138 research outputs found
The classification of non-local chiral CFT with c<1
All non-local but relatively local irreducible extensions of Virasoro chiral
CFTs with c<1 are classified. The classification, which is a prerequisite for
the classification of local c<1 boundary CFTs on a two-dimensional half-space,
turns out to be 1 to 1 with certain pairs of A-D-E graphs with distinguished
vertices.Comment: 13 pages. v3: additional material (concerning the Hilbert spaces)
adde
Ground state representations of loop algebras
Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point in
S^1 and identifying the real line with the punctured circle, we consider the
subalgebra Sg of Lg of rapidly decreasing elements on R. We classify the
translation-invariant 2-cocycles on Sg. We show that the ground state
representation of Sg is unique for each cocycle. These ground states correspond
precisely to the vacuum representations of Lg.Comment: 22 pages, no figur
How to add a boundary condition
Given a conformal QFT local net of von Neumann algebras B_2 on the
two-dimensional Minkowski spacetime with irreducible subnet A\otimes\A, where A
is a completely rational net on the left/right light-ray, we show how to
consistently add a boundary to B_2: we provide a procedure to construct a
Boundary CFT net B of von Neumann algebras on the half-plane x>0, associated
with A, and locally isomorphic to B_2. All such locally isomorphic Boundary CFT
nets arise in this way. There are only finitely many locally isomorphic
Boundary CFT nets and we get them all together. In essence, we show how to
directly redefine the C* representation of the restriction of B_2 to the
half-plane by means of subfactors and local conformal nets of von Neumann
algebras on S^1.Comment: 20 page
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
Longo-Rehren subfactors arising from -induction
We study (dual) Longo-Rehren subfactors arising
from various systems of endomorphisms of M obtained from alpha-induction for
some braided subfactor . Our analysis provides useful tools to
determine the systems of R-R morphisms associated with such Longo-Rehren
subfactors, which constitute the ``quantum double'' systems in an appropriate
sense. The key to our analysis is that alpha-induction produces half-braidings
in the sense of Izumi, so that his general theory can be applied. Nevertheless,
alpha-induced systems are in general not braided, and thus our results allow to
compute the quantum doubles of (certain) systems without braiding. We
illustrate our general results by several examples, including the computation
of the quantum double systems for the asymptotic inclusion of the E_8 subfactor
as well as its three analogues arising from conformal inclusions of SU(3)_k.Comment: 31 pages, late
On -induction, chiral generators and modular invariants for subfactors
We consider a type III subfactor of finite index with a finite
system of braided - morphisms which includes the irreducible constituents
of the dual canonical endomorphism. We apply -induction and, developing
further some ideas of Ocneanu, we define chiral generators for the double
triangle algebra. Using a new concept of intertwining braiding fusion
relations, we show that the chiral generators can be naturally identified with
the -induced sectors. A matrix is defined and shown to commute with
the S- and T-matrices arising from the braiding. If the braiding is
non-degenerate, then is a ``modular invariant mass matrix'' in the usual
sense of conformal field theory. We show that in that case the fusion rule
algebra of the dual system of - morphisms is generated by the images of
both kinds of -induction, and that the structural information about its
irreducible representations is encoded in the mass matrix . Our analysis
sheds further light on the connection between (the classifications of) modular
invariants and subfactors, and we will construct and analyze modular invariants
from loop group subfactors in a forthcoming publication, including
the treatment of all modular invariants.Comment: 66 pages, latex, epic, eepic; minor changes, typos fixed, references
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Classification of Subfactors with the Principal Graph D1n
AbstractWe show that the number of the conjugacy classes of the AFD type II1 subfactors with the principal graph D1n is n â 2. This gives the last missing number in the complete classfication list of subfactors with index 4 by S. Popa. This also disproves an announcement of A. Ocneanu that such a subfactor is unique for each n. We give two different proofs. One is by an application of an idea of an orbifold model in solvable lattice model theory to OcneanuâČs paragroup theory and the other is by reduction to classification of dihedral group actions. The latter also shows that the AFD type III1 subfactors with the principal graph D1n split as type II1 subfactors tensored with the common AFD type III1 factor. We also discuss a relation between these proofs and a construction of subfactors using Cuntz algebra endomorphisms
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
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
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