Chiral, Vector, Linear, and Field-Strength Superfields

Standard four-dimensional $\mathcal N=1$ superfields are distinguished by two kinds of data: differential constraints and gauge equivalences. Chirality leaves the off-shell components $(A,\psi,F)$; a real vector prepotential contains gauge-removable components and yields the chiral field strength $W_\alpha$; a real linear constraint leaves a scalar, a Weyl fermion, and a divergence-free vector dual to a two-form gauge field. The same convention must be used for every expansion, projection, Bianchi identity, and component count.

Required background. Supercovariant Derivatives, Chirality, and Integrability supplies $D$, $\bar D$, chiral coordinates, and projection conventions. Gauge Fields, Redundancy, and Observable Content supplies the distinction between a prepotential and its gauge-invariant content.

Helpful background. Coupling to Background Gauge Fields and Bundles explains how gauge fields and currents are coupled without confusing a local potential with global bundle data.

Chiral superfields

In four-dimensional Lorentzian superspace, a chiral scalar satisfies

$$\bar D_{\dot\alpha}\Phi=0, \qquad \Phi(y,\theta) =A(y)+\sqrt2\,\theta\psi(y)+\theta\theta F(y),$$

where $y^\mu=x^\mu+i\theta\sigma^\mu\bar\theta$. In ordinary coordinates,

$$\begin{aligned} \Phi=\;& A+\sqrt2\,\theta\psi+\theta\theta F +i\theta\sigma^\mu\bar\theta\,\partial_\mu A\\ &-\frac{i}{\sqrt2}\theta\theta\, \partial_\mu\psi\,\sigma^\mu\bar\theta +\frac14\theta\theta\bar\theta\bar\theta\,\Box A. \end{aligned}$$

The independent projections are

$$A=\Phi|, \qquad \psi_\alpha=\frac1{\sqrt2}D_\alpha\Phi|, \qquad F=-\frac14D^2\Phi|.$$

For a canonically normalized matter multiplet,

$$[A]=1, \qquad [\psi]=\frac32, \qquad [F]=2, \qquad [\Phi]=1.$$

The off-shell real count is $2+2=4$ bosonic components and four fermionic components. Products and holomorphic functions of chiral superfields are chiral because $\bar D$ obeys the graded Leibniz rule. This elementary fact underlies superpotentials, but action construction belongs to the next chapter.

A real vector prepotential

An Abelian vector multiplet is described by a dimensionless real scalar superfield

$$V=V^\dagger.$$

Before gauge fixing, a convenient component expansion is

$$\begin{aligned} V=\;&C+i\theta\chi-i\bar\theta\bar\chi +\frac{i}{2}\theta\theta(M+iN) -\frac{i}{2}\bar\theta\bar\theta(M-iN)\\ &-\theta\sigma^\mu\bar\theta\,A_\mu\\ &+i\theta\theta\bar\theta \left(\bar\lambda+\frac{i}{2}\bar\sigma^\mu\partial_\mu\chi\right) -i\bar\theta\bar\theta\theta \left(\lambda+\frac{i}{2}\sigma^\mu\partial_\mu\bar\chi\right)\\ &+\frac12\theta\theta\bar\theta\bar\theta \left(D+\frac12\Box C\right). \end{aligned}$$

Here $C,M,N,D,A_\mu$ are real, while $\chi$ and $\lambda$ are Weyl spinors. The derivative terms are chosen so that the named component projections transform simply. An unconstrained real superfield has $8+8$ real components.

The Abelian supergauge equivalence is

$$V\sim V+i(\Lambda-\Lambda^\dagger), \qquad \bar D_{\dot\alpha}\Lambda=0.$$

This is the supergauge construction of Salam and Strathdee 1974, pp. 477–482.

The scalar, spinor, and auxiliary components of $\Lambda$ remove $C,\chi,M,N$. The resulting Wess–Zumino representative is

$$V_{\rm WZ} =-\theta\sigma^\mu\bar\theta\,A_\mu +i\theta\theta\bar\theta\bar\lambda -i\bar\theta\bar\theta\theta\lambda +\frac12\theta\theta\bar\theta\bar\theta D.$$

Its residual transformation is the ordinary gauge transformation

$$A_\mu\longmapsto A_\mu+\partial_\mu\omega,$$

with $\lambda$ and $D$ invariant in the Abelian theory. Wess–Zumino gauge is a choice of representative, not a supersymmetry-invariant subspace. A supersymmetry transformation must be followed by a field-dependent chiral supergauge transformation to restore it.

