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

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 $w$ givesThis operator product expansion says that $\partial X$ is a holomorphic primary operator of conformal weight $(1,0)$.