Component Multiplets and Closure Records

A component-field supermultiplet is not just a list of spins. Its specification includes the fields and their reality conditions, transformation laws, gauge equivalences, auxiliary variables, and the exact result of applying two transformations. The four-dimensional $\mathcal N=1$ chiral multiplet closes off shell after adding one complex auxiliary scalar; the vector multiplet closes off shell modulo a gauge transformation. Eliminating the auxiliaries preserves the classical dynamics but generally changes the closure class to on shell.

Required background. The Four-Dimensional N=1 Super-Poincaré Algebra supplies the algebra and conjugation used below. Spinors, Conjugations, Bilinears, and Fierz Identities supplies the two-component identities needed to reduce the commutators.

Helpful background. Gauge Fields, Redundancy, and Observable Content explains why a gauge term is not a failure of the algebra on physical configurations.

A field multiplet is transformation data

Work in four-dimensional Lorentzian spacetime with the site metric $(+---)$. Use

$$\sigma^\mu_{\alpha\dot\alpha} =(\mathbf1,\boldsymbol\sigma), \qquad \bar\sigma^{\mu\dot\alpha\alpha} =(\mathbf1,-\boldsymbol\sigma), \qquad \epsilon^{12}=\epsilon_{21}=1.$$

The translation generator is $P_\mu=i\partial_\mu$, and

$$\{Q_\alpha,\bar Q_{\dot\alpha}\} =2\sigma^\mu_{\alpha\dot\alpha}P_\mu.$$

All spinor derivatives and component calculations use left Grassmann differentiation. Define the even transformation

$$\delta_\epsilon=\epsilon^\alpha Q_\alpha +\bar\epsilon_{\dot\alpha}\bar Q^{\dot\alpha},$$

where $\epsilon$ and $\bar\epsilon$ are constant Grassmann-odd parameters related by Lorentzian conjugation. With

$$\xi^\mu =2i\left( \epsilon_2\sigma^\mu\bar\epsilon_1 -\epsilon_1\sigma^\mu\bar\epsilon_2 \right),$$

the target algebra on an ordinary nongauge field is

$$[\delta_1,\delta_2]\varphi=\xi^\mu\partial_\mu\varphi.$$

A reproducible component specification answers six questions for every field:

Entry	Required statement
Field space	Lorentz representation, statistics, engineering dimension, and reality
Redundancy	Gauge transformations or other equivalences
Transformation	All coefficients, signs, and conjugates
Auxiliary status	Whether a field is algebraic in a named action
Closure	Translation, gauge term, constraint term, and equation-of-motion term separately
Count	Independent real components off shell and physical modes on shell

The field count is a diagnostic, not a proof. Equal bosonic and fermionic counts do not establish the transformation algebra.

The chiral multiplet closes without field equations

Let $(A,\psi_\alpha,F)$ consist of a complex scalar, a left Weyl spinor, and a complex scalar auxiliary field. Their engineering dimensions for a canonical four-dimensional action are $1$, $3/2$, and $2$. Choose

$$\begin{aligned} \delta A &=\sqrt2\,\epsilon\psi,\\ \delta\psi_\alpha &=i\sqrt2\,(\sigma^\mu\bar\epsilon)_\alpha \partial_\mu A +\sqrt2\,\epsilon_\alpha F,\\ \delta F &=i\sqrt2\,\bar\epsilon\bar\sigma^\mu \partial_\mu\psi, \end{aligned}$$

together with the complex-conjugate rules. These are the component projections of a chiral superfield in the convention developed later in this chapter.

The scalar check is immediate:

$$\begin{aligned} [\delta_1,\delta_2]A &= 2i\left( \epsilon_2\sigma^\mu\bar\epsilon_1 -\epsilon_1\sigma^\mu\bar\epsilon_2 \right)\partial_\mu A\\ &\quad +2(\epsilon_2\epsilon_1-\epsilon_1\epsilon_2)F =\xi^\mu\partial_\mu A. \end{aligned}$$

The displayed ordering of odd parameters matters. Because the spinor metric is antisymmetric while the parameters themselves anticommute, the contraction of two Grassmann-odd Weyl spinors is symmetric: $\epsilon_2\epsilon_1=\epsilon_1\epsilon_2$. The apparent $F$ term therefore cancels.

