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

Define the worldsheet stress-energy tensor byUsing $\delta \sqrt{|h|}=-\sqrt{|h|}{h}_{ab}\delta h^{ab}/2$ in the Polyakov action givesBecause the worldsheet is two-dimensional,This classical tracelessness is the Noether identity for Weyl invariance.

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.

For an infinitesimal Weyl transformation $\delta h^{ab}=-2\omega h^{ab}$, the definition of the stress tensor givesThe insertions are Weyl invariant by assumption, so varying the normalized path integral givesConsequently the stated condition $⟨h^{ab}{T}_{ab}{\varphi }_1\cdots {\varphi }_n⟩=0$ makes the correlation function Weyl invariant. Quantum failure of this condition is the worldsheet Weyl anomaly.

Changing the target metric by $\delta G_{\mu \nu }={ϵ}_{\mu \nu }$ changes the action bySince the path-integral weight is $e^{iS}$, the leading change is the insertionwhereA change of worldsheet metric within a gauge orbit changes only the redundant description and cannot alter gauge-invariant observables. The target-space metric is instead a physical coupling of the string nonlinear sigma model, so changing it changes the theory and its correlation functions.