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

The free equations of motion are $\bar{\partial }\psi =0$ and $\partial \bar{\partial }X=0$, henceThus the superconformal current $G=i\psi \partial X$ is holomorphic. Differentiating the boson OPE gives $\partial X(z)\partial X(w)\sim -1/(z-w{)}^2$. The double contraction in $G(z)G(w)$ is consequently $1/(z-w{)}^3$, while the single contractions combine into twice the full stress tensor. ThereforeThe leading coefficient equals $2c/3=1$ for $c=3/2$.