Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 1 / a

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 $S_{\mathrm{P}}=S_{\mathrm{NG}}$.