Scale and M\"obius covariance in two-dimensional Haag-Kastler net

TL;DR

This paper proves that certain two-dimensional quantum field theory models with Poincaré-dilation symmetry can be extended to models with full M"obius symmetry, under specific conditions, without relying on stress-energy tensors or scaling dimensions.

Contribution

It establishes conditions under which Poincaré-dilation covariant nets can be extended to M"obius covariant nets in two dimensions, without assuming stress-energy tensors.

Findings

Extension is possible under modular covariance or strong additivity.

Examples are provided of nets that cannot be extended to M"obius covariance.

Obstructions to extension are discussed.

Abstract

Given a two-dimensional Haag-Kastler net which is Poincar\'e-dilation covariant with additional properties, we prove that it can be extended to a M\"obius covariant net. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity. The proof relies neither on the existence of stress-energy tensor nor any assumption on scaling dimensions. We exhibit some examples of Poincar\'e-dilation covariant net which cannot be extended to a M\"obius covariant net, and discuss the obstructions.