$N=2$ Fermion-Shift Ward Contraction

Expanding the two terms in $W^{iAB}$ from [F1], the $k_{\mathrm{\Lambda }}^i\overline{L^{\mathrm{\Lambda }}}$ product with its conjugate gives the singlet ${\delta }^A{}_B{g}_{i\bar{j}}{k}_{\mathrm{\Lambda }}^i{k}_{\mathrm{\Sigma }}^{\bar{j}}\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}$. Write $s_x$ for the ordinary row-first Pauli matrix in the source and $t_x=s_x^{\mathsf{T}}$ for its mixed-index component matrix, so $(t_x{)}^A{}_B=({\sigma }^x{)}^A{}_B$ in [F3]. The source row-first identity is $s_x{s}_y={\delta }^{xy}I+i{ϵ}^{xyz}{s}_z$; hence the two distinct ordered identities are $t_y{t}_x={\delta }^{xy}I+i{ϵ}^{xyz}{t}_z$ and $t_x{t}_y={\delta }^{xy}I-i{ϵ}^{xyz}{t}_z$. To see the actual contraction order, put $E=({ϵ}_{AB})=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, $a^i=k_{\mathrm{\Lambda }}^i\overline{L^{\mathrm{\Lambda }}}$ and $b_x^i=P_{\mathrm{\Lambda }}^x{g}^{i\bar{j}}{\bar{f}}_{\bar{j}}^{\mathrm{\Lambda }}$. Then the displayed shift is the matrix $W^i=E(a^{i}I+i{s}_x{b}_x^i)$, and $E{s}_x{E}^{\mathsf{T}}=-t_x$. Thus $g_{i\bar{j}}{W}^i{\overline{W^j}}^{\mathsf{T}}$ has a mixed term $i{t}_x{g}_{i\bar{j}}(a^i\overline{b_x^j}-b_x^i\overline{a^j})$ and a moment-map triplet $-i{ϵ}^{xyz}{t}_z{g}_{i\bar{j}}{b}_x^i\overline{b_y^j}$. The latter coefficient is $g_{i\bar{j}}{b}_x^i\overline{b_y^j}=P_{\mathrm{\Lambda }}^x{P}_{\mathrm{\Sigma }}^y{U}^{\mathrm{\Sigma }\mathrm{\Lambda }}$. Relabeling $\mathrm{\Lambda },\mathrm{\Sigma }$ and then $x,y$ converts its triplet to $+i{ϵ}^{xyz}{P}_{\mathrm{\Lambda }}^x{P}_{\mathrm{\Sigma }}^y{U}^{\mathrm{\Lambda }\mathrm{\Sigma }}{t}_z$. This explicit relabeling is why the moment-map term written in the source coefficient order $U^{\mathrm{\Lambda }\mathrm{\Sigma }}$ uses $(s_x{s}_y{)}^{\mathsf{T}}=t_y{t}_x$. The singlet part is $U^{\mathrm{\Lambda }\mathrm{\Sigma }}{P}_{\mathrm{\Lambda }}^x{P}_{\mathrm{\Sigma }}^x$ by the definition of $U$. For the triplet part, the symmetric vector-coupling matrix in the identity of [F3] cancels after antisymmetrizing the moment-map indices; the canonical symplectic term vanishes by the electric locality constraint, and the remaining section term is $-{ϵ}^{xyz}{P}_{\mathrm{\Lambda }}^y{P}_{\mathrm{\Sigma }}^z\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}$. The mixed $k$–$P$ terms reduce by the exact special-Kähler contraction identity in [F3] to $-2X_{\mathrm{\Lambda }\mathrm{\Sigma }}{}^{\mathrm{\Gamma }}{P}_{\mathrm{\Gamma }}^x\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}$. Collecting these coefficients yields $g_{i\bar{j}}{W}^{iAC}{\bar{W}}^{\bar{j}}{}_{BC}={\delta }^A{}_B\text{\hspace{0.17em}⁣}(g_{i\bar{j}}{k}_{\mathrm{\Lambda }}^i{k}_{\mathrm{\Sigma }}^{\bar{j}}\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}+U^{\mathrm{\Lambda }\mathrm{\Sigma }}{P}_{\mathrm{\Lambda }}^x{P}_{\mathrm{\Sigma }}^x)+i({\sigma }^x{)}^A{}_B\text{\hspace{0.17em}⁣}(-2X_{\mathrm{\Lambda }\mathrm{\Sigma }}{}^{\mathrm{\Gamma }}{P}_{\mathrm{\Gamma }}^x-{ϵ}^{xyz}{P}_{\mathrm{\Lambda }}^y{P}_{\mathrm{\Sigma }}^z)\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}\mathrm{.}$ The relative sign between the two terms in $W$ is the corrected sign in [F1], and is needed for these mixed terms.

