Build the SQED-XYZ duality dictionary

Can a gauge-invariant composite and a magnetic-flux insertion become elementary fields in another description? The simplest SQED duality says that they can. It relates the infrared limit of a gauge theory to that of three chiral multiplets with a cubic interaction. We can derive the latter theory’s vacuum equations directly, then use the proposed equivalence to interpret the strongly coupled gauge theory.

Use the monopole charge dictionary, superpotential interactions, and superconformal shortening. Both theories live in flat three-dimensional spacetime and have $\mathcal N=2$ supersymmetry: four real Lorentzian Poincaré supercharges. We use their physical scalar potential to discuss vacua. Euclidean chiral and antichiral variables remain independent before choosing the physical bosonic contour.

Derive three branches from one interaction

Start on the side without a gauge field. The XYZ model contains three neutral chiral multiplets, denoted $X,Y,Z$, with scalar components $x,y,z$. Choose canonical ultraviolet kinetic terms and

$$W=\lambda XYZ,\qquad \lambda\ne0. \tag{S15.1}$$

Here $\lambda$ is a complex coupling. Differentiation gives

$$W_X=\lambda yz,\qquad W_Y=\lambda xz,\qquad W_Z=\lambda xy. \tag{S15.2}$$

To see why these derivatives determine vacua, retain the three auxiliary fields. With the component convention of the superpotential lesson, their Euclidean Lagrangian is

$$\mathcal L_F =\sum_{i=X,Y,Z} \left(-\widetilde F_iF_i+W_iF_i+\widetilde W_i\widetilde F_i\right).$$

The stationary equations give $\widetilde F_i=W_i$ and $F_i=\widetilde W_i$. On the physical bosonic contour, $\widetilde W_i=\overline{W_i}$, so elimination yields

$$U=\sum_i|W_i|^2 =|\lambda|^2\left(|yz|^2+|xz|^2+|xy|^2\right). \tag{S15.3}$$

This uses the auxiliary contour explained in the earlier lesson; it does not integrate $-\widetilde F F$ over an unjustified positive real Gaussian. There is no gauge multiplet here and hence no gauge D-term equation or gauge quotient.

For a constant supersymmetric vacuum, every nonnegative term in $U$ must vanish:

$$xy=xz=yz=0. \tag{S15.4}$$

If $x\ne0$, the first two equations force $y=z=0$. If $y\ne0$ or $z\ne0$, the corresponding argument applies. Thus the vacuum set is

$$\{(x,0,0):x\in\mathbb C\} \ \cup\ \{(0,y,0):y\in\mathbb C\} \ \cup\ \{(0,0,z):z\in\mathbb C\}. \tag{S15.5}$$

These are three complex lines meeting at the origin. The word “line” specifies their complex coordinates, not their quantum kinetic metric.

There is a related statement about operators. The chiral ring identifies chiral operators modulo terms that are exact under an antichiral supercharge. The supersymmetric field equation makes each $W_i$ such an exact term. Consequently the XYZ chiral ring has the presentation

$$\mathcal R_{\rm XYZ} =\frac{\mathbb C[X,Y,Z]}{(XY,XZ,YZ)}. \tag{S15.6}$$

The denominator means that a polynomial containing any of those three factors is identified with zero. This is a relation in the chiral ring, not a claim that all ordinary products or correlation functions of the underlying fields vanish. For example, $X^n$ survives for every positive integer $n$, whereas $X^2Y$ does not. The ring therefore retains three families of holomorphic operators sharing a single identity operator.

The displayed potential and its vacuum equations are an elementary calculation in the XYZ Lagrangian. Identifying this ring with that of SQED is an additional quantum-dynamics claim.

Match the operators and their charges

The gauge theory is the fixed one-flavor model: compact $U(1)$, chirals $Q,\widetilde Q$ of gauge charges $+1,-1$, Chern–Simons level $k=0$, and vanishing superpotential, masses, and FI parameter. Retain the paired regulator and background-counterterm convention of the monopole lesson.

