177 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
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
A Quantum Analogue of the Algebra
We define a natural quantum analogue for the algebra, and which we
refer to as the algebra, by modding out the Heisenberg algebra
from the quantum affine algebra with level . We discuss
the representation theory of this algebra. In particular, we
exhibit its reduction to a group algebra, and to a tensor product of a group
algebra with a quantum Clifford algebra when , and , and thus, we
recover the explicit constructions of \uq-standard modules as achieved by
Frenkel-Jing and Bernard, respectively. Moreover, for arbitrary nonzero level
, we show that the explicit basis for the simplest -generalized
Verma module as constructed by Lepowsky and primc is also a basis for its
corresponding -module, i.e., it is invariant under the
q-deformation for generic q. We expect this algebra (associated
with \uq at level ), to play the role of a dynamical symmetry in the
off-critical statistical models.Comment: 32 pages, LATEX, minor change
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
Vertex-algebraic structure of the principal subspaces of certain A_1^(1)-modules, II: higher level case
We give an a priori proof of the known presentations of (that is,
completeness of families of relations for) the principal subspaces of all the
standard A_1^(1)-modules. These presentations had been used by Capparelli,
Lepowsky and Milas for the purpose of obtaining the classical Rogers-Selberg
recursions for the graded dimensions of the principal subspaces. This paper
generalizes our previous paper.Comment: 26 pages; v2: minor revisions, to appear in Journal of Pure and
Applied Algebr
Nonmeromorphic operator product expansion and C_2-cofiniteness for a family of W-algebras
We prove the existence and associativity of the nonmeromorphic operator
product expansion for an infinite family of vertex operator algebras, the
triplet W-algebras, using results from P(z)-tensor product theory. While doing
this, we also show that all these vertex operator algebras are C_2-cofinite.Comment: 21 pages, to appear in J. Phys. A: Math. Gen.; the exposition is
improved and one reference is adde
Logarithmic intertwining operators and W(2,2p-1)-algebras
For every , we obtained an explicit construction of a family of
-modules, which decompose as direct sum of simple Virasoro
algebra modules. Furthermore, we classified all irreducible self-dual
-modules, we described their internal structure, and
computed their graded dimensions. In addition, we constructed certain hidden
logarithmic intertwining operators among two ordinary and one logarithmic
-modules. This work, in particular, gives a mathematically
precise formulation and interpretation of what physicists have been referring
to as "logarithmic conformal field theory" of central charge
. Our explicit construction can be
easily applied for computations of correlation functions. Techniques from this
paper can be used to study the triplet vertex operator algebra
and other logarithmic models.Comment: 22 pages; v2: misprints corrected, other minor changes. Final version
to appear in Journal of Math. Phy
An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}
By extending the methods used in our earlier work, in this paper, we present
an explicit realization of logarithmic \mathcal{W}_{p,p'}-modules that have
L(0) nilpotent rank three. This was achieved by combining the techniques
developed in \cite{AdM-2009} with the theory of local systems of vertex
operators \cite{LL}. In addition, we also construct a new type of extension of
, denoted by . Our results confirm several
claims in the physics literature regarding the structure of projective covers
of certain irreducible representations in the principal block. This approach
can be applied to other models defined via a pair screenings.Comment: 18 pages, v2: one reference added, other minor change
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