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From Vertex Operator Algebras to Conformal Nets and Back

About this Title

Sebastiano Carpi, Dipartimento di Economia, Università di Chieti-Pescara “G. d’Annunzio”, Viale Pindaro, 42, I-65127 Pescara, Italy, Yasuyuki Kawahigashi, Department of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan and Kavli IPMU (WPI), the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan, Roberto Longo, Dipartimento di Matematica, Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 1, I-00133 Roma, Italy and Mihály Weiner, Mathematical Institute, Department of Analysis, Budapest University of Technology & Economics (BME), Müegyetem rk. 3-9, H-1111 Budapest, Hungary

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 254, Number 1213
ISBNs: 978-1-4704-2858-7 (print); 978-1-4704-4742-7 (online)
Published electronically: March 29, 2018

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Table of Contents


  • 1. Introduction
  • 2. Preliminaries on von Neumann algebras
  • 3. Preliminaries on conformal nets
  • 4. Preliminaries on vertex algebras
  • 5. Unitary vertex operator algebras
  • 6. Energy bounds and strongly local vertex operator algebras
  • 7. Covariant subnets and unitary subalgebras
  • 8. Criteria for strong locality and examples
  • 9. Back to vertex operators
  • A. Vertex algebra locality and Wightman locality
  • B. On the Bisognano-Wichmann property for representations of the Mobius̈ group


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 $V$.

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