Worldsheet ghost action

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.