Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 313 / 2 / i

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- Solution i

Solution

The Hodge star operator is defined by

α \land ⋆ β = ⟨ α, β ⟩_{η} vol .

On two-forms in oriented Euclidean four-space,

⋆^{2} = 1

and the wedge product is symmetric. Therefore

⋆ ω \land ⋆ ω = ω \land ⋆^{2} ω = ω \land ω .

For an anti-self-dual curvature

F = - ⋆ F

, the Yang-Mills instanton action becomes purely topological. With

c_{2} = - \frac{1}{8 π ^{2}} \int_{R^{4}} Tr (F \land F),

the standard positive-action convention gives

S_{YM} = 8 π^{2} c_{2} .

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