A local operator $\mathcal O(w,\bar w)$ is a <primary operator> of <conformal weights> $(h,\widetilde h)$ when its <operator product expansions> with the holomorphic and antiholomorphic stress tensors are
$$
T(z)\mathcal O(w,\bar w)\sim
\frac{h\mathcal O(w,\bar w)}{(z-w)^2}
+\frac{\partial\mathcal O(w,\bar w)}{z-w},
$$
$$
\bar T(\bar z)\mathcal O(w,\bar w)\sim
\frac{\widetilde h\mathcal O(w,\bar w)}{(\bar z-\bar w)^2}
+\frac{\bar\partial\mathcal O(w,\bar w)}{\bar z-\bar w},
$$
with no more singular poles. Equivalently, under a local <conformal transformation> it transforms covariantly with holomorphic exponent $h$ and antiholomorphic exponent $\widetilde h$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 306 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-306/3/a/solution