Supersymmetric Yang–Mills Actions

Four-dimensional $\mathcal N=1$ super-Yang–Mills theory packages a gauge connection, one adjoint Weyl gaugino, and one real adjoint auxiliary field into a vector multiplet. Its chiral field-strength superfield produces the Yang–Mills kinetic term, the gaugino kinetic term, $D^2/2$ , and the theta term in one supersymmetric integral. The compact formula is safe only after the generator trace, coupling placement, global gauge group, theta periodicity, Wess–Zumino-gauge compensation, and spinor conventions have been fixed. The preceding superfield chapter owns the construction and closure of this vector multiplet. Here those data are imported as one convention package and used to normalize and verify the action.

Required background. Chiral, Vector, Linear, and Field-Strength Superfields constructs $V$ and $\mathcal W_\alpha$ . Supersymmetric Action Principles and Component Reduction supplies the projection and off-shell check.

Helpful background. Gauge Fields, Redundancy, and Observable Content separates local connection data from the global gauge group.

Fix the gauge and coupling normalization first

Let $G$ be a compact gauge group with Hermitian generators $[T^a,T^b]=if^{abc}T^c$ and $\operatorname{tr}_R(T^aT^b)=T(R)\delta^{ab}$ . It is useful to distinguish two normalizations.

In the holomorphic normalization, the connection $\mathcal A_\mu=\mathcal A_\mu^aT^a$ enters $\mathcal D_\mu=\partial_\mu-i\mathcal A_\mu$ and

\mathcal F_{\mu\nu} =\partial_\mu\mathcal A_\nu-\partial_\nu\mathcal A_\mu -i[\mathcal A_\mu,\mathcal A_\nu].

The coupling multiplies the action. In the site’s canonical normalization,

\mathcal A_\mu=gA_\mu, \qquad \mathcal F_{\mu\nu}=gF_{\mu\nu}, \qquad D_\mu=\partial_\mu-igA_\mu.

The same rescaling applies to the gaugino and auxiliary field. This translation explains why some sources display $1/g^2$ in front of the whole vector-multiplet action while the site’s canonical component action does not. Mixing the two normalizations is a common source of wrong theta and instanton factors.

The imported field strength fixes the action

In Wess–Zumino gauge, the holomorphically normalized real superfield can be written

V= -\theta\sigma^\mu\bar\theta\,\mathcal A_\mu +i\theta^2\bar\theta\,\bar\lambda_h -i\bar\theta^2\theta\,\lambda_h +\frac12\theta^2\bar\theta^2D_h.

For a non-Abelian group, define

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

It is covariantly chiral and transforms by conjugation. Its component expansion has the schematic but normalization-complete form

\mathcal W_\alpha =-i\lambda_{h\alpha} +\left[ \delta_\alpha{}^\beta D_h -i(\sigma^{\mu\nu})_\alpha{}^\beta \mathcal F_{\mu\nu} \right]\theta_\beta -\theta^2 \sigma^\mu_{\alpha\dot\alpha} \mathcal D_\mu\bar\lambda_h^{\dot\alpha}.

The ordering is the non-Abelian translation compatible with $\mathcal D_\mu=\partial_\mu-i\mathcal A_\mu$ . Its Abelian linearization is $\mathcal W_\alpha=-\bar D^2D_\alpha V/4$ , exactly the Chapter 3 field-strength convention. Thus the signs in $V$ , $\mathcal W_\alpha$ , the commutator term in $\mathcal F_{\mu\nu}$ , and the matter exponential on the next page form one package.

Here $\sigma^{\mu\nu}=\tfrac14(\sigma^\mu\bar\sigma^\nu- \sigma^\nu\bar\sigma^\mu)$ , $\epsilon^{0123}=+1$ , and $\widetilde{\mathcal F}^{\mu\nu} =\epsilon^{\mu\nu\rho\sigma}\mathcal F_{\rho\sigma}/2$ . The selected top component is

\begin{aligned} \left[\mathcal W^{a\alpha}\mathcal W^a_\alpha\right]_{\theta^2} ={}&D_h^aD_h^a -\frac12\mathcal F^a_{\mu\nu}\mathcal F^{a\mu\nu} +\frac{i}{2}\mathcal F^a_{\mu\nu} \widetilde{\mathcal F}^{a\mu\nu}\\ &+2i\lambda_h^a\sigma^\mu \mathcal D_\mu\bar\lambda_h^a. \end{aligned}

After a covariant integration by parts, the last term is $+2i\bar\lambda_h\bar\sigma^\mu\mathcal D_\mu\lambda_h$ . This directly checks the relative signs of the vector, gaugino, and auxiliary kinetic terms. A different factor in the exponential changes the factors in $\mathcal W_\alpha$ and must be translated as a unit. The gauge-invariant chiral field-strength construction originates in Ferrara and Zumino 1974, pp. 413–421.

