The worldsheet ghosts form an anticommuting <bc system> of weights $(2,-1)$. Its double contractions give central charge
$$
c_{bc}=-2(6\cdot2^2-6\cdot2+1)=-26.
$$
Therefore
$$
\delta_vT_{\rm gh}=v\partial T_{\rm gh}+2(\partial v)T_{\rm gh}
-\frac{26}{12}\partial^3v,
$$
and hence
$$
\boxed{c_{\rm gh}=-\frac{13}{6}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 2 / 2 / Solution

{title2=$bc$ system}

A bc system is the anticommuting counterpart of a <beta-gamma system>. For weights $h$ and $1-h$, its central charge is $-2(6h^2-6h+1)$.

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