The free equations of motion are $\bar\partial\psi=0$ and $\partial\bar\partial X=0$, hence
$$
\bar\partial G=i(\bar\partial\psi)\partial X+i\psi\bar\partial\partial X=0.
$$
Thus 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. Therefore
$$
\boxed{G(z)G(w)\sim
\frac1{(z-w)^3}+\frac{2T(w)}{z-w}}.
$$
The leading coefficient equals $2c/3=1$ for $c=3/2$.

Solution

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

 Solution

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