# From vertex operator algebras to conformal nets and back

March 04, 2015

We consider unitary simple vertex operator algebras whose vertex operators
satisfy certain energy bounds and a strong form of locality and call them
strongly local. We present a general procedure which associates to every
strongly local vertex operator algebra $V$ a conformal net $\mathcal{A}_V$ acting on the
Hilbert space completion of $V$ and prove that the isomorphism class of $\mathcal{A}_V$ does
not depend on the choice of the scalar product on $V$. We show that the class of
strongly local vertex operator algebras is closed under taking tensor products
and unitary subalgebras and that, for every strongly local vertex operator
algebra $V$, the map $W\mapsto \mathcal{A}_W$ gives a one-to-one correspondence between the
unitary subalgebras $W$ of $V$ and the covariant subnets of $\mathcal{A}_V$. Many known
examples of vertex operator algebras such as the unitary Virasoro vertex
operator algebras, the unitary affine Lie algebras vertex operator algebras,
the known $c=1$ unitary vertex operator algebras, the moonshine vertex operator
algebra, together with their coset and orbifold subalgebras, turn out to be
strongly local. We give various applications of our results. In particular we
show that the even shorter Moonshine vertex operator algebra is strongly local
and that the automorphism group of the corresponding conformal net is the Baby
Monster group. We prove that a construction of Fredenhagen and Jörss gives
back the strongly local vertex operator algebra $V$ from the conformal net $\mathcal{A}_V$
and give conditions on a conformal net $\mathcal{A}$ implying that $\mathcal{A}= \mathcal{A}_V$ for some strongly
local vertex operator algebra $V$.

Keywords:

conformal qft, vertex operator algebras