For the fermion, the variation of the derivative of $A$ and the variation of $F$ both produce derivatives of $\psi$. The convention-sensitive identity needed below is

$$\chi^\beta(\sigma^\mu\bar\eta)_\alpha +\chi_\alpha(\bar\eta\bar\sigma^\mu)^\beta =\delta_\alpha{}^\beta(\chi\sigma^\mu\bar\eta).$$

Here every bilinear is ordered with the unbarred odd spinor first. Keeping the two variation routes separate gives

$$\begin{aligned} [\delta_1,\delta_2]\psi_\alpha =2i\Big[& (\sigma^\mu\bar\epsilon_2)_\alpha\epsilon_1{}^\beta -(\sigma^\mu\bar\epsilon_1)_\alpha\epsilon_2{}^\beta\\ &+\epsilon_{2\alpha}\bar\epsilon_{1\dot\alpha} \bar\sigma^{\mu\dot\alpha\beta} -\epsilon_{1\alpha}\bar\epsilon_{2\dot\alpha} \bar\sigma^{\mu\dot\alpha\beta} \Big]\partial_\mu\psi_\beta. \end{aligned}$$

The second row is precisely the contribution from $\delta F$. Before using the identity, move the unbarred parameter in the first row to the left; for example,

$$(\sigma^\mu\bar\epsilon_2)_\alpha\epsilon_1{}^\beta =-\epsilon_1{}^\beta(\sigma^\mu\bar\epsilon_2)_\alpha.$$

The four terms then group into two complete spinor matrices:

$$\begin{aligned} [\delta_1,\delta_2]\psi_\alpha =2i\Big\{& \big[ \epsilon_2{}^\beta(\sigma^\mu\bar\epsilon_1)_\alpha +\epsilon_{2\alpha}(\bar\epsilon_1\bar\sigma^\mu)^\beta \big]\\ &- \big[ \epsilon_1{}^\beta(\sigma^\mu\bar\epsilon_2)_\alpha +\epsilon_{1\alpha}(\bar\epsilon_2\bar\sigma^\mu)^\beta \big] \Big\}\partial_\mu\psi_\beta\\ =2i\big(& \epsilon_2\sigma^\mu\bar\epsilon_1 -\epsilon_1\sigma^\mu\bar\epsilon_2 \big)\partial_\mu\psi_\alpha\\ =\;&\xi^\mu\partial_\mu\psi_\alpha. \end{aligned}$$

For the top component, applying $\delta_1$ to $\delta_2F$ and subtracting the reverse order gives

$$\begin{aligned} [\delta_1,\delta_2]F =\;&-2\Big( \bar\epsilon_2\bar\sigma^\mu\sigma^\nu\bar\epsilon_1 -\bar\epsilon_1\bar\sigma^\mu\sigma^\nu\bar\epsilon_2 \Big)\partial_\mu\partial_\nu A\\ &+2i\Big( \bar\epsilon_2\bar\sigma^\mu\epsilon_1 -\bar\epsilon_1\bar\sigma^\mu\epsilon_2 \Big)\partial_\mu F. \end{aligned}$$

Only the symmetric part of $\bar\sigma^\mu\sigma^\nu$ survives against $\partial_\mu\partial_\nu$. It is proportional to $\eta^{\mu\nu}$, and its coefficient vanishes because the contraction of the two dotted odd spinors is symmetric. Finally,

$$\bar\epsilon_2\bar\sigma^\mu\epsilon_1 -\bar\epsilon_1\bar\sigma^\mu\epsilon_2 =\epsilon_2\sigma^\mu\bar\epsilon_1 -\epsilon_1\sigma^\mu\bar\epsilon_2,$$

so $[\delta_1,\delta_2]F=\xi^\mu\partial_\mu F$.

