From Vertex Operator Algebras to Conformal Nets and Back

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 A(V) acting on the Hilbert space completion of V and prove that the isomorphism class of 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 (sic) A(W) gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of 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 Jor beta gives back the strongly local vertex operator algebra V from the conformal net A(V) and give conditions on a conformal net A implying that A = A(V) for some strongly local vertex operator algebra V.