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. These checks are exact protected data. Their agreement is necessary and highly nontrivial, but it does not determine the unprotected spectrum by itself. The original operator and anomaly tests appear in Seiberg 1995, §§3–4, with a systematic review in Intriligator and Seiberg 1996, §5.3.

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 protected operator map

Use the electric and magnetic charges of the preceding page. The basic chiral operators map as

$$\begin{aligned} Q^i\widetilde Q_j&\longleftrightarrow M^i{}_j,\\ \varepsilon Q^{N_c}&\longleftrightarrow C_B\,\varepsilon q^{\widetilde N_c},\\ \varepsilon\widetilde Q^{N_c}&\longleftrightarrow C_{\widetilde B}\,\varepsilon\widetilde q^{\widetilde N_c}. \end{aligned}$$

Flavor epsilon tensors convert an $N_c$-index antisymmetric representation into the complementary $\widetilde N_c=N_f-N_c$ representation. The constants $C_B,C_{\widetilde B}$ contain powers of $\mu$, $\Lambda$, and convention-dependent phases. Charges and relations fix their scaling; a choice of composite normalization fixes their numerical values.

The R-charges agree:

$$R(B)=N_c\left(1-\frac{N_c}{N_f}\right) =\widetilde N_c\frac{N_c}{N_f} =R(b).$$

The baryon charges agree because $B(q)=N_c/\widetilde N_c$. These two equalities are independent of strong coupling.

Chiral-ring relations and rank strata

Classically, the electric meson matrix satisfies

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

Electric baryons are nonzero only on strata where $Q$ has rank $N_c$, and meson–baryon relations follow from antisymmetrization. In magnetic variables,

$$W=\frac{1}{\mu}M^i{}_j q_i\widetilde q^j$$

gives

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

These equations reproduce complementary rank conditions. If $M$ has rank $r$, only $N_f-r$ magnetic flavors can participate in magnetic Higgsing; the remaining gauge quotient supplies the same stratum dimension as the electric description.

At a generic fully Higgsed electric point with $N_f\ge N_c$, the complex dimension is

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

This count subtracts the complexified $SU(N_c)$ orbit. At special ranks the stabilizer grows, so the formula must be applied to the correct stratum. Matching only the generic dimension would not establish equality of the singular stratification.

Quantum relations change at special $N_f$. For $N_f=N_c$,

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

in a standard normalization. For $N_f=N_c+1$, the confined composites have a generated superpotential. These are not obtained by imposing the generic magnetic F-terms with a fictitious $SU(1)$ gauge group.

Cubic nonabelian anomalies

Normalize the cubic anomaly of a left-handed Weyl fermion in the fundamental of $SU(N_f)$ to $+1$ and in the antifundamental to $-1$. On the electric side,

$$\mathcal A[SU(N_f)_L^3]=N_c, \qquad \mathcal A[SU(N_f)_R^3]=-N_c.$$

On the magnetic side, $q$ contributes $-\widetilde N_c$ to the left anomaly and the $N_f$ columns of $M$ contribute $+N_f$:

$$-\widetilde N_c+N_f=N_c.$$

Similarly, $\widetilde q$ contributes $+\widetilde N_c$ on the right and $M$ contributes $-N_f$, giving $-N_c$. The singlet meson is essential; without it the first anomaly check fails immediately. The underlying requirement that the massless infrared theory reproduce the ultraviolet global anomalies is the anomaly-matching condition of ’t Hooft 1980, §§III.10–III.12, pp. 149–151.

Mixed flavor, baryon, and R anomalies

For quadratic flavor anomalies use $T(\mathbf{N_f})=1/2$. The electric result is

$$\mathcal A[SU(N_f)_L^2U(1)_B]=\frac{N_c}{2}.$$

Magnetic quarks give

$$\widetilde N_c\,T(\mathbf{N_f}) \frac{N_c}{\widetilde N_c}=\frac{N_c}{2}.$$

For an R-anomaly, remember that the Weyl fermion in a chiral multiplet has charge $R_{\mathrm{fermion}}=R_{\mathrm{scalar}}-1$. Hence

$$\mathcal A[SU(N_f)_L^2U(1)_R]_{\mathrm{el}} =-\frac{N_c^2}{2N_f}.$$

Magnetic $q$ and $M$ contribute

$$-\frac{\widetilde N_c^2}{2N_f} +\frac{N_f-2N_c}{2} =-\frac{N_c^2}{2N_f}.$$

The baryon anomalies also match:

$$\operatorname{Tr}B^2R=-2N_c^2, \qquad \operatorname{Tr}B^3=\operatorname{Tr}B=0,$$

in the chosen $B(Q)=1$ normalization. The first equality includes both quarks and antiquarks; the last two cancel between them.

Gravitational and cubic R anomalies

Include the adjoint gaugino, whose R-charge is one. Electric fermions give

$$\operatorname{Tr}R =(N_c^2-1)+2N_fN_c\left(-\frac{N_c}{N_f}\right) =-N_c^2-1.$$

On the magnetic side,

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

Likewise,

$$\operatorname{Tr}R^3 =N_c^2-1-\frac{2N_c^4}{N_f^2}$$

on both sides after substituting $\widetilde N_c=N_f-N_c$. These two traces determine the candidate SCFT central charges $a$ and $c$ when the displayed R-symmetry is the exact superconformal one.

Faithful symmetry and discrete information

The continuous notation overcounts the symmetry because common center elements act identically on all gauge-invariant operators. The faithful group is a quotient of

$$SU(N_f)_L\times SU(N_f)_R\times U(1)_B$$

by discrete subgroups correlated with the $SU(N_c)$ center. Background bundles for the quotient can contain fractional fluxes that are invisible if each factor is treated separately.

Matching ordinary anomaly polynomials is therefore not the end of the global test. One should compare discrete anomalies, allowed background bundles, baryon normalization, and any line or surface operators created by gauging a quotient. With fundamental dynamical matter the gauge-center one-form symmetry is broken, but flavor-center quotients can still carry nontrivial global information.

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.

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

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

Matching only one anomaly coefficient. The meson is forced by the full nonabelian, mixed, baryon, and R-anomaly system together with the operator map.

Ignoring special-rank quantum relations. The generic chiral ring changes at $N_f=N_c$ and $N_f=N_c+1$.

Exercises

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.

References

• Intriligator, Kenneth, and Nathan Seiberg. “Lectures on Supersymmetric Gauge Theories and Electric–Magnetic Duality.” Nuclear Physics B Proceedings Supplements 45BC (1996): 1–28. arXiv:hep-th/9509066.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. arXiv:hep-th/9411149.
• ’t Hooft, Gerard. “Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking.” In Recent Developments in Gauge Theories, 135–157. Plenum Press, 1980. doi:10.1007/978-1-4684-7571-5_9.