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

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 classical pure $SU(2)$ gauge model on oriented and time-oriented Minkowski $M={\mathbb{R}}^{1,3}$ with $\eta =\mathrm{diag}(-1,1,1,1)$, one selected smooth right-principal bundle $P\to M$, connection $A$, curvature $F_A=dA+\frac{1}{2}[A\wedge A]$, invariant form $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$, and positive dimensionless $g_{\mathrm{Y}\mathrm{M}}$ from the non-Abelian Yang–Mills model. Use natural units $\mathrm{\hslash }=c=1$, so $[x^{\mu }]=-1$, $[g_{\mathrm{Y}\mathrm{M}}]=0$, $[A_{\mu }]=1$, and $[F_{A,\mu \nu }]=2$. Let $g=g_{\mathrm{Y}\mathrm{M}}$.

Choose a finite index set $I$ with $\mathrm{\mid }I\mathrm{\mid }\ge 1$. For each $a\in I$, choose a finite-dimensional complex Hermitian space $V_a$, a smooth unitary representation $R_a:SU(2)\to U(V_a)$, and its associated vector bundle $E_a=P{\times }_{R_a}{V}_a$. Define $V_{\mathrm{m}\mathrm{a}\mathrm{t}}={⨁}_{a\in I}{V}_a$ and $R_{\mathrm{m}\mathrm{a}\mathrm{t}}(h)={⨁}_{a\in I}{R}_a(h)$. The matter field ${\mathrm{\Phi }}^a$ is a smooth chiral superfield with local components in $E_a$ and unique full even scalar, odd left Weyl, and even auxiliary components $(a^a,{\psi }^a,{\mathcal{F}}^a)$ in the coefficient convention below; their engineering dimensions are $1$, $3\mathrm{/}2$, and $2$. Let $V$ be the full real vector prepotential of the local vector-superfield definition, with connection-normalized component $A_{\mu }=-ig{B}_{\mu }$ in a local unitary frame, and let ${\mathcal{W}}_{\alpha }$ be its abstract connection-normalized Lie-algebra-valued chiral field strength. Its represented matrix curvature is ${\mathcal{C}}_{\alpha ,R}=-\frac{1}{4}{\bar{D}}^{\dot{\beta }}{\bar{D}}_{\dot{\beta }}(e^{-2g{V}_R}{D}_{\alpha }{e}^{2g{V}_R})$; the dictionary is ${\rho }_{R\ast }({\mathcal{W}}_{\alpha })={\mathcal{C}}_{\alpha ,R}\mathrm{/}(2i)$, so in the defining representation ${\kappa }_{\mathbb{C}}({\mathcal{W}}^{\alpha },{\mathcal{W}}_{\alpha })={\mathrm{Tr}}_2({\mathcal{C}}^{\alpha }{\mathcal{C}}_{\alpha })\mathrm{/}2$. The matter covariant derivative is $D_{\mu }{\varphi }^a={\mathrm{\partial }}_{\mu }{\varphi }^a+{\rho }_{a\ast }(A_{\mu }){\varphi }^a={\mathrm{\partial }}_{\mu }{\varphi }^a-ig{B}_{\mu }^b{t}_{b,a}{\varphi }^a$, where $t_{b,a}=i{\rho }_{a\ast }(T_b)$ and $T_b=-i{\sigma }_b\mathrm{/}2$. Take smooth field-dependent component fluctuations with compact support in the local trivializing regions used for the action; noncompact backgrounds are treated only through compactly supported relative action differences.

Choose a holomorphic superpotential $W_G:V_{\mathrm{m}\mathrm{a}\mathrm{t}}\to \mathbb{C}$ of engineering dimension three, where holomorphic means that every Wirtinger derivative ${\mathrm{\partial }}_{{\bar{z}}^A}{W}_G$ vanishes in complex-linear coordinates $z^A$ on $V_{\mathrm{m}\mathrm{a}\mathrm{t}}$. Require invariance under the induced $SU(2)$ action, $W_G(R_{\mathrm{m}\mathrm{a}\mathrm{t}}(h)v)=W_G(v)$ for all $h\in SU(2)$ and $v\in V_{\mathrm{m}\mathrm{a}\mathrm{t}}$. Its complex differential $d{W}_{G,v}:V_{\mathrm{m}\mathrm{a}\mathrm{t}}\to \mathbb{C}$ is the complex-linear map with coordinate expression $d{W}_{G,v}(w)={\sum }_A({\mathrm{\partial }}_{z^A}{W}_G)(v)w^A$. Write ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{t}\ast }(X)={⨁}_{a\in I}{\rho }_{a\ast }(X)$ for the infinitesimal representation. Differentiating the invariance condition along $h=\exp (tX)$ gives $d{W}_{G,v}({\rho }_{\mathrm{m}\mathrm{a}\mathrm{t}\ast }(X)v)={\sum }_{a\in I}d{W}_{G,v}(0,\dots ,{\rho }_{a\ast }(X)v^a,\dots ,0)=0$ for every $X\in \mathfrak{s}\mathfrak{u}(2)$.

