Local energy bounds and strong locality in chiral CFT

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.

arXiv comment27 pages, no figure. The constant C' in Proposition 3.3 replaced, main results unaffected

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Cited by in corpus (4)

• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Unitary representations of the ${\mathcal{W}}_3$-algebra with $c\ge 2$
• Wightman fields for two-dimensional conformal field theories with pointed representation category
• Unitary vertex algebras and Wightman conformal field theories