Vertex algebras generated by primary fields of low conformal weight

We classify a certain class of vertex algebras, finitely generated by a Virasoro field and even (resp. odd) primary fields of conformal weight 1 (resp 3/2). This is the first interesting case to consider when looking at finitely generated vertex algebras containing a Virasoro field (the most interesting from the point of view of physics). By the axioms of vertex algebras it follows that the space g of fields with conformal weight 1 is a Lie algebra, and the space U of fields with conformal weight 3/2 is a g--module with a symmetric invariant bilinear form. One of the main observations is that, under the assumption of existence of a quasi--classical limit (which basically translates to the existence of a one parameter family of vertex algebras, the free parameter being the Kac--Moody level k), the complex connected algebraic group G corresponding to the Lie algebra g acts transitively on the quadric S^2={u in U s.t. (u,u)=1}. This generalizes a similar result of Kac in the case of conformal algebras. Using this observation, we will classify vertex algebras satisfying the above assumptions, by using the classification of connected compact subgroups of SO_N acting transitively on the unit sphere. The solution is given by the following list: - g=so_n, U=C^n, for n geq 3, - g=gl_n, U=C^n+C^{n,*}, for n geq 1, n not 2, - g=sl_2, U=C^2+C^{2,*}, - g=sp_n+sp_2, U=C^n+C^2, n geq 2, - g=B_3, U=spin_7, - g=G_2, U=V_{\pi_1}. However, if one removes the assumption of existence of quasi--classical limit, the above argument fails and the problem of classification has to be studied using different techniques. In the case in which g is a simple Lie algebra and $U$ an irreducible g-module, we will prove, under some weak technical assumption, that no examples with ``discrete'' values of the Kac--Moody level appear.