Build a Seiberg-duality dictionary

The preceding lesson described a strongly coupled gauge theory using mesons and baryons. Sometimes those composites alone do not provide the useful infrared description: a second gauge theory does. Its gauge group can even have a different rank.

We will construct a concrete dictionary between an electric $SU(3)$ theory with five flavors and a magnetic $SU(2)$ theory. The task is to identify fields, interactions, and gauge-invariant operators precisely enough that the proposed equivalence can be tested.

Specify what is being compared

Both theories live in flat four-dimensional Lorentzian spacetime and have $\mathcal N=1$ supersymmetry: four real supercharges, all manifest in $\mathcal N=1$ superspace. There are no adjoint chiral multiplets. Keep the ordinary trace convention $\operatorname{Tr}(t^At^B)=\delta^{AB}/2$.

The electric theory has gauge group $SU(3)$, rank two, with five fundamental chirals $Q_i^a$ and five independent antifundamentals $\widetilde Q^{\,j}_a$. Here $a=1,2,3$ and $i,j=1,\ldots,5$. Its massless tree superpotential vanishes.

The proposed magnetic theory has gauge group

$$SU(N_f-N_c)=SU(2),$$

of rank one. It contains five fundamental chirals $q^{i\alpha}$, five independent antifundamentals $\widetilde q_{j\alpha}$, and twenty-five gauge-singlet chirals $M^j{}_i$. The magnetic color index is $\alpha=1,2$. Each entry of $M$ is a field; “singlet” means a singlet under magnetic color, not under flavor.

Seiberg duality identifies the long-distance quantum physics near the massless symmetric vacuum, with operators and sources appropriately mapped. This example has an interacting infrared limit. The magnetic gauge field is part of another description of that limit; it is not an extra microscopic field added to the electric Lagrangian. Different gauge redundancies and different numbers of elementary fields can describe the same physical operators. Seiberg, section 3

The two descriptions need not both be weakly coupled. Their gauge beta-function coefficients are $3(3)-5=4$ and $3(2)-5=1$: both gauge sectors are asymptotically free, rather than free in the deep infrared. These coefficients alone do not establish the common interacting limit or control its coupling at these small ranks. Nor does duality assert equality of the two microscopic actions at all energies.

Distinguish the two meson normalizations

Write the unrescaled electric composite with a different symbol:

$$\mathcal M^j{}_i=\widetilde Q^{\,j}_aQ_i^a, \qquad [\mathcal M]=2 .$$

As before, its row is $j$ and its column is $i$. The elementary magnetic field has ultraviolet engineering dimension $[M]=1$. Introduce a nonzero complex matching scale $\mu_*$ of dimension one and write the operator dictionary

$$\mathcal M^j{}_i\ \longleftrightarrow\ \mu_* M^j{}_i. \tag{Q09.1}$$

The arrow identifies renormalized operators flowing to the same infrared operator. It is not an identity between numerical elementary fields in the two path integrals. Engineering dimensions at their respective ultraviolet descriptions need not coincide before inserting this conversion factor. $\mu_*$ fixes an operator normalization; it is not an additional flavor mass.

The magnetic superpotential is

$$W_{\mathrm m} =h\sum_{i,j,\alpha}M^j{}_i q^{i\alpha}\widetilde q_{j\alpha}, \qquad h\ne0 . \tag{Q09.2}$$

Here $h$ is a dimensionless holomorphic coupling. It is not determined numerically by the charge table below. A nonzero interaction is part of the proposed dual dynamics.

Intriligator–Seiberg write the singlet in electric-composite units and use $W=\mathcal M q\widetilde q/\mu$. To translate their formulas while retaining our explicit $h$, define

$$\mu=\frac{\mu_*}{h}, \qquad W_{\mathrm m}=\frac1\mu\sum_{i,j,\alpha} \mathcal M^j{}_i q^{i\alpha}\widetilde q_{j\alpha}. \tag{Q09.3}$$

Their holomorphic scale relation specializes to

$$\Lambda_{\mathrm e}^{\,4}\Lambda_{\mathrm m} =\mu^5 =\left(\frac{\mu_*}{h}\right)^5. \tag{Q09.4}$$

The general source phase is $(-1)^{N_f-N_c}$, which is $+1$ here. The exponents $4$ and $1$ are the two beta-function coefficients. Both sides have dimension five; phases matter as well as magnitudes. This is a sourced matching convention, not an equation obtained by equating two canonical running couplings. Intriligator–Seiberg, (5.5)–(5.6)

A rescaling of the elementary singlet changes $\mu_*$ and $h$ together. Setting $h=1$ by a holomorphic field redefinition need not leave its kinetic term canonical. The unknown wavefunction factors must be included when discussing physical couplings or masses.

