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

Extract a mode usingIn the radially ordered double contour for the anticommutator, the simple pole $2T(w)/(z-w)$ gives $2L_{m+n}$. Expanding $z^{m+1/2}$ about $w$, the third-order pole contributes one half of its second derivative,The remaining contour is nonzero only for $m+n=0$. Restoring the general leading OPE coefficient $2c/3$ givesThis is the fermionic relation in the N=1 super-Virasoro algebra.