BRST operator ($Q_B$)

The BRST operator is the Grassmann-odd conserved charge that combines a gauge transformation with its ghost field. Quantum gauge consistency requires $Q_B^2=0$, and physical states are its cohomology classes.

If $L_m=\{b_m,Q\}$, nilpotence gives $[L_m,Q]=0$. The graded Jacobi identity and $[b_m,L_n]=(m-n)b_{m+n}$ then imply the displayed algebra. Thus a quantum central extension in the total generators obstructs BRST nilpotence.

For a matter state tensored with the ghost oscillator vacuum, the BRST charge produces the independent ghost states $c_0|0⟩$ and $c_{-m}|0⟩$ with coefficients $(L_0-1)|\Psi ⟩$ and $L_m|\Psi ⟩$. Thus BRST closure is equivalent to $(L_0-1)|\Psi ⟩=0$ and $L_{m>0}|\Psi ⟩=0$ in this sector. It reproduces old covariant string quantization and the open bosonic string mass spectrum.

The BRST current is the conserved worldsheet current whose contour integral is the BRST operator. For the bosonic string it is built from the reparameterization ghost and the matter and ghost stress tensors; its self-OPE is nonsingular only when the total central charge vanishes.