545 research outputs found
On the fundamental representation of Borcherds algebras with one imaginary simple root
Borcherds algebras represent a new class of Lie algebras which have almost
all the properties that ordinary Kac-Moody algebras have, and the only major
difference is that these generalized Kac-Moody algebras are allowed to have
imaginary simple roots. The simplest nontrivial examples one can think of are
those where one adds ``by hand'' one imaginary simple root to an ordinary
Kac-Moody algebra. We study the fundamental representation of this class of
examples and prove that an irreducible module is given by the full tensor
algebra over some integrable highest weight module of the underlying Kac-Moody
algebra. We also comment on possible realizations of these Lie algebras in
physics as symmetry algebras in quantum field theory.Comment: 8 page
Algebraic orbifold conformal field theories
We formulate the unitary rational orbifold conformal field theories in the
algebraic quantum field theory framework. Under general conditions, we show
that the orbifold of a given unitary rational conformal field theories
generates a unitary modular category. Many new unitary modular categories are
obtained. We also show that the irreducible representations of orbifolds of
rank one lattice vertex operator algebras give rise to unitary modular
categories and determine the corresponding modular matrices, which has been
conjectured for some time.Comment: 24 pages, Amste
From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory
This is an expository article invited for the ``Commentary'' section of PNAS
in connection with Y.-Z. Huang's article, ``Vertex operator algebras, the
Verlinde conjecture, and modular tensor categories,'' appearing in the same
issue of PNAS. Huang's solution of the mathematical problem of constructing
modular tensor categories from the representation theory of vertex operator
algebras is very briefly discussed, along with background material. The
hypotheses of the theorems entering into the solution are very general, natural
and purely algebraic, and have been verified in a wide range of familiar
examples, while the theory itself is heavily analytic and geometric as well as
algebraic.Comment: latex file, 4 page
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