45 research outputs found
The Virasoro algebra and sectors with infinite statistical dimension
We show that the sectors with lowest weight , , j\in
{1/2}\ZZ of the local net of von Neumann algebras on the circle generated by
the Virasoro algebra with central charge c=1 have infinite statistical
dimension.Comment: 14 pages, minor changes, one reference adde
On the representation theory of Virasoro Nets
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 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
is in a certain subset of , including , and , the irreducible representation with lowest weight of the
corresponding Virasoro net has infinite statistical dimension. As a consequence
we show that if the central charge is in the above set and satisfies then the corresponding Virasoro net has no proper local extensions of
compact type.Comment: 34 page
Classification of Subsystems for Local Nets with Trivial Superselection Structure
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
the fixed point net under a compact group action on one component in a suitable
tensor product decomposition of F. Then we discuss some application of our
result, including free field models and certain theories with at most countably
many sectors.Comment: 31 pages, LaTe
Intersecting Jones projections
Let M be a von Neumann algebra on a Hilbert space H with a cyclic and
separating unit vector \Omega and let \omega be the faithful normal state on M
given by \omega(\cdot)=(\Omega,\cdot\Omega). Moreover, let {N_i :i\in I} be a
family of von Neumann subalgebras of M with faithful normal conditional
expectations E_i of M onto N_i satisfying \omega=\omega\circ E_i for all i\in I
and let N=\bigcap_{i\in I} N_i. We show that the projections e_i, e of H onto
the closed subspaces \bar{N_i\Omega} and \bar{N\Omega} respectively satisfy
e=\bigwedge_{i\in I}e_i.This proves a conjecture of V.F.R. Jones and F. Xu in
\cite{JonesXu04}
Conformal nets and KK-theory
Given a completely rational conformal net A on the circle, its fusion ring
acts faithfully on the K_0-group of a certain universal C*-algebra associated
to A, as shown in a previous paper. We prove here that this action can actually
be identified with a Kasparov product, thus paving the way for a fruitful
interplay between conformal field theory and KK-theory
Structure and Classification of Superconformal Nets
We study the general structure of Fermi conformal nets of von Neumann
algebras on the circle, 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.Comment: 49 pages. Section 8 has been removed. More details concerning the
diffeomorphism covariance are give
From vertex operator algebras to conformal nets and back
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\mapsto 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 J\"{o}rss 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.Comment: Minor correction
Energy bounds for vertex operator algebra extensions
Let V be a simple unitary vertex operator algebra and U be a (polynomially)
energy-bounded unitary subalgebra containing the conformal vector of V. We give
two sufficient conditions implying that V is energy-bounded. The first
condition is that U is a compact orbifold for some compact group G of unitary
automorphisms of V. The second condition is that V is exponentially
energy-bounded and it is a finite direct sum of simple U-modules. As
consequence of the second condition, we prove that if U is a regular
energy-bounded unitary subalgebra of a simple unitary vertex operator V, then
is energy-bounded. In particular, every simple unitary extension (with the
same conformal vector) of a simple unitary affine vertex operator algebra
associated with a semisimple Lie algebra is energy-bounded.Comment: 19 page
On the uniqueness of diffeomorphism symmetry in Conformal Field Theory
A Moebius covariant net of von Neumann algebras on S^1 is diffeomorphism
covariant if its Moebius 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 Moebius
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 Moebius 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).Comment: 25 pages, LaTe