Local $N=1$ Wilsonian Superspace Action Data

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 flat four-dimensional $N=1$ superspace conventions in natural units $c=\mathrm{\hslash }=1$, with $M={\mathbb{R}}^{1,3}$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, and the ordered Grassmann generators, Berezin measures, chiral constraint, and covariant derivatives fixed by the earlier superspace definitions. Fix the global gauge group as a compact closed matrix subgroup $G\subset U(N)$; this chosen subgroup, rather than only its Lie algebra, is part of the data. Give $U(N)$ its usual finite-dimensional real matrix Lie-group structure and give $G$ its unique inherited smooth embedded Lie-group structure as a closed subgroup. The group and tangent facts here use only the stated countable-choice premise ${\mathrm{A}\mathrm{C}}_{\omega }$ from the mathematical suppliers. Put $\mathfrak{g}=\mathrm{Lie}(G)=T_{I}G\subseteq \mathfrak{u}(N)$, where $\mathfrak{u}(N)=\{X\in M_N(\mathbb{C}):X^{†}=-X\}$, with bracket $[X,Y{]}_{\mathfrak{g}}=XY-YX$. This is the inherited tangent bracket: the left-invariant matrix field generated by $X$ is $u\mapsto uX$, and its bracket with the field generated by $Y$ has value $XY-YX$ at $I$. These statements cover compact closed matrix subgroups including Abelian groups such as $U(1)$; no non-Abelian global-group hypothesis is imposed. Let ${\mathcal{H}}_i$ be finite-dimensional Hermitian spaces, and let ${\rho }_i:G\to U({\mathcal{H}}_i)$ be smooth unitary representations for a finite family of chiral superfields ${\mathrm{\Phi }}_i$. Define their differentials by ${\rho }_{i\ast }=d({\rho }_i{)}_I:T_{I}G\to T_{I}U({\mathcal{H}}_i)$; equivalently, for $X\in \mathfrak{g}$ and any smooth curve $\gamma$ in $G$ with $\gamma (0)=I$ and ${\gamma }^{\mathrm{\prime }}(0)=X$, ${\rho }_{i\ast }(X)={\frac{d}{dt}\mid }_{t=0}{\rho }_i(\gamma (t))$. Thus ${\rho }_{i\ast }$ takes values in $\mathfrak{u}({\mathcal{H}}_i)$ and preserves Lie brackets. Write ${\mathfrak{g}}_{\mathbb{C}}=\mathfrak{g}{\otimes }_{\mathbb{R}}\mathbb{C}$ and let ${\rho }_{i\ast }^{\mathbb{C}}:{\mathfrak{g}}_{\mathbb{C}}\to {\mathrm{End}}_{\mathbb{C}}({\mathcal{H}}_i)$ be the complex-linear extension; all represented Lie-algebra parameters below use this map. Work on the trivial bundle $M\times G$ and expand in the topologically trivial sector. Let $A_{\mu }$ be the connection component of the vector prepotential in a local trivialization. For the odd ghost $c$, use the non-Abelian BRST transformations $s{A}_{\mu }={\mathrm{\partial }}_{\mu }c+[A_{\mu },c]$, $sc=-\frac{1}{2}[c,c]$, and $s{\mathrm{\Phi }}_i=-{\rho }_{i\ast }(c){\mathrm{\Phi }}_i$ on component fields and jets.

A vector prepotential $V$ is a dimensionless even Hermitian matrix-valued superfield with values in $i\mathfrak{g}$ in the chosen faithful representation. Set $V_i={\rho }_{i\ast }^{\mathbb{C}}(V)$. Let $\mathrm{\Lambda }$ be an even chiral ${\mathfrak{g}}_{\mathbb{C}}$-valued parameter, set ${\mathrm{\Lambda }}_i={\rho }_{i\ast }^{\mathbb{C}}(\mathrm{\Lambda })$ and $h_i=e^{-{\mathrm{\Lambda }}_i}$, and define the Hermitian matter kernel ${\mathcal{H}}_i=e^{-V_i}$. The representation maps are differentials of the group representations, complex-linearly extended; a group representation is not evaluated on a Lie-algebra element. The supergauge laws are ${\mathrm{\Phi }}_i\mapsto h_i{\mathrm{\Phi }}_i$ and ${\mathcal{H}}_i\mapsto (h_i^{†}{)}^{-1}{\mathcal{H}}_i{h}_i^{-1}$. Define the curvature using this SAME matter kernel:

