88 research outputs found
Classification of Two-dimensional Local Conformal Nets with c<1 and 2-cohomology Vanishing for Tensor Categories
We classify two-dimensional local conformal nets with parity symmetry and
central charge less than 1, up to isomorphism. The maximal ones are in a
bijective correspondence with the pairs of A-D-E Dynkin diagrams with the
difference of their Coxeter numbers equal to 1. In our previous classification
of one-dimensional local conformal nets, Dynkin diagrams D_{2n+1} and E_7 do
not appear, but now they do appear in this classification of two-dimensional
local conformal nets. Such nets are also characterized as two-dimensional local
conformal nets with mu-index equal to 1 and central charge less than 1. Our
main tool, in addition to our previous classification results for
one-dimensional nets, is 2-cohomology vanishing for certain tensor categories
related to the Virasoro tensor categories with central charge less than 1.Comment: 40 pages, LaTeX 2
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
-induction for bi-unitary connections
The tensor functor called -induction arises from a Frobenius algebra
object, or a Q-system, in a braided unitary fusion category. In the operator
algebraic language, it gives extensions of endomorphism of to arising
from a subfactor of finite index and finite depth giving a braided
fusion category of endomorpshisms of . It is also understood in terms of
Ocneanu's graphical calculus. We study this -induction for bi-unitary
connections, which give a characterization of finite-dimensional nondegenerate
commuting squares and gives certain 4-tensors appearing in recent studies of
2-dimensional topological order. We show that the resulting -induced
bi-unitary connections are flat if we have a commutative Frobenius algebra, or
a local Q-system. Examples related to chiral conformal field theory and the
Dynkin diagrams are presented.Comment: 32 pages, more explanations have been adde
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