Fundamental SU(2) chiral matter and its covariant derivative

Choose the fundamental representation $R_{\mathrm{f}}:SU(2)\to U({\mathbb{C}}^2)$ defined by $R_{\mathrm{f}}(h)w=hw$ for $h\in SU(2)\subset M_2(\mathbb{C})$ and $w\in {\mathbb{C}}^2$, with the standard Hermitian form $w^{†}z$ on ${\mathbb{C}}^2$. Its differential is ${\rho }_{\mathrm{f}\ast }(X)=X$ for $X\in \mathfrak{s}\mathfrak{u}(2)$, and its Hermitian generators are $t_a={\sigma }_a\mathrm{/}2$. In a local trivialization write a scalar matter component as $\varphi =(u,v{)}^T$ and the Hermitian vector potential as $B_{\mu }=B_{\mu }^a{t}_a$, so the supplied connection is $A_{\mu }=-i{g}_{\mathrm{Y}\mathrm{M}}{B}_{\mu }$. The fundamental covariant derivative $D_{\mu }\varphi ={\mathrm{\partial }}_{\mu }\varphi +{\rho }_{\mathrm{f}\ast }(A_{\mu })\varphi$ has components

$D_{\mu }u={\mathrm{\partial }}_{\mu }u-\frac{i{g}_{\mathrm{Y}\mathrm{M}}}{2}(B_{\mu }^{3}u+(B_{\mu }^1-i{B}_{\mu }^2)v),D_{\mu }v={\mathrm{\partial }}_{\mu }v-\frac{i{g}_{\mathrm{Y}\mathrm{M}}}{2}((B_{\mu }^1+i{B}_{\mu }^2)u-B_{\mu }^{3}v)\mathrm{.}$

On an overlap with transition $h(x)\in SU(2)$, the associated-bundle component is ${\varphi }^{\mathrm{\prime }}=h^{-1}\varphi$ and the right-connection local form is $A_{\mu }^{\mathrm{\prime }}=h^{-1}{A}_{\mu }h+h^{-1}{\mathrm{\partial }}_{\mu }h$. Consequently $D_{\mu }^{\mathrm{\prime }}{\varphi }^{\mathrm{\prime }}=h^{-1}{D}_{\mu }\varphi$, so the scalar kinetic density $-(D_{\mu }\varphi {)}^{†}{D}^{\mu }\varphi$ agrees in both local frames.

Stephen P. Martin, A Supersymmetry Primer, §4.9.

Adel Bilal, Introduction to Supersymmetry, §5.2.

Rigid $N=1$ SU(2) gauge-matter superspace action

Four-dimensional $N=1$ non-Abelian vector superfield in connection normalization

Non-Abelian gauge data and bundle sectors

Classical pure SU(2) Yang–Mills model