= Solution
Use the classically equivalent <Polyakov action>, fix <worldsheet diffeomorphisms> and <Weyl invariance>, and perform a <Wick rotation> of the <string worldsheet>. In complex coordinates, the <worldsheet ghost action> for the <closed string> is
$$
\boxed{S_{bc}=\frac1{2\pi}\int d^2z\,
\bigl(b_{zz}\bar\partial c^z+\bar b_{\bar z\bar z}\partial\bar c^{\bar z}\bigr).}
$$
The <worldsheet ghost fields> are anticommuting. Their <conformal weights> are $b:(2,0)$, $c:(-1,0)$ and $\bar b:(0,2)$, $\bar c:(0,-1)$. The two chiral sectors are counted separately.
In either chiral sector, each embedding coordinate is a <free boson conformal field theory> with <central charge> one. For the <bc system>, putting $J=2$ in $c_{bc}=-2(6J^2-6J+1)$ gives $c_{bc}=-26$. Cancellation of the <worldsheet Weyl anomaly> requires
$$
\boxed{c_{\mathrm{tot}}=D-26=0,\qquad D=26.}
$$
The antiholomorphic sector gives the same equation, rather than doubling the required <critical dimension of string theory>. This dimension count assumes flat target spacetime with just the embedding fields; more general internal <conformal field theories> change the matter accounting.
For the <spinning string>, the <worldsheet Majorana fermions> have <conformal weight> $1/2$. At $J=1/2$, the complex anticommuting <bc system> has $c=1$; a real <Majorana spinor> contributes half of this. Hence $D$ real <worldsheet Majorana fermions> contribute $D/2$ per chiral sector. <Ramond sector> versus <Neveu–Schwarz sector> changes their mode numbers and vacuum energies, not this local <central charge>.
The additional local <worldsheet supersymmetry> has a fermionic gauge parameter, so its <superconformal ghosts> are commuting fields forming a <beta-gamma system>, with <conformal weights> $\beta:3/2$ and $\gamma:-1/2$ in the holomorphic sector. Commuting statistics reverse the sign of the <bc system> formula, giving
$$
c_{\beta\gamma}=2\left(6\left(\frac32\right)^2-6\left(\frac32\right)+1\right)=11.
$$
Together with the reparameterization <worldsheet ghost fields>, their total <central charge> is $-26+11=-15$. Thus
$$
\boxed{c_{\mathrm{tot}}=D+\frac D2-26+11=0,\qquad D=10.}
$$
These cancellations make the <string worldsheet> gauge symmetries compatible with quantization; in the <BRST symmetry> formulation they enter the nilpotence condition for the <BRST operator>.
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