Supersymmetric Actions, Supercurrents, and Quantum Effective Theory

Supersymmetric actions are assembled in layers. A superspace measure selects an invariant component; the Kähler potential, superpotential, gauge kinetic function, representations, and Fayet–Iliopoulos data specify the theory; and component reduction exposes the kinetic terms, Yukawa couplings, auxiliary equations, and scalar potential. Quantum calculations then require one more choice: whether the object is Wilsonian or one-particle irreducible. Questions about currents require a different record again—the supercurrent multiplet and the global improvements that are actually available.

This chapter teaches how to produce that complete record. It works primarily with rigid four-dimensional $\mathcal N=1$ theories in Lorentzian signature. General action principles and effective actions are imported from Foundations; ordinary Ward identities from Symmetry and Gauge Structure; amplitudes from Scattering; and dynamical supergravity from its own volume.

Helpful background. The Action Principle and Field Equations supplies the variational logic, The 1PI Effective Action and Mean-Field Equations fixes the quantum-functional distinction, and Current Sources and Generating Functionals explains how currents couple to backgrounds.

Parent volume. Supersymmetry and Duality

Jump to: choose a route · follow the construction · export on shell · use the chapter guide · review the chapter

Diagnose your preparation

Use these checks to choose an entry point. An “unsure” answer points to the background that will make the corresponding route easier to follow.

Superspace algebra. Ready: you can apply $D_\alpha$ and $\bar D_{\dot\alpha}$ to a chiral superfield, track Grassmann parity, and explain why a top component varies by a spacetime derivative. Enter Superspace Measures. Unsure: review Supercovariant Derivatives, Chirality, and Integrability and Grassmann Functional Integrals for Free Fermions.

Off-shell closure. Ready: you can distinguish an auxiliary equation from a propagating equation of motion and can say whether a transformation closes exactly or only on shell. Enter Component Reduction. Unsure: repair this with Off-Shell Closure and Auxiliary Fields.

Differential geometry. Ready: you recognize a Kähler metric $g_{i\bar j}=\partial_i\partial_{\bar j}K$ , its Levi–Civita connection, and a Hamiltonian group action with moment map. Enter Kähler Sigma Models or Gauge–Matter Systems. Unsure: use Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields and Hamiltonian Group Actions and Moment Maps.

Gauge conventions. Ready: you can distinguish the Lie algebra from the global gauge group, state a trace normalization, and identify a gauge-invariant operator. Enter Supersymmetric Yang–Mills Actions. Unsure: use Gauge Fields, Redundancy, and Observable Content.

Quantum functionals. Ready: you can distinguish a Wilsonian action at a finite infrared cutoff from the 1PI Legendre transform and can name the regulator and subtraction scheme of a loop calculation. Enter Supergraphs and Quantum Effective Actions. Unsure: use the 1PI background page linked above.

Currents and improvements. Ready: you can explain how adding an identically conserved improvement changes a local current without changing its charge under suitable boundary conditions. Enter Supercurrent Multiplets. Unsure: use Spacetime Currents, Stress Tensors, and Charge Algebras.

Choose a route

Construct a first interacting model. Read superspace measures → component reduction → the Wess–Zumino model. This route ends with a complete off-shell action, its auxiliary elimination, its vacua, and its boson–fermion mass check.
Build a nonlinear chiral theory. Establish the measure and reduction rules, then read Kähler sigma models. Continue to gauge–matter systems only when a group action and moment map are part of the problem.
Build a supersymmetric gauge theory. Read supersymmetric Yang–Mills actions after securing vector superfields and component reduction, then assemble matter, $F$ terms, $D$ terms, and FI data on the gauge–matter page.
Organize loop corrections. Read supergraphs, $D$ -algebra, and quantum effective actions. The result is a method for reducing a supergraph and labeling its output as Wilsonian or 1PI; the nonrenormalization theorem itself belongs to the next chapter.
Choose a current package or rigid-background input. Read supercurrent multiplets, improvements, and obstructions. This route ends with an $S$ -, Ferrara–Zumino-, or $R$ -multiplet decision whose global assumptions remain visible.

These arrows give a useful reading order. They do not imply that every theory admits a Lagrangian superspace description or a globally defined preferred supercurrent multiplet.

From a measure to a quantum current record

The construction has four logically distinct layers.

