Under $\Phi'=e^{-2i\Lambda}\Phi$, the conjugate field transforms as $\Phi'{}^\dagger=\Phi^\dagger e^{2i\Lambda^\dagger}$. Invariance of the canonical Kähler term therefore requires
$$
e^{2i\Lambda^\dagger}e^{2V'}e^{-2i\Lambda}=e^{2V},
$$
or
$$
\boxed{e^{2V'}=e^{-2i\Lambda^\dagger}e^{2V}e^{2i\Lambda}}.
$$
This is the finite non-Abelian <supergauge transformation> of the vector superfield.

Solution

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

A supergauge transformation acts on a charged chiral superfield and vector superfield together. In one common convention, $\Phi'=e^{-2i\Lambda}\Phi$ and $e^{2V'}=e^{-2i\Lambda^\dagger}e^{2V}e^{2i\Lambda}$.


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