Local Nets of Observables through Longo-Witten Endomorphisms. II

In the first part, we have constructed several families of interacting wedge-local nets of von Neumann algebras. In particular, we discovered a family of models based on the endomorphisms of the U(1)-current algebra

Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models

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Homomorphisms of von Neumann Algebras

of Longo-Witten.In this second part, we further investigate endomorphisms and interacting models. The key ingredient is the free massless fermionic net, which contains the U(1)-current net as the fixed point subnet with respect to the U(1) gauge action. Through the restriction to the subnet, we construct a new family of Longo-Witten endomorphisms on

Frobenius Algebras, Q-Systems and Modules

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Characterization of 2D Rational Local Conformal Nets and Its Boundary Conditions: the Maximal Case

and accordingly interacting wedge-local nets in two-dimensional spacetime. The U(1)-current net admits the structure of particle numbers and the S-matrices of the models constructed here do mix the spaces with different particle numbers of the bosonic Fock space.