Operator Dictionaries, Chiral Rings, and Anomaly Matching

The Seiberg-duality dictionary is constrained simultaneously by flavor representations, baryon number, R-charge, chiral-ring relations, moduli-space strata, and ’t Hooft anomalies. This page treats the generic left–right flavor card with $N_c\ge3$ and $N_f\ge N_c+2$, then states the $N_f=N_c,N_c+1$ endpoint relations separately. The protected agreements are necessary and highly nontrivial, but they do not determine the unprotected spectrum or prove the duality.

Required background. Seiberg duality in SQCD fixes both theory cards, and ’t Hooft anomaly matching fixes anomaly conventions. Helpful background. Quantum chiral rings and Konishi anomalies explains operator relations beyond the classical ring.

The shared theory-card and deformation figure gives the visual overview. Its semantic SQCD dictionary preserves the field and check data in a linear form.

The normalized protected operator map

Write $\widetilde N_c=N_f-N_c$ and use epsilon contractions without additional factorials. The electric meson and baryons map to

$$\begin{aligned} Q^i\widetilde Q_j&\longleftrightarrow M^i{}_j,\\ B^{i_1\cdots i_{N_c}} &=C\,\epsilon^{i_1\cdots i_{N_c}j_1\cdots j_{\widetilde N_c}} b_{j_1\cdots j_{\widetilde N_c}},\\ \widetilde B_{i_1\cdots i_{N_c}} &=C\,\epsilon_{i_1\cdots i_{N_c}j_1\cdots j_{\widetilde N_c}} \widetilde b^{j_1\cdots j_{\widetilde N_c}}. \end{aligned}$$

In the convention $M=Q\widetilde Q$ and $W=Mq\widetilde q/\mu$, the same holomorphic convention that gives

$$\Lambda^{3N_c-N_f}\widetilde\Lambda^{3\widetilde N_c-N_f} =(-1)^{\widetilde N_c}\mu^{N_f}$$

fixes

$$C^2=-(-\mu)^{N_c-N_f}\Lambda^{3N_c-N_f}.$$

A square-root branch and the orientations of the flavor epsilon tensors are part of the convention. Changing those choices changes the phase of $C$, not the invariant operator correspondence. This normalization and its consistency under dualizing twice are derived in Intriligator and Seiberg 1996, §5.3, pp. 21–23, eqs. (5.6)–(5.10), Open PDF.

Three immediate checks are

$$\begin{aligned} \dim C &=\frac12\bigl[(N_c-N_f)+(3N_c-N_f)\bigr] =2N_c-N_f,\\ B(b)&=\widetilde N_c\frac{N_c}{\widetilde N_c}=N_c=B(B),\\ R(b)&=\widetilde N_c\frac{N_c}{N_f} =N_c\left(1-\frac{N_c}{N_f}\right)=R(B). \end{aligned}$$

Thus $C b$ has the engineering dimension, baryon number, and R-charge of the electric baryon. These checks do not fix a unit-normalized physical correlator, which also depends on Kähler normalization.

Classical relations and moduli strata

Classically, the electric invariants obey

$$\operatorname{rank}M\le N_c,$$

together with baryon–meson and baryon–antibaryon relations. In a common antisymmetrization convention, representative relations are

$$\begin{aligned} B^{[i_1\cdots i_{N_c}}M^{i_{N_c+1}]}{}_j&=0,\\ M^i{}_{[j_1}\widetilde B_{j_2\cdots j_{N_c+1}]}&=0,\\ B^{i_1\cdots i_{N_c}}\widetilde B_{j_1\cdots j_{N_c}} &=M^{[i_1}{}_{[j_1}\cdots M^{i_{N_c}]}{}_{j_{N_c}]}. \end{aligned}$$

The baryons separately satisfy their Plücker relations. A rank-$r$ stratum of the meson determinantal variety has complex dimension $r(2N_f-r)$, but this is not by itself the dimension of every baryonic stratum: nonzero baryons record that $Q$ or $\widetilde Q$ has full color rank.

