Classification of irreducible modules of W_3 algebra with c = -2

Abstract

We construct irreducible modules V_α, α\in \C over W_3 algebra with c = -2 in terms of a free bosonic field. We prove that these modules exhaust all the irreducible modules of W_3 algebra with c = -2. Highest weights of modules V_α, α\in \C with respect to the full (two-dimensional) Cartan subalgebra of W_3 algebra are (α(α-1)/2, α(α-1)(2α-1)/6). They are parametrized by points (t, w) on a rational curve w^2 - t^2 (8t + 1)/9 = 0. Irreducible modules of vertex algebra W_{1+\infty} with c = -1 are also classified.