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

Let $\mathcal{A}={\mathcal{A}}_{\bar{0}}\oplus {\mathcal{A}}_{\bar{1}}$ be the finite-dimensional supercommutative complex coefficient algebra, and let ${\mathrm{\Lambda }}_4$ have ordered generators ${\theta }^1,{\theta }^2,{\bar{\theta }}^{\dot{1}},{\bar{\theta }}^{\dot{2}}$ and the product, parity, derivatives and conjugation specified in the four-dimensional superspace Grassmann algebra. Let $\mathcal{S}=C^{\mathrm{\infty }}({\mathbb{R}}^{1,3},\mathcal{A}\otimes {\mathrm{\Lambda }}_4)$ be the space of smooth coefficient functions; ${\mathrm{\partial }}_{\mu }$ differentiates the real coordinate $x^{\mu }$ and acts trivially on the finite Grassmann basis. Use the mostly-plus metric and the matrices ${\sigma }_{\alpha \dot{\alpha }}^{\mu }$ specified by the four-dimensional super-Poincare algebra. A chiral superfield is an even element $\mathrm{\Phi }\in \mathcal{S}$ satisfying the two equations

${\bar{D}}_{\dot{\alpha }}\mathrm{\Phi }=0,{\bar{D}}_{\dot{\alpha }}:=-{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{\alpha }}}^L-i{\sum }_{\alpha ,\mu }{\theta }^{\alpha }{\sigma }_{\alpha \dot{\alpha }}^{\mu }{\mathrm{\partial }}_{\mu },\dot{\alpha }\in \{\dot{1},\dot{2}\}\mathrm{.}$

Define the nilpotent coordinate shift $\mathrm{\Delta }{y}^{\mu }=i{\sum }_{\alpha ,\dot{\alpha }}{\theta }^{\alpha }{\sigma }_{\alpha \dot{\alpha }}^{\mu }{\bar{\theta }}^{\dot{\alpha }}$ and $y^{\mu }=x^{\mu }+\mathrm{\Delta }{y}^{\mu }$. For a smooth coefficient function $f$, $f(y)$ means the finite Taylor substitution ${\sum }_{k=0}^2(\mathrm{\Delta }{y}^{{\mu }_1}\cdots \mathrm{\Delta }{y}^{{\mu }_k}\mathrm{/}k!){\mathrm{\partial }}_{{\mu }_1}\cdots {\mathrm{\partial }}_{{\mu }_k}f(x)$, with the $k=0$ term equal to $f(x)$ and repeated spacetime indices summed; higher terms vanish in ${\mathrm{\Lambda }}_4$. The chiral components are the unique smooth coefficient functions $A,F\in C^{\mathrm{\infty }}({\mathbb{R}}^{1,3},{\mathcal{A}}_{\bar{0}})$ and ${\psi }_1,{\psi }_2\in C^{\mathrm{\infty }}({\mathbb{R}}^{1,3},{\mathcal{A}}_{\bar{1}})$ in

$\mathrm{\Phi }(y,\theta )=A(y)+{\theta }^1{\psi }_1(y)+{\theta }^2{\psi }_2(y)+{\theta }^1{\theta }^{2}F(y)\mathrm{.}$

The total parity of every term is even. Define the remaining covariant derivatives by $D_{\alpha }={\mathrm{\partial }}_{{\theta }^{\alpha }}^L+i{\sum }_{\mu ,\dot{\alpha }}{\sigma }_{\alpha \dot{\alpha }}^{\mu }{\bar{\theta }}^{\dot{\alpha }}{\mathrm{\partial }}_{\mu }$. Define the supersymmetry generators on $\mathcal{S}$ by

$Q_{\alpha }={\mathrm{\partial }}_{{\theta }^{\alpha }}^L-i{\sum }_{\mu ,\dot{\alpha }}{\sigma }_{\alpha \dot{\alpha }}^{\mu }{\bar{\theta }}^{\dot{\alpha }}{\mathrm{\partial }}_{\mu },{\bar{Q}}_{\dot{\alpha }}=-{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{\alpha }}}^L+i{\sum }_{\mu ,\alpha }{\theta }^{\alpha }{\sigma }_{\alpha \dot{\alpha }}^{\mu }{\mathrm{\partial }}_{\mu }\mathrm{.}$