No action or field equation has entered. This is exact off-shell closure. The independent real count is

$$\underbrace{2_A+2_F}_{4\ {\rm bosonic}} = \underbrace{4_\psi}_{4\ {\rm fermionic}}.$$

The classic direct construction and its auxiliary completion are given in Weinberg 2000, § 26.1, pp. 55–59; the original four-dimensional multiplet construction appears in Wess and Zumino 1974, pp. 39–50.

Eliminating the auxiliary changes the closure class

For the free massless action, the algebraic field equation is $F=0$. If one deletes $F$ and uses only

$$\delta A=\sqrt2\,\epsilon\psi, \qquad \delta\psi_\alpha =i\sqrt2(\sigma^\mu\bar\epsilon)_\alpha\partial_\mu A,$$

then the scalar still closes by translation. The fermion commutator is

$$[\delta_1,\delta_2]\psi_\alpha = 2i\left[ (\sigma^\mu\bar\epsilon_2)_\alpha\epsilon_1{}^\beta -(\sigma^\mu\bar\epsilon_1)_\alpha\epsilon_2{}^\beta \right]\partial_\mu\psi_\beta.$$

Subtracting the translation and applying the same completeness identity gives the exact remainder

$$\mathcal R_\alpha =2i\left( \epsilon_{1\alpha}\bar\epsilon_{2\dot\alpha} -\epsilon_{2\alpha}\bar\epsilon_{1\dot\alpha} \right) \bar\sigma^{\mu\dot\alpha\beta} \partial_\mu\psi_\beta.$$

Thus $\mathcal R_\alpha$ is explicitly a parameter-dependent linear combination of

$$\bar\sigma^{\mu\dot\alpha\beta}\partial_\mu\psi_\beta,$$

which is the free Weyl equation. Thus

$$[\delta_1,\delta_2]\psi_\alpha =\xi^\mu\partial_\mu\psi_\alpha+\mathcal R_\alpha, \qquad \mathcal R_\alpha=0\quad\text{only on shell}.$$

For one chiral field with canonical Kähler metric, an interacting Wess–Zumino model replaces $F$ by $-\overline{W'(A)}$ in this convention. With a nontrivial Kähler metric, its inverse and connection-dependent fermion terms enter the auxiliary equation. In either case, the corresponding remainder is proportional to the interacting fermion equation, not merely the free Weyl operator. Algebraic elimination therefore preserves the action-level theory while removing an off-shell representation on the reduced fields. Weinberg 2000, § 26.4, pp. 75–82 makes this distinction explicit.

The vector multiplet closes modulo gauge

For an Abelian vector multiplet in Wess–Zumino gauge, take a real gauge potential $A_\mu$, a Weyl gaugino $\lambda_\alpha$, and a real auxiliary scalar $D$. Define

$$\sigma^{\mu\nu} =\frac14\left( \sigma^\mu\bar\sigma^\nu-\sigma^\nu\bar\sigma^\mu \right)$$

and use

$$\begin{aligned} \delta A_\mu &=i\epsilon\sigma_\mu\bar\lambda -i\lambda\sigma_\mu\bar\epsilon,\\ \delta\lambda_\alpha &=(\sigma^{\mu\nu}\epsilon)_\alpha F_{\mu\nu} +i\epsilon_\alpha D,\\ \delta D &=-\epsilon\sigma^\mu\partial_\mu\bar\lambda -\partial_\mu\lambda\,\sigma^\mu\bar\epsilon. \end{aligned}$$

The conjugate gaugino transformation needed below is

$$\delta\bar\lambda_{\dot\alpha} =(\bar\sigma^{\mu\nu}\bar\epsilon)_{\dot\alpha}F_{\mu\nu} -i\bar\epsilon_{\dot\alpha}D, \qquad \bar\sigma^{\mu\nu} :=\frac14\left( \bar\sigma^\mu\sigma^\nu-\bar\sigma^\nu\sigma^\mu \right).$$

