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

Use the physical action brackets $[U{]}_D=-I_D(U)\mathrm{/}4$ and $[Y{]}_F=I_F(Y)\mathrm{/}2$ from Physical superspace action brackets and canonical chiral components, with raw mathematical coefficient integrals $I_D,I_F$ unchanged. Physical chiral components obey ${\psi }_{\mathrm{r}\mathrm{a}\mathrm{w}}=\sqrt{2}\psi$ and $F_{\mathrm{r}\mathrm{a}\mathrm{w}}=2F$. All physical action and coupling formulas below use these brackets and physical component names; adding Hermitian conjugation applies only to the F action and occurs once.

Use the selected smooth right-principal $SU(2)$ bundle $P\to M$ and its connection $A$ from the non-Abelian Yang–Mills model, with $M={\mathbb{R}}^{1,3}$, $\eta =\mathrm{diag}(-1,1,1,1)$, $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$, and positive dimensionless coupling $g_{\mathrm{Y}\mathrm{M}}$. For each matter representation, let ${\mathcal{H}}_R$ be a finite-dimensional complex Hermitian space and $R:SU(2)\to U({\mathcal{H}}_R)$ a smooth unitary representation; write ${\rho }_{R\ast }$ for its Lie-algebra differential. In the fundamental representation set $T_a=-i{\sigma }_a\mathrm{/}2\in \mathfrak{s}\mathfrak{u}(2)$ and $t_a=i{T}_a={\sigma }_a\mathrm{/}2$, so ${\mathrm{Tr}}_2(t_a{t}_b)=\frac{1}{2}{\delta }_{ab}$. In representation $R$, set $t_{a,R}=i{\rho }_{R\ast }(T_a)$. In a local trivialization write $V=V^a{t}_a$ and let $V_R=V^a{t}_{a,R}$ be its represented matrix superfield in ${\mathrm{End}}_{\mathbb{C}}({\mathcal{H}}_R)$, and set $A_{\mu }=-i{g}_{\mathrm{Y}\mathrm{M}}{B}_{\mu }$ when $B_{\mu }=B_{\mu }^a{t}_a$ is its vector component. Thus the matter covariant derivative is $D_{\mu }={\mathrm{\partial }}_{\mu }+{\rho }_{R\ast }(A_{\mu })={\mathrm{\partial }}_{\mu }-i{g}_{\mathrm{Y}\mathrm{M}}{B}_{\mu }^a{t}_{a,R}$.

A vector superfield prepotential is a dimensionless smooth even real matrix-valued superfield $V$ in the finite Grassmann-valued superspace of the $N=1$ chiral-superfield convention. Under a global rigid supersymmetry variation it transforms as $\delta V=({ϵ}^{\alpha }{Q}_{\alpha }+{\overline{ϵ}}^{\dot{\alpha }}{\bar{Q}}_{\dot{\alpha }})V$, with the supercharges defined in the chiral-superfield convention. Its local gauge transformations are ${\mathrm{\Phi }}_R\mapsto e^{i{\mathrm{\Lambda }}_R}{\mathrm{\Phi }}_R$ and $e^{2g_{\mathrm{Y}\mathrm{M}}{V}_R}\mapsto e^{i{\mathrm{\Lambda }}_R^{†}}{e}^{2g_{\mathrm{Y}\mathrm{M}}{V}_R}{e}^{-i{\mathrm{\Lambda }}_R}$, where ${\mathrm{\Phi }}_R$ is chiral, $V_R=V^a{t}_{a,R}$, and ${\mathrm{\Lambda }}_R={\mathrm{\Lambda }}^a{t}_{a,R}$ is a dimensionless chiral gauge-parameter superfield in the represented complexified Lie algebra $\mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}}=\mathfrak{s}\mathfrak{u}(2){\otimes }_{\mathbb{R}}\mathbb{C}$. The associated represented matrix curvature is defined by

${\mathcal{C}}_{\alpha ,R}=-\frac{1}{4}{\bar{D}}^{\dot{\beta }}{\bar{D}}_{\dot{\beta }}\text{\hspace{0.17em}⁣}\left(e^{-2g_{\mathrm{Y}\mathrm{M}}{V}_R}{D}_{\alpha }{e}^{2g_{\mathrm{Y}\mathrm{M}}{V}_R}\right),$

where $D_{\alpha }={\mathrm{\partial }}_{{\theta }^{\alpha }}^L+i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\mathrm{\partial }}_{\mu }$ and ${\bar{D}}_{\dot{\alpha }}=-{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{\alpha }}}^L-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\mathrm{\partial }}_{\mu }$. Admissible vector-superfield data require ${\bar{D}}_{\dot{\gamma }}{\mathcal{C}}_{\alpha ,R}=0$ and the adjoint transition law ${\mathcal{C}}_{\alpha ,j}={\mathrm{Ad}}_{g_{ij}^{-1}}{\mathcal{C}}_{\alpha ,i}$ on each overlap with principal transition function $g_{ij}$.