Derive the global charges

Use the electric flavor convention of the SQCD lesson: $Q$ is a fundamental of $SU(5)_L$, while $\widetilde Q$ is an antifundamental of $SU(5)_R$. To make (Q09.2) invariant, $q$ must have the conjugate left representation and $\widetilde q$ the conjugate right representation:

Chiral field	Gauge representation	Left flavor	Right flavor	Baryon charge	Scalar R-charge
Electric $Q$	$\mathbf3$	$\mathbf5$	$\mathbf1$	$1$	$2/5$
Electric $\widetilde Q$	$\overline{\mathbf3}$	$\mathbf1$	$\overline{\mathbf5}$	$-1$	$2/5$
Magnetic $q$	$\mathbf2$	$\overline{\mathbf5}$	$\mathbf1$	$3/2$	$3/5$
Magnetic $\widetilde q$	$\overline{\mathbf2}$	$\mathbf1$	$\mathbf5$	$-3/2$	$3/5$
Magnetic $M$	$\mathbf1$	$\mathbf5$	$\overline{\mathbf5}$	$0$	$4/5$

The fractional baryon charges will follow from the baryon map. First derive the R column. A chiral scalar of R-charge $r$ has a Weyl partner of charge $r-1$; each gaugino has charge one. Gauge consistency therefore requires

$$\begin{aligned} k_{SU(3)^2R}&=3+5(r_{\mathrm e}-1)=0, &r_{\mathrm e}&=\frac25,\\ k_{SU(2)^2R}&=2+5(r_{\mathrm m}-1)=0, &r_{\mathrm m}&=\frac35 . \end{aligned} \tag{Q09.5}$$

Each flavor pair contributes two fundamental indices $T(\mathbf{fund})=1/2$, explaining the factor five. Gauge-singlet mesons contribute nothing to the second equation. The electric composite has R-charge $2r_{\mathrm e}=4/5$, fixing the singlet charge when $\mu_*$ is neutral. The interaction then has

$$R(W_{\mathrm m})=\frac45+\frac35+\frac35=2.$$

The matching scales and $h$ are neutral under this anomaly-free R assignment. Gauge invariance of an R current is different from matching its global anomalies between descriptions; that further test comes next.

We have chosen the assignment symmetric between each quark and antiquark. Mixing with baryon number remains possible: $R\mapsto R+sB$ gives electric quark charges $2/5\pm s$ and magnetic charges $3/5\pm3s/2$. The mixed gauge anomalies and the superpotential still allow this. Identifying the superconformal R-symmetry, and using it to determine infrared scaling dimensions, requires a further argument. We have not made that identification here.

Although magnetic color is $SU(2)$ and its fundamental is pseudoreal, the whole theory does not acquire the $SU(10)$ symmetry of ten doublets with only gauge interactions. The twenty-five singlets and their nonzero superpotential select the displayed five-plus-five flavor split. Seiberg, (3.1)–(3.3)

Map baryons using complementary flavors

An electric baryon uses three colors:

$$B_{ijk}=\varepsilon_{abc}Q_i^aQ_j^bQ_k^c, \qquad \widetilde B^{ijk}= \varepsilon^{abc}\widetilde Q^{\,i}_a \widetilde Q^{\,j}_b\widetilde Q^{\,k}_c .$$

For increasing flavor indices these are determinants, with no factorial. Each side has $\binom53=10$ independent components. A magnetic baryon instead uses two colors:

$$b^{lm}=\varepsilon_{\alpha\beta}q^{l\alpha}q^{m\beta}, \qquad \widetilde b_{lm} =\varepsilon^{\alpha\beta}\widetilde q_{l\alpha} \widetilde q_{m\beta}, \qquad \varepsilon_{12}=\varepsilon^{12}=1.$$

These epsilons orient color determinants; they are not the Lorentz spinor raising/lowering tensors. Again there are $\binom52=10$ components. With flavor orientation $\varepsilon_{12345}=1$, the map is

$$B_{ijk}\ \longleftrightarrow\ \frac{\kappa}{2}\varepsilon_{ijklm}b^{lm}, \qquad \widetilde B^{ijk}\ \longleftrightarrow\ \frac{\widetilde\kappa}{2}\varepsilon^{ijklm}\widetilde b_{lm}. \tag{Q09.6}$$

The factor $1/2$ compensates the sum over both orders of $l,m$. In particular $B_{123}\leftrightarrow\kappa b^{45}$. The flavor epsilon converts the two conjugate-flavor indices into three electric-flavor indices; equality of component counts is not by itself the reason the map transforms correctly.

