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A Faddeev-Popov ghost field is a Grassmann-valued scalar field whose Gaussian functional integral represents the Faddeev-Popov determinant. Ghosts occur only on internal lines and cancel unphysical gauge-field contributions.

For axial gauge,so the Faddeev-Popov determinant is $\det (n^{\mu }{D}_{\mu })$. Representing it with a Faddeev-Popov ghost field pair givesThe delta functional sets $\omega =n\cdot A$ in its Gaussian weight, and henceSubstitution yieldsIn the strict axial-gauge limit $n\cdot A=0$, $n\cdot D=n\cdot \partial$, so the ghost determinant is independent of $A$ and can be absorbed into $N$.

Solved by gpt-5.6-sol high.

Choose a reference metric in each conformal class. An infinitesimal worldsheet diffeomorphism generated by $c^{\alpha }$ and a Weyl transformation decompose the metric variation into its trace and the traceless operatorInserting the gauge condition and its Faddeev-Popov determinant cancels the formal gauge-orbit volume. Representing $\det P_1$ by anticommuting worldsheet ghost fields giveswhere $b_{\alpha \beta }$ is symmetric and traceless and $c^{\alpha }$ is a vector. The gauge-fixed integral is consequentlyOn higher-genus worldsheets one must additionally integrate over moduli and treat conformal-Killing and ghost zero modes separately; these finite-dimensional factors are suppressed in the displayed formal expression.

Solved by gpt-5.6-sol high.