Distinguish this represented curvature from the abstract connection-normalized Lie-algebra-valued curvature ${\mathcal{W}}_{\alpha }$. With canonical adjoint component field strength $W_{\mathrm{c}\mathrm{a}\mathrm{n},\alpha }^a$, the matrix convention is ${\mathcal{C}}_{\alpha ,R}=2g_{\mathrm{Y}\mathrm{M}}{t}_{a,R}{W}_{\mathrm{c}\mathrm{a}\mathrm{n},\alpha }^a$, whereas ${\mathcal{W}}_{\alpha }=g_{\mathrm{Y}\mathrm{M}}{T}_a{W}_{\mathrm{c}\mathrm{a}\mathrm{n},\alpha }^a$. Since $t_{a,R}=i{\rho }_{R\ast }(T_a)$, their precise dictionary is

${\rho }_{R\ast }({\mathcal{W}}_{\alpha })=\frac{{\mathcal{C}}_{\alpha ,R}}{2i}\mathrm{.}$

In the fundamental representation this gives ${\kappa }_{\mathbb{C}}({\mathcal{W}}^{\alpha },{\mathcal{W}}_{\alpha })=\frac{1}{2}{\mathrm{Tr}}_2({\mathcal{C}}^{\alpha }{\mathcal{C}}_{\alpha })$, where ${\kappa }_{\mathbb{C}}=-2{\mathrm{Tr}}_2$ is evaluated on the Lie-algebra matrices $\mathcal{W}$, not on the raw $\mathcal{C}$. Thus the chiral action coefficient ${\kappa }_{\mathbb{C}}({\mathcal{W}}^{\alpha },{\mathcal{W}}_{\alpha })\mathrm{/}(4g_{\mathrm{Y}\mathrm{M}}^2)$ equals ${\mathrm{Tr}}_2({\mathcal{C}}^{\alpha }{\mathcal{C}}_{\alpha })\mathrm{/}(8g_{\mathrm{Y}\mathrm{M}}^2)$, before adding its Hermitian conjugate. This is a fixed real-coupling classical normalization identity; it is not a quantum-field rescaling or measure identity. Both curvatures are chiral and have the corresponding conjugation/transition law. The abstract $\mathcal{W}$ has the connection-normalized gaugino and auxiliary components specified below.

In a local Wess–Zumino-gauge representative, the canonical component fields in the prepotential are a Hermitian gauge potential $B_{\mu }$, an adjoint Grassmann-odd Weyl gaugino ${\lambda }_{B,\alpha }$, and a real adjoint auxiliary field $D_B$, with expansion $V_{\mathrm{W}\mathrm{Z}}=\theta {\sigma }^{\mu }\bar{\theta }{B}_{\mu }+i{\theta }^2\bar{\theta }{\bar{\lambda }}_B-i{\bar{\theta }}^2\theta {\lambda }_B+\frac{1}{2}{\theta }^2{\bar{\theta }}^2{D}_B$. Here ${\theta }^2={\theta }^{\alpha }{\theta }_{\alpha }$ and ${\bar{\theta }}^2={\bar{\theta }}_{\dot{\alpha }}{\bar{\theta }}^{\dot{\alpha }}$ use the fixed antisymmetric spinor tensors. Their dimensions are $[B_{\mu }]=1$, $[{\lambda }_B]=3\mathrm{/}2$, and $[D_B]=2$. The connection-normalized fields are $A_{\mu }=-i{g}_{\mathrm{Y}\mathrm{M}}{B}_{\mu }$, $\lambda =g_{\mathrm{Y}\mathrm{M}}{\lambda }_B$, and $D=g_{\mathrm{Y}\mathrm{M}}{D}_B$; hence $[A_{\mu }]=1$, $[\lambda ]=3\mathrm{/}2$, and $[D]=2$. The full prepotential is used for supersymmetry transformations; Wess–Zumino gauge is only a component representative and need not be preserved by an uncompensated supersymmetry variation.

Stephen P. Martin, A Supersymmetry Primer, §§4.5 and 4.9.

Adel Bilal, Introduction to Supersymmetry, §§4.4 and 5.1.

Classical pure SU(2) Yang–Mills model

Non-Abelian gauge data and bundle sectors

Four-dimensional $N=1$ chiral superfield on flat superspace

Chiral-coordinate solution and supersymmetry covariance

Four-dimensional $N=1$ superspace Grassmann algebra and Berezin integration

Physical superspace action brackets and canonical chiral components