# Local energy bounds and strong locality in chiral CFT

March 30, 2021

A family of quantum fields is said to be strongly local if it generates a
local net of von Neumann algebras. There are very limited 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. 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.

Keywords:

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