Its relevant gauge-invariant chirals are the meson $M=Q\widetilde Q$ and the elementary monopoles $V_+,V_-$ inserting flux $\pm2\pi$. Choose their phases and multiplicative normalizations so that the infrared dictionary reads

$$X=M,\qquad Y=V_+,\qquad Z=V_-, \qquad W=XYZ. \tag{S15.7}$$

Equalities here identify corresponding renormalized chiral operators. They are not identities between bare ultraviolet fields. The last expression uses holomorphic coordinates with the nonzero cubic coefficient absorbed into their normalization. When using canonical ultraviolet kinetic terms or calculating masses, we restore $\lambda$ as in (S15.1).

The original source writes $W=-MV_+V_-$. Replacing its $V_-$ by our $-V_-$ gives the plus sign in (S15.7), without changing any charge or branch. The source explicitly treats the sign and normalization as conventional. Aharony et al., v1, (3.2)

Let $A$ denote axial charge and $J$ topological charge. Write the common trial scalar R-charge of $Q,\widetilde Q$ as $r$. The previously derived charges become

XYZ field	SQED operator	$A$	$J$	Trial R-charge
$X$	$M$	$2$	$0$	$2r$
$Y$	$V_+$	$-1$	$+1$	$1-r$
$Z$	$V_-$	$-1$	$-1$	$1-r$

This immediately checks the interaction: its axial and topological charges are zero, while its R-charge is $2r+(1-r)+(1-r)=2$. It works for the entire trial family. Charge consistency alone therefore does not choose the superconformal R-symmetry.

Under the dictionary, the $X$ branch is the Higgs branch and the $Y,Z$ branches are the two quantum Coulomb branches. In particular, the ring relation $YZ=0$ becomes $V_+V_-=0$. It is incompatible with treating both monopoles as globally inverse exponentials of a single free dual-photon coordinate. The exponential coordinates in the semiclassical Coulomb calculation were local descriptions away from the strongly coupled junction.

What an infrared equivalence asserts

The duality identifies the interacting infrared theories at the common origin, including their appropriately matched local observables. On the SQED side the gauge coupling has $[g_{\rm YM}^2]=1$; on the canonical XYZ side $[\lambda]=1/2$. To approach that limit, energies must be small compared with the respective crossover scales $g_{\rm YM}^2$ and $|\lambda|^2$. No equality of these two ultraviolet parameters is required.

It is useful to separate three kinds of support for this statement:

• The cubic description reproduces the branch structure and symmetry assignments. AHISS proposed the common fixed point and checked discrete anomaly constraints.
• A formal Abelian mirror construction also leads from SQED to a three-chiral cubic theory. This is the $\mathcal N=2$ specialization in Kapustin–Strassler, not the distinct $\mathcal N=4$ duality to a free hypermultiplet.
• Supersymmetric sphere partition functions obey matching integral identities. They test a quantum observable beyond the algebraic vacuum equations.

These are substantial, mutually consistent reasons to use the duality. Differentiating $W$ alone does not prove it, and equality of a protected observable does not establish every correlator. AHISS, section 3.1, Kapustin–Strassler, v2, section III.D, Willett–Yaakov, v2, section 3.1

Dimensional analysis prevents another misleading interpretation. Canonical ultraviolet XYZ scalars have engineering dimension $1/2$, whereas the ultraviolet composite $Q\widetilde Q$ has engineering dimension $1$. A matching coordinate with engineering dimension $1/2$ could therefore be defined at a chosen scale $\mu_0$ by $X_{\rm match}=c_M\mu_0^{-1/2}M_{\rm ren}(\mu_0)$, with dimensionless normalization $c_M$. This specifies an operator normalization, not an equality of bare elementary fields or a classical change of variables between the actions. Monopole operators likewise require a renormalization prescription.

