Vertex Operator Algebras and Associative Algebras

Abstract

Let V be a vertex operator algebra. We construct a sequence of associative Ž . Ž . Ž . Ž . algebras A V n s 0, 1, 2, . . . such that A V is a quotient of A V and a n n n q 1 Ž . Ž . pair of functors between the category of A V -modules which are not A Vn n y 1 modules and the category of admissible V-modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is Ž . rational if and only if all A V are finite-dimensional semisimple algebras.

Modular invariance of vertex operator algebras satisfying C 2 -cofiniteness

“…is an A n (V )-module and every A n (V )-module is an n-th (or less) graded piece of an N-graded weak V -module, (see Theorem 4.2 in [DLiM2]). A n (V ) is called an n-th Zhu algebra, in particular, 0-th Zhu algebra is the original Zhu algebra.…”

Representations of vertex operator algebras

“…It is proved in [DLM3] that V is rational if and only if A n (V ) are finite dimensional semisimple associative algebras for all n. Since the construction of A n (V ) is very complicated, it is hard to compute A n (V ) for large n. It has been suspected for a long time that the semisimplicity of A(V ) is good enough to characterize the rationality of V.…”

On associative algebras, modules and twisted modules for vertex operator algebras

“…The functors we have constructed in this paper satisfy the same universal property as the functors constructed in [DLM1] and [DLM2]. The importance of our construction is that it allows us to use the method in the present paper to give constructions and prove results in the cases where there is no commutator formula.…”

“…Huang, the author gave a formula for the residues of certain formal series involving iterates of vertex operators obtained using the weak associativity and the lower truncation property of vertex operators. We proved that the weak associativity for an admissible module is equivalent to this residue formula together with a formula that expresses products of components of vertex operators as linear combinations of iterates of components of vertex operators given in [DLM1] and [L]. We applied this result to give a new construction of admissible modules for an N-graded vertex algebra V from modules for its Zhu algebra A(V ).…”