Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 2

The Green function for the normalization in the question isDifferentiating at distinct points gives the holomorphic two-point function

In $T(z)\partial X(w)$, a single Wick contraction can be made with either factor in $T$. Using the two-point function and expanding the remaining field around $w$ givesThis operator product expansion says that $\partial X$ is a holomorphic primary operator of conformal weight $(1,0)$.

A primary operator of conformal weight $(h,\tilde{h})$ transforms under $z\mapsto w(z)$ asThe derivative $\partial X$ is a primary of weight $(1,0)$, and $\bar{\partial }X$ is a primary of weight $(0,1)$.

Transforming the point-split normal ordering changes the subtraction as well as the two derivatives. Expanding $w(z\pm \delta /2)$ to third order giveswhere the Schwarzian derivative isThe inhomogeneous Schwarzian term means that $T$ is not a primary operator. It reflects the central charge $c=1$ of one free scalar.