From vertex operator algebras to conformal nets and back. (English) Zbl 1434.17001
Memoirs of the American Mathematical Society 1213. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2858-7/print; 978-1-4704-4742-7/ebook). vi, 90 p. (2018).
Publisher’s description: 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 \(V\).
MSC:
| 17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
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