Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 3

Under a finite conformal coordinate change $z^{\prime }=f(z)$ and ${\bar{z}}^{\prime }=\bar{f}(\bar{z})$, a primary operator of conformal weight $(h,\tilde{h})$ obeysEquivalently, its operator product expansion with the holomorphic stress-energy tensor has singular partThe antiholomorphic stress tensor gives the analogous formula with $\tilde{h}$.

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After integration by parts, the quadratic operator in the action is $K=-{\pi }^{-1}\partial \bar{\partial }$. Its Green function $G(z,w)=⟨X(z,\bar{z})X(w,\bar{w})⟩$ therefore satisfiesUsing the given distributional identity giveswhere the additive constant depends on the infrared convention and cancels from neutral correlators.

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For this normalization,Contracting once and twice withgivesIt is therefore a primary operator with

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Because normal ordering removes contractions within either exponential, Wick theorem sums all cross-contractions intoUnder $z\mapsto \lambda z$, this scales as ${\lambda }^{-k^2}$. Each operator has scaling dimension $\Delta =h+\tilde{h}=k^2/2$, so a two-point function of two such primaries must scale as ${\lambda }^{-2\Delta }={\lambda }^{-k^2}$. The two calculations agree.

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