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

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.

For the stereographic coordinate $Z=\tan (\theta /2)e^{i\phi }$, the round metric isSubstitution into the sigma-model action givesso

Set $Z=\lambda \delta Z$. Expansion about the north pole givesThus $⟨\delta Z(k)\delta \bar{Z}(-k)⟩=1/k^2$ up to the Fourier convention, and the leading four-point vertex is $-2{\lambda }^2$ times the scalar product of the two differentiated momenta, summed over assignments of differentiated legs.

Contracting the two undifferentiated fields in this vertex gives a one-loop tadpole proportional toIt produces the divergent metric correction $-2{\lambda }^{2}I\partial \delta Z\partial \delta \bar{Z}$, up to the allowed overall-normalization ambiguity. Geometrically this is the one-loop term ${\beta }_{\mu \nu }^G\propto {\alpha }^{\prime }{R}_{\mu \nu }$. Since the round sphere has positive Ricci curvature, the isolated $S^2$ sigma model is not conformal. Therefore $S^2\times N$ is not a bosonic-string background unless contributions from $N$, a dilaton, an antisymmetric tensor, or further corrections cancel the sphere beta function.