SQCD Fields, Symmetries, Global Data, and Classical Moduli

Four-dimensional $\mathcal N=1$ SQCD is not specified by the pair $(N_c,N_f)$ alone. The canonical theory used in this chapter has gauge group $SU(N_c)$, $N_f$ massless pairs $Q, \widetilde Q$, vanishing tree superpotential, a definite baryon normalization, and a holomorphic scale. Those data determine its faithful symmetries, anomalies, invariant coordinates, and classical branches before any strong-coupling claim is made.

Required background. Gauge-invariant coordinates and classical moduli varieties supplies the complexified quotient used below. ’t Hooft anomaly matching supplies the ultraviolet anomaly coefficients that later infrared proposals must reproduce.

Helpful background. Global form and faithful gauge groups explains the finite quotients and line-operator statements in the theory card.

The SQCD theory card

Work in four-dimensional Lorentzian spacetime with the site-wide $(+---)$ convention. Let $N_c\geq2$ and $N_f\geq0$. The microscopic chiral multiplets are

$$Q^a{}_i\in(\mathbf{N_c},\mathbf{N_f},\mathbf1), \qquad \widetilde Q_a{}^{\tilde\imath} \in(\overline{\mathbf{N_c}},\mathbf1,\overline{\mathbf{N_f}}),$$

where $a=1,\ldots,N_c$ is color and $i,\tilde\imath=1,\ldots,N_f$ are left and right flavor indices. The gauge algebra is $\mathfrak{su}(N_c)$ and the gauge group is $SU(N_c)$. Fundamental matter does not define a representation of $SU(N_c)/\mathbb Z_k$ for nontrivial $k$, so replacing the group by a quotient changes or invalidates the theory.

The defining superspace terms are

$$\int d^4\theta\, \left(Q^\dagger e^{2V}Q +\widetilde Qe^{-2V}\widetilde Q^\dagger\right) +\frac{1}{4}\int d^2\theta\,\tau\,\operatorname{Tr}W^\alpha W_\alpha +\text{h.c.},$$

with $W_{\rm tree}=0$. There is no Fayet–Iliopoulos parameter for the simple non-Abelian group. A mass deformation will mean $W_{\rm tree}=m^i{}_{\tilde\jmath}M_i{}^{\tilde\jmath}$, with no hidden factor of $1/2$.

We normalize $T(\mathbf{N_c})=1/2$ and baryon number by

$$B(Q)=+1, \qquad B(\widetilde Q)=-1.$$

Thus a baryon made from $N_c$ quarks has charge $N_c$. This choice matters: anomaly coefficients involving $U(1)_B$ rescale if the baryon itself is instead assigned unit charge.

For the Wilsonian holomorphic coupling, define

$$\Lambda^{b_0}=\mu^{b_0} \exp\!\left[-\frac{8\pi^2}{g_h^2(\mu)}+i\theta\right], \qquad b_0=3N_c-N_f.$$

An overall multiplicative redefinition of $\Lambda$ is a scheme convention; every exact coefficient below assumes this convention and the composite definitions given here.

Because the dynamical quarks carry unit center charge, the electric $\mathbb Z_{N_c}$ one-form symmetry is explicitly broken and a fundamental Wilson line can end on a quark. This fact will later prevent an asymptotic fundamental area law from serving as an order parameter for confinement.

Faithful global symmetry and R-charges

At the Lie-algebra level the continuous symmetry is

$$\mathfrak{su}(N_f)_L\oplus\mathfrak{su}(N_f)_R \oplus\mathfrak u(1)_B\oplus\mathfrak u(1)_R.$$

The axial $U(1)_A$ is anomalous. Requiring the mixed $SU(N_c)^2U(1)_R$ anomaly to vanish fixes the scalar R-charges to

$$R(Q)=R(\widetilde Q)=1-\frac{N_c}{N_f} \qquad (N_f>0),$$

so the quark Weyl fermions have $R=-N_c/N_f$ and the gaugino has $R=1$.

