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We discuss some basic problems and conjectures in a program to construct general orbifold conformal field theories using the representation theory of vertex operator algebras. We first review a program to construct conformal field theories.…
Generalizing some of our earlier work, we prove natural presentations of the principal subspaces of the level one standard modules for the untwisted affine Lie algebras of types A, D and E, and also of certain related spaces. As a…
We prove the Verlinde conjecture in the following general form: Let V be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0, the homogeneous subspace of V of…
We study the properties of shifted vertex operator algebras, which are vertex algebras derived from a given theory by shifting the conformal vector. In this way, we are able to exhibit large numbers of vertex operator algebras which are…
Vertex operators are constructed providing representations of the exchange relations containing either the S-matrix of a real coupling (simply-laced) affine Toda field theory, or its minimal counterpart. One feature of the construction is…
Given a modular tensor category $\mathscr{C}$, we construct an associative algebra $\mathrm{Tor({\mathscr{C}}})$, which we call the torus algebra. We prove that the torus algebra is semisimple by explicitly constructing all the simple…
In the references [HL1]--[HL5] and [H1], a theory of tensor products of modules for a vertex operator algebra is being developed. To use this theory, one first has to verify that the vertex operator algebra satisfies certain conditions. We…
The rationality of the parafermion vertex operator algebra associated to any finite dimensional simple Lie algebra and any nonnegative integer is established and the irreducible modules are determined.
This paper is the first of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we present an elementary, progressive and self-contained approch to vertex operator superalgebra.…
We prove a conjecture stated in a previous paper by the author about the existence of canonical filtrations for a family of vertex operator algebras in rational levels.
We give a general criterion for conformal embeddings of vertex operator algebras associated to affine Lie algebras at arbitrary levels. Using that criterion, we construct new conformal embeddings at admissible rational and negative integer…
We revisit the construction of integral forms for vertex (operator) algebras $V_L$ based on even lattices $L$ using generators instead of bases, and we construct integral forms for $V_L$-modules. We construct integral forms for vertex…
We present a general vertex operator construction based on the Fock space for an affine Lie algebras of type $A$. This construction allows us to give a unified treatment for both the homogeneous and principle realizations of the affine Lie…
The notion of vertex operator coalgebra is presented which corresponds to the family of correlation functions of one string propagating in space-time splitting into n strings in conformal field theory. This notion is in some sense dual to…
After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in…
Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V^{\otimes k}. We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the…
This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give interesting…
We analyze the non-semisimple category of line operators in Chern-Simons gauge theories based off the Lie superalgebra $\mathfrak{gl}(1|1)$. Our proposal is that the category of line operators $\mathcal{C}$ can be identified with the…
The vertex operator algebras and modules associated to the highest weight modules for the Virasoro algebra over an arbitrary field F whose characteristic is not equal to 2 are studied. The irreducible modules of vertex operator algebra…
Concrete semi-realistic string/M theory constructions often predict the existence of new physics at the TeV scale, which may be different in character from the bottom-up ideas that are motivated by specific problems of the standard model. I…