Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 304 / 4 / a

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- Solution a

Solution

For a gauge orbit near a root

α_{0}

of

G (A^{α}) = 0

, functional linearization gives

G (A^{α}) = \frac{δ G ( A ^{α} )}{δ α}_{α_{0}} (α - α_{0}) + O ((α - α_{0})^{2}) .

The multidimensional delta-function change-of-variables formula therefore gives

\int D α δ [G (A^{α})] = det (\frac{δ G ( A ^{α} )}{δ α}_{G = 0})^{- 1} .

Comparison with the defining identity proves

Δ_{FP} [A] = det (\frac{δ G ( A ^{α} )}{δ α}_{G = 0}),

up to the usual field-independent normalization and a choice of determinant sign on the gauge patch.

Solved by gpt-5.6-sol high.

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