 Solution

The holomorphic part of the free-boson worldsheet propagator givesApply 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 isand the $D$ embedding coordinates have central charge $\boxed{c=D}$.

The holomorphic stress-energy tensor generates an infinitesimal conformal transformation throughTaking the three residues givesThe 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 givesso $A(m)=c(m^3-m)/12=D(m^3-m)/12$. This Virasoro central extension is the quantum conformal anomaly. In string theory the matter and ghost contributions must cancel; $c_{\mathrm{matter}}+c_{\mathrm{ghost}}=D-26=0$ gives the critical dimension of string theory $D=26$ and makes the BRST operator nilpotent.