To specify external odd constant parameters, let $\mathcal{E}$ be a second copy of the four-generator exterior algebra, with ordered generators ${ϵ}^1,{ϵ}^2,{\overline{ϵ}}^{\dot{1}},{\overline{ϵ}}^{\dot{2}}$ and the same signed subset-basis multiplication as ${\mathrm{\Lambda }}_4$. Equip it with the conjugate-linear, order-reversing involution $({ϵ}^{\alpha }{)}^{\ast }=-{\overline{ϵ}}^{\dot{\alpha }}$ and $({\overline{ϵ}}^{\dot{\alpha }}{)}^{\ast }=-{ϵ}^{\alpha }$. The minus signs are the conjugation-compatible convention for parameters placed on the left: $\delta {\theta }^{\alpha }={ϵ}^{\alpha }$ and $\delta {\bar{\theta }}^{\dot{\alpha }}=-{\overline{ϵ}}^{\dot{\alpha }}$ then respect $({\theta }^{\alpha }{)}^{\ast }={\bar{\theta }}^{\dot{\alpha }}$. The four generators remain independent over $\mathbb{C}$. Set ${\mathcal{A}}_{ϵ}=\mathcal{A}\widehat{\otimes }\mathcal{E}$, where $\widehat{\otimes }$ denotes the graded tensor product: $(a\otimes u)(b\otimes v)=(-1{)}^{\mathrm{\mid }u\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ab\otimes uv$ on homogeneous factors, with total parity and the order-reversing conjugation specified by the coefficient-algebra convention. This is again a finite-dimensional supercommutative coefficient algebra. Set ${\mathcal{S}}_{ϵ}=C^{\mathrm{\infty }}({\mathbb{R}}^{1,3},{\mathcal{A}}_{ϵ}\widehat{\otimes }{\mathrm{\Lambda }}_4)$ and embed $\mathcal{S}$ into it by $a\mapsto a\otimes 1$. The parameters commute with even elements and anticommute with odd elements of both $\mathcal{A}$ and ${\mathrm{\Lambda }}_4$.

Extend the displayed coordinate derivatives to ${\mathcal{S}}_{ϵ}$ by the same left-derivative rule, vanishing on ${\mathcal{A}}_{ϵ}$; extend $Q_{\alpha }$ and ${\bar{Q}}_{\dot{\alpha }}$ by their displayed formulas. In particular, an odd derivative or supercharge acquires a minus sign when moved past an external odd parameter. Define the even complex-linear variation ${\delta }_{ϵ,\overline{ϵ}}:\mathcal{S}\to {\mathcal{S}}_{ϵ}$ by

${\delta }_{ϵ,\overline{ϵ}}\mathrm{\Phi }={\sum }_{\alpha }{ϵ}^{\alpha }{Q}_{\alpha }\mathrm{\Phi }+{\sum }_{\dot{\alpha }}{\overline{ϵ}}^{\dot{\alpha }}{\bar{Q}}_{\dot{\alpha }}\mathrm{\Phi }\mathrm{.}$

For a chiral $\mathrm{\Phi }$, the component transformation is the unique tuple $(\delta A,\delta {\psi }_1,\delta {\psi }_2,\delta F)$ of smooth ${\mathcal{A}}_{ϵ}$-valued functions in the displayed $y,\theta$ expansion of ${\delta }_{ϵ,\overline{ϵ}}\mathrm{\Phi }$, with $\delta A,\delta F$ even and $\delta {\psi }_{\alpha }$ odd. Chirality and uniqueness in this extended coefficient algebra are established by the local chiral-superspace identities. The same variation formula acts on ${\mathcal{S}}_{ϵ}$ if repeated variations with these parameters are desired; adjoining further independent parameter copies uses the same graded tensor construction. The space of fields is classical and smooth: the coordinate differential operators act on all of $\mathcal{S}$, or all of ${\mathcal{S}}_{ϵ}$ after scalar extension; no quantum-field operator domain or field equation is assumed. In natural units $[x^{\mu }]=-1$ and $[\theta ]=[\bar{\theta }]=-\frac{1}{2}$. If $[\mathrm{\Phi }]=d$ is assigned, the formal component dimensions are $[A]=d$, $[{\psi }_{\alpha }]=d+\frac{1}{2}$, and $[F]=d+1$; no value of $d$ is fixed. In the same natural-unit convention, $D_{\alpha }$, ${\bar{D}}_{\dot{\alpha }}$, $Q_{\alpha }$ and ${\bar{Q}}_{\dot{\alpha }}$ have mass dimension $+\frac{1}{2}$, while $[{ϵ}^{\alpha }]=[{\overline{ϵ}}^{\dot{\alpha }}]=-\frac{1}{2}$, so ${\delta }_{ϵ,\overline{ϵ}}$ is dimensionless.

Martin, §§4.2–4.4.

Gates et al., §§3.4 and 3.6.

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

Associativity and coefficient calculus for the finite superspace Grassmann algebra: Associativity and coefficient calculus for the finite superspace Grassmann algebra.

Four-dimensional real extended super-Poincare algebra: Four-dimensional real extended super-Poincare algebra.