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

LetThe exponential is a primary operator with conformal weights $({\alpha }^{\prime }{k}^2/4,{\alpha }^{\prime }{k}^2/4)$. In the $T(z)V(w,\bar{w})$ OPE, contracting one derivative in $T$ with $\partial X^{\mu }$ and the other with the exponential produces a third-order pole proportional to $k^{\mu }{ϵ}_{\mu \nu }$. Its antiholomorphic counterpart is proportional to ${ϵ}_{\mu \nu }{k}^{\nu }$. Therefore $V$ is primary exactly when its polarization tensor is transverse in both indices,The remaining second-order poles give