14,396 research outputs found
Nets of Subfactors
A subtheory of a quantum field theory specifies von~Neumann subalgebras
\aa(\oo) (the `observables' in the space-time region \oo) of the
von~Neumann algebras \bb(\oo) (the `fields' localized in \oo). Every local
algebra being a (type \III_1) factor, the inclusion \aa(\oo) \subset
\bb(\oo) is a subfactor. The assignment of these local subfactors to the
space-time regions is called a `net of subfactors'. The theory of subfactors is
applied to such nets. In order to characterize the `relative position' of the
subtheory, and in particular to control the restriction and induction of
superselection sectors, the canonical endomorphism is studied. The crucial
observation is this: the canonical endomorphism of a local subfactor extends to
an endomorphism of the field net, which in turn restricts to a localized
endomorphism of the observable net. The method allows to characterize, and
reconstruct, local extensions \bb of a given theory in terms of the
observables. Various non-trivial examples are given.Comment: Plain TeX, 32 pages. Several unnecessarily restrictive assumptions
have been relaxed. Proposition 4.10. has been reformulated in a more natural
way. Sect. 3 has been rearranged and a too general statement has been
adjusted. Some further minor change
How to remove the boundary in CFT - an operator algebraic procedure
The relation between two-dimensional conformal quantum field theories with
and without a timelike boundary is explored.Comment: 18 pages, 2 figures. v2: more precise title, reference correcte
On local boundary CFT and non-local CFT on the boundary
The holographic relation between local boundary conformal quantum field
theories (BCFT) and their non-local boundary restrictions is reviewed, and
non-vacuum BCFT's, whose existence was conjectured previously, are constructed.Comment: 16 pages. Contribution to "Rigorous Quantum Field Theory", Symposium
in honour of J. Bros, Paris, July 2004. Based on joint work math-ph/0405067
with R. Long
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
The Conformal Spin and Statistics Theorem
We prove the equality between the statistics phase and the conformal
univalence for a superselection sector with finite index in Conformal Quantum
Field Theory on . A relevant point is the description of the PCT symmetry
and the construction of the global conjugate charge.Comment: plain tex, 22 page
Some computations in the cyclic permutations of completely rational nets
In this paper we calculate certain chiral quantities from the cyclic
permutation orbifold of a general completely rational net. We determine the
fusion of a fundamental soliton, and by suitably modified arguments of A. Coste
, T. Gannon and especially P. Bantay to our setting we are able to prove a
number of arithmetic properties including congruence subgroup properties for
matrices of a completely rational net defined by K.-H. Rehren .Comment: 30 Pages Late
Gravitational wave scintillation by a stellar cluster
The diffraction effects on gravitational waves propagating through a stellar
cluster are analyzed in the relevant approximation of Fresnel diffraction
limit. We find that a gravitational wave scintillation effect - similar to the
radio source scintillation effect - comes out naturally, implying that the
gravitational wave intensity changes in a characteristic way as the observer
moves.Comment: 9 pages, in press in IJMP
On intermediate subfactors of Goodman-de la Harpe-Jones subfactors
In this paper we present a conjecture on intermediate subfactors which is a
generalization of Wall's conjecture from the theory of finite groups. Motivated
by this conjecture, we determine all intermediate subfactors of
Goodman-Harpe-Jones subfactors, and as a result we verify that
Goodman-Harpe-Jones subfactors verify our conjecture. Our result also gives a
negative answer to a question motivated by a conjecture of
Aschbacher-Guralnick.Comment: To appear in Comm. Math. Phy
Representations of Conformal Nets, Universal C*-Algebras and K-Theory
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
On the extension of stringlike localised sectors in 2+1 dimensions
In the framework of algebraic quantum field theory, we study the category
\Delta_BF^A of stringlike localised representations of a net of observables O
\mapsto A(O) in three dimensions. It is shown that compactly localised (DHR)
representations give rise to a non-trivial centre of \Delta_BF^A with respect
to the braiding. This implies that \Delta_BF^A cannot be modular when
non-trival DHR sectors exist. Modular tensor categories, however, are important
for topological quantum computing. For this reason, we discuss a method to
remove this obstruction to modularity.
Indeed, the obstruction can be removed by passing from the observable net
A(O) to the Doplicher-Roberts field net F(O). It is then shown that sectors of
A can be extended to sectors of the field net that commute with the action of
the corresponding symmetry group. Moreover, all such sectors are extensions of
sectors of A. Finally, the category \Delta_BF^F of sectors of F is studied by
investigating the relation with the categorical crossed product of \Delta_BF^A
by the subcategory of DHR representations. Under appropriate conditions, this
completely determines the category \Delta_BF^F.Comment: 36 pages, 1 eps figure; v2: appendix added, minor corrections and
clarification
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