Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 2 / d / Solution

The $D$ free embedding scalars contribute $c_X=D$. The ordinary diffeomorphism ghosts contribute $c_{bc}=-26$. The fermionic $(m,n)$ pair has $h_n=0$ and contributes $-2$. Each of the $D/2$ fermionic $({\chi }_a,{\psi }^a)$ pairs has $h_{\psi }=1/2$ and contributes $+1$. Each bosonic $({\beta }_r,{\gamma }^r)$ pair has $h_{\gamma }=-1/2$ and contributes $+11$, and there are two such pairs. The total holomorphic central charge is thereforeCancellation of the worldsheet Weyl anomaly requires $c_{\mathrm{tot}}=0$, soThe antiholomorphic sector gives the same condition.