Seiberg Duality in SQCD

Seiberg duality proposes that two asymptotically different four-dimensional $\mathcal N=1$ gauge theories flow to the same infrared physics. The canonical pair is completely specified by its gauge groups, matter, singlets, superpotential, global charges, scale relation, and rank regime. Omitting any of these turns the claim into an ambiguous mnemonic.

Required background. SQCD fields, symmetries, and classical moduli defines the electric theory, and duality claims and dictionaries fixes the comparison standard. Helpful background. Infrared phases and the conformal window explains the rank-dependent regimes.

Electric and magnetic theory cards

The electric theory has gauge group $SU(N_c)$, $N_f$ pairs of chiral multiplets,

$$Q^i\in\mathbf{N_c}, \qquad \widetilde Q_j\in\overline{\mathbf{N_c}}, \qquad i,j=1,\ldots,N_f,$$

and $W_{\mathrm{el}}=0$. Use the global symmetry notation

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

with the understanding that the faithful group includes quotients by common centers. Normalize $B(Q)=+1$ and $B(\widetilde Q)=-1$. The anomaly-free R-charge is

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

For $N_f>N_c+1$, the generic magnetic theory has

$$\widetilde N_c=N_f-N_c, \qquad G_{\mathrm{mag}}=SU(\widetilde N_c),$$

magnetic quarks $q_i,\widetilde q^j$, a gauge-singlet matrix $M^i{}_j$, and

$$W_{\mathrm{mag}}=\frac{1}{\mu}M^i{}_j q_i\widetilde q^j.$$

Here $M$ is normalized to map to the dimension-two electric composite $Q\widetilde Q$, so the matching parameter $\mu$ has mass dimension one. Equivalently, define a dimension-one elementary singlet $\Phi=M/\mu$ and write $W=\Phi q\widetilde q$.

The charge table is

Field	Gauge	$SU(N_f)_L$	$SU(N_f)_R$	$U(1)_B$	$U(1)_R$
$Q$	$\mathbf{N_c}$	$\mathbf{N_f}$	$\mathbf1$	$+1$	$1-N_c/N_f$
$\widetilde Q$	$\overline{\mathbf{N_c}}$	$\mathbf1$	$\overline{\mathbf{N_f}}$	$-1$	$1-N_c/N_f$
$q$	$\mathbf{\widetilde N_c}$	$\overline{\mathbf{N_f}}$	$\mathbf1$	$N_c/\widetilde N_c$	$N_c/N_f$
$\widetilde q$	$\overline{\mathbf{\widetilde N_c}}$	$\mathbf1$	$\mathbf{N_f}$	$-N_c/\widetilde N_c$	$N_c/N_f$
$M$	$\mathbf1$	$\mathbf{N_f}$	$\overline{\mathbf{N_f}}$	$0$	$2(1-N_c/N_f)$

Every term in $W_{\mathrm{mag}}$ is a flavor singlet, has zero baryon charge, and has R-charge two. Gauge anomalies cancel because $q$ and $\widetilde q$ are vectorlike. The magnetic theory, charge assignments, operator map, and deformation tests are constructed in Seiberg 1995, §§2–4; complementary pedagogical derivations appear in Intriligator and Seiberg 1996, §5.3 and Weinberg 2000, ch. 29.

Operator map and the role of μ

The basic map is

$$Q^i\widetilde Q_j\longleftrightarrow M^i{}_j.$$

Electric baryons are

$$B^{[i_1\cdots i_{N_c}]} =\varepsilon Q^{i_1}\cdots Q^{i_{N_c}},$$

while magnetic baryons contain $\widetilde N_c=N_f-N_c$ magnetic quarks. Flavor epsilon tensors identify the complementary antisymmetric representations. Their baryon charges agree because

$$\widetilde N_c\,B(q)=N_c.$$

The precise baryon map contains powers of $\mu$ and the holomorphic scales. Those coefficients depend on composite-operator normalization, but charge, flavor representation, and chiral-ring relations do not. A convention must be fixed before comparing numerical correlators or mass deformations.

The magnetic F-term

$$q_i\widetilde q^j=0$$

sets the magnetic quark bilinear to zero in the chiral ring. This is consistent because that bilinear is not an additional independent electric operator; $M$ already represents the electric meson. The $q$ and $\widetilde q$ F-terms impose $M\widetilde q=0$ and $qM=0$, reproducing rank-stratified moduli relations.

