= Worldsheet integration count after conformal gauge fixing
{title2=$d_g+n-k_g=3g-3+n$}
For a stable $n$-point <closed-string scattering amplitude> on a <genus>-$g$ surface, let $d_g$ count complex unmarked <worldsheet moduli> and $k_g$ count complex dimensions of the <worldsheet conformal Killing group>. Integrating the $n$ vertex positions and moduli, then fixing $k_g$ positions, leaves $d_g+n-k_g$ complex integrations. The <punctured Riemann surface moduli dimension> identifies this as $3g-3+n$. <Faddeev-Popov ghost fields> and antighost insertions implement the removed zero modes and the surviving modulus measure; they do not add independently integrated worldsheet coordinates.
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