$W_{\alpha }=-\frac{1}{4}{\bar{D}}^{\dot{\beta }}{\bar{D}}_{\dot{\beta }}({\mathcal{H}}^{-1}{D}_{\alpha }\mathcal{H})=-\frac{1}{4}{\bar{D}}^{\dot{\beta }}{\bar{D}}_{\dot{\beta }}(e^V{D}_{\alpha }{e}^{-V})\mathrm{.}$

It is chiral and transforms by $W_{\alpha }\mapsto h{W}_{\alpha }{h}^{-1}$. Indeed, with $D_{\alpha }(h^{†}{)}^{-1}=0$, the transformed logarithmic derivative is $h{\mathcal{H}}^{-1}(D_{\alpha }\mathcal{H})h^{-1}+h{D}_{\alpha }{h}^{-1}$. The chiral projection of the second term vanishes: ${\bar{D}}_{\dot{\beta }}h=0$ and the superspace derivative anticommutator makes ${\bar{D}}^2{D}_{\alpha }{h}^{-1}=0$. Chiral projection of the first term gives the stated conjugation law. The matter density is invariant because ${\mathrm{\Phi }}_i^{\mathrm{\prime }†}{\mathcal{H}}_i^{\mathrm{\prime }}{\mathrm{\Phi }}_i^{\mathrm{\prime }}={\mathrm{\Phi }}_i^{†}{\mathcal{H}}_i{\mathrm{\Phi }}_i$.

In addition to the displayed chiral supergauge transformations, include constant transformations $k\in G$ from every connected component: ${\mathrm{\Phi }}_i\mapsto {\rho }_i(k){\mathrm{\Phi }}_i$, ${\mathcal{H}}_i\mapsto {\rho }_i(k){\mathcal{H}}_i{\rho }_i(k{)}^{-1}$, and $W_{\alpha }\mapsto k{W}_{\alpha }{k}^{-1}$. Unitarity proves invariance of the canonical matter term under these transformations as well. Here gauge invariance of $F$ and of the integrated additional D-term means invariance under these transformations as well as the chiral supergauge laws.

The gauge-kinetic datum is a fixed symmetric complex-bilinear quadratic form on ${\mathfrak{g}}_{\mathbb{C}}$ invariant under the full group: $Q({\mathrm{Ad}}_{k}X,{\mathrm{Ad}}_{k}Y)=Q(X,Y)$ for every $k\in G$. Equivalently choose a basis $Q_1,\dots ,Q_s$ of this finite-dimensional invariant-form space and use ${\sum }_a{\tau }_a{Q}_a(W^{\alpha },W_{\alpha })\mathrm{/}(8\pi i)$ as the quadratic chiral density; this includes invariant Abelian mixing. For each simple factor choose a fixed invariant quadratic trace functional ${\mathrm{Tr}}_{\mathrm{h}}$ (including its normalization and sign), and set $\tau =\vartheta \mathrm{/}(2\pi )+4\pi i\mathrm{/}{g}_{\mathrm{h}}^2$. The gauge density uses $\tau {\mathrm{Tr}}_{\mathrm{h}}(W^{\alpha }{W}_{\alpha })\mathrm{/}(8\pi i)$; the quadratic functional and $g_{\mathrm{h}}$ together, not a symbol $g$ alone, determine its normalization. For connected product groups these choices may be independent per simple factor. If disconnected components permute simple ideals, the total quadratic form must satisfy the full-group invariance just stated, so the corresponding coefficients cannot be chosen independently unless that invariance permits it. For a fixed normalized trace, $g_{\mathrm{h}}$ occurs in $\tau$, not in the prepotential exponential.

