450 research outputs found
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
A logarithmic generalization of tensor product theory for modules for a vertex operator algebra
We describe a logarithmic tensor product theory for certain module categories
for a ``conformal vertex algebra.'' In this theory, which is a natural,
although intricate, generalization of earlier work of Huang and Lepowsky, we do
not require the module categories to be semisimple, and we accommodate modules
with generalized weight spaces. The corresponding intertwining operators
contain logarithms of the variables.Comment: 39 pages. Misprints corrected. Final versio
Twisted modules for vertex operator algebras and Bernoulli polynomials
Using general principles of the theory of vertex operator algebras and their
twisted modules, we obtain a bosonic, twisted construction of a certain central
extension of a Lie algebra of differential operators on the circle, for an
arbitrary twisting automorphism. The construction involves the Bernoulli
polynomials in a fundamental way. This is explained through results in the
general theory of vertex operator algebras, including a new identity, which we
call ``modified weak associativity.'' This paper is an announcement. The
detailed proofs will appear elsewhere.Comment: 15 pages, LaTeX, Revised version (to appear in I.M.R.N.
Vertex-algebraic structure of the principal subspaces of certain A_1^(1)-modules, I: level one case
This is the first in a series of papers in which we study vertex-algebraic
structure of Feigin-Stoyanovsky's principal subspaces associated to standard
modules for both untwisted and twisted affine Lie algebras. A key idea is to
prove suitable presentations of principal subspaces, without using bases or
even ``small'' spanning sets of these spaces. In this paper we prove
presentations of the principal subspaces of the basic A_1^(1)-modules. These
convenient presentations were previously used in work of
Capparelli-Lepowsky-Milas for the purpose of obtaining the classical
Rogers-Ramanujan recursion for the graded dimensions of the principal
subspaces.Comment: 20 pages. To appear in International J. of Mat
- …