Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 3 / 2

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- Solution 2

Solution

The zero mode of the free embedding field enforces momentum conservation, while contractions of normal-ordered exponentials give the Koba-Nielsen factor:

⟨ \prod_{i = 1}^{n} : e^{i k_{i} \cdot X (z_{i}, \overset{z}{ˉ}_{i})} : ⟩ = (2 π)^{26} δ^{26} (\sum_{i} k_{i}) \prod_{i < j} ∣ z_{i} - z_{j} ∣^{α^{'} k_{i} \cdot k_{j}} .

Combining this with the supplied ghost correlator yields, up to the conventional overall normalization,

F_{n} = g_{c}^{, n - 2} ∣ z_{12} z_{23} z_{31} ∣^{2} i < j \prod ∣ z_{i} - z_{j} ∣^{α^{'} k_{i} \cdot k_{j}} .

Therefore

A_{n} = δ^{26} (\sum_{i} k_{i}) \int d^{2} z_{4} \dots d^{2} z_{n} F_{n},

with the factor

(2 π)^{26}

absorbed into the amplitude normalization used by the question.

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