Four-Dimensional $N=2$ Gauged Scalar Potential

The local Ward contraction in [F1] equals $\mathcal{V}$ before the gauge-coupling factor is restored. The adopted supersymmetry transformations in [F3] multiply every shift by the same $g$, so each quadratic shift term in the Ward identity, and hence the scalar potential, acquires a common factor $g^2$.

In the electric polarization of [F2], the magnetic components of the embedding tensor, and hence the magnetic gauging components, vanish; the section ${\mathcal{V}}^M=(L^{\mathrm{\Lambda }},M_{\mathrm{\Lambda }})$ still has both electric and magnetic components. Each section factor in the potential of [F1] is contracted against gauging data in its symplectic slot, so only its electric components $L^{\mathrm{\Lambda }}$ occur. The contraction of $U^{MN}$ against the electric moment maps selects its electric block $U^{\mathrm{\Lambda }\mathrm{\Sigma }}$. Substitute $U^{\mathrm{\Lambda }\mathrm{\Sigma }}=g^{i\bar{j}}{f}_i^{\mathrm{\Lambda }}{\bar{f}}_{\bar{j}}^{\mathrm{\Sigma }}$ and relabel the dummy electric indices to obtain exactly the Statement. The $4h_{uv}{k}_{\mathrm{\Lambda }}^u{k}_{\mathrm{\Sigma }}^v$ factor is the hyperino contraction, $U^{\mathrm{\Lambda }\mathrm{\Sigma }}{P}_{\mathrm{\Lambda }}^x{P}_{\mathrm{\Sigma }}^x$ is the gaugino contribution, and $-3\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}{P}_{\mathrm{\Lambda }}^x{P}_{\mathrm{\Sigma }}^x$ is the gravitino contribution already separated in [F1].