The Lorentzian coefficient can now be fixed rather than inferred. Define

\tau=\frac{\vartheta}{2\pi}+\frac{4\pi i}{g^2}.

With $\operatorname{tr}_F(T^aT^b)=\delta^{ab}/2$ , the holomorphically normalized action is

\begin{aligned} S_{\mathrm{SYM},h} &=\int d^4x\left[ \int d^2\theta\, \frac{\tau}{8\pi i}\, \operatorname{tr}_F (\mathcal W_h^\alpha\mathcal W_{h\alpha}) +\text{h.c.}\right]\\ &=\frac1{4\pi}\operatorname{Im} \int d^4x\,d^2\theta\, \tau\,\operatorname{tr}_F (\mathcal W_h^\alpha\mathcal W_{h\alpha}). \end{aligned}

Because $\operatorname{tr}_F\mathcal W_h^2=\mathcal W_h^{a\alpha} \mathcal W^a_{h\alpha}/2$ , the untraced coefficient is

\frac{\tau}{16\pi i} =\frac1{4g^2}-\frac{i\vartheta}{32\pi^2}.

Equivalently, the gauge kinetic function is $f=\tau/(4\pi i)=g^{-2}-i\vartheta/(8\pi^2)$ in $\int d^2\theta\,f\mathcal W_h^a\mathcal W_h^a/4+\text{h.c.}$ Directly inserting the top component gives

\begin{aligned} \mathcal L_{\mathrm{SYM},h}={}& -\frac1{4g^2}\mathcal F^a_{\mu\nu} \mathcal F^{a\mu\nu} +\frac{i}{g^2}\bar\lambda_h^a\bar\sigma^\mu \mathcal D_\mu\lambda_h^a +\frac1{2g^2}D_h^aD_h^a\\ &+\frac{\vartheta}{32\pi^2} \mathcal F^a_{\mu\nu}\widetilde{\mathcal F}^{a\mu\nu}. \end{aligned}

Rescaling

(\mathcal A,\lambda_h,D_h)=g(A,\lambda,D), \qquad \mathcal W_h=g\mathcal W_c,

gives the site’s canonical form

\begin{aligned} \mathcal L_{\mathrm{SYM}}={}& -\frac14F^a_{\mu\nu}F^{a\mu\nu} +i\bar\lambda^a\bar\sigma^\mu D_\mu\lambda^a +\frac12D^aD^a\\ &+\frac{\vartheta g^2}{32\pi^2} F^a_{\mu\nu}\widetilde F^{a\mu\nu}. \end{aligned}

The theta density is a local total derivative but can integrate to nonzero topological charge. Integer normalization and $2\pi$ periodicity in this form hold for the standard simply connected $SU(N)$ bundle sectors and trace choice. Quotients of the gauge group, background two-form fields, spin structure, or restricted bundles can change the allowed topological charges and theta periodicity Aharony, Seiberg, and Tachikawa 2013, §§2–3, arXiv v5, Open PDF. Those are global data, not modifications of the local superfield algebra.

The component reduction and the Abelian/non-Abelian constructions are given in Weinberg 2000, §§27.2–27.3, pp. 122–131.

The imported transformations verify off-shell action invariance

In canonical normalization, the component transformations imported from the vector-multiplet construction are

\begin{aligned} \delta A_\mu^a &=i\epsilon\sigma_\mu\bar\lambda^a -i\lambda^a\sigma_\mu\bar\epsilon,\\ \delta\lambda_\alpha^a &=(\sigma^{\mu\nu}\epsilon)_\alpha F^a_{\mu\nu} +i\epsilon_\alpha D^a,\\ \delta D^a &=-\epsilon\sigma^\mu D_\mu\bar\lambda^a -(D_\mu\lambda^a)\sigma^\mu\bar\epsilon, \end{aligned}

with the conjugate transformation for $\bar\lambda$ . Varying the kinetic terms, using the Bianchi identity and integrating by parts yields a spacetime divergence. The $D$ -dependent variations cancel without setting $D=0$ . Hence the pure action is supersymmetric off shell.