The chiral field strength

Define

$$W_\alpha=-\frac14\bar D^2D_\alpha V.$$

It is chiral:

$$\bar D_{\dot\alpha}W_\beta=0,$$

because a third barred derivative vanishes in two-component spinor space. Under the Abelian supergauge transformation,

$$\delta W_\alpha =-\frac{i}{4}\bar D^2D_\alpha(\Lambda-\Lambda^\dagger)=0.$$

For $\Lambda$ chiral, commuting $D_\alpha$ through $\bar D^2$ leaves spacetime derivatives annihilated by the remaining barred derivative; for $\Lambda^\dagger$ antichiral, $D_\alpha\Lambda^\dagger=0$. Thus $W_\alpha$ depends only on the gauge-equivalence class of $V$.

In Wess–Zumino gauge its chiral-coordinate expansion is

$$\begin{aligned} W_\alpha(y,\theta) =\;&-i\lambda_\alpha +\theta_\alpha D -\frac{i}{2} (\sigma^\mu\bar\sigma^\nu\theta)_\alpha F_{\mu\nu}\\ &-\theta\theta\, \sigma^\mu_{\alpha\dot\alpha} \partial_\mu\bar\lambda^{\dot\alpha}. \end{aligned}$$

The antisymmetric part of $\sigma^\mu\bar\sigma^\nu$ is selected by $F_{\mu\nu}$, so no separate antisymmetrization is required in that term. The minus sign of the $\theta^2$ descendant follows from applying the declared left-acting superspace derivatives to $V_{\rm WZ}$; with the site’s $(+---)$ convention it is also the sign that gives the positive barred-first gaugino kinetic term in the field-strength action. The dimensions are

$$[V]=0, \qquad [W_\alpha]=\frac32, \qquad [A_\mu]=1, \qquad [\lambda]=\frac32, \qquad [D]=2.$$

The Abelian superspace Bianchi identity is

$$D^\alpha W_\alpha =\bar D_{\dot\alpha}\bar W^{\dot\alpha}.$$

At component level its vector part is $\partial_{[\mu}F_{\nu\rho]}=0$; its lowest projections also encode the reality of $D$. This identity shows why a chiral spinor $W_\alpha$ is more constrained than an arbitrary chiral spinor superfield.

For a non-Abelian group, with the connection and coupling absorbed into $V$ for this formula,

$$W_\alpha =\frac18\bar D^2 \left(e^{2V} D_\alpha e^{-2V}\right).$$

It transforms covariantly rather than invariantly, and its Abelian linearization is again $-\bar D^2D_\alpha V/4$. Restoring the coupling, generator normalization, and global gauge form is mandatory before constructing an action or discussing genuine line operators. Weinberg 2000, §§ 27.1–27.3, pp. 113–131 develops the prepotential, Wess–Zumino gauge, field strength, and Bianchi identity in a convention that must be translated as a complete package.

Real linear superfields

A real linear superfield obeys

$$L=L^\dagger, \qquad D^2L=0, \qquad \bar D^2L=0.$$

Rather than hide convention-sensitive derivative coefficients in a long coordinate expansion, define its independent components by projections:

$$\begin{aligned} C&=L|,\\ \chi_\alpha&=\frac1{\sqrt2}D_\alpha L|,\\ H_{\alpha\dot\alpha} &=-\frac12[D_\alpha,\bar D_{\dot\alpha}]L|. \end{aligned}$$

Here the bracket in the last line is the ordinary operator commutator, not the graded anticommutator. The linear constraints determine every higher coefficient from derivatives of $C$, $\chi$, and $H_\mu$. In particular,

$$\begin{aligned} \partial^{\alpha\dot\alpha}H_{\alpha\dot\alpha} &=-\frac12\partial^{\alpha\dot\alpha} [D_\alpha,\bar D_{\dot\alpha}]L\big|\\ &=\frac{i}{8} \left(D^2\bar D^2-\bar D^2D^2\right)L\big|=0. \end{aligned}$$

The second line follows from $\{D_\alpha,\bar D_{\dot\alpha}\}=-2i\partial_{\alpha\dot\alpha}$; the last equality uses both linear constraints. Thus $\partial_\mu H^\mu=0$ is a kinematic consequence of linearity, not a field equation. The corresponding tensor-multiplet projections and Bianchi identity are derived in Gates, Grisaru, Roček, and Siegel 1983, § 4.4(c.1), pp. 187–190.