At a generic fully Higgsed electric point, quotienting the $2N_cN_f$ chiral coordinates by $SL(N_c,\mathbb C)$ gives

$$\dim_{\mathbb C}\mathcal M_{\mathrm{generic}} =2N_cN_f-(N_c^2-1).$$

On lower strata the stabilizer grows, so its dimension must be restored in the quotient count. Equality of the generic dimensions would not establish equality of the singular stratification.

How magnetic quantum dynamics restores the rank bound

The classical magnetic F-terms are

$$q_i\widetilde q^j=0, \qquad q_iM^i{}_j=0, \qquad M^i{}_j\widetilde q^j=0.$$

They constrain the kernels of $M$, but they do not impose $\operatorname{rank}M\le N_c$: setting $q=\widetilde q=0$ would classically allow any $M$. Let $r=\operatorname{rank}M$. The $r$ nonzero eigenvalues make $r$ magnetic flavors massive, leaving

$$F_{\mathrm{mag}}=N_f-r$$

flavors of $SU(\widetilde N_c)$. Quantum dynamics now distinguishes three cases.

• If $r<N_c$, then $F_{\mathrm{mag}}>\widetilde N_c$. Supersymmetric magnetic Higgs branches remain and map to the allowed electric strata.
• If $r=N_c$, then $F_{\mathrm{mag}}=\widetilde N_c$. With $N=q\widetilde q$, the low-energy magnetic theory has the quantum constraint $$\det N-b\widetilde b=\widetilde\Lambda_L^{2\widetilde N_c}.$$ The $M$ equation sets $N=0$; the scale and baryon maps then give the electric maximal-minor relation $$B^I\widetilde B_J=\det M^I{}_J.$$ Here $I$ and $J$ are $N_c$-element left- and right-flavor subsets, and $\det M^I{}_J$ is the corresponding $N_c\times N_c$ minor. The chosen epsilon orientation and baryon normalization carry the convention-dependent sign through $C$.
• If $r>N_c$, then $F_{\mathrm{mag}}<\widetilde N_c$. The resulting ADS or pure-glue superpotential has no stationary point compatible with $N=0$, so this putative classical magnetic point is not a supersymmetric vacuum.

This is the mechanism—not the classical cubic F-terms—by which the magnetic description recovers the electric rank bound Intriligator and Seiberg 1996, §5.5, pp. 26–27, eqs. (5.19)–(5.20), Open PDF.

Special flavor numbers and color rank

For $N_f=N_c$, the exact moduli space is quantum modified:

$$\det M-B\widetilde B=\Lambda^{2N_c}.$$

For $N_f=N_c+1$, the infrared variables are the confined mesons and baryons with

$$W_{N_c+1} =\frac{B_iM^i{}_j\widetilde B^j-\det M} {\Lambda^{2N_c-1}}.$$

The second formula follows from the completely Higgsed magnetic $SU(2)$ flow, including its broken-group instanton Intriligator and Seiberg 1996, §5.5, pp. 24–25, eqs. (5.15)–(5.17), Open PDF. Neither endpoint is obtained by pretending that $SU(1)$ is an ordinary magnetic gauge group.

For $N_c=2$, the $2N_f$ electric doublets are pseudoreal and the flavor symmetry enhances to $SU(2N_f)$. The left–right table below correctly tests a subgroup but is not the complete enhanced-symmetry anomaly card; claims about the full $N_c=2$ theory require that separate card.

Complete ordinary anomaly table

All traces are over left-handed Weyl fermions, and $R=R_0$ denotes the charge-conjugation-symmetric, baryon-unmixed current on the preceding theory card. Normalize $T(\mathbf{N_f})=1/2$ and the cubic coefficient of an $SU(N_f)$ fundamental to $+1$. A chiral multiplet with scalar charge $R_\Phi$ contributes a fermion with $R_\psi=R_\Phi-1$; the adjoint gaugino has R-charge one.

