Local energy bounds and strong locality in chiral CFT

TL;DR

This paper introduces a new method to establish strong locality in chiral conformal field theories by linking energy bounds to local properties, with applications to $ ext{W}_3$-algebra-based models.

Contribution

It provides a novel approach connecting energy bounds of degree d-1 to strong locality, expanding the class of models known to be strongly local.

Findings

Proves that energy bounds of degree d-1 imply strong locality in chiral CFTs.

Shows that certain $ ext{W}_3$-algebra models are strongly local and yields new conformal nets.

Demonstrates these nets are not strongly additive and not completely rational.

Abstract

A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. There are few methods of showing directly strong locality of a quantum field. Among them, linear energy bounds are the most widely used, yet a chiral conformal field of conformal weight $d>2$ cannot admit linear energy bounds. In this paper we give a new direct method to prove strong locality in two-dimensional conformal field theory. We prove that if a chiral conformal field satisfies an energy bound of degree $d-1$, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry. As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the $\mathcal{W}_3$-algebra is strongly local. For central charge $c > 2$, this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.