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 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 coupling absorbed into $V$ for this formula,

$$W_\alpha =-\frac14\bar D^2 \left(e^{-V}D_\alpha e^V\right).$$

It transforms covariantly rather than invariantly. 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.

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,

$$\partial_\mu H^\mu=0.$$

Locally on a contractible patch,

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

where

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

The off-shell bosonic count is one from $C$ plus three from a divergence-free $H_\mu$, matching the four real components of $\chi$. 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.

Constraint and closure comparison

Object	Defining data	Independent off-shell content	Gauge status	Closure
Chiral $\Phi$	$\bar D_{\dot\alpha}\Phi=0$	$A,\psi,F$	No intrinsic gauge equivalence	Exact
Real vector $V$	$V=V^\dagger$	$8+8$ before gauge	$V\sim V+i(\Lambda-\Lambda^\dagger)$	Exact on the gauge class
Vector in Wess–Zumino gauge	Gauge-fixed $V$	$A_\mu,\lambda,D$	Residual ordinary gauge symmetry	Exact modulo gauge; compensator required
Field strength $W_\alpha$	Chiral projection of $V$ plus Bianchi identity	Same vector multiplet, not a new one	Abelian invariant; non-Abelian covariant	Inherited from $V$
Real linear $L$	$L=L^\dagger$, $D^2L=\bar D^2L=0$	$C,\chi,H_\mu$ with $\partial_\mu H^\mu=0$	Two-form gauge description locally	Exact on the constrained field space

This is a comparison of field multiplets. It is not a table of one-particle irreducible representations, and the counts do not include equations of motion.

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.