Holomorphic scale matching

Let $\Lambda$ and $\widetilde\Lambda$ be the electric and magnetic holomorphic scales in a common scheme. A standard convention gives

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

The phase depends on definitions of the scales and baryons, but it cannot be discarded after a convention is fixed: it ensures consistent decoupling and moduli-space matching. Since

$$3\widetilde N_c-N_f=2N_f-3N_c,$$

the magnetic theory is infrared free when $N_f<3N_c/2$ and asymptotically free when $N_f>3N_c/2$. Electric asymptotic freedom requires $N_f<3N_c$.

The two descriptions therefore have complementary weak-coupling regions. Their strong scales are not independent parameters; changing the arbitrary matching normalization $\mu$ rescales the magnetic singlet and shifts $\widetilde\Lambda$ so physical predictions remain invariant.

Rank regimes and special endpoints

Range	Infrared description and qualification
$N_f>3N_c$	Electric theory is not asymptotically free; the canonical ultraviolet setup changes.
$3N_c/2<N_f<3N_c$	Candidate interacting conformal window; both descriptions can be strongly coupled in the interior.
$N_c+2\le N_f<3N_c/2$	Free magnetic phase: magnetic gauge coupling is infrared free, with interacting superpotential effects treated at the endpoint.
$N_f=N_c+1$	s-confining description in mesons and baryons with a generated superpotential; the formal magnetic rank is one.
$N_f=N_c$	Quantum-modified moduli space; no generic magnetic gauge group.
$N_f<N_c$	Affleck–Dine–Seiberg dynamics and, without masses, a runaway for the standard theory.

At equality $N_f=3N_c/2$ or $3N_c$, beta-function coefficients vanish at leading order and logarithmic or accidental effects require separate analysis. Open-window formulas should not be asserted at endpoints without that analysis.

Small ranks also matter. $SU(1)$ is not an ordinary gauge group, $SU(2)$ fundamentals are pseudoreal and enhance flavor symmetry, and baryon representations can coincide with mesonic structures. Each such case deserves its own theory card.

Global form and line data

With dynamical fundamental quarks, the electric center one-form symmetry is broken because a fundamental Wilson line can end on $Q$. The magnetic center is likewise broken by $q$. This removes a common source of line-lattice mismatch, but it does not make global data automatic. The faithful flavor symmetry is a quotient involving baryon number and flavor centers, and background bundles can obey correlated flux conditions.

A complete duality statement maps those background bundles and all discrete anomalies. If one changes the gauge group, gauges baryon number, or quotients a flavor center, the resulting dual pair can acquire nontrivial one-form symmetries and topological sectors. It is a new operation, not an innocuous notation change.

Evidence and status

The duality is supported by matched continuous and discrete anomalies, chiral rings and moduli, holomorphic scale relations, mass and Higgs flows, special-rank limits, and protected partition functions. These checks are powerful but correlated: several use the same global charges and holomorphy.

The safe statement is that Seiberg duality is a strongly supported infrared equivalence in its specified regimes, with many exact protected consequences. It is not a proven equality of ultraviolet Lagrangians, and a protected-index identity alone is not a proof of the full infrared claim.

Common pitfalls

Writing only the magnetic rank. Without $M$, $W_{\mathrm{mag}}$, charges, and scale matching, one has not defined the magnetic theory.

Using $M$ with inconsistent dimension. If $M\sim Q\widetilde Q$ has dimension two in the ultraviolet normalization, $1/\mu$ belongs in the cubic superpotential.

Extending the generic card through special ranks. $N_f=N_c+1$ and $N_f=N_c$ have confining and quantum-modified descriptions that must be written separately.

Exercises

For $N_c=4$ and $N_f=7$:

1. find the magnetic rank;
2. compute $R(Q)$, $R(q)$, and $R(M)$;
3. verify that $Mq\widetilde q/\mu$ has R-charge two;
4. determine whether the magnetic gauge coupling is infrared free at one loop.

Solution

$\widetilde N_c=3$. The charges are

$$R(Q)=\frac37, \qquad R(q)=\frac47, \qquad R(M)=\frac67.$$

Their superpotential sum is $6/7+4/7+4/7=2$. The magnetic one-loop coefficient is $3\widetilde N_c-N_f=9-7=2>0$, so the magnetic theory is asymptotically free, not infrared free. Since $3N_c/2=6<N_f=7<12=3N_c$, the pair lies in the candidate interacting conformal window.

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.
• Weinberg, Steven. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge University Press, 2000, ch. 29. doi:10.1017/CBO9781139644198.