Locally on a contractible patch,

$$H^\mu =\frac12\epsilon^{\mu\nu\rho\sigma} \partial_\nu B_{\rho\sigma},$$

where

$$B_{\mu\nu} \sim B_{\mu\nu}+\partial_\mu\Lambda_\nu-\partial_\nu\Lambda_\mu.$$

Locally, and for generic nonzero momentum, the off-shell bosonic count is one from $C$ plus three from a divergence-free $H_\mu$, matching the four real components of $\chi$. Zero modes and global flux sectors require separate boundary and cohomology data. There is no independent algebraic scalar auxiliary in this minimal real-linear multiplet. Globally, writing $H=*\,\mathrm dB$ can fail in nontrivial cohomology; the superfield constraint gives a closed field strength, while a single global potential requires additional topology.

Linear superfields can describe tensor multiplets and conserved-current multiplets. These uses share the same kinematic constraint but not necessarily the same engineering dimension, gauge interpretation, or action.

From supertranslations to constrained components

The figure follows the common construction once, then separates the chiral, vector, and real-linear branches. Inspect where a differential constraint restricts the field space, where a gauge equivalence instead identifies representatives, and where component projection reads off independent fields. The Euclidean fork is deliberately outside that Lorentzian chain because it changes the real structure and integration contour.

The same four-dimensional Lorentzian $\mathcal N=1$ convention package carries $Q$ to covariant $D$ constraints and then to component projections. Chirality, a vector prepotential modulo chiral supergauge transformations, and real linearity are different reductions; Euclidean tilded fields require an independently declared real structure and contour rather than another Lorentzian projection.

Superfield constraints and closure at a glance

The table is generated from one frozen set of the expansions and projections developed above. Counts are real local off-shell functional counts in four-dimensional Lorentzian $\mathcal N=1$ supersymmetry; the real-linear count additionally assumes generic nonzero momentum. A gauge-removable coefficient is not called auxiliary, a Bianchi identity is not called an equation of motion, and $W_\alpha$ is not counted independently of its prepotential.

Object and reality	Defining constraint	Gauge equivalence	Independent fields and real count $B/F$	Auxiliary status	Supersymmetry realization	Closure class	Equations of motion used
Chiral scalar $\Phi$; complex	$\bar D_{\dot\alpha}\Phi=0$	None intrinsically	$A,\psi_\alpha,F$; $4/4$	One complex $F$—two real algebraic auxiliary components—in the canonical chiral action	Apply $\epsilon Q+\bar\epsilon\bar Q$ to $\Phi$ and project with $1,D_\alpha/\sqrt2,-D^2/4$	Exact off shell on the chiral field space	None
Real vector prepotential $V=V^\dagger$	Reality only before gauge fixing	$V\sim V+i(\Lambda-\Lambda^\dagger)$, $\bar D_{\dot\alpha}\Lambda=0$	$C,\chi,M,N,A_\mu,\lambda,D$; $8/8$ before the quotient	$C,\chi,M,N$ are gauge-removable, not auxiliary; $D$ becomes the one real auxiliary in Wess–Zumino gauge	Apply $\epsilon Q+\bar\epsilon\bar Q$ to the whole real superfield	Exact off shell on the supergauge class	None
Wess–Zumino representative $V_{\rm WZ}$; real	Wess–Zumino gauge condition	Residual $A_\mu\sim A_\mu+\partial_\mu\omega$	$A_\mu,\lambda_\alpha,D$; $(3+1)/4=4/4$ after the residual gauge quotient	One real algebraic auxiliary $D$ in the standard vector action	Raw supertranslation followed by a field-dependent chiral supergauge compensator	Exact off shell modulo ordinary gauge	None
Field strength $W_\alpha$; chiral spinor with conjugate $\bar W_{\dot\alpha}$	$\bar D_{\dot\alpha}W_\beta=0$ and $D^\alpha W_\alpha=\bar D_{\dot\alpha}\bar W^{\dot\alpha}$	Abelian invariant; non-Abelian covariant	$F_{\mu\nu},\lambda_\alpha,D$ subject to the Bianchi identity; the same $4/4$ gauge-class data as $V$, not a second multiplet	The same one real $D$	Induced from $W_\alpha=-\bar D^2D_\alpha V/4$	Abelian: exact off shell; non-Abelian: exact off shell modulo gauge covariance	None; the displayed relation is a Bianchi identity
Real linear scalar $L=L^\dagger$	$D^2L=\bar D^2L=0$	Locally $H=*\,\mathrm dB$ with $B\sim B+\mathrm d\Lambda$; global flux sectors are separate	$C,\chi_\alpha,H_\mu$ with $\partial_\mu H^\mu=0$; locally $4/4$ at generic nonzero momentum	No independent algebraic scalar auxiliary in the minimal real-linear description	Apply $\epsilon Q+\bar\epsilon\bar Q$ to $L$; the derivative algebra preserves both linear constraints	Exact off shell on the constrained field space	None; transversality is kinematic

This is a comparison of field multiplets, not of one-particle irreducible representations. “Exact” therefore means closure on the stated off-shell field space, with its displayed constraint or gauge quotient, before imposing dynamics. The extended-superspace page uses this finite $\mathcal N=1$ baseline to identify when an auxiliary sector becomes an infinite harmonic or projective tower.