The exact classical bridge to Four-dimensional $N=1$ non-Abelian vector superfield in connection normalization is at fixed real $g=g_{\mathrm{Y}\mathrm{M}}>0$: set $V=-2g{V}_{30}$, $\mathrm{\Lambda }=-i{\mathrm{\Lambda }}_{30}$ and $\mathcal{H}=e^{2g{V}_{30}}$. Then $h=e^{i{\mathrm{\Lambda }}_{30}}$, the matter kernel and its gauge law agree, and $W_{\alpha }={\mathcal{C}}_{\alpha ,30}$ is the earlier REPRESENTED matrix curvature by substitution. The earlier abstract Lie-algebra-valued curvature instead satisfies ${\rho }_{\ast }({\mathcal{W}}_{\alpha ,30})=W_{\alpha }\mathrm{/}(2i)$, because $W_{\alpha }=2g{t}_a{W}_{\mathrm{c}\mathrm{a}\mathrm{n},\alpha }^a$ and ${\mathcal{W}}_{\alpha ,30}=g{T}_a{W}_{\mathrm{c}\mathrm{a}\mathrm{n},\alpha }^a$ with $t_a=i{T}_a$. The connection remains $A_{\mu }=-ig{B}_{\mu }$, with the same component BRST law. For this selected $SU(2)$ bridge use the ordinary defining trace ${\mathrm{Tr}}_{\mathrm{h}}={\mathrm{Tr}}_2$. Then

$\frac{1}{4g^2}{\kappa }_{\mathbb{C}}({\mathcal{W}}_{30}^{\alpha },{\mathcal{W}}_{\alpha ,30})=\frac{1}{8g^2}{\mathrm{Tr}}_2(W^{\alpha }{W}_{\alpha })\mathrm{.}$

The holomorphic datum writes this coefficient as $\tau {\mathrm{Tr}}_2(W^{\alpha }{W}_{\alpha })\mathrm{/}(8\pi i)$. Therefore at zero topological angle the exact parameter match is $\tau =\pi i\mathrm{/}{g}^2$, or $g_{\mathrm{h}}=2g$ in the displayed definition $\tau =4\pi i\mathrm{/}{g}_{\mathrm{h}}^2$. This factor is derived from the matrix-versus-Lie curvature dictionary, not removed by changing the trace sign. Hermitian conjugation is added once on both sides of the chiral equality; the real D-term is not doubled. The selected earlier models have zero theta term, so no global theta-period or D'Hoker prepotential conversion is asserted here. These are identities of classical densities at fixed real parameters, with no quantum change of variables, anomalous Jacobian, holomorphic canonical normalization or equality of regulated measures.

A local rigid $N=1$ action datum has the fixed canonical matter kinetic term ${\sum }_i{\mathrm{\Phi }}_i^{†}{e}^{-V_i}{\mathrm{\Phi }}_i$, an additional real formal D-term integrand $G(\mathrm{\Phi },\bar{\mathrm{\Phi }},V,{\mathrm{\partial }}_{\mu },D_{\alpha },{\bar{D}}_{\dot{\alpha }})$ whose integrated D-term is gauge invariant under the stated boundary conditions, and a holomorphic gauge-invariant formal power series $F(\mathrm{\Phi },W)$. The additional $G$ may contain local spacetime or superspace derivatives and may transform by terms with zero D-integral; constant Abelian Fayet–Iliopoulos terms described below are included in this convention. The F-term integrand contains no such derivatives, anti-chiral variables, or spacetime dependence. Its formal ultraviolet-cutoff density is

${\mathcal{L}}_{{\mathrm{\Lambda }}_0}=[{\sum }_i{\mathrm{\Phi }}_i^{†}{e}^{-V_i}{\mathrm{\Phi }}_i+G{]}_D+2\mathrm{Re}[\frac{\tau }{8\pi i}{\mathrm{Tr}}_{\mathrm{h}}(W^{\alpha }{W}_{\alpha })+F(\mathrm{\Phi },W){]}_F\mathrm{.}$