At the infrared fixed point the matched operators have the same scaling dimensions. Their ultraviolet engineering dimensions need not agree. Nor must their ultraviolet metrics: the SQED Higgs metric was $dM\,d\overline M/(2|M|)$, while canonical XYZ starts with $dx\,d\overline x$. Quantum evolution and operator normalization matter before comparing infrared observables.

Fix the superconformal R-symmetry

Choose the XYZ ultraviolet kinetic terms symmetrically among its three fields. The cubic interaction respects permutations of $X,Y,Z$. At the symmetric interacting fixed point the superconformal R-symmetry respects this permutation symmetry, so write

$$R_{\rm SC}(X)=R_{\rm SC}(Y)=R_{\rm SC}(Z)=\rho.$$

The nonzero interaction has R-charge two. Therefore $3\rho=2$. Scalar chiral superconformal primaries obey $\Delta=R_{\rm SC}$, giving

$$\Delta_X=\Delta_Y=\Delta_Z=\frac23. \tag{S15.8}$$

Matching the charges in the table now fixes

$$2r_*=1-r_*=\frac23,\qquad r_*=\frac13. \tag{S15.9}$$

The value $r_*$ specifies the R-current combination acting on the ultraviolet SQED fields that flows to the superconformal current. It is not obtained by declaring an arbitrary trial R-charge to be a dimension. The independent partition-function analysis gives the same value: in Willett–Yaakov’s notation $r=1/2+\delta$ and $\delta_*=-1/6$. Willett–Yaakov, v2, section 5, (5.2)

The scalar unitarity bound in a three-dimensional CFT is $\Delta\ge1/2$ for a nonidentity physical scalar primary. The three gauge-invariant operators all satisfy it: $2/3>1/2$. There is no contradiction in assigning $R(Q)=1/3$: $Q$ is gauge charged and is not an ordinary gauge-invariant local operator creating a physical radial-quantization state. Applying this bound to $Q$ as though it were such a state is the error.

Finally, $R(W)=2$ does not by itself make $\lambda$ an exactly marginal physical parameter. The kinetic normalization runs, and $XYZ$ itself vanishes in the chiral ring. The protected statements here concern the operator relations and the dimensions at the specified fixed point.

Worked example: move along one branch

Consider the canonical XYZ description at the supersymmetric vacuum

$$x=v\ne0,\qquad y=z=0.$$

Set $x=v+\xi$. Expanding the potential to quadratic order gives

$$U^{(2)}=|\lambda v|^2(|y|^2+|z|^2). \tag{S15.10}$$

Thus the complex fluctuation $\xi$ is massless, while the four real components of $y,z$ have mass squared $|\lambda v|^2$. The superpotential Hessian in the order $(X,Y,Z)$ is

$$W_{ij}\big|_v= \begin{pmatrix} 0&0&0\\ 0&0&\lambda v\\ 0&\lambda v&0 \end{pmatrix}. \tag{S15.11}$$

Its singular values are $0,|\lambda v|,|\lambda v|$, giving the same fermion masses. Singular values are appropriate for a complex mass matrix; a complex eigenvalue is not itself a physical mass.

For $|v|\gg|\lambda|$, the dimensionless interaction strength at the massive threshold is of order $|\lambda|^2/|\lambda v|=|\lambda|/|v|\ll1$. The canonical tree-level calculation is then controlled. It illustrates one massless tangent chiral multiplet and massive transverse multiplets. Near $v=0$ this approximation fails: all three become light and the junction is interacting. Neither the weak-branch mass formula nor the ultraviolet flat metric is an exact metric prediction at the infrared origin.

Exercises

1. Count the surviving chiral polynomials

Give each of $X,Y,Z$ formal polynomial degree one. Find a basis of the ring through degree three and then its generating function $H(t)=\sum_{n\ge0}d_n t^n$, where $d_n$ counts independent degree-$n$ polynomials. Explain why there are three branches but only one identity.

Hint 1

Any monomial involving two distinct variables contains one of the three vanishing pair products.

