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

In this article we give new examples of models in boundary quantum field theory, i.e. local time-translation covariant nets of von Neumann algebras, using a recent construction of Longo and Witten, which uses a local conformal net A on the real line together with an element of a unitary semigroup associated with A. Namely, we compute elements of this semigroup coming from Hölder continuous symmetric inner functions for a family of (completely rational) conformal nets which can be obtained by starting with nets of real subspaces, passing to its second quantization nets and taking local extensions of the former. This family is precisely the family of conformal nets associated with lattices, which as we show contains as a special case the level 1 loop group nets of simply connected, simply laced groups. Further examples come from the loop group net of Spin(n) at level 2 using the orbifold construction.