Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 1 / 3 / Solution

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.