Invariant selection. Full superspace and chiral superspace integrals select $D$ and $F$ components. Their variations are total derivatives only after Berezin orientation, conjugation, chirality, and boundary conditions have been fixed.
Classical theory data. A Kähler potential $K$ , holomorphic superpotential $W$ , gauge kinetic function $f_{ab}$ , gauge representation, and admissible FI parameters determine the two-derivative rigid action. Eliminating $F^i$ and $D^a$ produces a nonnegative scalar potential only when the Kähler and gauge kinetic matrices have the required positivity.
Quantum destination. Supergraphs keep supersymmetry manifest, but a local term in a Wilsonian action and a momentum-dependent term in the 1PI functional answer different questions. Infrared singularities can make the distinction decisive.
Current and background compatibility. The stress tensor and supersymmetry current live in a supermultiplet. Improvements are allowed only when their operator is local, gauge invariant, and globally defined; FI data and Kähler patching can obstruct a Ferrara–Zumino representative Komargodski and Seiberg 2010, arXiv:1002.2228.

The shared check is therefore

\text{measure and conventions} \longrightarrow \text{off-shell action} \longrightarrow \text{auxiliary equations} \longrightarrow \text{physical interactions} \longrightarrow \text{quantum/current destination}.

A successful calculation can be run backward through this chain. The final potential must reproduce the auxiliary equations; the component action must reassemble into the declared superspace integrals; and an improved current must generate the same conserved charges when the boundary term vanishes.

The on-shell handoff has three gates

The chapter does not recompute amplitudes, but it must say when its action and current data are sufficient inputs for an on-shell supersymmetry constraint. Three checks make that handoff precise.

Choose an asymptotic vacuum and normalize its particles. The vacuum must be Poincaré invariant, the quadratic action must have positive residues, and its mass and helicity eigenstates must be identified. The auxiliary fields are not external states. In a gauge theory, only the physical polarizations and the declared color representation are exported.
Establish an asymptotic supercharge. The chosen vacuum must preserve supersymmetry, the supercurrent must define a conserved charge with no boundary flux, and the quantum Ward identity must be anomaly free in the chosen regulator. Under those hypotheses, LSZ reduction gives $[Q,S]=0$ . A broken vacuum instead requires Goldstino soft identities, not the unbroken relation.
Import the canonical on-shell package. On-Shell Supermultiplets and Supersymmetric Ward Identities owns the external-state superwavefunctions and, for a massless $\mathcal N=1$ package, the operators
$\mathsf q_\alpha=\sum_i\lambda_{i\alpha}\eta_i, \qquad \bar{\mathsf q}_{\dot\alpha} =\sum_i\widetilde\lambda_{i\dot\alpha} \frac{\partial}{\partial\eta_i},$
together with $\mathsf q_\alpha\mathscr A_n=0$ and $\bar{\mathsf q}_{\dot\alpha}\mathscr A_n=0$ . The present chapter exports the normalized model, vacuum, current, anomaly, and infrared assumptions needed to decide whether those equations apply.

The destination in Scattering is therefore a compact input record: external supermultiplets and CPT completion; masses, little-group and internal representations; all-outgoing and coupling conventions; the preserved charge; and regulator, anomaly, and infrared status. On-Shell States and Little-Group Scaling and the later amplitude pages determine the kinematic coefficient functions, factorization, cuts, and observables. Supersymmetry constrains those functions; it does not calculate them by itself.

Exact chapter guide

Superspace Measures, F-Terms, D-Terms, and Component Extraction derives what the full and chiral measures select and how invariance becomes a boundary statement.
Supersymmetric Action Principles and Component Reduction gives a reproducible expansion, integration-by-parts, auxiliary-elimination, and closure check.
Wess–Zumino Models turns that workflow into a complete interacting chiral model and checks masses around each vacuum.
Kähler Sigma Models and Supersymmetric Target Geometry derives the metric, connection, curvature interaction, patching rule, and EFT ceiling of a nonlinear chiral theory.
Supersymmetric Yang–Mills Actions fixes the gauge, trace, theta-angle, gaugino, auxiliary, and compensating-transformation conventions.
Gauge–Matter Systems, F- and D-Term Potentials, and FI Data combines chiral and vector sectors and derives the complete scalar potential and vacuum equations.
Supergraphs, D-Algebra, and Quantum Effective Actions reduces a representative loop graph and diagnoses gauge fixing, locality, infrared behavior, and Wilsonian-versus-1PI scope.
Supercurrent Multiplets, Improvements, Anomalies, and Background Sources gives the current conservation equations and a decision table for globally admissible improvements and later background couplings.

