Construction of vertex operator algebras from commutative associative algebras

Given a commutative associative algebra A with an associative form (,), we construct a vertex operator algebra V with the weight two space V-2 congruent to A. If in addition the form (,) is nondegenerate, we show that there is a simple vertex operator algebra with V-2 congruent to A. We also show that if A is semisimple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.