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, 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 Z{\cal Z} Algebra

We define a natural quantum analogue for the ${\cal Z}$ algebra, and which we refer to as the ${\cal Z}_q$ algebra, by modding out the Heisenberg algebra from the quantum affine algebra $U_q(\hat{sl(2)})$ with level $k$. We discuss the representation theory of this ${\cal Z}_q$ 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 $k=1$, and $k=2$, and thus, we recover the explicit constructions of \uq-standard modules as achieved by Frenkel-Jing and Bernard, respectively. Moreover, for arbitrary nonzero level $k$, we show that the explicit basis for the simplest ${\cal Z}$-generalized Verma module as constructed by Lepowsky and primc is also a basis for its corresponding ${\cal Z}_q$-module, i.e., it is invariant under the q-deformation for generic q. We expect this ${\cal Z}_q$ algebra (associated with \uq at level $k$), to play the role of a dynamical symmetry in the off-critical $Z_k$ statistical models.Comment: 32 pages, LATEX, minor change

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