The holomorphic part of the <free-boson worldsheet propagator> gives
$$
\partial X^\mu(z)\partial X^\nu(w)\sim-\frac{\alpha'}2\frac{\eta^{\mu\nu}}{(z-w)^2}.
$$
Apply <Wick theorem> to $T(z)T(w)$. The two double contractions give $D/[2(z-w)^4]$, while the single contractions reconstruct $T$ and its derivative. Thus the <stress-tensor operator-product expansion> is
$$
\boxed{
T(z)T(w)\sim\frac{D/2}{(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{z-w}
}
$$
and the $D$ embedding coordinates have <central charge> $\boxed{c=D}$.

The <holomorphic stress-energy tensor> generates an infinitesimal conformal transformation through
$$
\delta_vT(z)=\frac1{2\pi i}\oint_zdw\,v(w)T(w)T(z).
$$
Taking the three residues gives
$$
\boxed{\delta_vT=\frac{c}{12}\partial^3v+2(\partial v)T+v\partial T}.
$$
The third derivative is the anomalous term that prevents $T$ from transforming as an ordinary weight-two <Virasoro primary operator>.

With <Virasoro algebra> modes $L_n=(2\pi i)^{-1}\oint dz\,z^{n+1}T(z)$, a second contour calculation gives
$$
\boxed{[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1)\delta_{m+n,0}},
$$
so $A(m)=c(m^3-m)/12=D(m^3-m)/12$. This <Virasoro central extension> is the quantum conformal anomaly. In string theory the matter and ghost contributions must cancel; $c_{\rm matter}+c_{\rm ghost}=D-26=0$ gives the <critical dimension of string theory> $D=26$ and makes the <BRST operator> nilpotent.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 306 / 1 / Solution

For the normalization $S=(2\pi\alpha')^{-1}\int d^2z\,\partial X\bar\partial X$, the free-boson worldsheet propagator has singular part
$$
X^\mu(z,\bar z)X^\nu(w,\bar w)\sim-\frac{\alpha'}2\eta^{\mu\nu}\log|z-w|^2.
$$
Differentiation gives the singular operator-product expansions of the embedding currents.


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