Gauge–Matter Systems, F- and D-Term Potentials, and FI Data

A rigid four-dimensional $\mathcal N=1$ gauge–matter theory is determined at two derivatives by a Kähler target with a holomorphic gauge-group action, a gauge-invariant superpotential, a holomorphic gauge kinetic matrix, and any globally admissible Fayet–Iliopoulos data. The chiral auxiliaries measure failure of F-flatness; the vector auxiliaries are shifted moment maps. Eliminating both gives

V=V_F+V_D =g^{i\bar j}W_i\bar W_{\bar j} +\frac12(\operatorname{Re}f)^{-1,ab} \mathcal P_a\mathcal P_b,

with $\mathcal P_a=g\mu_a+\xi_a$ in the convention below. Positivity requires positive Kähler and gauge kinetic matrices. The formula is not complete until representations, global gauge form, moment-map normalization, anomaly status, and FI admissibility are recorded.

Required background. Supersymmetric Yang–Mills Actions fixes the vector normalization, Kähler Sigma Models fixes $g_{i\bar j}$ , and Hamiltonian Group Actions and Moment Maps supplies the geometric moment map.

Helpful background. Global Form, Matter Representations, and the Faithful Gauge Group explains why the Lie algebra alone does not determine the model.

Gauge invariance is a condition on all defining data

Let $G$ act on chiral coordinates $z^i$ through holomorphic Killing vectors $k_a^i(z)$ ,

\delta_\alpha z^i=\alpha^a k_a^i(z).

For a linear unitary representation, $k_a^i=i(T_a)^i{}_jz^j$ . Define the real moment map by

\partial_{\bar j}\mu_a =-i g_{i\bar j}k_a^i,

with the conjugate relation. For canonical $K$ this gives $\mu_a=z^\dagger T_a z$ . An additive constant is possible only along an Abelian generator; that is the local origin of FI data.

Write the gauge-covariant Kähler completion as $K_{\mathrm{gauged}}(\Phi,\Phi^\dagger,V)$ . In the present moment-map convention it obeys

K_{\mathrm{gauged}}\big|_{V=0}=K, \qquad \left.\frac{\partial K_{\mathrm{gauged}}}{\partial V^a}\right|_{V=0} =-2g\mu_a.

For canonical fields in a linear unitary representation this is simply $K_{\mathrm{gauged}}=\Phi^\dagger e^{-2gV}\Phi$ . A general nonlinear Kähler target instead needs the equivariant $V$ -dependent completion determined by its holomorphic Killing vectors and moment maps; the linear matrix exponential is not a formula for an arbitrary target.

The complete superspace action is then schematically

\begin{aligned} S={}&\int d^4x\,d^4\theta\, K_{\mathrm{gauged}}(\Phi,\Phi^\dagger,V)\\ &+\int d^4x\left[ \int d^2\theta\left( W(\Phi)+\frac14f_{ab}(\Phi)\mathcal W^{a\alpha} \mathcal W^b_\alpha \right)+\text{h.c.}\right]\\ &-2\int d^4x\,d^4\theta\,\xi_aV^a, \end{aligned}

For the canonical linear model, the minus signs in $e^{-2gV}$ and in the FI integral are the translation from the imported prepotential $V=-\theta\sigma^\mu\bar\theta A_\mu+\cdots+\theta^2\bar\theta^2D/2$ to the site convention $D_\mu=\partial_\mu-igA_\mu$ . They fix the gaugino phase and the sign of the $D$ equation at the same time. The final term is present only for admissible Abelian factors. Gauge invariance requires:

the Kähler metric and symplectic form to be preserved by the action;
$W$ to obey $k_a^iW_i=0$ for an ordinary rigid gauge symmetry;
$f_{ab}$ to transform as an invariant symmetric tensor, including any chiral-field dependence;
the matter representation to descend to the declared global group; and
perturbative and global gauge anomalies to cancel.

A Kähler potential may transform by a holomorphic plus antiholomorphic term while the metric and rigid action remain invariant. That local freedom must not be confused with a globally defined FI constant or with an allowed supergravity coupling.

