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

The target metric is a coupling of the string nonlinear sigma model. Quantum consistency requires the gauge-fixed worldsheet theory to preserve Weyl invariance, so its sigma-model beta functions must vanish. With no B-field or varying dilaton, the metric beta function begins asTherefore vanishing of the worldsheet Weyl anomaly requiresto leading order in ${\alpha }^{\prime }$: the target-space metric must be Ricci-flat at this order.