The two routes through the gaugino commutator have distinct roles. The part obtained by varying $F_{\mu\nu}$ reduces by the three-sigma identities to

$$\big[\delta_1,\delta_2\big]_{F\text{-route}}\lambda_\alpha =\xi^\mu\partial_\mu\lambda_\alpha -\Delta^{(D)}_\alpha,$$

where the contribution from varying the explicit $i\epsilon D$ term is

$$\begin{aligned} \Delta^{(D)}_\alpha :={}&i\epsilon_{2\alpha}\delta_1D -i\epsilon_{1\alpha}\delta_2D\\ =\;&-i\epsilon_{2\alpha}\Big( \epsilon_1\sigma^\mu\partial_\mu\bar\lambda +\partial_\mu\lambda\,\sigma^\mu\bar\epsilon_1 \Big)\\ &+i\epsilon_{1\alpha}\Big( \epsilon_2\sigma^\mu\partial_\mu\bar\lambda +\partial_\mu\lambda\,\sigma^\mu\bar\epsilon_2 \Big). \end{aligned}$$

The two pieces cancel without a gaugino equation, leaving $[\delta_1,\delta_2]\lambda_\alpha=\xi^\mu\partial_\mu\lambda_\alpha$.

The calculation on $D$ separates just as cleanly. The part containing $F_{\rho\sigma}$ is

$$\begin{aligned} \mathcal B:=\Big[& -\epsilon_2\sigma^\mu\bar\sigma^{\rho\sigma}\bar\epsilon_1 -(\sigma^{\rho\sigma}\epsilon_1)\sigma^\mu\bar\epsilon_2\\ &+\epsilon_1\sigma^\mu\bar\sigma^{\rho\sigma}\bar\epsilon_2 +(\sigma^{\rho\sigma}\epsilon_2)\sigma^\mu\bar\epsilon_1 \Big]\partial_\mu F_{\rho\sigma}. \end{aligned}$$

The metric terms in the three-sigma identities cancel pairwise, while the remaining totally antisymmetric term is proportional to $\partial_{[\mu}F_{\rho\sigma]}$ and vanishes by the Bianchi identity. The variations of the $i\epsilon D$ and $-i\bar\epsilon D$ terms supply $\xi^\mu\partial_\mu D$. Thus

$$[\delta_1,\delta_2]D=\xi^\mu\partial_\mu D \qquad\text{without a field equation}.$$

On the connection,

$$\begin{aligned} [\delta_1,\delta_2]A_\mu &=\xi^\nu F_{\nu\mu}\\ &=\xi^\nu\partial_\nu A_\mu +\partial_\mu\Omega, \qquad \Omega=-\xi^\nu A_\nu. \end{aligned}$$

No equation of motion was used. The algebra closes off shell on the gauge-equivalence class and modulo an Abelian gauge transformation on a representative. The original vector multiplet and its auxiliary completion are exhibited in Wess and Zumino 1974, pp. 42–48; Weinberg 2000, §§ 27.2–27.3, pp. 122–131 gives the corresponding two-component construction.

For the non-Abelian extension, retain the site’s Hermitian-generator convention and place the gauge coupling in the connection:

$$\begin{aligned} F_{\mu\nu} &:=\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu],\\ D_\mu X&:=\partial_\mu X-ig[A_\mu,X],\\ \delta_\Omega A_\mu&:=D_\mu\Omega. \end{aligned}$$

Then

$$\xi^\nu F_{\nu\mu} =\xi^\nu\partial_\nu A_\mu +D_\mu(-\xi^\nu A_\nu),$$

so $\Omega=-\xi^\nu A_\nu$ is Lie-algebra valued. This is a translation plus a field-dependent gauge transformation in the stated convention.

The off-shell count is

$$\underbrace{(4-1)_{A_\mu}+1_D}_{4\ {\rm bosonic}} = \underbrace{4_\lambda}_{4\ {\rm fermionic}}.$$

This local count is made for a generic nonzero Fourier mode, equivalently at the level of the principal symbol: the subtraction by one is a local gauge function, not an on-shell polarization count. Global zero modes and boundary data require a separate count. After imposing the Maxwell equation and quotienting residual gauge transformations, the photon has two physical helicities; the on-shell Weyl gaugino also has two real physical modes.

