Adding the matter and ghost transformations gives
$$
\boxed{\delta_vT_{\rm tot}=v\partial T_{\rm tot}
+2(\partial v)T_{\rm tot}+\frac{D-26}{12}\partial^3v}.
$$
For the Laurent modes $L_n=\oint dz\,z^{n+1}T_{\rm tot}(z)/(2\pi i)$, the corresponding <Virasoro algebra> is
$$
\boxed{[L_m,L_n]=(m-n)L_{m+n}
+\frac{D-26}{12}m(m^2-1)\delta_{m+n,0}}.
$$
At the bosonic-string <critical dimension of string theory>, $D=26$, the total central charge and central term vanish. The total stress tensor then transforms as a genuine conformal tensor of weight two, and the quantum worldsheet gauge anomaly cancels.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 2 / 3 / Solution

{c}
{title2=$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1)\delta_{m+n,0}$}
{wiki}

The Virasoro algebra is the central extension of the algebra of holomorphic vector fields on a circle. Its generators are the Laurent modes of the holomorphic stress-energy tensor, and its central term is controlled by the <central charge> $c$.

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