Modular Structure and Duality in Conformal Quantum Field Theory

TL;DR

This paper establishes an algebraic framework for conformal quantum field theories, demonstrating geometric interpretations of modular groups, duality properties, and the existence of PCT symmetry, extending to Poincaré covariant theories under certain conditions.

Contribution

It provides an algebraic version of the Bisognano-Wichmann theorem for conformal QFTs and explores duality, PCT symmetry, and the uniqueness of Poincaré representations.

Findings

Modular groups correspond to Lorentz transformations preserving wedges.

Essential duality holds on Minkowski space with extensions to the superworld.

PCT symmetry exists for algebraic conformal field theories in even dimensions.

Abstract

Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector concides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski space $M$, and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworld $\tilde M$, i.e. the universal covering of the Dirac-Weyl compactification of $M$. As a consequence a PCT symmetry exists for any algebraic conformal field theory in even space-time dimension. Analogous results hold for a Poincar\'e covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In particular the Poincar\'e representation is unique in this case.