The group is a finite quotient, not a direct product. Ignoring the additional finite identifications involving rationally normalized $U(1)_R$, the faithful non-R connected group is

$$\frac{SU(N_f)_L\times SU(N_f)_R\times U(1)_B} {\mathbb Z_{N_f}\times\mathbb Z_{N_c}}.$$

Here is an operational definition that removes convention ambiguity. If $z\in\mathbb Z_{N_f}$ and $\omega\in\mathbb Z_{N_c}$, the tuple

$$(z\mathbf1,z\mathbf1,e^{i\beta}=\omega z^{-1})$$

acts on $Q,\widetilde Q$ exactly as the gauge-center element $(\omega,\omega^{-1})$ and is therefore divided out. The fully faithful group including $U(1)_R$ is obtained by dividing by the entire finite kernel whose action on both elementary fields equals a gauge transformation. Stating the kernel this way remains correct under any integral rescaling of $B$ or $R$.

For $N_c=2$, the fundamental is pseudoreal and the flavor symmetry enhances to $SU(2N_f)$; mesons and baryons reorganize into its antisymmetric tensor. Formulas written with separate left and right flavor groups must therefore be translated before using this special rank.

Ultraviolet anomaly data

The following coefficients use a left-handed Weyl basis, cubic index $A(\mathbf{N_f})=+1$, and $T(\mathbf{N_f})=1/2$.

Anomaly	Coefficient
$SU(N_f)_L^3$	$N_c$
$SU(N_f)_R^3$	$-N_c$
$SU(N_f)_L^2U(1)_B$	$N_c/2$
$SU(N_f)_R^2U(1)_B$	$-N_c/2$
$SU(N_f)_L^2U(1)_R$	$-N_c^2/(2N_f)$
$SU(N_f)_R^2U(1)_R$	$-N_c^2/(2N_f)$
$\operatorname{Tr}B$ and $\operatorname{Tr}B^3$	$0$
$\operatorname{Tr}RB^2$	$-2N_c^2$
$\operatorname{Tr}R$	$-N_c^2-1$
$\operatorname{Tr}R^3$	$N_c^2-1-2N_c^4/N_f^2$

As a quick internal check,

$$\mathcal A_{SU(N_c)^2R} =N_c+N_f\left[\frac12\!\left(-\frac{N_c}{N_f}\right) +\frac12\!\left(-\frac{N_c}{N_f}\right)\right]=0.$$

These are anomalies of the presentation group. A background-field calculation for the faithful quotient must also impose compatible bundles and may expose torsion data invisible in the polynomial coefficients. The continuous coefficients nevertheless provide the standard matching tests used in exact SQCD. The field and charge assignment agrees with Intriligator and Seiberg 1996, §§ 3–4, pp. 9–19.

D-flat quotient and invariant coordinates

With $W=0$, F-flatness is automatic. D-flatness is

$$QQ^\dagger-\widetilde Q^\dagger\widetilde Q -\frac{\mathbf1_{N_c}}{N_c} \operatorname{Tr}\!\left(QQ^\dagger-\widetilde Q^\dagger\widetilde Q\right)=0.$$

The classical moduli variety is equivalently the affine quotient by $SL(N_c,\mathbb C)$. Its basic holomorphic invariants are

$$M_i{}^{\tilde\jmath} =\widetilde Q_a{}^{\tilde\jmath}Q^a{}_i,$$

and, when $N_f\geq N_c$,

$$B_{[i_1\cdots i_{N_c}]} =\epsilon_{a_1\cdots a_{N_c}} Q^{a_1}{}_{i_1}\cdots Q^{a_{N_c}}{}_{i_{N_c}},$$

with an analogous $\widetilde B$. The meson has engineering dimension $2$, baryon number zero, and R-charge $2(1-N_c/N_f)$. The baryon has dimension $N_c$, baryon number $N_c$, and R-charge $N_c(1-N_c/N_f)$.