Electric baryons have baryon charge three. Thus $2B(q)=3$, explaining $B(q)=3/2$; the tilde charges follow similarly. The R-charges also agree: $3(2/5)=2(3/5)=6/5$. The engineering dimensions do not: $[B]=3$, $[b]=2$. Consequently $[\kappa]=[\widetilde\kappa]=1$, while both coefficients are baryon- and R-neutral.

These checks do not fix their numerical normalization or phase. In the holomorphic determinant convention compatible with (Q09.4), the sourced product is

$$\kappa\widetilde\kappa =-\frac{\Lambda_{\mathrm e}^{\,4}}{\mu^2}. \tag{Q09.7}$$

A convenient local convention sets $\kappa=\widetilde\kappa$ and chooses a square root of the right side. Rephasing baryons changes the displayed dictionary; the root choice is not a physical extra vacuum. This implements Intriligator–Seiberg, (5.10) with explicitly normalized component minors.

A worked check fixes the baryon-product sign

Consider an electric vacuum patch with

$$\mathcal M=\operatorname{diag}(x_1,x_2,x_3,0,0), \qquad x_1x_2x_3\ne0 .$$

The first three flavors can span color. Their ordinary minor identity gives $B_{123}\widetilde B^{123}=x_1x_2x_3$. In the magnetic theory these three meson entries give holomorphic masses $x_a/\mu$ to three magnetic flavor pairs. Removing them leaves the two-flavor $SU(2)$ theory already studied. Its scale is

$$\mathcal L_{\mathrm{low}} =\frac{x_1x_2x_3}{\mu^3}\Lambda_{\mathrm m}.$$

Begin where those magnetic thresholds are large enough for the usual scale matching, then continue the holomorphic relation. This does not require both descriptions to have canonical weakly coupled metrics at the same point.

Let $N$ be the two-by-two meson made from the remaining magnetic flavors $4,5$. The light singlet equations set $N=0$, while their magnetic quantum constraint gives

$$\det N-b^{45}\widetilde b_{45}=\mathcal L_{\mathrm{low}} \quad\Longrightarrow\quad b^{45}\widetilde b_{45}=-\mathcal L_{\mathrm{low}}.$$

This is a consistent low-energy vacuum: introduce a multiplier for that constraint, set it and the light singlet block to zero, and choose the displayed nonzero baryon product. Every remaining F equation then vanishes. As in the preceding lesson, these are quantum composite coordinates, not numerical quark matrices required to violate a classical determinant identity.

Applying (Q09.6) and requiring the electric minor identity yields

$$\kappa\widetilde\kappa =-\frac{\mu^3}{\Lambda_{\mathrm m}} =-\frac{\Lambda_{\mathrm e}^{\,4}}{\mu^2},$$

which is (Q09.7). The minus sign comes from the quantum constraint, not from an arbitrary claim that operators with matching charges have equal coefficients. This check is the specialization of Intriligator–Seiberg, (5.19)–(5.20).

Why the magnetic interaction is essential

The gauge-invariant magnetic product $N^i{}_j=\sum_\alpha q^{i\alpha}\widetilde q_{j\alpha}$ has the flavor representation conjugate to the electric meson. It is not the image of $\mathcal M$: that image is the elementary singlet $\mu_*M$.

Varying the full singlet gives

$$\frac{\partial W_{\mathrm m}}{\partial M^j{}_i}=hN^i{}_j . \tag{Q09.8}$$

The singlet equation of motion makes this magnetic product a supersymmetry descendant, hence zero in the chiral ring when no source is present. In a Wilsonian description its F equation is also a necessary vacuum condition. The remaining equations are $\sum_jM^j{}_i\widetilde q_{j\alpha}=0$ and $\sum_iM^j{}_iq^{i\alpha}=0$. Thus the interaction removes an unwanted independent chiral operator. Setting $h=0$ changes the proposed dual theory.

One must still include quantum dynamics when comparing entire vacuum spaces. Classically the magnetic equations permit $q=\widetilde q=0$ with an arbitrary singlet matrix, whereas the electric meson has rank at most three. The magnetic dynamics of the remaining flavors restore the required restrictions; the rank-three example above already uses their quantum constraint. A field list and classical F equations alone are not the whole duality. Intriligator–Seiberg, section 5.5

Exercises

1. Check a baryon component and a fractional charge

Find the magnetic image of $B_{124}$, including its sign, and check its baryon and R-charges. Under a baryon rotation of angle $2\pi$, what happens to the magnetic quarks? Does this contradict the operator dictionary?

