Seiberg duality and matching tests

The electric theory is four-dimensional N equals one SU(N c) gauge theory with N f massless vectorlike flavors Q and Q tilde in fundamental and antifundamental color representations, and no tree superpotential. Its proposed magnetic description has gauge group $SU(N_f-N_c)$, N f dual flavors q and q tilde, gauge-singlet meson fields M, and a cubic meson-dual-quark superpotential. In the usual conformal window $3N_c/2<N_f<3N_c$, the descriptions are expected to approach the same interacting infrared fixed point at the origin of moduli space.

The meson singlet corresponds to the electric composite Q Q tilde, while baryons map to magnetic baryons with appropriate scale factors. Ultraviolet engineering dimensions require care: a canonically normalized magnetic elementary M has dimension one, while the electric composite has dimension two. A matching scale supplies the difference in an ultraviolet operator dictionary. At the interacting fixed point the corresponding operators have equal scaling dimensions. The duality does not assert equal ultraviolet couplings, identical elementary quark spectra, or weak coupling in both descriptions simultaneously.

Gauge groups are descriptive redundancies and need not match, but global symmetries and their 't Hooft anomalies must agree if the theories have the same infrared limit. Normalize the cubic SU(N f) left-flavor anomaly so one left-handed Weyl fundamental contributes plus one and an antifundamental minus one. The electric Q fermions supply N c copies of the flavor fundamental. In the magnetic theory the q fermions are flavor antifundamentals with N f minus N c color copies, while M supplies N f flavor fundamentals through its right-flavor index.

The magnetic quarks contribute a negative color multiplicity, while the meson adds its N f copies, giving $-(N_f-N_c)+N_f=N_c$. This is a nontrivial match involving the meson singlet: omitting that field fails. R-charge assignments likewise make the magnetic superpotential charge two. Mixed anomalies involving R require fermion charges, not the scalar charges printed in the superfield table. Consistency requires a whole collection of matching tests; one successful cubic anomaly does not establish the entire operator algebra or every infrared correlation function.

Further checks compare moduli spaces, chiral-ring relations, baryon charges, and responses to mass deformations. Giving an electric flavor a supersymmetric mass must produce the corresponding magnetic deformation and low-energy flavor reduction, including the appropriate strong-scale matching. Limits where one side is weakly coupled can explain dynamics obscure on the other. These interlocking tests are powerful precisely because a mistaken matter content or operator map usually cannot satisfy all of them. They remain statements about specified supersymmetric theories and their relevant parameter regimes.

The original proposal and these subsequent matching analyses provide substantial evidence for the infrared duality conjecture. The checks performed here do not constitute a general proof of equality of all infrared correlation functions. Protected quantities constrain the proposed infrared equivalence without determining unprotected Kähler metrics or arbitrary correlation functions by themselves. The phase structure also changes outside the quoted conformal window, so formulas for that fixed point cannot simply be extended through its endpoints. Studying this framework teaches how strongly coupled gauge dynamics can admit different useful variables. It does not establish that ordinary nonsupersymmetric QCD has this dual description or that supersymmetric partners have been detected.