In a two-derivative scalar sector with scalar values $\varphi =({\varphi }_1,\dots ,{\varphi }_n)$ in an open set $\mathcal{M}\subseteq {\mathbb{C}}^n$, let $K\in C^{\mathrm{\infty }}(\mathcal{M},\mathbb{R})$. Its complex Hessian defines the Hermitian tensor $g_{i\bar{j}}={\mathrm{\partial }}^{2}K\mathrm{/}(\mathrm{\partial }{\varphi }_i\mathrm{\partial }{\bar{\varphi }}_j)$ and the real $(1,1)$-form ${\omega }_K=i{g}_{i\bar{j}}d{\varphi }_i\wedge d{\bar{\varphi }}_j$. Equality of mixed partials gives $d{\omega }_K=0$. With the standard complex structure on $\mathcal{M}$, a Kähler metric is a positive-definite Hermitian tensor whose associated real $(1,1)$-form is closed; hence on any open region where $g_{i\bar{j}}{v}^i{\bar{v}}^{\bar{j}}>0$ for every nonzero $v\in {\mathbb{C}}^n$, $g$ is a Kähler metric and $K$ is its Kähler potential. A nondegenerate indefinite Hermitian tensor with closed associated form is pseudo-Kähler data, not a positive-definite Kähler metric. With $[{\varphi }_i]=1$, the scalar kinetic potential has mass dimension $[K]=2$ and its Hessian has $[g_{i\bar{j}}]=0$, with coefficient dimensions assigned accordingly. Both kinds of data belong to the real D-term, not the holomorphic F-term.

Here $[{\mathrm{\Phi }}_i]=1$, $[W_{\alpha }]=3\mathrm{/}2$, $[g]=0$, $[\tau ]=0$, the F-term integrand has mass dimension three, and the D-term integrand has mass dimension two; dimensionful coefficients in nonrenormalizable interactions carry the dimensions needed to make the density have mass dimension four. A Wilsonian prescription with cutoffs $0<\lambda <{\mathrm{\Lambda }}_0$ formally separates Euclidean momentum modes with $\mathrm{\mid }q\mathrm{\mid }\ge \lambda$ from modes with $\mathrm{\mid }p\mathrm{\mid }<\lambda$. Let $S_{\lambda }^{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}}$ denote the coefficientwise functional assigned by the selected cutoff prescription. For this definition, its perturbative vertex coefficients are assumed to be sums of one-particle-irreducible graphs in that prescribed expansion: cutting a single internal propagator line cannot disconnect a graph counted as one vertex. This is an explicit property of the selected formal vertex functional, not a theorem about generic partial integration in an interacting theory. Assume also that $S_{\lambda }^{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}}$ has a local derivative expansion in local terms with finitely many derivatives at each formal perturbative order for external momenta $\mathrm{\mid }p\mathrm{\mid }\ll \lambda$. Write its density using $G_{\lambda }$, $F_{\lambda }$, and ${\tau }_{\lambda }$ in the same form as ${\mathcal{L}}_{{\mathrm{\Lambda }}_0}$.

A concrete component graph completion of a selected prescription is linear Lorenz gauge. Its integrated physical variables are the ordinary Wess–Zumino representative's connection, gaugino, auxiliary, and matter components; redundant supergauge components are not integration variables. Add adjoint-valued odd scalar ghosts $c,\bar{c}$ and an even multiplier $b$. Extend the component BRST differential by $s\bar{c}=b$, $sb=0$, with $sX=-[c,X]$ on adjoint gaugino/auxiliary fields and $su=-{\rho }_{i\ast }(c)u$ on every matter component; brackets are graded and $s$ is an odd derivation. Choose a nondegenerate real invariant gauge-kinetic bilinear form $B$ on $\mathfrak{g}$ and a nonzero real gauge parameter $\alpha$. With $\mathcal{F}(A)={\mathrm{\partial }}^{\mu }{A}_{\mu }$, use $\mathrm{\Psi }=\int d^{4}xB(\bar{c},\mathcal{F}(A)-\alpha b\mathrm{/}2)$. Then $s\mathrm{\Psi }=\int d^{4}x\{B(b,\mathcal{F})-\alpha B(b,b)\mathrm{/}2-B(\bar{c},{\mathrm{\partial }}^{\mu }{D}_{\mu }c)\}$. In the formal Euclidean $e^{-S}$ Gaussian prescription the multiplier contour is parallel to $i\mathfrak{g}$, with fixed field-independent Gaussian normalization. Completing the square gives $-\alpha B(b-\mathcal{F}\mathrm{/}\alpha ,b-\mathcal{F}\mathrm{/}\alpha )\mathrm{/}2+B(\mathcal{F},\mathcal{F})\mathrm{/}(2\alpha )$, so eliminating $b$ adds only quadratic gauge fixing. The ghost interaction is $-B(\bar{c},{\mathrm{\partial }}^{\mu }[A_{\mu },c])$, with two ghost legs and one gauge-boson leg, and no gaugino, matter, or auxiliary leg. Supply an infrared boundary/inverse prescription with invertible free gauge and ghost kernels on the selected modes; residual gauge zero modes are fixed rather than integrated as additional interaction variables.

