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

$$\int d^4\theta\, \left(Q^\dagger e^{2V}Q +\widetilde Qe^{-2V}\widetilde Q^\dagger\right) +\frac{\tau}{8\pi i}\int d^2\theta\, \operatorname{Tr}_{\mathbf F}W_h^\alpha W_{h\alpha} +\text{h.c.},$$

where $W_h$ is the holomorphically normalized field-strength superfield and

$$\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g_h^2}.$$

Here $W_{\rm tree}=0$. 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 the generators and baryon number by

$$\operatorname{Tr}_{\mathbf F}(T^aT^b)=\frac12\delta^{ab}, \qquad T(\mathbf{N_c})=\frac12,$$

and

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

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

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

The perturbative definition still allows a finite multiplicative redefinition of $\Lambda$. We remove that freedom operationally: at $N_f=N_c-1$, the constrained one-instanton calculation is normalized to give

$$W=\frac{\Lambda^{2N_c+1}}{\det M}.$$

Every exact coefficient in this chapter uses this unit-instanton convention together with the composite definitions below. A different subtraction scheme must rescale all appearances of $\Lambda$ coherently. The anchor and its scheme-normalized coefficient are reviewed in Intriligator and Seiberg 1996, § 4.1, arXiv PDF pp. 13–14.

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

For $N_f>0$ and generic $N_c\geq3$, the continuous symmetry algebra is

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

A trivial $\mathfrak{su}(1)$ factor is omitted when $N_f=1$. The axial $U(1)_A$ is anomalous. Requiring the mixed $SU(N_c)^2U(1)_R$ anomaly to vanish fixes only the sum of the quark R-charges. We choose the baryon-unmixed, left–right-symmetric representative:

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

The general anomaly-free basis is $R_t=R_0+tB$, so $R_t(Q)=1-N_c/N_f+t$ and $R_t(\widetilde Q)=1-N_c/N_f-t$. The table and anomaly coefficients below use $t=0$, so the quark Weyl fermions have $R=-N_c/N_f$ and the gaugino has $R=1$. Keeping scalar and fermion charges in separate columns prevents a common one-unit error in anomaly calculations:

Multiplet	$SU(N_c)$	$SU(N_f)_L$	$SU(N_f)_R$	$B$	Scalar $R$	Left-handed fermion $R$
$Q$	$\mathbf{N_c}$	$\mathbf{N_f}$	$\mathbf1$	$+1$	$1-N_c/N_f$	$-N_c/N_f$
$\widetilde Q$	$\overline{\mathbf{N_c}}$	$\mathbf1$	$\overline{\mathbf{N_f}}$	$-1$	$1-N_c/N_f$	$-N_c/N_f$
Vector multiplet $(\lambda)$	$\mathrm{adj}$	$\mathbf1$	$\mathbf1$	$0$	—	$1$

The group is a finite quotient, not a direct product. For $N_c\geq3$ and $N_f>0$, the faithful connected non-R 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 $SU(1)$ and $\mathbb Z_1$ are simply omitted. An operational description of the kernel 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. Including $U(1)_R$ requires further finite identifications, including its overlap with fermion parity and hence with the spacetime spin group. The convention-independent prescription is to divide the combined spin–internal group by every finite element whose action on all fields agrees with a gauge transformation; the displayed bosonic quotient alone should not be read as that full fermionic symmetry group.

For $N_c=2$ and $N_f>0$, 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. For $N_f=0$, there is no continuous flavor or baryon symmetry and no anomaly-free continuous $U(1)_R$; the classical gaugino rotation is reduced by the anomaly to $\mathbb Z_{2N_c}$.

Ultraviolet anomaly data

For $N_c\geq3$ and $N_f>0$, the following coefficients use the left–right presentation above, a left-handed Weyl basis, cubic index $A(\mathbf{N_f})=+1$, and $T(\mathbf{N_f})=1/2$. Rows containing $SU(N_f)$ are absent when $N_f=1$; the perturbative cubic rows exist only for $N_f\geq3$.

Anomaly	Coefficient
$SU(N_f)_L^3$ for $N_f\geq3$	$N_c$
$SU(N_f)_R^3$ for $N_f\geq3$	$-N_c$
$SU(N_f)_L^2U(1)_B$	$N_c/2$
$SU(N_f)_R^2U(1)_B$	$-N_c/2$
$SU(N_f)_L^2U(1)_R$	$-N_c^2/(2N_f)$
$SU(N_f)_R^2U(1)_R$	$-N_c^2/(2N_f)$
$\operatorname{Tr}B$ 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,

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

For $N_f=2$, the perturbative $SU(2)^3$ invariant vanishes, but there is a mod-2 Witten anomaly: each of $SU(2)_L$ and $SU(2)_R$ sees $N_c$ Weyl doublets, so its anomaly class is $N_c\bmod 2$. This is a global-symmetry matching datum, not a gauge inconsistency; the odd-doublet criterion is derived in Witten 1982, pp. 324–328. The $N_c=2$ theory should instead be analyzed using its enhanced $SU(2N_f)$ symmetry.

The table records 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, arXiv PDF 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 $N_f\geq N_c$,

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

and

$$\widetilde B^{[\tilde\imath_1\cdots\tilde\imath_{N_c}]} =\epsilon^{a_1\cdots a_{N_c}} \widetilde Q_{a_1}{}^{\tilde\imath_1}\cdots \widetilde Q_{a_{N_c}}{}^{\tilde\imath_{N_c}}.$$

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 $N_f>N_c$, choose ordered $N_c$-element flavor subsets $I$ and $\widetilde J$. With the displayed epsilon-tensor normalizations, the exact minor relation is

$$B_I\widetilde B^{\widetilde J} =\det M_I{}^{\widetilde J}.$$

Compatibility with the meson is expressed by

$$B_{[i_1\cdots i_{N_c}}M_{i_{N_c+1}]}{}^{\tilde\jmath}=0, \qquad M_i{}^{[\tilde\jmath_1} \widetilde B^{\tilde\jmath_2\cdots\tilde\jmath_{N_c+1}]}=0.$$

Each baryon is also a decomposable $N_c$-form and therefore obeys its Plücker relations, for example

$$B_{i_1\cdots i_{N_c-1}[j_1} B_{j_2\cdots j_{N_c+1}]}=0,$$

with the corresponding equation for $\widetilde B$. These relations, together with $\operatorname{rank}M\leq N_c$, make clear why a list of unconstrained invariant symbols is 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 $SL(N_c-N_f,\mathbb C)$ as the complexified stabilizer. The quotient dimension is therefore

$$2N_cN_f- \left[(N_c^2-1)-\bigl((N_c-N_f)^2-1\bigr)\right] =N_f^2.$$

For $N_f\geq N_c-1$, a generic D-flat pair has trivial stabilizer, so the full $SL(N_c,\mathbb C)$ dimension is subtracted. 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 trivial (sometimes denoted $SU(1)$), 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, arXiv PDF pp. 3–7. Quantum dynamics will preserve the chiral-coordinate logic while changing the superpotential or the defining relation. For the dynamical classification built on this theory card, consult the SQCD phase figure and the accompanying regime, exact-result, and evidence-limit table.

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. For $N_f\geq2$, 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.
• Witten, Edward. “An $SU(2)$ Anomaly.” Physics Letters B 117 (1982): 324–328. doi:10.1016/0370-2693(82)90728-6.