Auxiliary elimination produces the full scalar potential

Suppressing fermion-dependent auxiliary shifts, the component auxiliary sector is

\mathcal L_{\mathrm{aux}} =g_{i\bar j}F^i\bar F^{\bar j} +F^iW_i+\bar F^{\bar j}\bar W_{\bar j} +\frac12h_{ab}D^aD^b -D^a\mathcal P_a,

where $h_{ab}=\operatorname{Re}f_{ab}$ and $\mathcal P_a=g\mu_a+\xi_a$ . Varying independent barred and unbarred fields gives

F^i=-g^{i\bar j}\bar W_{\bar j}, \qquad D^a=h^{-1,ab}\mathcal P_b.

Completing both squares yields

\mathcal L_{\mathrm{aux,on}} =-g^{i\bar j}W_i\bar W_{\bar j} -\frac12h^{-1,ab}\mathcal P_a\mathcal P_b,

so the potential is the positive expression in the opening when $g$ and $h$ are positive. The classical supersymmetric-vacuum equations are

W_i=0, \qquad \mathcal P_a=0,

modulo gauge equivalence, provided the metric matrices are nonsingular. These equations define a candidate classical vacuum space; stability, singular strata, invariant coordinates, and quantum corrections are treated in the next chapter.

The complete reduction flow places this $D=h^{-1}\mathcal P$ branch beside the chiral $F$ branch and makes the sign and positivity checks visible in one transcript.

For canonical matter, the principal fermion interactions are

\mathcal L_{\mathrm{ferm,int}} =-\frac12W_{ij}\psi^i\psi^j -i\sqrt2g\,\phi^\dagger T^a\psi\,\lambda^a +\text{h.c.},

in addition to gauge-covariant kinetic terms. A nontrivial Kähler metric replaces $W_{ij}$ by $\nabla_iW_j$ and covariantizes the gauge Yukawa term; field-dependent $f_{ab}$ adds gaugino–matter couplings. These interactions are not optional decorations: their coefficients are fixed by the same superspace action as $V_F$ and $V_D$ . The canonical construction is worked out in Weinberg 2000, §§27.1 and 27.4, pp. 113–121 and 132–143.

An anomaly-free Abelian model checks every coefficient

Take $G=U(1)$ with chiral fields $\Phi_+$ and $\Phi_-$ of charges $+1$ and $-1$ , canonical $K$ , constant canonical gauge kinetic term, and

W=m\Phi_+\Phi_-.

The opposite charges cancel the cubic and mixed gravitational $U(1)$ anomalies in this two-field sector, and $W$ is gauge invariant. With FI parameter $\xi$ ,

\begin{aligned} F_+&=-m^*\phi_-^*, &F_-&=-m^*\phi_+^*,\\ D&=g(|\phi_+|^2-|\phi_-|^2)+\xi. \end{aligned}

Therefore

V=|m|^2(|\phi_+|^2+|\phi_-|^2) +\frac12\left[g(|\phi_+|^2-|\phi_-|^2)+\xi\right]^2.

The corresponding Yukawa terms are

\mathcal L_{\mathrm{Yukawa}} =-m\psi_+\psi_- -i\sqrt2g(\phi_+^*\psi_+-\phi_-^*\psi_-)\lambda +\text{h.c.}

Three checks are immediate.

Setting $g=\xi=0$ leaves two massive chiral multiplets and a decoupled free vector multiplet. If $g=0$ but $\xi\ne0$ , the decoupled vector–FI sector remains and contributes the vacuum energy $\xi^2/2$ .
Setting $m=0$ leaves the D-flat equation $g(|\phi_+|^2-|\phi_-|^2)+\xi=0$ modulo $U(1)$ .
For $m\neq0$ , F-flatness requires $\phi_+=\phi_-=0$ ; a nonzero $\xi$ then leaves $D=\xi$ , so no supersymmetric classical vacuum exists in this model.

For $m=0$ and $g>0$ , the sign of $\xi$ selects which charged field must be nonzero on the D-flat locus:

\begin{array}{c|c|c} \text{FI regime} & \text{D-flat equation} & \text{gauge realization}\\ \hline \xi>0 &|\phi_-|^2=|\phi_+|^2+\xi/g &\phi_-\ne0\text{ everywhere; }U(1)\text{ Higgsed}\\ \xi<0 &|\phi_+|^2=|\phi_-|^2-\xi/g &\phi_+\ne0\text{ everywhere; }U(1)\text{ Higgsed}\\ \xi=0 &|\phi_+|=|\phi_-| &\text{Higgsed away from the unbroken origin} \end{array}

The invariant $M=\phi_+\phi_-$ is a complex coordinate on the regular quotient. This sign-level analysis identifies the classical Higgs regimes; the metric near the unbroken stratum, invariant-coordinate construction, and the full stratified quotient belong to the next chapter.

