For an infinitesimal <Weyl transformation> $\delta h^{ab}=-2\omega h^{ab}$, the definition of the stress tensor gives
$$
\delta_\omega S=\frac1{2\pi}\int d^2\sigma\sqrt{|h|}\,
\omega h^{ab}T_{ab}.
$$
The insertions are Weyl invariant by assumption, so varying the normalized path integral gives
$$
\delta_\omega\langle\phi_1\cdots\phi_n\rangle
=\frac{i}{2\pi}\int d^2\sigma\sqrt{|h|}\,
\omega\langle h^{ab}T_{ab}\phi_1\cdots\phi_n\rangle_{\rm conn}.
$$
Consequently the stated condition $\langle h^{ab}T_{ab}\phi_1\cdots\phi_n\rangle=0$ makes the correlation function Weyl invariant. Quantum failure of this condition is the <worldsheet Weyl anomaly>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-306/1/3/solution