Ghost oscillator Virasoro generators ($L_m^{\mathrm{gh}}={\sum }_k(m-k):b_{m+k}{c}_{-k}:$)

Contracting an antighost with the cubic FP ghost term of the BRST operator produces the bilinear generators. They give $[L_m,b_n]=(m-n)b_{m+n}$ and $[L_m,c_n]=-(2m+n)c_{m+n}$. Their zero-mode additive constant depends on the vacuum and normal ordering convention; it must agree with the corresponding $c_0$ term in the charge.