In each local unitary frame, use the finite Grassmann convention of the Wess–Zumino model postulate for this complete finite matter family: take the finite exterior coefficient algebra $\mathcal{G}$ on $m$ odd generators with generator-fixing conjugate-linear, parity-preserving involutive anti-automorphism $(c{\zeta }_{i_1}\cdots {\zeta }_{i_r}{)}^{\ast }=\bar{c}{\zeta }_{i_r}\cdots {\zeta }_{i_1}$ and its degree-zero augmentation, exactly as defined there, and let $\mathfrak{J}$ be the zero-body ideal in $\mathcal{G}\otimes {\mathrm{\Lambda }}_4$, so ${\mathfrak{J}}^{m+5}=0$. Local full scalar and auxiliary components are smooth maps $a^a,{\mathcal{F}}^a$ into ${\mathcal{G}}_{\bar{0}}\otimes V_a$, and fermions into ${\mathcal{G}}_{\bar{1}}\otimes V_a$, with coefficientwise compact support on $M$ in the stated frames. Complex conjugation combines the specified $\mathcal{G}$ involution with the Hermitian dual in the unitary frame. The ordinary bosonic restriction sets all positive coefficient-algebra degrees and fermions to zero, giving complex fields ${\varphi }^a=b(a^a)$ and $F^a=b({\mathcal{F}}^a)$; it is not the formal field space used for off-shell closure. Use the full $\mathcal{G}$-valued action and coefficientwise supersymmetry variation, and the conjugation-compatible parameter convention of the Wess–Zumino postulate. Let $v(x)=({\varphi }^a(x){)}_{a\in I}$ be the body and $\nu (x,\theta ,\bar{\theta })=\mathrm{\Phi }(x,\theta ,\bar{\theta })-v(x)\in \mathfrak{J}{\otimes }_{\mathbb{C}}{V}_{\mathrm{m}\mathrm{a}\mathrm{t}}$ its nilpotent increment. Evaluate $W_G$ by the finite Taylor rule of the Wess–Zumino model postulate: after choosing complex-linear coordinates on $V_{\mathrm{m}\mathrm{a}\mathrm{t}}$, write $d={\dim }_{\mathbb{C}}{V}_{\mathrm{m}\mathrm{a}\mathrm{t}}$ and let $\gamma =({\gamma }_A{)}_{A=1}^d\in {\mathbb{N}}^d$ be a multi-index, with $\gamma !={\prod }_A{\gamma }_A!$, ${\mathrm{\partial }}_z^{\gamma }={\prod }_A{\mathrm{\partial }}_{z^A}^{{\gamma }_A}$, and ${\nu }^{\gamma }={\prod }_A({\nu }^A{)}^{{\gamma }_A}$.

$W_G(\mathrm{\Phi })(x,\theta ,\bar{\theta }):={\sum }_{\mathrm{\mid }\gamma \mathrm{\mid }\le m+4}\frac{({\mathrm{\partial }}_z^{\gamma }{W}_G)(v(x))}{\gamma !}\nu (x,\theta ,\bar{\theta }{)}^{\gamma },$

where the finite Grassmann algebra and its nilpotent ideal are chosen for this finite component family as in that postulate. Thus the local expression is chiral and, by the stated $SU(2)$ invariance, defines the same scalar superpotential on overlapping associated-bundle frames. For a superspace superfield $X$, use the normalized physical brackets $[X{]}_D=-I_D(X)\mathrm{/}4$ and, for chiral $Y$, $[Y{]}_F=I_F(Y)\mathrm{/}2$. Let ${\kappa }_{\mathbb{C}}$ denote the complex-bilinear extension of $\kappa$. Adopt the action

$S_{\mathrm{G}\mathrm{M}}={\int }_M{d}^{4}x\{{\sum }_{a\in I}[{\bar{\mathrm{\Phi }}}^a{e}^{2g{V}_{R_a}}{\mathrm{\Phi }}^a{]}_D+({[\frac{1}{4g^2}{\kappa }_{\mathbb{C}}({\mathcal{W}}^{\alpha },{\mathcal{W}}_{\alpha })+W_G(\mathrm{\Phi })]}_F+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.})\},$

where $V_{R_a}$ is the represented vector prepotential from the local definition. Its coupling convention is fixed by requiring the bosonic gauge-curvature component to be exactly $-\kappa (F_A\wedge {\ast }_{\eta }{F}_A)\mathrm{/}(2g^2)$; the same $g$ occurs in $e^{2g{V}_{R_a}}$ and in $D_{\mu }={\mathrm{\partial }}_{\mu }+{\rho }_{a\ast }(A_{\mu })$, with no second gauge-coupling rescaling. The matter fields have $[{\mathrm{\Phi }}^a]=1$, $[{\psi }^a]=3\mathrm{/}2$, and $[F^a]=2$; $W_G$ has engineering dimension three. The model is defined locally in bundle trivializations with the transition laws induced by $P$; the D/F densities are required to obey the displayed supergauge law, with $\kappa$ invariant and $W_G$ gauge invariant, so the local action densities patch. Take configurations for which the action is integrable, or use compactly supported relative action differences with the same boundary convention as the supplied pure-gauge model. This is an adopted classical action, not a derivation that nature realizes it or that a quantum continuum theory exists. The gauge auxiliary field contributes its separate $D$-term sector; the $F$-term theorem on this page concerns only the chiral auxiliary contribution.

Stephen P. Martin, A Supersymmetry Primer, §4.5 and §§4.8–4.9.

Adel Bilal, Introduction to Supersymmetry, §§4.4 and 5.1–5.2.

Rigid four-dimensional $N=1$ Wess–Zumino sigma-model action

Classical pure SU(2) Yang–Mills model

Non-Abelian gauge data and bundle sectors

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

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

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

Physical superspace action brackets and canonical chiral components