88 research outputs found
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
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
N=2 superconformal nets
We provide an Operator Algebraic approach to N=2 chiral Conformal Field
Theory and set up the Noncommutative Geometric framework. Compared to the N=1
case, the structure here is much richer. There are naturally associated nets of
spectral triples and the JLO cocycles separate the Ramond sectors. We construct
the N=2 superconformal nets of von Neumann algebras in general, classify them
in the discrete series c<3, and we define and study an operator algebraic
version of the N=2 spectral flow. We prove the coset identification for the N=2
super-Virasoro nets with c<3, a key result whose equivalent in the vertex
algebra context has seemingly not been completely proved so far. Finally, the
chiral ring is discussed in terms of net representations.Comment: 42 pages. Final version to be published in Communications in
Mathematical Physic
Classification of subsystems for graded-local nets with trivial superselection structure
We classify Haag-dual Poincar\'e covariant subsystems \B\subset \F of a
graded-local net \F on 4D Minkowski spacetime which satisfies standard
assumptions and has trivial superselection structure. The result applies to the
canonical field net \F_\A of a net \A of local observables satisfying
natural assumptions. As a consequence, provided that it has no nontrivial
internal symmetries, such an observable net \A is generated by (the abstract
versions of) the local energy-momentum tensor density and the observable local
gauge currents which appear in the algebraic formulation of the quantum Noether
theorem. Moreover, for a net \A of local observables as above, we also
classify the Poincar\'e covariant local extensions \B \supset \A which
preserve the dynamics.Comment: 38 pages, LaTe
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→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
Thermal States in Conformal QFT. II
We continue the analysis of the set of locally normal KMS states w.r.t. the
translation group for a local conformal net A of von Neumann algebras on the
real line. In the first part we have proved the uniqueness of KMS state on
every completely rational net. In this second part, we exhibit several
(non-rational) conformal nets which admit continuously many primary KMS states.
We give a complete classification of the KMS states on the U(1)-current net and
on the Virasoro net Vir_1 with the central charge c=1, whilst for the Virasoro
net Vir_c with c>1 we exhibit a (possibly incomplete) list of continuously many
primary KMS states. To this end, we provide a variation of the
Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework:
if there is an inclusion of split nets A in B and A is the fixed point of B
w.r.t. a compact gauge group, then any locally normal, primary KMS state on A
extends to a locally normal, primary state on B, KMS w.r.t. a perturbed
translation. Concerning the non-local case, we show that the free Fermi model
admits a unique KMS state.Comment: 36 pages, no figure. Dedicated to Rudolf Haag on the occasion of his
90th birthday. The final version is available under Open Access. This paper
contains corrections to the Araki-Haag-Kaster-Takesaki theorem (and to a
proof of the same theorem in the book by Bratteli-Robinson). v3: a reference
correcte
Spectral triples and the super-Virasoro algebra
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 theta-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 theta-summable generalised spectral triples
where there is no Dirac operator but only a superderivation.Comment: 27 pages; v2: a comment concerning the difficulty in defining cyclic
cocycles in the NS case have been adde
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