Covariant Sectors with Infinite Dimension and Positivity of the Energy

TL;DR

This paper investigates the energy positivity in Moebius covariant sectors of local conformal nets, establishing conditions under which sectors have positive energy, especially in infinite-dimensional cases, with implications for sector stability.

Contribution

It provides a criterion for positive energy in infinite index sectors and analyzes the stability of positive energy sectors under various operations.

Findings

Finite index sectors automatically have positive energy.

Infinite index sectors have spectrum containing positive reals but may include negatives.

Existence of an unbounded Moebius covariant left inverse characterizes positive energy sectors.

Abstract

We consider a Moebius covariant sector, possibly with infinite dimension, of a local conformal net of von Neumann algebras on the circle. If the sector has finite index, it has automatically positive energy. In the infinite index case, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on the real line, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded Moebius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition.