Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 307 / 1 / vii

Table of contents
- Solution vii

Solution

The relation in part vi gives

Φ^{'}^{†} e^{2 V^{'}} Φ^{'} = Φ^{†} e^{2 i Λ^{†}} e^{- 2 i Λ^{†}} e^{2 V} e^{2 i Λ} e^{- 2 i Λ} Φ = Φ^{†} e^{2 V} Φ,

so the full-superspace term has gauge invariance. The chiral field-strength superfield transforms covariantly,

W_{α}^{'} = e^{- 2 i Λ} W_{α} e^{2 i Λ} .

Cyclicity of the matrix trace then gives

Tr (W^{'}^{α} W_{α}^{'}) = Tr (W^{α} W_{α})

. Both superspace integrals, and hence the entire Lagrangian density, are invariant.

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