Worldsheet ghost field ($b_{\alpha \beta },c^{\alpha }$)

Gauge fixing worldsheet diffeomorphism and Weyl symmetry introduces an anticommuting vector ghost $c^{\alpha }$ and a symmetric traceless antighost $b_{\alpha \beta }$. In complex coordinates they form holomorphic and antiholomorphic $bc$ systems.

On the Riemann sphere, three holomorphic reparameterization ghost zero modes correspond to the three infinitesimal Möbius transformations. Three $c$ insertions saturate their Grassmann integral. In a basis $1,z,z^2$, their integral is the Vandermonde determinant, giving $z_{12}{z}_{13}{z}_{23}$ up to an ordering-dependent sign. Including the antiholomorphic factor gives $|z_{12}{z}_{13}{z}_{23}{|}^2$, the Jacobian in sphere gauge fixing for four string vertices.

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.

With the ghost oscillator vacuum, one can define $L_0^{\mathrm{gh}}={\sum }_{k>0}k(b_{-k}{c}_k+c_{-k}{b}_k)$ with vacuum eigenvalue zero. The corresponding Virasoro algebra contains a linear shift: $[L_m^{\mathrm{gh}},L_n^{\mathrm{gh}}]=(m-n)(L_{m+n}^{\mathrm{gh}}-{\delta }_{m+n,0})-\frac{26}{12}(m^3-m){\delta }_{m+n,0}$. Shifting the zero mode by minus one removes the linear term but changes the stated vacuum eigenvalue. A BRST charge must use a consistent choice throughout.

For the anticommuting bc system, $\{b_m,c_n\}={\delta }_{m+n,0}$. A useful oscillator vacuum obeys $b_n|0⟩=c_n|0⟩=0$ for $n>0$ and $b_0|0⟩=0$. Its partner $c_0|0⟩$ is nonzero. This specifies a ghost zero-mode doublet and differs from imposing annihilation by both zero modes.

With Hermitian $b_0,c_0$ and $\{b_0,c_0\}=1$, a ket annihilated by $b_0$ has zero bare full-ghost norm. Nonzero amplitudes need FP ghost zero-mode saturation. Low-level Virasoro descendant calculations can instead normalize the highest-weight coefficient and compute $L_n{L}_{-n}|\Omega ⟩$ algebraically. Such normalized contravariant coefficients are not positive-definite Hilbert norms. The pairing convention must be stated before using FP ghost descendants in a norm argument.

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.

For the traceless diffeomorphism operatorthe ghost action is

In conformal gauge, let the odd left-acting BRST differential act by $sX=c^{+}{\partial }_{+}X+c^{-}{\partial }_{-}X$ and $s{c}^{\pm }=c^{\pm }{\partial }_{\pm }{c}^{\pm }$. The graded Leibniz rule gives the displayed expression for $s^{2}X$. It vanishes using the ghost equations of motion ${\partial }_{-}{c}^{+}={\partial }_{+}{c}^{-}=0$, but it is not zero for arbitrary off-shell fields. For independent Grassmann variables $\xi ,\rho$, take $c^{+}=\xi$, $c^{-}=\rho {\sigma }^{+}$ and $X={\sigma }^{-}$; then $s^{2}X=-\xi \rho \ne 0$. Each ghost itself satisfies $s^2{c}^{\pm }=0$ algebraically. With the eliminated-antighost transformation $s{b}_{\pm \pm }=i{\theta }_{\pm \pm }/{\alpha }^{\prime }$ and $s{\theta }_{\pm \pm }=0$ on the solution space, the antighosts are nilpotent too. This on-shell closure is compatible with an off-shell total-derivative variation of the gauge-fixed action. A full off-shell BRST quantization uses the complete gauge transformation and, where required, auxiliary fields rather than silently dropping mixed ghost derivatives.

In two dimensions the metric variation splits into a scalar trace and the trace-free diffeomorphism operator $(P_{1}v{)}_{ab}={\nabla }_a{v}_b+{\nabla }_b{v}_a-g_{ab}\nabla \cdot v$. Integrating the Weyl parameter in the inverse Faddeev-Popov determinant removes the trace delta function. The remaining determinant is exponentiated by an anticommuting vector ghost and symmetric trace-free antighost, yielding the two chiral $b\bar{\partial }c$ terms. Conformal Killing vectors and moduli are zero modes treated separately from this local determinant.

The trace-free metric variation is $(P_{1}c{)}_{ab}={\nabla }_a{c}_b+{\nabla }_b{c}_a-{\gamma }_{ab}{\nabla }_d{c}^d$. A symmetric trace-free antighost exponentiates the Faddeev-Popov determinant. In light-cone worldsheet coordinates its two chiral kinetic terms are $b_{++}{\partial }_{-}{c}^{+}$ and $b_{--}{\partial }_{+}{c}^{-}$.

Use the half-normalized operator $(P_{1}v{)}_{ab}=({\nabla }_a{v}_b+{\nabla }_b{v}_a-h_{ab}{\nabla }_c{v}^c)/2$. On a closed worldsheet, a symmetric trace-free tensor obeysThis integration by parts identifies the displayed formal adjoint operator. A convention with twice $P_1$ has twice this adjoint, with identical kernels. The worldsheet conformal Killing group corresponds to the kernel of $P_1$, while infinitesimal worldsheet moduli correspond to the trace-free kernel of its adjoint.