Hint 1

The missing flavors are $3,5$. Compare the ordering $1,2,4,3,5$ with the chosen orientation.

Hint 2

A magnetic quark has charge $3/2$, but is gauge-variant. The center of $SU(2)$ contains $-\mathbf1$.

Solution

The relevant permutation has one interchange, so

$$B_{124}\longleftrightarrow-\kappa b^{35}.$$

Both sides have baryon charge $3$ and R-charge $6/5$; the dimension-one coefficient supplies the engineering-dimension difference. Under a $2\pi$ baryon rotation, $q\mapsto e^{3\pi i}q=-q$ and $\widetilde q\mapsto e^{-3\pi i}\widetilde q=-\widetilde q$. This is the magnetic gauge-center transformation. Gauge-invariant operators are unchanged, so fractional charges of the gauge-variant quarks cause no contradiction. This checks that particular finite transformation; it is not a specification of every quotient in the full global symmetry group.

2. Change the elementary meson normalization

Set $M'=cM$ with a nonzero dimensionless complex constant $c$, leaving magnetic quarks fixed. Rewrite the dictionary and superpotential, and determine whether $\mu$ in (Q09.3) changes. If the kinetic term has coefficient $Z_M$, what happens to it? Finally map an electric source $J^i{}_j\mathcal M^j{}_i$ and check that the source interaction is unchanged.

Hint 1

Replace every old $M$ by $M'/c$, including in the operator map.

Hint 2

The source is held fixed while changing field coordinates. Its engineering dimension is one.

Solution

The new parameters are

$$\mu_*'=\frac{\mu_*}{c},\qquad h'=\frac hc,\qquad \mu'=\frac{\mu_*'}{h'}=\mu,\qquad Z_M'=\frac{Z_M}{|c|^2}.$$

Thus the holomorphic scale relation and baryon-product coefficient are unchanged. A general holomorphic rescaling does not preserve a specified canonical kinetic coefficient.

The electric source maps to $J^i{}_j\mu_*M^j{}_i=J^i{}_j\mu_*'M'^j{}_i$. It has dimension three and R-charge two: $[J]=1$, $R(J)=6/5$, while $\mu_*$ is neutral and $R(M)=4/5$. The R charge of $J$ is a spurion assignment; fixing a nonzero source need not preserve the original R symmetry. Forgetting $\mu_*$ would give both the wrong engineering dimension and a normalization-dependent source map.

3. Remove the interaction and test a candidate vacuum

At the level of the magnetic classical action, set $M=0$ and take only

$$q^{4\alpha}=(w,0),\qquad \widetilde q_{4\alpha}=(w,0),\qquad w\ne0$$

to be nonzero. Check D-flatness and decide whether the configuration is F-flat for $h=0$ and for $h\ne0$. What does this teach about trying to infer duality from the field content and gauge-anomaly-free charge assignment alone?

Hint 1

The color moment map uses $qq^\dagger-\widetilde q^\dagger\widetilde q$. The tilde scalar is independently chosen equal here, not generally defined by conjugation.

Hint 2

The singlet whose derivative can be nonzero is $M^4{}_4$.

Solution

Both terms in the moment map are $\operatorname{diag}(|w|^2,0)$, so every $SU(2)$ D equation vanishes. If $h=0$ there is no superpotential and all classical F equations vanish as well.

For $h\ne0$,

$$\frac{\partial W_{\mathrm m}}{\partial M^4{}_4}=h w^2\ne0 .$$

The other superpotential derivatives vanish on the stated configuration, but one nonzero F equation is enough to exclude it as a supersymmetric vacuum of that classical action. With a positive nonsingular singlet kinetic metric its F contribution to the energy is positive; the metric sets the numerical coefficient.

Removing the interaction changes the equations and leaves an additional magnetic meson operator unconstrained. The same field list and the charge assignment used in (Q09.5) do not by themselves establish the proposed infrared equivalence. This classical diagnostic is not a claim that classical equations determine either theory’s complete quantum vacuum space.

We now have a dictionary with specified fields, interactions, charges, operator normalizations and scale phases. Continue with Check a duality with anomalies and a mass deformation to compare the global anomalies and follow an electric flavor mass into magnetic Higgsing. These tests probe the dictionary in different ways, without turning any single check into a proof of the entire duality.