By the typed vielbein identity and conjugate lower-hyperino frame in [F3], ${\mathcal{U}}^A{}_{u\alpha }{\bar{\mathcal{U}}}^{\alpha }{}_{vB}=\frac{1}{2}{h}_{uv}{\delta }^A{}_B+\frac{i}{2}{K}_{uv}^x({\sigma }^x{)}^A{}_B$. The corrected shift in [F1] is $N_{\alpha }{}^A=2{\mathcal{U}}^A{}_{u\alpha }{k}_{\mathrm{\Lambda }}^u\overline{L^{\mathrm{\Lambda }}}$, whose conjugate is ${\bar{N}}^{\alpha }{}_B=2{\bar{\mathcal{U}}}^{\alpha }{}_{vB}{k}_{\mathrm{\Sigma }}^v{L}^{\mathrm{\Sigma }}$; the Killing vectors are real on the quaternionic target. Each hyperino shift has coefficient 2 by [F1], so the product with the Ward factor 2 gives an overall coefficient $2\cdot 2\cdot 2\cdot \frac{1}{2}=4$ in each contraction: $2N_{\alpha }{}^A{\bar{N}}^{\alpha }{}_B=4{\delta }^A{}_B{h}_{uv}{k}_{\mathrm{\Lambda }}^u{k}_{\mathrm{\Sigma }}^v\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}+4i({\sigma }^x{)}^A{}_B{K}_{uv}^x{k}_{\mathrm{\Lambda }}^u{k}_{\mathrm{\Sigma }}^v\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}\mathrm{.}$ This gives the hyperino singlet and triplet terms without using field equations.

Multiplying the electric gravitino shift of [F1] by its conjugate gives the ordinary mixed-index order $t_x{t}_y={\delta }^{xy}I-i{ϵ}^{xyz}{t}_z$ from step 1.1, with prefactor $-12(1\mathrm{/}2)(1\mathrm{/}2)=-3$. Electric support of the gauging selects $L^{\mathrm{\Lambda }}$ in this contraction; it does not set the magnetic section components $M_{\mathrm{\Lambda }}$ to zero. This order therefore gives a negative singlet and a positive triplet: $-12{\bar{S}}^{AC}{S}_{CB}=-3{\delta }^A{}_B{P}_{\mathrm{\Lambda }}^x{P}_{\mathrm{\Sigma }}^x\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}+3i({\sigma }^x{)}^A{}_B{ϵ}^{xyz}{P}_{\mathrm{\Lambda }}^y{P}_{\mathrm{\Sigma }}^z\overline{L^{\mathrm{\Lambda }}}{L}^{\mathrm{\Sigma }}\mathrm{.}$ The factor $-3$ follows from $-12$ times the two shift coefficients $1\mathrm{/}2$ and $1\mathrm{/}2$.

Adding the singlets in 1.1, 1.2, and 2.1 gives $\mathcal{V}$. Adding their triplets gives the displayed $Z^x$. By [F3], $2Kkk+ϵPP=XP$; hence its triplet coefficient is $-2XP+2ϵPP+4Kkk=-2XP+2(ϵPP+2Kkk)=-2XP+2XP=0$. Thus $Z^x=0$, leaving the singlet Ward contraction.