 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-306/3/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

After integration by parts, the quadratic operator in the action is

K = - π^{- 1} \partial \overset{ˉ}{\partial}

G (z, w) = ⟨ X (z, \overset{z}{ˉ}) X (w, \overset{w}{ˉ})⟩

therefore satisfies

- \frac{1}{π} \partial \overset{ˉ}{\partial} G (z, w) = δ^{(2)} (z - w) .

⟨ X (z, \overset{z}{ˉ}) X (w, \overset{w}{ˉ})⟩ = - \frac{1}{2} lo g ∣ z - w ∣^{2} + C,

where the additive constant depends on the infrared convention and cancels from neutral correlators.

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