Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 342 / 1 / b

Table of contents
- Solution b

Solution

Under

a_{α}^{I} \mapsto a_{α}^{I} + \partial_{α} Λ^{I}

, the Chern--Simons term changes by

\frac{1}{4 π ℏ} K_{I J} ϵ^{α β γ} \partial_{α} Λ^{I} \partial_{β} a_{γ}^{J} = \partial_{α} [\frac{1}{4 π ℏ} K_{I J} ϵ^{α β γ} Λ^{I} \partial_{β} a_{γ}^{J}],

because the contraction with two commuting derivatives vanishes. The source term changes by

- j_{I}^{α} \partial_{α} Λ^{I} = - \partial_{α} (j_{I}^{α} Λ^{I}) + Λ^{I} \partial_{α} j_{I}^{α} .

Current conservation removes the last term, so

δ L = a total derivative .

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