From Vertex Operator Algebras to Conformal Nets and Back.
- 1st ed.
- 1 online resource (97 pages)
- Memoirs of the American Mathematical Society Series ; v.254 .
- Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Preliminaries on von Neumann algebras -- 2.1. Von Neumann algebras -- 2.2. Unbounded operators affiliated with von Neumann algebras -- 2.3. Tomita-Takesaki modular theory -- Chapter 3. Preliminaries on conformal nets -- 3.1. \diff and its subgroup \mob -- 3.2. Positive-energy projective unitary representations of \diff and of \difftilde and positive-energy representations of -- 3.3. Möbius covariant nets and conformal nets on \s1 -- 3.4. Covariant subnets -- Chapter 4. Preliminaries on vertex algebras -- 4.1. Vertex algebras -- 4.2. Conformal vertex algebras -- 4.3. Vertex operator algebras and invariant bilinear forms -- Chapter 5. Unitary vertex operator algebras -- 5.1. Definition of unitarity -- 5.2. An equivalent approach to unitarity -- 5.3. Unitary automorphisms and essential uniqueness of the unitary structure -- 5.4. Unitary subalgebras -- Chapter 6. Energy bounds and strongly local vertex operator algebras -- Chapter 7. Covariant subnets and unitary subalgebras -- Chapter 8. Criteria for strong locality and examples -- Chapter 9. Back to vertex operators -- Appendix A. Vertex algebra locality and Wightman locality -- Appendix B. On the Bisognano-Wichmann property for representations of the Mobius group -- Acknowledgments -- Bibliography -- Back Cover.
The authors consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. They 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. They 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.