What's the Virasoro module structure of free boson,. etc?

Question

I'm reading Chapter 2. of "Vertex Algebras and Algebraic Curves". In which they give some examples of 2d CFTs. For example, the Hilbert space of 2d massless free boson are generated by $b_{-n}^{j_n}\dots b_{-1}^{j_i}|{0}\rangle$. In which $b_{-i}$ are creation operators. This Hilbert space has action of Virasoro algebra. So how this space decompose as irreducible modules (maybe highest weight module or integrable highest weight module) of Virasoro algebra? And what about WZW models, free fermion, etc?

Answer 1

(At least for those case) it's only about finding the primary states, which are vectors annihilated by $L_n(n>0)$.

For massless free boson (with $\lambda=0$), one can verify that $L_n|0\rangle=0 $ and $L_na_{-1}|0\rangle=0$, so it split into two modules generated by $|0\rangle$ and $a_{-1}|0\rangle$.