Rational vertex operator algebras are finitely generated

It is proved that any vertex operator algebra for which the image of the Virasoro element in Zhu’s algebra is algebraic over complex numbers is finitely generated. In particular, any vertex operator algebra with a finite dimensional Zhu’s algebra is finitely generated. As a result, any rational vertex operator algebra is finitely generated. Although many well known vertex operator algebras are finitely generated, but whether or not an arbitrary rational vertex operator algebra is finitely generated has been a basic problem in the theory of vertex operator algebra. In this paper we give a positive answer to this problem and our result justifies the assumption in the physics literature that any rational conformal field theory is finitely generated. A systematic study of generators for an arbitrary vertex operator algebra was initiated in [L], [KL]. A vertex operator algebra V is called C1-cofinite if V = V0 + V1 · · ·+ with V0 being 1-dimensional and V/C1(V ) is finite dimensional where C1(V ) is a subspace of V spanned by vectors u−1v, u−21 for u, v ∈ V + = ∑ n>0 Vn and un is the component operator of Y (u, z) = ∑ n∈Z unz . It is proved in [L] that if a vertex operator algebra V is C1-cofinite then it is finitely generated. In fact, if {xi, i ∈ I} is a set of vectors of V such that xi + C1(V ) for i ∈ I form a basis of V/C1(V ), then V is generated by this set of vectors and V has a PBW-like spanning set [KL]. Another important finiteness for a vertex operator algebra is the C2-cofiniteness introduced by Zhu [Z] in the proof of modular invariance of the q-characters of irreducible modules for a rational vertex operator algebra. A vertex operator algebra V is called C2cofinite if V/C2(V ) is finite dimensional where C2(V ) is a subspace of V spanned by u−2v for u, v ∈ V. It is proved in [Z] that the span of the q-characters of irreducible modules for a rational, C2-cofinite vertex operator algebra affords a representation of the modular group SL(2, Z). So many results in the theory of vertex operator algebra using the modularity of the q-characters of the irreducible modules need both C2-cofiniteness and rationality (see [DLM3], [DM1], [DM2], [DM3]). It is shown in [GN] that a C2-cofinite vertex operator algebra V is finitely generated with a better PBW-like spanning set. Again one can choose a set X of homogeneous vectors of V such that x+ C2(V ) for x ∈ X form a basis of V/C2(V ), then V is spanned by x1−n1 · · ·x k −nk 1 Supported by NSF grants and a research grant from the Committee on Research, UC Santa Cruz ([email protected]).