41,502 research outputs found
Classification of Local Conformal Nets. Case c < 1
We completely classify diffeomorphism covariant local nets of von Neumann
algebras on the circle with central charge c less than 1. The irreducible ones
are in bijective correspondence with the pairs of A-D_{2n}-E_{6,8} Dynkin
diagrams such that the difference of their Coxeter numbers is equal to 1. We
first identify the nets generated by irreducible representations of the
Virasoro algebra for c<1 with certain coset nets. Then, by using the
classification of modular invariants for the minimal models by
Cappelli-Itzykson-Zuber and the method of alpha-induction in subfactor theory,
we classify all local irreducible extensions of the Virasoro nets for c<1 and
infer our main classification result. As an application, we identify in our
classification list certain concrete coset nets studied in the literature.Comment: 30 pages, LaTeX2
Algebraic conformal quantum field theory in perspective
Conformal quantum field theory is reviewed in the perspective of Axiomatic,
notably Algebraic QFT. This theory is particularly developped in two spacetime
dimensions, where many rigorous constructions are possible, as well as some
complete classifications. The structural insights, analytical methods and
constructive tools are expected to be useful also for four-dimensional QFT.Comment: Review paper, 40 pages. v2: minor changes and references added, so as
to match published versio
Symmetries of the Kac-Peterson Modular Matrices of Affine Algebras
The characters of nontwisted affine algebras at fixed level define
in a natural way a representation of the modular group . The
matrices in the image are called the Kac-Peterson modular
matrices, and describe the modular behaviour of the characters. In this paper
we consider all levels of , and for
each of these find all permutations of the highest weights which commute with
the corresponding Kac-Peterson matrices. This problem is equivalent to the
classification of automorphism invariants of conformal field theories, and its
solution, especially considering its simplicity, is a major step toward the
classification of all Wess-Zumino-Witten conformal field theories.Comment: 16 pp, plain te
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
The Rank four Heterotic Modular Invariant Partition Functions
In this paper, we develop several general techniques to investigate modular
invariants of conformal field theories whose algebras of the holomorphic and
anti-holomorphic sectors are different. As an application, we find all such
``heterotic'' WZNW physical invariants of (horizontal) rank four: there are
exactly seven of these, two of which seem to be new. Previously, only those of
rank have been completely classified. We also find all physical modular
invariants for , for , and ,
, completing the classification of ref.{} \SUSU.Comment: 25 pp., plain te
Of McKay Correspondence, Non-linear Sigma-model and Conformal Field Theory
The ubiquitous ADE classification has induced many proposals of often
mysterious correspondences both in mathematics and physics. The mathematics
side includes quiver theory and the McKay Correspondence which relates finite
group representation theory to Lie algebras as well as crepant resolutions of
Gorenstein singularities. On the physics side, we have the graph-theoretic
classification of the modular invariants of WZW models, as well as the relation
between the string theory nonlinear -models and Landau-Ginzburg
orbifolds. We here propose a unification scheme which naturally incorporates
all these correspondences of the ADE type in two complex dimensions. An
intricate web of inter-relations is constructed, providing a possible guideline
to establish new directions of research or alternate pathways to the standing
problems in higher dimensions.Comment: 35 pages, 4 figures; minor corrections, comments on toric geometry
and references adde
Renormalization of gauge theories without cohomology
We investigate the renormalization of gauge theories without assuming
cohomological properties. We define a renormalization algorithm that preserves
the Batalin-Vilkovisky master equation at each step and automatically extends
the classical action till it contains sufficiently many independent parameters
to reabsorb all divergences into parameter-redefinitions and canonical
transformations. The construction is then generalized to the master functional
and the field-covariant proper formalism for gauge theories. Our results hold
in all manifestly anomaly-free gauge theories, power-counting renormalizable or
not. The extension algorithm allows us to solve a quadratic problem, such as
finding a sufficiently general solution of the master equation, even when it is
not possible to reduce it to a linear (cohomological) problem.Comment: 29 pages; v2: references updated, EPJ
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