For pure Abelian super-Yang–Mills theory, the auxiliary sector is $\mathcal L_D=\tfrac12D^2$, so the algebraic equation is $D=0$. Deleting $D$ also deletes the $i\epsilon D$ term in $\delta\lambda$. The connection still closes modulo gauge, but the reduced gaugino transformation obeys

$$\begin{aligned} [\delta^{(0)}_1,\delta^{(0)}_2]\lambda_\alpha &=\xi^\mu\partial_\mu\lambda_\alpha +\mathcal R^{(D)}_\alpha,\\ \mathcal R^{(D)}_\alpha &=-\Delta^{(D)}_\alpha\\ &=i\epsilon_{2\alpha}\Big( \epsilon_1\sigma^\mu\partial_\mu\bar\lambda +\partial_\mu\lambda\,\sigma^\mu\bar\epsilon_1 \Big)\\ &\quad-i\epsilon_{1\alpha}\Big( \epsilon_2\sigma^\mu\partial_\mu\bar\lambda +\partial_\mu\lambda\,\sigma^\mu\bar\epsilon_2 \Big). \end{aligned}$$

The remainder vanishes only after imposing the conjugate free Weyl equations $\sigma^\mu\partial_\mu\bar\lambda=0$ and $(\partial_\mu\lambda)\sigma^\mu=0$. Evaluating the already-closed full commutator at $D=0$ would hide this change of representation, just as evaluating the full chiral commutator at $F=0$ hides the reduced fermion remainder.

Wess–Zumino gauge requires a compensator

An unconstrained real vector superfield contains additional scalar and spinor components. A supergauge transformation removes them to reach Wess–Zumino gauge. A bare superspace translation generally regenerates the removed components, so the component transformation above is actually

$$\delta_{\rm WZ} =\delta_{\rm SUSY} +\delta_{\rm supergauge}(\Lambda_{\rm comp}),$$

where the compensating chiral parameter depends on the supersymmetry parameter and the fields. Consequently the commutator contains the ordinary gauge transformation displayed above. Saying that “supersymmetry does not close in Wess–Zumino gauge” is incomplete: it closes on the gauge orbit, and the compensator is part of the transformation law. The prepotential and compensator structure are developed in Gates, Grisaru, Roček, and Siegel 1983, §§ 3.9–3.10, pp. 108–119, and § 4.2, pp. 159–177.

Field multiplets are not state multiplets

Object being counted	Chiral example	Vector example
Covariant off-shell fields	$A,\psi,F$	$A_\mu,\lambda,D$ modulo gauge
Auxiliary variables	Complex $F$	Real $D$
On-shell one-particle content	Complex scalar plus Weyl fermion, with CPT completion as required	Helicity $\pm1$ gauge boson plus helicity $\pm\tfrac12$ gaugino
Closure statement	Exact before eliminating $F$; on shell afterward	Exact modulo gauge before eliminating $D$; on shell modulo gauge afterward

Off-shell component equality concerns functions on spacetime before equations of motion. On-shell state equality concerns physical polarizations at fixed momentum. Gauge fixing, constraints, CPT completion, and auxiliary elimination affect the two counts differently.

Common pitfalls

A translation plus gauge is not failed closure. The gauge potential is not itself a gauge-invariant observable. The correct representation space is the space of connections modulo gauge transformations.

An algebraic field is not automatically auxiliary. “Auxiliary” refers to its role in a specified action and transformation system. A Lagrange multiplier, Stückelberg field, compensator, and gauge-removable component have different functions even when none propagates.

Degree matching is necessary, not sufficient. Incorrect transformation coefficients can preserve a $4=4$ count while violating the commutator. Closure must be calculated field by field.

Exercises

1. Close the scalar

Using the chiral transformations, show explicitly that the two $F$ terms in $[\delta_1,\delta_2]A$ cancel.

Solution

The relevant contribution is

$$2\epsilon_2^\alpha\epsilon_{1\alpha}F -2\epsilon_1^\alpha\epsilon_{2\alpha}F.$$

Moving the odd parameters to the same order shows that $\epsilon_2\epsilon_1=\epsilon_1\epsilon_2$: the minus sign from exchanging the Grassmann-odd parameters cancels the minus sign from the antisymmetric spinor metric. The two $F$ terms are therefore equal and opposite. The derivative terms combine into $\xi^\mu\partial_\mu A$.