These coordinates are constrained. Always $\operatorname{rank}M\leq N_c$. At $N_f=N_c$ the sole classical relation is

$$\det M-B\widetilde B=0.$$

For $N_f>N_c$, the baryons obey Plücker relations and compatibility relations with $M$; for example, contracting $M$ into a flavor index already occupied by $B$ vanishes. A list of unconstrained symbols is therefore not a moduli-space description.

Branch dimensions and stabilizers

The generic stabilizer changes with rank. For $N_f<N_c$, generic quark expectation values leave $SU(N_c-N_f)$ unbroken. For $N_f\geq N_c-1$, a generic D-flat pair breaks the gauge group completely. Hence

$$\dim_{\mathbb C}\mathcal M_{\rm cl}= \begin{cases} N_f^2, & N_f\leq N_c-1,\\[2mm] 2N_cN_f-(N_c^2-1), & N_f\geq N_c-1. \end{cases}$$

The two expressions agree at $N_f=N_c-1$. The first equals the number of independent meson entries; the second subtracts the full complexified gauge-orbit dimension. At lower-rank loci the stabilizer grows, additional gauge multiplets become light, and the invariant-coordinate description can become singular. Those loci must be analyzed stratum by stratum rather than by extrapolating a generic Higgs description.

Worked check: $SU(3)$ with two flavors. There are $12$ elementary complex scalars and the generic $SL(3,\mathbb C)$ orbit has dimension $8$, leaving four moduli. No baryon exists because $N_f<N_c$, and the four entries of the $2\times2$ meson give exactly those coordinates. The generic stabilizer is $SU(1)$, so the gauge group is completely broken even though only two flavors are present.

The quotient construction and its SQCD coordinate relations are developed in Seiberg 1994, §§ 2–3, pp. 2–7. Quantum dynamics will preserve the chiral-coordinate logic while changing the superpotential or the defining relation.

Checks before making an infrared claim

• Global form: can every matter representation and every claimed line be defined for the chosen gauge group?
• Faithful symmetry: have center identifications and the baryon normalization been fixed before anomaly matching?
• Branch: is the calculation at a generic Higgs point, a singular stratum, or the origin?
• Scale: is $\Lambda$ holomorphic or canonical, and which scheme fixes an exact coefficient?
• Small rank: does $N_c=2$ enhance flavor symmetry, or does the proposed rank interval become empty?
• Tree superpotential: are masses, baryon sources, or other deformations genuinely absent?

Ignoring any one of these questions can turn a correct formula for one SQCD theory into a false statement about another.

Exercises

1. Verify the $SU(N_f)_L^2U(1)_R$ anomaly coefficient in the table.

Solution

Only the $N_c$ color copies of the $Q$ fermion transform under $SU(N_f)_L$. Each has $T(\mathbf{N_f})=1/2$ and fermionic R-charge $R(Q)-1=-N_c/N_f$. Therefore

$$\mathcal A=N_c\cdot\frac12\cdot\left(-\frac{N_c}{N_f}\right) =-\frac{N_c^2}{2N_f}.$$

2. Show that the classical $N_f=N_c$ relation has consistent engineering dimension, baryon number, and R-charge.

Solution

$\det M$ has dimension $2N_c$, baryon number zero, and R-charge $2N_c(1-N_c/N_f)=0$ at $N_f=N_c$. The product $B\widetilde B$ has dimension $N_c+N_c=2N_c$, baryon number $N_c-N_c=0$, and the same vanishing R-charge. Thus subtraction is allowed. The quantum deformation on the later page can consistently replace the right-hand side by $\Lambda^{2N_c}$.

References

• Intriligator, Kenneth, and Nathan Seiberg. “Lectures on Supersymmetric Gauge Theories and Electric-Magnetic Duality.” Nuclear Physics B - Proceedings Supplements 45BC (1996): 1–28. doi:10.1016/0920-5632(95)00626-5. Open PDF.
• Seiberg, Nathan. “Exact Results on the Space of Vacua of Four-Dimensional SUSY Gauge Theories.” Physical Review D 49 (1994): 6857–6863. doi:10.1103/PhysRevD.49.6857. Open PDF.