A convention bridge for the chapter

The chapter inherits the site’s $(+---)$ metric, Hermitian gauge generators, $D_\mu=\partial_\mu-igA_\mu$ , and $T(F)=1/2$ for the fundamental of $SU(N)$ . For four-dimensional two-component spinors it uses $\sigma^\mu=(\mathbf 1,\boldsymbol\sigma)$ and $\bar\sigma^\mu=(\mathbf 1,-\boldsymbol\sigma)$ , so $\sigma^\mu\bar\sigma^\nu+\sigma^\nu\bar\sigma^\mu=2\eta^{\mu\nu}$ . Berezin orientation is declared on the first leaf.

The vector convention is one indivisible sign package:

\begin{gathered} V=-\theta\sigma^\mu\bar\theta A_\mu +i\theta^2\bar\theta\bar\lambda -i\bar\theta^2\theta\lambda +\frac12\theta^2\bar\theta^2D,\\ D_\mu=\partial_\mu-igA_\mu, \qquad \Phi^\dagger e^{-2gV}\Phi, \qquad S_{\mathrm{FI}}=-2\int d^4x\,d^4\theta\,\xi V. \end{gathered}

For that canonically normalized prepotential, the corresponding non-Abelian field strength is

\mathcal W_{c\alpha} =\frac1{8g}\bar D^2 \left(e^{2gV}D_\alpha e^{-2gV}\right),

whose Abelian linearization is $-\bar D^2D_\alpha V/4$ . Equivalently, setting $V_h=gV$ and $\mathcal W_h=g\mathcal W_c$ gives the coupling-free formula $\mathcal W_{h\alpha}=\bar D^2(e^{2V_h}D_\alpha e^{-2V_h})/8$ used in the holomorphic normalization. Reversing only one exponential or one auxiliary sign would change the covariant derivative, gauge Yukawa, or $D$ equation; the Yang–Mills and gauge–matter pages derive the complete translation.

Many standard supersymmetry references use the mostly-plus metric or absorb $g$ into the vector superfield. Never import a component sign or an instanton normalization by inspection. Translate the sigma-matrix identity and the covariant derivative first; the invariant checks are positivity of the Hamiltonian, equality of boson and fermion pole masses in an unbroken multiplet, and gauge invariance of the final action.

Review the chapter

Reconstruct the action. Starting from $K=\Phi^\dagger\Phi$ and a holomorphic $W(\Phi)$ , identify the selected Grassmann components, write the off-shell auxiliary terms, and eliminate $F$ . A successful reconstruction recovers $V=|W'|^2$ without using a propagating equation of motion.

Check a gauge–matter model. Given $G$ , the matter representation, $K$ , $W$ , $f_{ab}$ , and a proposed FI term, list the checks required before writing $V_F+V_D$ . The answer must include global form, gauge invariance of $W$ , positivity of $K_{i\bar j}$ and $\operatorname{Re}f$ , moment-map normalization, anomaly status, and FI admissibility.

Classify a loop result. Suppose a massless supergraph produces a nonanalytic function of external momentum. Explain why writing it as a local superpotential correction would be wrong. A complete answer names the 1PI functional, the infrared origin of the nonlocality, and the separate hypotheses behind a Wilsonian holomorphy statement.

Choose a supercurrent. For a sigma model with a non-exact Kähler form and no continuous $R$ symmetry, decide which current package is guaranteed and why the usual smaller multiplets can fail. The check is global: a formal local improvement is insufficient unless its operator patches and is gauge invariant.

If any answer depends on an unstated sign convention, trace normalization, auxiliary branch, regulator, or boundary condition, repair that record before continuing to Supersymmetric Vacua, Moduli Geometry, and BPS Sectors or Holomorphy, Anomalies, and Exact Quantum Constraints.

References

Komargodski, Zohar, and Nathan Seiberg. “Comments on Supercurrent Multiplets, Supersymmetric Field Theories and Supergravity.” Journal of High Energy Physics 2010, no. 7 (2010): 017. DOI. Open PDF, arXiv v4.