In this component completion assign $R(c)=R(\bar{c})=R(b)=R(A_{\mu })=0$ and engineering dimensions $[c]=0$, $[\bar{c}]=[b]=2$, with $s$ of dimension zero. Under uniform real scaling of all gauge-kinetic coefficients by $t$, use $B\mapsto tB$, fixing $\alpha$ and all coefficient ratios. The supplied regulated gauge and ghost inverses are homogeneous of degree $-1$ in $t$; the quadratic gauge-fixing kernel and the ghost interaction have degree $1$. The cutoff and infrared prescriptions preserve these homogeneities and the local spurion and Ward hypotheses when this completion is used. There are no additional interacting regulator fields. These are operational conditions on a supplied formal regulator, not its existence theorem. Coefficients belong to its fixed component/source coordinates; this completion asserts no equality of arbitrary gauge-fixed off-shell vertex representatives.

For the component completion, supersymmetry means the following explicit induced variation, rather than raw supercharges acting outside the Wess–Zumino slice. Work in the external-parameter graded coefficient extension of the chiral-superfield definition, with its conjugation-compatible odd parameters. Put $K=\log \mathcal{H}=-V$ and write its Wess–Zumino representative as $K=\theta {\sigma }^{\mu }\bar{\theta }{k}_{\mu }+i{\theta }^2\bar{\theta }\bar{\mathrm{\ell }}-i{\bar{\theta }}^2\theta \mathrm{\ell }+\frac{1}{2}{\theta }^2{\bar{\theta }}^{2}d$. Define $A_{\mu }=k_{\mu }\mathrm{/}(2i)$, $\lambda =\mathrm{\ell }\mathrm{/}2$, and $D=d\mathrm{/}2$, using the chosen faithful representation. This agrees exactly with the fixed bridge $K=2g{V}_{30}$: $k_{\mu }=2g{B}_{\mu }$, $\mathrm{\ell }=2g{\lambda }_B$, $d=2g{D}_B$, so $A_{\mu }=-ig{B}_{\mu }$, $\lambda =g{\lambda }_B$, $D=g{D}_B$. The same linear inverse coefficient map defines their variations; no field is rescaled in the graph expansion.

Let ${\delta }_0={ϵ}^{\alpha }{Q}_{\alpha }+{\overline{ϵ}}^{\dot{\alpha }}{\bar{Q}}_{\dot{\alpha }}$, set $U={\delta }_{0}K$, and first evaluate $U(x,\theta ,0)$ in the ordered superspace coordinates. Its constant coefficient is zero because $K$ has minimum Grassmann degree two. Define the even chiral parameter ${\mathrm{\Lambda }}_{ϵ}(y,\theta )=-U(y,\theta ,0)$ by finite Taylor substitution $y=x+i\theta \sigma \bar{\theta }$ in its coefficient functions; its scalar component is zero, fixing the residual ordinary gauge parameter. With ${\mathrm{ad}}_{K}X=[K,X]$, define ${\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}K=U+{\mathrm{\Lambda }}_{ϵ}^{†}+{\mathrm{\Lambda }}_{ϵ}+\frac{1}{2}[K,{\mathrm{\Lambda }}_{ϵ}-{\mathrm{\Lambda }}_{ϵ}^{†}]+\frac{1}{12}[K,[K,{\mathrm{\Lambda }}_{ϵ}^{†}+{\mathrm{\Lambda }}_{ϵ}]]\mathrm{.}$ Indeed the infinitesimal supergauge law is ${\delta }_{\mathrm{\Lambda }}{e}^K={\mathrm{\Lambda }}^{†}{e}^K+e^K\mathrm{\Lambda }$; right multiplication by $e^{-K}$ and inversion of $(e^{{\mathrm{ad}}_K}-1)\mathrm{/}{\mathrm{ad}}_K$ gives the displayed correction, since the inverse is $1-{\mathrm{ad}}_K\mathrm{/}2+{\mathrm{ad}}_K^2\mathrm{/}12$. Higher terms vanish by the four-generator Grassmann degree. Here the displayed double commutator also vanishes, since $K$ has minimum degree two and ${\mathrm{\Lambda }}_{ϵ}$ has minimum degree one. At $\bar{\theta }=0$ the compensator cancels $U$ and the commutator vanishes; conjugation gives the cancellation at $\theta =0$. Thus only mixed monomials remain, exactly the Wess–Zumino coefficient space. The correction preserves Hermitian reality, total parity and the complexified gauge Lie algebra. For matter set ${\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}{\mathrm{\Phi }}_i={\delta }_0{\mathrm{\Phi }}_i-{\rho }_{i\ast }^{\mathbb{C}}({\mathrm{\Lambda }}_{ϵ}){\mathrm{\Phi }}_i$ and conjugate this rule for barred fields. Both terms are chiral, so their physical component variations are uniquely obtained in the ordered $y,\theta$ expansion: the constant, linear, and ${\theta }^1{\theta }^2$ coefficients are respectively $\delta a_i$, $\sqrt{2}\delta {\psi }_i$, and $2\delta F_i$. Even scalar/auxiliary and odd fermion variations belong to the total parity spaces of the external-parameter extension. This constructs a smooth local component variation without equations of motion or an identification with the bosonic slice.

