Deformations of Half-Sided Modular Inclusions and Non-local Chiral Field Theories

Abstract

We construct explicit examples of half-sided modular inclusions $$\mathcal {N}\subset \mathcal {M}$$ N ⊂ M of von Neumann algebras with trivial relative commutants. After stating a general criterion for triviality of the relative commutant in terms of an algebra localized at infinity, we consider a second quantization inclusion $$\mathcal {N}\subset \mathcal {M}$$ N ⊂ M with large relative commutant and construct a one-parameter family $$\mathcal {N}_\kappa \subset \mathcal {M}_\kappa $$ N κ ⊂ M κ , $$\kappa \ge 0$$ κ ≥ 0 , of half-sided inclusions such that $$\mathcal {N}_0=\mathcal {N}$$ N 0 = N , $$\mathcal {M}_0=\mathcal {M}$$ M 0 = M and $$\mathcal {N}_\kappa '\cap \mathcal {M}_\kappa =\mathbb {C}1$$ N κ ′ ∩ M κ = C 1 for $$\kappa >0$$ κ > 0 . The technique we use is an explicit deformation procedure (warped convolution), and we explain the relation of this result to the construction of chiral conformal quantum field theories on the real line and on the circle.