The third conclusion is model-specific. It does not say that every FI theory breaks supersymmetry; additional charged fields or another superpotential can solve all auxiliary equations.

FI terms carry global and current-theoretic conditions

A constant term $-\xi_aD^a$ linear in the auxiliary is gauge invariant only when $a$ labels an Abelian direction admitting an invariant linear functional. A semisimple non-Abelian algebra has no such invariant vector. Even for $U(1)$ , one must record:

the compact or noncompact global group and charge normalization;
whether the FI parameter is compatible with large gauge transformations, boundaries, and the allowed bundles;
cancellation of gauge and mixed anomalies;
whether the $U(1)$ is an ordinary symmetry or an $R$ symmetry; and
which supercurrent multiplet and background coupling remain globally available.

In this convention, the rigid local term $-2\int d^4\theta\,\xi V$ is supersymmetric and gauge invariant up to the usual chiral/full-superspace identity. But the Ferrara–Zumino multiplet obtained by a standard improvement is not gauge invariant in the presence of a genuine FI term. This obstruction and its consequences are established in Komargodski and Seiberg 2010, §§1, 3.1, pp. 2–4 and 11–13, Open PDF. It is therefore wrong to infer background or supergravity compatibility from the local scalar potential alone.

A gauge–matter checklist

Before accepting a potential or vacuum equation, verify:

field	required record
gauge data	global group, Lie algebra basis, $T(R)$ , charges, coupling placement, theta sectors
chiral data	target patch, positive $g_{i\bar j}$ , representation and faithful quotient
holomorphic data	gauge invariance and dimensions of $W$ ; covariance and positivity of $f_{ab}$
moment map	sign, additive constants, coupling factor, and equivariance
FI data	Abelian direction, global admissibility, anomaly and supercurrent implications
elimination	independent $F,\bar F,D$ , algebraic Hessian, branch and positivity
quantum scope	regulator, scheme, Wilsonian/1PI destination and possible anomalies

The independent round trip is to recover the same $F$ and $D$ equations from the component action, from the superfield equations, and from the derivative of the final potential with respect to the auxiliary completion.

Common pitfalls

Writing $V_D$ from charges but omitting g. Whether $g$ appears in the moment map or in the kinetic matrix depends on normalization. Translate the whole vector multiplet, not just $D$ .

Adding an FI constant to a non-Abelian generator. A constant shift must be invariant under the adjoint action; semisimple factors do not permit it.

Checking $W$ only infinitesimally. The representation and operator must also be well-defined under the global gauge group, including quotient identifications and large transformations.

Exercises

1. Positivity by completing the square. Complete the $D$ square for a positive matrix $h_{ab}$ .

Solution

\frac12D^{\mathsf T}hD-D^{\mathsf T}\mathcal P =\frac12(D-h^{-1}\mathcal P)^{\mathsf T}h (D-h^{-1}\mathcal P) -\frac12\mathcal P^{\mathsf T}h^{-1}\mathcal P.

Thus $D=h^{-1}\mathcal P$ and $V_D=\mathcal P^{\mathsf T}h^{-1}\mathcal P/2\geq0$ .

2. Gauge invariance of the mass term. Verify the $U(1)$ charge of $\Phi_+\Phi_-$ and explain why $\Phi_+^2$ is forbidden.

Solution

$q(\Phi_+\Phi_-)=+1-1=0$ , so the mass term is invariant. $q(\Phi_+^2)=+2$ , so it is not an operator of the compact $U(1)$ gauge theory without an additional field or spurion carrying charge $-2$ .

Where the equations lead

F- and D-Flatness and Gauge Quotients constructs the quotient and its singular strata. Supercurrent Multiplets, Improvements, Anomalies, and Background Sources records the FI obstruction, and SQCD: Fields, Symmetries, Invariants, and Classical Moduli Space applies the framework to non-Abelian matter.

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.
Weinberg, Steven. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge: Cambridge University Press, 2000, §§27.1 and 27.4, pp. 113–121 and 132–143. DOI.