Anomaly	Electric theory	Magnetic theory
$SU(N_f)_L^3$	$N_c$	$-\widetilde N_c+N_f=N_c$
$SU(N_f)_R^3$	$-N_c$	$\widetilde N_c-N_f=-N_c$
$SU(N_f)_L^2U(1)_B$	$N_c/2$	$(\widetilde N_c/2)(N_c/\widetilde N_c)=N_c/2$
$SU(N_f)_R^2U(1)_B$	$-N_c/2$	$-(\widetilde N_c/2)(N_c/\widetilde N_c)=-N_c/2$
$SU(N_f)_L^2U(1)_R$	$-N_c^2/(2N_f)$	$-\widetilde N_c^2/(2N_f)+(N_f-2N_c)/2$
$SU(N_f)_R^2U(1)_R$	$-N_c^2/(2N_f)$	$-\widetilde N_c^2/(2N_f)+(N_f-2N_c)/2$
$\operatorname{Tr}B^2R$	$-2N_c^2$	$-2N_c^2$
$\operatorname{Tr}BR^2$	$0$	$0$
$\operatorname{Tr}B^3$	$0$	$0$
$\operatorname{Tr}B$	$0$	$0$
$\operatorname{Tr}R$	$-N_c^2-1$	$-N_c^2-1$
$\operatorname{Tr}R^3$	$N_c^2-1-2N_c^4/N_f^2$	$N_c^2-1-2N_c^4/N_f^2$

For example, the magnetic linear and cubic R traces are

$$\begin{aligned} \operatorname{Tr}R={}&(\widetilde N_c^2-1) +2N_f\widetilde N_c\left(-\frac{\widetilde N_c}{N_f}\right) +N_f^2\left(1-\frac{2N_c}{N_f}\right),\\ \operatorname{Tr}R^3={}&(\widetilde N_c^2-1) +2N_f\widetilde N_c\left(-\frac{\widetilde N_c}{N_f}\right)^3 +N_f^2\left(1-\frac{2N_c}{N_f}\right)^3. \end{aligned}$$

Substituting $\widetilde N_c=N_f-N_c$ gives the electric entries. The $N_f^2$ meson fermions are indispensable; without them even $SU(N_f)_L^3$ fails. These are the anomaly-matching conditions of ’t Hooft 1980, §§III.10–III.12, pp. 149–151, applied to the Seiberg pair as in Seiberg 1995, §3, pp. 6–8, Open PDF.

The displayed abelian currents are exact gauge-theory symmetries because

$$\begin{aligned} \mathcal A[SU(N_c)^2U(1)_B]&=0,& \mathcal A[SU(N_c)^2U(1)_R]&=N_c+N_f\left(-\frac{N_c}{N_f}\right)=0,\\ \mathcal A[SU(\widetilde N_c)^2U(1)_B]&=0,& \mathcal A[SU(\widetilde N_c)^2U(1)_R]&=\widetilde N_c+N_f\left(-\frac{\widetilde N_c}{N_f}\right)=0. \end{aligned}$$

When the displayed R-current is the exact superconformal current, $\operatorname{Tr}R$ and $\operatorname{Tr}R^3$ determine $a$ and $c$. Anomaly equality alone does not establish that fixed-point identification.

Faithful quotient, discrete data, and one-form symmetry

For the generic electric card, the connected faithful non-R symmetry acting on gauge-invariant operators is

$$G_{\mathrm{faithful}}^{(0)} =\frac{SU(N_f)_L\times SU(N_f)_R\times U(1)_B} {\mathbb Z_{N_f}\times\mathbb Z_{N_c}}.$$

To see the kernel directly, let $z\in\mathbb Z_{N_f}$ and $\omega\in\mathbb Z_{N_c}$. The global element

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

acts on $Q$ and $\widetilde Q$ exactly as the gauge-center element $\omega$ and its inverse. Quotienting by all such pairs gives the denominator above. The rational magnetic-quark baryon charges define the same action on gauge-invariant operators only after their correlation with the magnetic gauge center is retained. The full group including $U(1)_R$, spin, and fermion parity requires an additional declared quotient and is not being silently inferred here.

