From Vertex Operator Algebras to Conformal Nets and Back

Abstract

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 → 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 Jörß 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 .if S ⊂ B(H) is self-adjoint then S ′ is a self-adjoint subalgebra of B(H) which is also unital, i.e. 1 H ∈ S ′ .A self-adjoint subalgebra M ⊂ B(H) is called a von Neumann algebra if M = M ′′ . Accordingly, (S ∪ S * ) ′ is a von Neumann algebra for all subsets S ⊂ B(H) and W * (S) ≡ (S ∪ S * ) ′′ is the smallest von Neumann algebra containing S.A von Neumann algebra M is said to be a factor if M ′ ∩ M = C1 H , i.e. M has a trivial center. B(H) is a factor for any Hilbert space H. Its isomorphism class as an abstract complex * -algebra only depends on the Hilbertian dimension of H. A von Neumann algebra M isomorphic to some B(H) (here H is not necessarily the same Hilbert space on which M acts) is called a type I factor. If H has dimension n ∈ Z >0 then M is called a type I n factor while if H is infinite-dimensional then M is called a type I ∞ factor.There exist factors which are not of type I. They are divided in two families: the type II factors (type II 1 or type II ∞ ) and type III factors (type III λ , λ ∈ [0, 1], cf.[23]).If M and N are von Neumann algebras and N ⊂ M then N is called a von Neumann subalgebra of M. If M is a factor then a von Neumann subalgebra N ⊂ M which is also a factor is called a subfactor. The theory of subfactors is a central topic in the theory of operator algebras and in its applications to quantum field theory. Subfactor theory was initiated in the seminal work [56] where V. Jones introduced and studied an index [M : N] for type II 1 factors. Subfac...

Entanglement properties of boundary state and thermalization

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“…Specially as shown in paper [27], the smeared chiral vertex operators of 2D scalar field f (z)e −iαφ(z) dz is a bounded operator. In paper [26] the authors discuss how to construct the local observables by the vertex algebras. Here we won't discuss the details of the construction, but only assume the local bounded operators exists, they can be constructed by (qusai-)primary operators and suitable smearing functions.…”

Some exceptional extensions of Virasoro vertex operator algebras

“…It is well-known that the central charges of rational Virasoro vertex operator algebras belong to a discrete set (see the formula (3.1)), and there is an important subclass of rational Virasoro vertex operator algebras which are unitary vertex operator algebras [15]. Connections between unitary vertex operator algebras and conformal nets have been studied recently in [7]. In particular, each chiral half of a unitary rational conformal field theory can be also controlled by a conformal net.…”

Classification of unitary vertex subalgebras and conformal subnets for rank-one lattice chiral CFT models

“…It is generally believed that these two framework should be more or less equivalent in the unitary case but for a long time a direct general connection between those two approaches had not been clear. However, a significant step forward has been achieved recently in [10] where a map from a suitable class of unitary VOAs, the class of strongly local VOAs, to the class of conformal nets has been defined. Moreover, many examples of unitary VOAs have been shown to be strongly local and it is conjectured that the map gives in fact a one-to-one correspondence between (simple) unitary VOAs Date: August 26, 2019.…”

“…These subtheories are called conformal subnets or unitary vertex subalgebras (or simply unitary subalgebras) depending on the chosen framework. For a conformal net A V that comes from a strongly local VOA V , it has been shown in [10] that the conformal subnets A ⊂ A V are in one-to-one correspondence with the unitary subalgebras W ⊂ V . The study and the classification of these subtheories is a very natural problem in either of the two approaches.…”