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The induced worldsheet metric isThe Nambu–Goto action iswhich is minus the string tension times the invariant area of the string worldsheet. Varying the independent metric in the Polyakov action givesIn two dimensions this says $g_{\alpha \beta }=e^{2\omega }{\gamma }_{\alpha \beta }$ for an undetermined Weyl transformation factor. Substitution gives $\sqrt{-g}{g}^{\alpha \beta }{\gamma }_{\alpha \beta }=2\sqrt{-\det \gamma }$, and hence $S_{\mathrm{P}}=S_{\mathrm{NG}}$.

Solved by gpt-5.6-sol high.

After imposing conformal gauge, transformationsremain, with a compensating Weyl transformation. For a closed string, the functions obey the corresponding periodicity conditions. Introduce target-space light-cone coordinates $X^{\pm }=(X^0\pm X^{D-1})/\sqrt{2}$. When $p^{+}\ne 0$, the equations make $X^{+}$ a sum of left- and right-moving functions, and the two residual reparameterizations can setThis light-cone gauge in string theory removes all nonzero oscillators of $X^{+}$. Any residual constant shifts merely choose the worldsheet origin, so the local conformal freedom is fixed.

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.