The target metric is a coupling of the <string nonlinear sigma model>. Quantum consistency requires the gauge-fixed worldsheet theory to preserve <Weyl transformation>[Weyl invariance], so its <sigma-model beta functions> must vanish. With no B-field or varying dilaton, the metric beta function begins as
$$
\beta^G_{\mu\nu}=\alpha'R_{\mu\nu}+O(\alpha'^2).
$$
Therefore vanishing of the <worldsheet Weyl anomaly> requires
$$
\boxed{R_{\mu\nu}=0}
$$
to leading order in $\alpha'$: the target-space metric must be Ricci-flat at this order.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 306 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-306/4/a/solution