The holomorphic free boson contributes <central charge> $c_X=1$, while one real chiral free fermion contributes $c_\psi=1/2$. Equivalently, the double contractions in $T(z)T(w)$ give
$$
T(z)T(w)\sim\frac{3/4}{(z-w)^4}
+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{z-w}.
$$
Since the fourth-order coefficient is $c/2$,
$$
\boxed{c=1+\frac12=\frac32}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 3 / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-306/3/iv/solution