The closure class is established on Chiral, Vector, Linear, and Field-Strength Superfields. For completeness, Wess–Zumino gauge itself is not preserved by a bare supersymmetry translation. The transformation must be followed by a field-dependent gauge transformation that restores the removed superfield components. Consequently the commutator on $A_\mu$ is

[\delta_1,\delta_2]A_\mu =v^\nu\partial_\nu A_\mu+D_\mu\Lambda_{12},

where $v^\nu$ is the standard spinor bilinear and $\Lambda_{12}$ contains $-v^\nu A_\nu$ in this gauge. On gauge-invariant operators the second term vanishes; on the gauge potential it is essential. Calling this “exact translation closure” would be incorrect; it is exact off shell modulo gauge. The present page uses that imported result only as a check that the normalized action and its auxiliary term form the same convention package.

For pure SYM the algebraic equation is $D^a=0$ . Eliminating it changes the field realization but not the classical bosonic potential. Once charged matter or an FI term is added, $D^a$ instead enforces a moment-map equation and produces a nontrivial positive potential.

A gauge kinetic function adds field dependence

For several gauge factors or neutral chiral fields, the most general local two-derivative chiral term is

\frac14\int d^2\theta\, f_{ab}(\Phi)\,\mathcal W^{a\alpha}\mathcal W^b_\alpha +\text{h.c.},

with symmetric holomorphic $f_{ab}$ . Its real part is the gauge kinetic matrix and must be positive in the convention where it multiplies the quadratic kinetic form. Its imaginary part supplies generalized theta couplings. Derivatives of $f_{ab}$ generate scalar–gaugino interactions and additional auxiliary couplings. A field-dependent $f$ therefore cannot be treated as a mere replacement $g\mapsto g(\phi)$ in the bosonic action.

At the quantum level, the holomorphic coupling appearing in a Wilsonian action and the canonically normalized physical coupling are related by field rescalings and anomalies. That distinction belongs to Holomorphic Couplings and Background Superfields, not to the classical normalization derived here.

A normalization cross-check

Before importing any SYM formula, check:

datum	question	invariant check
generators	Hermitian or anti-Hermitian? what is $T(R)$ ?	covariant derivative and commutator agree
coupling	in $D_\mu$ , in $V$ , or in front of the action?	canonical two-point residue
theta term	which trace and dual-tensor orientation?	topological charge in an allowed bundle
global form	which line operators and bundle sectors exist?	charge lattice and theta periodicity
gaugino	Weyl/Majorana phase and kinetic normalization?	hermiticity and pole residue
closure	exact, modulo gauge, or on shell?	commutator on $A_\mu$ and on $\operatorname{tr}F^2$

The round trip $\mathcal A=gA\to\mathcal F=gF\to\mathcal L_h\to\mathcal L_{\rm can}$ must recover both the kinetic term and the theta coefficient.

Common pitfalls

Using the local Lie algebra as the full theory. $SU(N)$ and $SU(N)/\mathbb Z_N$ share a Lie algebra but differ in line operators, bundle sectors, and sometimes theta periodicity.

Forgetting the compensating gauge transformation. Wess–Zumino gauge is a partial gauge choice, not a supersymmetry-invariant subspace by itself.

Rescaling only the gauge field. Supersymmetry requires the connection, gaugino, and auxiliary normalization to be translated together.

Exercises

1. Translate the theta term. Starting from $\mathcal F=gF$ , show how the holomorphic normalization becomes the canonical one.

Solution

Substitution gives

\frac{\vartheta}{32\pi^2}\mathcal F^a\widetilde{\mathcal F}^a =\frac{\vartheta g^2}{32\pi^2}F^a\widetilde F^a.

The same substitution changes $-\mathcal F^2/(4g^2)$ to $-F^2/4$ . Translating one term without the other would mix normalizations.

2. Why keep D? Identify the cancellation that would be obscured by setting $D=0$ before varying the action.

Solution

$\delta\lambda$ contains $i\epsilon D$ , while $\delta D$ contains a derivative of the gaugino. The variation of the gaugino kinetic term proportional to $D$ cancels the variation of $D^2/2$ . Setting $D=0$ first removes the independent field that makes this off-shell cancellation and closure visible.

Where gauge dynamics continues

Gauge–Matter Systems, F- and D-Term Potentials, and FI Data couples charged chiral multiplets. Pure Super-Yang–Mills Vacua and Domain Walls studies the strong quantum dynamics. Generalized Symmetries, Global Forms, and Anomalies under Duality keeps the global gauge data in later equivalence claims.

References

Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 2013, no. 8 (2013): 115. DOI. Open PDF, arXiv v5.
Ferrara, Sergio, and Bruno Zumino. “Supergauge Invariant Yang–Mills Theories.” Nuclear Physics B 79, no. 3 (1974): 413–421. DOI.
Weinberg, Steven. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge: Cambridge University Press, 2000, §§27.2–27.3, pp. 122–131. DOI.