Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 4 / a / Solution

The string embedding coordinates form a nonlinear sigma model whose target-space metric is the spacetime metric $G_{\mu \nu }$. A consistent quantum string background must preserve worldsheet Weyl invariance, so all sigma-model beta functions must vanish, and the total matter-plus-ghost central charge must cancel. For a bosonic string with only a metric, this requires $D=26$ and, to leading order in ${\alpha }^{\prime }$, ${\beta }_{\mu \nu }^G={\alpha }^{\prime }{R}_{\mu \nu }+O({\alpha }^{\prime 2})=0$; thus the target metric must be Ricci-flat at leading order.