Why Wess–Zumino gauge hides manifest closure

Let $\delta_\epsilon^{\rm raw}$ denote the superspace translation of $V_{\rm WZ}$. Its lowest scalar and spinor components are generally nonzero, so it exits the gauge slice. Choose a chiral parameter $\Lambda_{\rm comp}(\epsilon;A,\lambda,D)$ that removes those regenerated components and define

$$\delta_\epsilon^{\rm WZ} =\delta_\epsilon^{\rm raw} +\delta_{\Lambda_{\rm comp}}.$$

Two such transformations close as

$$[\delta_1^{\rm WZ},\delta_2^{\rm WZ}] =\xi^\mu\partial_\mu+\delta_{\rm gauge}(\Omega), \qquad \Omega=-\xi^\mu A_\mu$$

in the Abelian theory. The compensator is not optional bookkeeping: without it, the displayed component fields are not mapped into the chosen representative. The role of physical, auxiliary, gauge, and compensator components is analyzed in Gates, Grisaru, Roček, and Siegel 1983, §§ 3.9–3.10 and 4.2.

Common pitfalls

A prepotential is not a field strength. $V$ is gauge redundant and depends on a choice of representation. $W_\alpha$ is constrained and carries the gauge-covariant curvature data.

The Bianchi identity is not a field equation. $D^\alpha W_\alpha=\bar D_{\dot\alpha}\bar W^{\dot\alpha}$ follows from the prepotential definition. A dynamical equation such as $D^\alpha W_\alpha=0$ is stronger.

A real linear field is not an unconstrained real field with two coefficients deleted. The differential constraints relate higher components and impose $\partial_\mu H^\mu=0$; the two-form gauge interpretation is part of the count.

Exercises

1. Prove field-strength chirality

Show that $\bar D_{\dot\beta}W_\alpha=0$.

Solution

By definition,

$$\bar D_{\dot\beta}W_\alpha =-\frac14\bar D_{\dot\beta}\bar D^2D_\alpha V.$$

Any product of three barred spinor derivatives vanishes because there are only two independent dotted components and the derivatives anticommute. Hence the result is zero without an equation of motion.

2. Count Wess–Zumino gauge

Why does $A_\mu$ contribute three, rather than two, real off-shell components?

Solution

Before field equations, a four-component gauge potential is quotiented by one arbitrary scalar gauge function, leaving three functional components. The reduction to two photon polarizations additionally uses the Maxwell equation and residual gauge conditions, so it is an on-shell count. Adding the real $D$ gives four bosonic off-shell components.

3. Distinguish identity from dynamics

Compare the statements

$$D^\alpha W_\alpha=\bar D_{\dot\alpha}\bar W^{\dot\alpha} \quad\text{and}\quad D^\alpha W_\alpha=0.$$

Solution

The first is the superspace Bianchi identity implied by writing $W_\alpha$ in terms of a real prepotential. The second is the source-free field equation of the free vector action. Imposing the second puts the vector multiplet on shell.

Continue

Off-Shell Closure and Auxiliary Fields uses these component counts to diagnose auxiliary completions. Superspace Measures, F-Terms, D-Terms, and Component Extraction builds invariant actions from the superfields defined here.

References

• Gates, S. James, Jr., Marcus T. Grisaru, Martin Roček, and Warren Siegel. Superspace, or One Thousand and One Lessons in Supersymmetry. Frontiers in Physics 58. Reading, MA: Benjamin/Cummings, 1983. Corrected open edition, 2001. arXiv:hep-th/0108200.

• Salam, Abdus, and John Strathdee. “Super-Gauge Transformations.” Nuclear Physics B 76 (1974): 477–482. DOI.

• Weinberg, Steven. The Quantum Theory of Fields. Volume III: Supersymmetry. Cambridge: Cambridge University Press, 2000, §§ 26.2–26.3 and §§ 27.1–27.3. DOI.