Rational vertex operator algebras are finitely generated

Regularity of Rational Vertex Operator Algebras

A bstract . We discuss the foundations of vertex operator algebras and their representations, concentrating on rational vertex operator algebras. We use the Witt-Grothendieck group of conformal field theories as a vehicle to describe results, in particular the connections with modular forms. INTRODUCTION Vertex operator algebras suggest themselves as objects that could play a role in a geometric description of elliptic cohomology and related topics. As linear spaces with lots of symmetry they can participate in K-theoretic type constructions, and many (but not all) vertex operator algebras are endowed with a natural modular form as part of their structure. The incorporation of vertex operator algebras into topology is well under way (e.g. [Bo], [MS], [MSV], [T]), but it seems true to say that the underlying algebraic theory is not well understood by many potential users of the subject. What follows is an attempt to convey some of the basic ideas about vertex operator algebras, in particular the rapidly advancing theory of rational vertex operator algebras. These are the algebras most naturally associated to modular forms, and in the guise of RCFT (rational conformal field theory) they constitute an active area of research in physics ([FMS]) as well as mathematics. For more information and background the reader may refer to the following: [DM4], [FLM], [G], [K1], [KR], [QFS]. Additional references will be mentioned below. In my talk at the Newton Institute I emphasized questions about group actions (orbifold theory) and in particular how one can get information about finite group cohomology (e.g., for the Monster group) by looking at maps of equivariant Witt-Grothendieck groups of vertex operator algebras into group cohomology.

We prove the rationality of all the minimal series principal W -algebras discovered by Frenkel, Kac and Wakimoto, thereby giving a new family of rational and C2-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu's algebra of simple W -algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu's algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu's algebras of vertex operator algebras associated with admissible representations of affine Kac-Moody algebras as well.

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.

Cohomological varieties associated to vertex operator algebras

The Kac-Wakimoto admissible modules for \(\hat{sl}_2\) are studied from the point of view of vertex operator algebras. It is shown that the vertex operator algebra L(l,0) associated to irreducible highest weight modules at admissible level \(l={p\over q}-2\) is not rational if l is not a positive integer. However, a suitable change of the Virasoro algebra makes L(l,0) a rational vertex operator algebra whose irreducible modules are exactly these admissible modules for \(\hat{sl}_2\) and for which the fusion rules are calculated. It is also shown that the q-dimensions with respect to the new Virasoro algebra are modular functions.

The relationship between skew group algebras and orbifold theory

The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A criterion for simple currents of a rational vertex operator algebra is given and a full Galois theory for rational vertex operator algebras is established using the quantum dimensions.

Introduction The Conway-Norton conjectures about the Monster-modular connection, later established by Borcherds [B] following the work of Frenkel-Lepowsky-Meurman [FLM], set the stage for an intensive study of the origins of the relations between finite groups and modular functions. Norton also introduced generalized moonshine which associates q -expansions to pairs of commuting elements in M . His conjecture that the nonconstant functions that one obtains in this way are also hauptmoduln remains open. By now it is clear that the principal mathematical idea underlying the general study of such phenomena is that of a vertex operator algebra . For an extensive class of vertex operator algebras, so-called rational orbifold models , one expects a theory completely parallel to Monstrous Moonshine whereby one associates modular functions to automorphisms of finite order. Furthermore, this theory naturally accommodates generalized moonshine. From this perspective, the Conway-Norton phenomena would be a particularly interesting spradic example of a general theory, just as the Monster itself is a particularly interesting sporadic example of a finite simple group. In the spirit of the Edinburgh Conference, this paper is mainly devoted to a review of some of the main results currently available concerning the structure of rational vertex operator algebras and their orbifolds. We sometimes use the Frenkel-Lepowsky-Meurman Moonshine Module V ♭ [FLM] to illustrate the ideas. We also suggest open problems - many of them well-known - whose solution would contribute to a more complete theory.

formula omitted]-code vertex operator algebras

Vertex operator algebras with positive central charges whose dimensions of weight one spaces are 8 and 16

For a rational and C2-cofinite vertex operator algebra V with an automorphism group G of prime order, the fusion rules for twisted V-modules are studied, a twisted Verlinde formula which relates fusion rules for g-twisted modules to the S-matrix in the orbifold theory is established. As an application of the twisted Verlinde formula, a twisted analogue of the Kac-Walton formula is proved, which gives fusion rules between twisted modules of affine vertex operator algebras at positive integer levels in terms of Clebsch–Gordan coefficients associated to the corresponding finite dimensional simple Lie algebras.

The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters, and the interpretation of its category of modules as a modular tensor category. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and C2-cofinite vertex operator algebras. We suggest logarithmic variants of those pillars and of Verlinde’s formula. We illustrate our ideas with the -triplet algebras and the symplectic fermions.

The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type.

The commutant of [formula omitted] in the vertex operator algebra [formula omitted