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

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Solution

The Green function for the normalization in the question is

⟨ X (z, \overset{z}{ˉ}) X (w, \overset{w}{ˉ})⟩ = - \frac{1}{2} lo g ∣ z - w ∣^{2} + e x t co n s t an t .

Differentiating at distinct points gives the holomorphic two-point function

⟨ \partial_{z} X (z) \partial_{w} X (w)⟩ = - \frac{1}{2 ( z - w ) ^{2}} .

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