In $T(z)\partial X(w)$, a single Wick contraction can be made with either factor in $T$. Using the two-point function and expanding the remaining field around $w$ gives
$$
T(z)\partial X(w)
\sim\frac{\partial X(w)}{(z-w)^2}
+\frac{\partial^2X(w)}{z-w}.
$$
This <operator product expansion> says that $\partial X$ is a holomorphic <primary operator> of conformal weight $(1,0)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-306/2/b/solution