An operator $\mathcal O(w,\bar w)$ is a <primary operator> of <conformal weights> $(h,\widetilde h)$ when its OPEs with the stress tensors are
$$
T(z)\mathcal O(w,\bar w)\sim
\frac{h\mathcal O(w,\bar w)}{(z-w)^2}
+\frac{\partial_w\mathcal O(w,\bar w)}{z-w},
$$
$$
\bar T(\bar z)\mathcal O(w,\bar w)\sim
\frac{\widetilde h\mathcal O(w,\bar w)}{(\bar z-\bar w)^2}
+\frac{\partial_{\bar w}\mathcal O(w,\bar w)}{\bar z-\bar w},
$$
with no more singular terms. Its <scaling dimension> and two-dimensional spin are
$$
\boxed{\Delta=h+\widetilde h,
\qquad s=h-\widetilde h}.
$$

Solution

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

 Solution

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