2. Close the top chiral component

Starting from $\delta F=i\sqrt2\,\bar\epsilon\bar\sigma^\mu\partial_\mu\psi$, identify separately the terms proportional to $\partial_\mu\partial_\nu A$ and $\partial_\mu F$ in $[\delta_1,\delta_2]F$.

Solution

The direct calculation gives

$$\begin{aligned} [\delta_1,\delta_2]F =\;&-2\Big( \bar\epsilon_2\bar\sigma^\mu\sigma^\nu\bar\epsilon_1 -\bar\epsilon_1\bar\sigma^\mu\sigma^\nu\bar\epsilon_2 \Big)\partial_\mu\partial_\nu A\\ &+2i\Big( \bar\epsilon_2\bar\sigma^\mu\epsilon_1 -\bar\epsilon_1\bar\sigma^\mu\epsilon_2 \Big)\partial_\mu F. \end{aligned}$$

The antisymmetric part of $\bar\sigma^\mu\sigma^\nu$ is killed by the symmetric derivative, while the remaining dotted-spinor contraction cancels between the two orderings. Reordering the last line puts the unbarred parameters first and gives $\xi^\mu\partial_\mu F$. No field equation was used.

3. Identify the gauge parameter

Rewrite $\xi^\nu F_{\nu\mu}$ as a translation plus an Abelian gauge transformation.

Solution

Because $\xi^\nu$ is constant,

$$\xi^\nu F_{\nu\mu} =\xi^\nu\partial_\nu A_\mu -\xi^\nu\partial_\mu A_\nu =\xi^\nu\partial_\nu A_\mu +\partial_\mu(-\xi^\nu A_\nu).$$

Thus $\Omega=-\xi^\nu A_\nu$. No Maxwell equation occurs, so the result is off shell modulo gauge.

4. Test the reduced chiral multiplet

Why is setting $F=0$ before computing the commutator different from computing the off-shell commutator and then evaluating it on the solution $F=0$?

Solution

In the full multiplet, $\delta F$ participates in the cancellation of terms in $[\delta_1,\delta_2]\psi$. If $F$ is deleted, that variation is unavailable and an equation-of-motion remainder remains. Evaluating the already-closed full commutator at $F=0$ hides this loss of a transformation variable.

5. Delete the vector auxiliary

Set $D=0$ in pure Abelian super-Yang–Mills theory and delete the $i\epsilon D$ term from $\delta\lambda$. Which part of the full commutator has been lost, and when does the reduced commutator close?

Solution

The lost route is

$$\Delta^{(D)}_\alpha =i\epsilon_{2\alpha}\delta_1D -i\epsilon_{1\alpha}\delta_2D.$$

The $F_{\mu\nu}$ route by itself therefore gives

$$[\delta^{(0)}_1,\delta^{(0)}_2]\lambda_\alpha =\xi^\mu\partial_\mu\lambda_\alpha-\Delta^{(D)}_\alpha.$$

Substituting the full transformation of $D$ shows that $-\Delta^{(D)}_\alpha$ is a parameter-dependent linear combination of $\sigma^\mu\partial_\mu\bar\lambda$ and $(\partial_\mu\lambda)\sigma^\mu$. It vanishes only on the free gaugino equations and their conjugates. Thus the reduced vector system closes on shell modulo gauge, even though the full $(A_\mu,\lambda,D)$ system closes off shell modulo gauge.

Continue

Superspace and Supertranslations packages these transformations as geometry. Superfield constraints and closure comparison organizes the same multiplets by constraint, gauge equivalence, component content, and closure class. The off-shell auxiliary closure map then compares the counting and elimination logic, including the limitations for extended supersymmetry.

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.

• Weinberg, Steven. The Quantum Theory of Fields. Volume III: Supersymmetry. Cambridge: Cambridge University Press, 2000, §§ 26.1–26.4. DOI.

• Wess, Julius, and Bruno Zumino. “Supergauge Transformations in Four Dimensions.” Nuclear Physics B 70 (1974): 39–50. DOI. CERN record.