Hint 2

For positive degree, restrict a proposed relation among pure powers to each coordinate axis separately.

Solution

The basis is $1$ at degree zero; $X,Y,Z$ at degree one; $X^2,Y^2,Z^2$ at degree two; and $X^3,Y^3,Z^3$ at degree three. The same pattern continues. Pure powers are independent because restricting to the $X$ axis kills the $Y,Z$ terms while leaving a polynomial in $X$, and similarly on the other axes. Thus

$$d_0=1,\quad d_n=3\ (n>0),\qquad H(t)=1+\frac{3t}{1-t}=\frac{1+2t}{1-t}.$$

A polynomial restricts to a triple of polynomials on the three axes whose constant values agree at their shared origin. There are not three independent constants. Equivalently,

$$(XY,XZ,YZ)=(Y,Z)\cap(X,Z)\cap(X,Y),$$

since a monomial belongs to that intersection precisely when it involves at least two distinct variables. This also checks the three-branch interpretation without relying on a picture. The grading variable $t$ counts polynomial degree; it is not a spacetime coordinate or an energy.

2. Separate a trial charge from a physical dimension

Allow a trial R-generator also to mix with the topological symmetry: $R_{r,s}=R_r+sJ$. Write the three XYZ charges. Find the necessary region in the $(r,s)$ plane obtained by provisionally applying $\Delta=R_{r,s}$ and scalar unitarity to all three generators. Does that region determine the actual R-symmetry? Impose the permutation symmetry to find the answer.

Hint 1

The meson has $J=0$, while the monopoles have $J=\pm1$.

Hint 2

Require each proposed physical charge to be at least $1/2$, then require all three to be equal.

Solution

The charges are $2r,1-r+s,1-r-s$. They sum to two for every $(r,s)$, so the interaction alone does not distinguish them. The provisional necessary inequalities are

$$r\ge\frac14,\qquad |s|\le\frac12-r.$$

In particular $r\le1/2$. This region is only a filter on a proposed physical assignment, not permission to identify every trial R with a dimension. The charge-exchange symmetry already requires $s=0$ for an invariant actual R-current. Full XYZ permutation symmetry additionally gives $2r=1-r$, hence $r=1/3$. Its three generator dimensions are $2/3$, strictly inside the bound. The identity is excluded from the nonidentity bound and retains dimension zero.

3. Turn off the cubic coupling

Keep canonical XYZ kinetic terms but set $\lambda=0$. Determine the vacuum space, chiral ring, and superconformal dimensions. Explain why this theory cannot retain the preceding SQED dictionary as an equivalence to the same interacting fixed point.

Hint 1

Recompute the F-term equations before dividing them by a coupling.

Hint 2

A free three-dimensional scalar has dimension $1/2$. Compare the cubic superpotential’s dimension with the required dimension two.

Solution

All three derivatives of $W$ vanish identically. There is no F-term constraint, the vacuum space is $\mathbb C^3$, and the ring is $\mathbb C[X,Y,Z]$. With canonical kinetic terms this is three free chirals with $\Delta_X=\Delta_Y=\Delta_Z=1/2$, saturating the scalar bound. In particular $XY$ is a nonzero chiral operator.

At this free point $XYZ$ has dimension $3/2$, so its superpotential coupling has positive engineering dimension $2-3/2=1/2$. It is a relevant deformation: an arbitrarily small nonzero coupling changes the far-infrared theory. The condition $\lambda\ne0$ was essential when passing from (S15.2) to the ring relations. Taking $\lambda\to0$ at fixed energy and taking the interacting infrared limit at fixed nonzero $\lambda$ therefore describe different limits. Matching charges cannot repair their different rings and vacuum spaces.

The dictionary now specifies operators, branch relations, and the actual infrared R-charge. A further test is to perturb the two descriptions by matching sources and follow how their vacua change; that requires keeping the phase convention in (S15.7) fixed.