Applying <Wick theorem> to the stated <holomorphic stress-energy tensor> gives
$$
T(z)\gamma(w)\sim
\frac{h\gamma(w)}{(z-w)^2}+\frac{\partial\gamma(w)}{z-w},
$$
and
$$
T(z)\beta(w)\sim
\frac{(1-h)\beta(w)}{(z-w)^2}+\frac{\partial\beta(w)}{z-w}.
$$
These are exactly the stress-tensor OPEs of <primary operators>. Hence $\gamma$ has holomorphic <conformal weight> $h$ and $\beta$ has weight $1-h$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-306/2/b/solution