With canonical normalization, the renormalizable <supersymmetric gauge theory> has
$$
\mathcal L=
\int d^4\theta\,\Phi^\dagger e^{2V}\Phi
+\left[\frac1{16g^2}\int d^2\theta\,
\operatorname{Tr}(W^\alpha W_\alpha)+\mathrm{h.c.}\right].
$$
The first term is the gauge-covariant <Kähler potential> and the second is the <Supersymmetric Yang-Mills action>. A holomorphic <superpotential> would also be integrated as $\int d^2\theta\,\mathcal W(\Phi)+\mathrm{h.c.}$, but one commuting field in the fundamental representation of $SU(3)$ admits no nonconstant renormalizable gauge-invariant superpotential: the apparent cubic $\epsilon_{ijk}\Phi^i\Phi^j\Phi^k$ vanishes.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-307/1/v/solution