There is no independent gauge-center one-form symmetry in either canonical simply connected theory: the dynamical fundamental quarks let the corresponding Wilson line end. This follows from the general center-screening criterion in Gaiotto, Kapustin, Seiberg, and Willett 2015, §1, pp. 3–4, Open PDF. Gauging baryon number, quotienting a flavor center, or changing the gauge-group global form is a new operation that can introduce one-form symmetries and topological sectors.

Ordinary supersymmetric $SU(N)$ QCD has no additional useful Abelian discrete symmetry beyond subgroups of its continuous symmetries; color conjugation exchanges quarks and antiquarks rather than furnishing an independent Abelian anomaly equation Csáki and Murayama 1998, §5 opening, Open PDF. Consequently, those subgroup anomalies do not provide independent evidence beyond the continuous table. Correlated quotient bundles can nevertheless probe global or torsion information. That full quotient-background comparison has not been completed here and must not be reported as a matched result.

Independence and limits of the checks

The anomaly matches above all derive from one charge table, so they are many exact equations but not many independent physical mechanisms. Chiral-ring and moduli matching add algebraic information; mass flows add dynamical information; global-form tests add topological information. The preceding chapter’s typed claim–dictionary–check matrix records these shared inputs and keeps the still-untested global SQCD row visible.

None of these protected checks computes generic long-multiplet dimensions or proves existence of the interacting fixed point throughout the proposed conformal window. That distinction is developed on the fixed-point page.

Common pitfalls

Deriving the meson-rank bound from classical magnetic F-terms. Those equations constrain $q$ and $\widetilde q$ for a chosen $M$ but leave $M$ itself classically arbitrary. The forbidden ranks disappear only after the low-energy magnetic quantum dynamics is included.

Using scalar R-charges in fermion traces. Anomalies use $R_{\psi}=R_\Phi-1$. The gaugino contributes with R-charge one.

Treating Lie-algebra matching as the full global test. The faithful quotient controls allowed background bundles. A quotient or gauging operation can also change genuine lines and introduce a topological sector.

Exercises

A mixed-anomaly check

Verify $\operatorname{Tr}B^2R$ on the magnetic side.

Solution

Only $q$ and $\widetilde q$ contribute. Each has $N_f\widetilde N_c$ Weyl fermions, baryon charge magnitude $N_c/\widetilde N_c$, and fermion R-charge

$$R(q)-1=\frac{N_c}{N_f}-1=-\frac{\widetilde N_c}{N_f}.$$

Therefore

$$\operatorname{Tr}B^2R =2N_f\widetilde N_c \left(\frac{N_c}{\widetilde N_c}\right)^2 \left(-\frac{\widetilde N_c}{N_f}\right) =-2N_c^2,$$

equal to the electric result.

A quantum rank check

Take $N_c=3$, $N_f=6$, and compare magnetic vacua with $\operatorname{rank}M=3$ and $4$.

Solution

The magnetic rank is $\widetilde N_c=3$. At $\operatorname{rank}M=3$, three magnetic flavors remain, so the low-energy $SU(3)$ theory has $F=\widetilde N_c$ and a quantum-modified constraint. With $q\widetilde q=0$, that constraint relates the magnetic baryons and maps to $B^I\widetilde B_J=\det M^I{}_J$, where $I,J$ select the corresponding $3\times3$ flavor minor.

At $\operatorname{rank}M=4$, only two magnetic flavors remain. Since $F=2<\widetilde N_c=3$, an ADS superpotential is generated and there is no supersymmetric vacuum satisfying $q\widetilde q=0$. Thus the electric bound $\operatorname{rank}M\le3$ is recovered quantum mechanically.

References

• Csáki, Csaba, and Hitoshi Murayama. “Discrete Anomaly Matching.” Nuclear Physics B 515 (1998): 114–162. Open PDF.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI. Open PDF.
• Intriligator, Kenneth, and Nathan Seiberg. “Lectures on Supersymmetric Gauge Theories and Electric–Magnetic Duality.” Nuclear Physics B Proceedings Supplements 45BC (1996): 1–28. DOI. Open PDF.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. DOI. Open PDF.
• ’t Hooft, Gerard. “Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking.” In Recent Developments in Gauge Theories, 135–157. Plenum Press, 1980. DOI.