For this specified even change of component variables set ${\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}c={\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}\bar{c}={\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}b=0$ and extend by the graded product and spacetime derivative rules. Let $S_{\lambda }^{\mathrm{g}\mathrm{f}}$ denote the full graph action including the gauge-fixing and ghost sector. The supersymmetry change-of-variables equation below uses ${\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$ and $S_{\lambda }^{\mathrm{g}\mathrm{f}}$; it retains the gauge-fixing and ghost variations, which generally do not vanish. Its regulator/Jacobian compatibility is an assumption of the selected prescription. The gauge-invariant superspace counterterm sector is required to obey the declared spurion and rigid-supersymmetry conditions; no invariance of the gauge-fixed action under raw supercharges or under ${\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$ is asserted. The BRST differential remains the separate nilpotent differential defined above.

A regulated formal expectation $⟨\mathcal{O}{⟩}_{\lambda }$ is the normalized coefficientwise functional assigned by a specified cutoff prescription; the notation asserts no convergence or physical continuum measure. A local insertion $\mathcal{O}$ is a polynomial or formal local function of the fields and finitely many spacetime derivatives, multiplied by a compactly supported test function. For an insertion $\mathcal{O}$, its component-completion Ward identity is the change-of-variables equation $⟨{\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}\mathcal{O}{⟩}_{\lambda }=⟨\mathcal{O}{\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}{S}_{\lambda }^{\mathrm{g}\mathrm{f}}{⟩}_{\lambda }$. The BRST Ward identity is $⟨s\mathcal{O}{⟩}_{\lambda }=0$ when $s{S}_{\lambda }^{\mathrm{g}\mathrm{f}}=0$. A local gauge anomaly is a nonzero local insertion in the latter identity. A regulator preserves these Ward identities when the equations hold coefficientwise at every perturbative order without an anomaly insertion; this definition does not assert that such a regulator exists for every field representation.

