On generators and relations for higher level Zhu algebras and applications

Abstract

We give some general results about the generators and relations for the higher level Zhu algebras for a vertex operator algebra. In particular, for any element u in a vertex operator algebra V, such that u has weight greater than or equal to −n for n∈N, we prove a recursion relation in the nth level Zhu algebra An(V) and give a closed formula for this relation. We use this and other properties of An(V) to reduce the modes of u that appear in the generators for An(V) as long as u∈V has certain properties (properties that apply, for instance, to the conformal vector for any vertex operator algebra or if u generates a Heisenberg vertex subalgebra), and we then prove further relations in An(V) involving such an element u. We present general techniques that can be applied once a set of reasonable generators is determined for An(V) to aid in determining the relations of those generators, such as using the relations of those generators in the lower level Zhu algebras and the zero mode actions on V-modules induced from those lower level Zhu algebras. We prove that the condition that (L(−1)+L(0))v acts as zero in An(V) for n∈Z+ and for all v in V is a necessary added condition in the definition of the Zhu algebra at level higher than zero. We discuss how these results on generators and relations apply to the level n Zhu algebras for the Heisenberg vertex operator algebra and the Virasoro vertex operator algebras at any level n∈N.