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

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

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Sincethe determinant variation givesIntegrating by parts and taking $\delta X^{\mu }$ to vanish on the boundary yieldsThe principle of stationary action therefore gives

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In static gauge, take $\sigma \in [0,2\pi )$ and use the circular ansatzIts Nambu–Goto Lagrangian isThe conserved energy givesfor $R(0)=R_0$ and $\dot{R}(0)=0$. Equivalently,whose solution isThe signed continuation is periodic; the geometric radius is $|R(\tau )|$, and the loop collapses whenever the cosine vanishes.

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For the circular embedding,On $R=R_0\cos (\tau /R_0)$ this becomesThus the worldsheet metric is conformally flat between collapse events and becomes degenerate at the instant when the loop shrinks to zero size.

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