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

Gandalf Lechner,Charley Scotford

doi:10.1007/s00220-022-04324-x

Gandalf Lechner, Charley Scotford

Open Access

https://doi.org/10.1007/s00220-022-04324-x

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Abstract

We construct explicit examples of half-sided modular inclusions mathcal {N}subset mathcal {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} with large relative commutant and construct a one-parameter family mathcal {N}_kappa subset mathcal {M}_kappa , kappa ge 0, of half-sided inclusions such that mathcal {N}_0=mathcal {N}, mathcal {M}_0=mathcal {M} and mathcal {N}_kappa 'cap mathcal {M}_kappa =mathbb {C}1 for kappa >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.

Highlights

In the operator-algebraic approach to quantum field theory [Haa96,Ara99], models of quantum field theories on a spacetime M are described by assigning to open regions O ⊂ M von Neumann algebras A(O) that act on a common Hilbert space and are subject to various interrelated inclusion, commutation, covariance, and spectral properties
After the free product construction of Longo, Tanimoto and Ueda [LTU19], the halfsided inclusions constructed in this paper provide further and arguably simpler examples of singular Borchers triples, giving further insight into the structure of the family of all half-sided modular inclusions
It became apparent that half-sided inclusions are very sensitive to deformations and their relative commutant can vary “discontinuously” with a deformation parameter

Summary

Introduction

In the operator-algebraic approach to quantum field theory [Haa96,Ara99], models of quantum field theories on a spacetime M are described by assigning to open regions O ⊂ M von Neumann algebras A(O) that act on a common Hilbert space and are subject to various interrelated inclusion, commutation, covariance, and spectral properties. A possible point of view is to fix one of the local algebras M = A(O0), typically isomorphic [BFD87] to the unique hyperfinite factor of type III1 [Haa87] and aim at constructing the net O → A(O) of all local algebras with the help of group actions, generating von Neumann algebras, and relative commutants – see [BW92,GLW98,BL04,BLS11,Tan14,BJM21] for various implementations of this and related ideas. In these approaches, it is often of central importance to make sure that the relative commutant of an inclusion A(O) ⊂ A(O ) is large in some sense or at least nontrivial.

Three Types of Half-Sided Modular Inclusions

Half-Sided Inclusions and the Algebra at Infinity

Definition of the deformed and undeformed inclusions

Two-dimensional nets and warped convolution

Analysis of the deformed inclusions

Conclusion and Outlook