Chongying Dong

Generalized Vertex Algebras and Relative Vertex Operators

1. Introduction. 2. The setting. 3. Relative untwisted vertex operators. 4. Quotient vertex operators. 5. A Jacobi identity for relative untwisted vertex operators. 6. Generalized vertex operator algebras and their modules. 7. Duality for generalized vertex operator algebras. 8. Monodromy representations of braid groups. 9. Generalized vertex algebras and duality. 10. Tensor products. 11. Intertwining operators. 12. Abelian intertwining algebras, third cohomology and duality. 13. Affine Lie algebras and vertex operator algebras. 14. Z-algebras and parafermion algebras. References. List of frequently-used symbols, in order of appearance.

Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

Abstract: The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms.Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise number of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlation functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representation of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge.In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway–Norton–Queen and to equivariant elliptic cohomology.

On quantum Galois theory

The goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the “operator content of orbifold models.” To explain what this might mean, and to clarify the title of the paper, we will assume that the reader is familiar with the algebraic formulation of 2-dimensional CFT in the guise of vertex operator algebras (VOA), see [B], [FLM] and [DM] for more information on this point. In the paper [DVVV], several ideas are proposed concerning the structure of a holomorphic orbifold. In other words, if V is a holomorphic VOA and if G is a finite group of automorphisms of V, then the sub VOA V G of G-invariants is itself a VOA and the subject of [DVVV] is very much concerned with speculation on the nature of the V -modules. It turns out to be more useful − at least for purpose of inductive proofs − to take V to be a simple VOA. We will then see that V G is also simple whenever G is a finite group of automorphisms of V. One consequence of our main results is the following:

Framed Vertex Operator Algebras, Codes and the Moonshine Module

Abstract:For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge ½, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge ½ are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.

Simple currents and extensions of vertex operator algebras

We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE8) and to get various extensions of the vertex operator algebras associated with integrable representations.

The algebraic structure of relative twisted vertex operators

Abstract Twisted vertex operators based on rational lattices have had many applications in vertex operator algebra theory and conformal field theory. In this paper, “relativized” twisted vertex operators are constructed in a general context based on isometries of rational lattices, and a generalized twisted Jacobi identity is established for them. This result generalizes many previous results. Relatived untwisted vertex operators had been studied in a monograph by the authors. The present paper includes as a special case the proof of the main relations among twisted vertex operators based on even lattices announced some time ago by the second author.

Rationality, regularity, and ₂-cofiniteness

We demonstrate that, for vertex operator algebras of CFT type, C 2 -cofiniteness and rationality is equivalent to regularity. For C 2 -cofinite vertex operator algebras, we show that irreducible weak modules are ordinary modules and C 2 -cofinite, V + L is C 2 -cofinite, and the fusion rules are finite.

Rational vertex operator algebras and the effective central charge

We establish that the Lie algebra of weight 1 states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank 1 is bounded above by the effective central charge c~. We show that lattice vertex operator algebras may be characterized by the equalities c~=l=c, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator algebras by the equality l = c.

Compact automorphism groups of vertex operator algebras

Let V be a simple vertex operator algebra which admits the continuous, faithful action of a compact Lie group G of automorphisms. We establish a Schur-Weyl type duality between the unitary, irreducible modules for G and the irreducible modules for V G which are contained in V where V G is the space of G-invariants of V. We also prove a concomitant Galois correspondence between vertex operator subalgebras of V which contain V G and closed Lie subgroups of G in the case that G is abelian. These results extend those of [DM1] and [DM2].

Representations of Vertex Operator Algebra VL+ for Rank One Lattice L

Abstract:We classify the irreducible modules for the fixed point vertex operator subalgebra VL+ of the vertex operator algebra VL associated to a positive definite even lattice of rank 1 under the automorphism lifted from the −1 isometry of L.