Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 1 / 2 / Solution

The Nambu–Goto action isThe metric equation of the Polyakov action is $T_{ab}=0$, which saysThus $h_{ab}$ is conformal to the induced worldsheet metric, $h_{ab}=e^{2\omega }{\gamma }_{ab}$. Substitution into the Polyakov action cancels the conformal factor and yieldsThe two actions are therefore classically equivalent after the auxiliary worldsheet metric is eliminated.