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...

A Möbius covariant net of von Neumann algebras on S 1 is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Möbius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Möbius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff + (S 1 ).

We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c = 1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge c is in a certain subset of (1, ∞), including [2, ∞), and h ≥ (c − 1)/24, the irreducible representation with lowest weight h of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge c is in the above set and satisfies c ≤ 25 then the corresponding Virasoro net has no proper local extensions of compact type.

We study the general structure of Fermi conformal nets of von Neumann algebras on S 1 , consider a class of topological representations, the general representations, that we characterize as Neveu-Schwarz or Ramond representations, in particular a Jones index can be associated with each of them. We then consider a supersymmetric general representation associated with a Fermi modular net and give a formula involving the Fredholm index of the supercharge operator and the Jones index. We then consider the net associated with the super-Virasoro algebra and discuss its structure. If the central charge c belongs to the discrete series, this net is modular by the work of F. Xu and we get an example where our setting is verified by considering the Ramond irreducible representation with lowest weight c/24. We classify all the irreducible Fermi extensions of any super-Virasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2.

Let F be a local net of von Neumann algebras in four spacetime dimensions satisfying certain natural structural assumptions. We prove that if F has trivial superselection structure then every covariant, Haag-dual subsystem B is of the form F G 1 ⊗ I for a suitable decomposition F = F 1 ⊗ F 2 and a compact group action. Then we discuss some application of our result, including free field models and certain theories with at most countably many sectors.

We construct infinite dimensional spectral triples associated with representations of the super-Virasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h = c/24 is graded and gives rise to a net of even θ-summable spectral triples with non-zero Fredholm index. The irreducible unitary positive energy representations of the Neveu-Schwarz algebra give rise to nets of even θ-summable generalised spectral triples where there is no Dirac operator but only a superderivation.

This paper provides a further step in the program of studying superconformal nets over S 1 from the point of view of noncommutative geometry. For any such net A and any family ∆ of localized endomorphisms of the even part A γ of A, we define the locally convex differentiable algebra A ∆ with respect to a natural Dirac operator coming from supersymmetry. Having determined its structure and properties, we study the family of spectral triples and JLO entire cyclic cocycles associated to elements in ∆ and show that they are nontrivial and that the cohomology classes of the cocycles corresponding to inequivalent endomorphisms can be separated through their even or odd index pairing with K-theory in various cases. We illustrate some of those cases in detail with superconformal nets associated to well-known CFT models, namely supercurrent algebra nets and super-Virasoro nets. All in all, the result allows us to encode parts of the representation theory of the net in terms of noncommutative geometry.

We study the representation theory of a conformal net A on the circle from a K-theoretical point of view using its universal C*-algebra C*(A). We prove that if A satisfies the split property then, for every representation \pi of A with finite statistical dimension, \pi(C*(A)) is weakly closed and hence a finite direct sum of type I_\infty factors. We define the more manageable locally normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C*_ln(A) is a direct sum of n type I_\infty factors. Its ideal K_A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C*(A) with finite statistical dimension act on K_A, giving rise to an action of the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.Comment: v2: we added some comments in the introduction and new references. v3: new authors' addresses, minor corrections. To appear in Commun. Math. Phys. v4: minor corrections, updated reference