The local background extension of this formal prescription promotes independent F coefficients and $\tau$ to nondynamical full off-shell chiral superfields. First separate every bare constant Abelian Fayet–Iliopoulos term $\xi [V_1{]}_D$ from the remaining D integrand; retain each $\xi$ as a fixed real number, and promote only the other independent real D coefficients to unconstrained real superfields. This keeps the constant FI terms in the action class without imposing a false gauge Ward identity for an arbitrary coefficient multiplying $V_1$. Indeed $V_1\mapsto V_1-{\mathrm{\Lambda }}_1-{\bar{\mathrm{\Lambda }}}_1$ gives the action variation $-\int d^{4}x[P_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(P){\mathrm{\Lambda }}_1{]}_F$ and its conjugate for a real coefficient background $P$, under the boundary condition of the physical projection identity. This need not vanish for unconstrained $P$. A separately supplied real linear background $L$, satisfying ${\bar{D}}^{\dot{\alpha }}{\bar{D}}_{\dot{\alpha }}L=0=D^{\alpha }{D}_{\alpha }L$, may instead couple as $[L{V}_1{]}_D$; it is constrained D-sector data, not one of the unconstrained coefficient spurions. A chiral coefficient background obeys the chiral constraint and has independently variable smooth scalar, Weyl-spinor, and auxiliary components; its conjugate is anti-chiral. The prescribed regulated counterterms obey the same superspace and gauge Ward identities for these backgrounds and have a local derivative expansion in them; constant scalar backgrounds are imposed only after those identities are applied. The derivative-free chiral sector is evaluated at constant scalar backgrounds after that extension; terms containing background superderivatives are assigned to the real D sector after chiral projection. The extension is interpreted coefficientwise in the formal local background-jet algebra before setting the backgrounds constant. At a fixed perturbative order and for any fixed monomial coefficient, let $P_1,\dots ,P_k$ be the finite set of independent unconstrained real coefficient spurions involved, and write $u_j=P_j-p_j^{(0)}$ for their formal deviations from the selected constant expansion values. A derivative-free coefficient is a formal power series in $u_1,\dots ,u_k$ over the characteristic-zero algebra of the remaining chiral fields and chiral coefficient spurions. The unconstrained real backgrounds have independent first anti-chiral jets ${\bar{D}}_{\dot{\alpha }}{P}_j$; these are free odd generators over that derivative-free coefficient algebra (with their conjugate jets fixed by reality). Their higher superderivative jets are likewise formal free generators subject only to the graded derivative algebra and conjugation/reality relations. Formal partial derivatives act monomial by monomial, and $\bar{D}$ acts by the graded chain rule. Thus chirality is an identity in this jet algebra, rather than an equation imposed only after constant-background evaluation. No convergence, positive scalar body, logarithm, or analytic continuation in the real coefficients is required. This local background-extension and regulator-compatibility assumption is not a consequence of classical BRST nilpotence or a regulator-existence theorem.

The component background fields are smooth on $M$, and admissible variations and insertions are compactly supported or satisfy decay conditions that make the stated total-derivative boundary terms vanish. Loop momenta are regulated separately by the Wilsonian high-mode prescription.

An allowed constant FI coefficient is a real linear functional on the central Abelian Lie algebra that is invariant under ${\mathrm{Ad}}_G$; thus disconnected components that permute or invert central directions constrain the FI coefficients as well. For a U(1) factor, normalize its gauge parameter as $e^{i\alpha }$ with period $2\pi$. Each chiral matter representation decomposes into integer charge weights $q_{i,a}$; the total charge trace ${\sum }_i{\sum }_a{q}_{i,a}$ counts each weight with its multiplicity. In the stated supergauge convention the real Abelian vector prepotential $V_1$ shifts by the negative of a chiral parameter and its conjugate. Under compact-support or vanishing-boundary-term conditions, the physical action $\int d^{4}x[\xi V_1{]}_D$ is gauge invariant for constant $\xi$ and is called a Fayet–Iliopoulos term; $\xi$ is its field-independent coefficient. For an Abelian fixed-$g$ component bridge, $V_1=-2g{V}_{30}$ gives $[\xi V_1{]}_D=-g\xi D_B$ in the auxiliary sector. Thus the coefficient ${\xi }_D$ of $+D_B$ is ${\xi }_D=-g\xi$ in this convention; no differently signed source FI symbol is silently identified. This definition specifies formal action and cutoff data only; it does not assert that a gauge-invariant supersymmetric regulator exists for every field representation or that the perturbative series converges.

Weinberg, equations (1)–(3)

Martin, §§4.7 and 4.10

earlier vector-prepotential convention and fixed-coupling bridge

$N=1$ chiral-superfield definition

$N=1$ superspace Grassmann algebra

finite-jet non-Abelian BRST complex

BRST nilpotence lemma

Physical superspace action brackets and canonical chiral components

Countable choice used by the Lie-group suppliers

Unitary matrix groups and their tangent Lie algebras

Inherited Lie-group structure on closed subgroups

Tangent Lie algebra at the identity

Differentials preserve Lie brackets