Deformations, Compactification, and Duality Flows

A duality that survives a deformation should lead to equivalent endpoint theories along both paths. Establishing that square requires the operator map, scale matching, vacuum choice, induced interactions, global sectors, and order of limits. Merely adding terms with the same name on both sides can hide Higgsing, runaways, accidental symmetries, or topological factors.

Required background. Duality claims and dictionaries supplies the source and target data. Helpful background. Compactification and semiclassical continuity and holomorphic decoupling provide two important classes of flow.

The deformation square

Suppose $D$ relates theories $\mathcal T_A$ and $\mathcal T_B$, with an operator map $\mathcal O_A\leftrightarrow\mathcal O_B$. Add a relevant deformation

$$\delta S_A=\lambda_A\int\mathcal O_A, \qquad \delta S_B=\lambda_B(\lambda_A)\int\mathcal O_B.$$

The proposed square is

$$\begin{array}{ccc} \mathcal T_A & \xleftrightarrow{\ D\ } & \mathcal T_B\\ \big\downarrow {\lambda_A} && \big\downarrow {\lambda_B}\\ \mathcal T'_A & \xleftrightarrow{\ D'\ } & \mathcal T'_B. \end{array}$$

It closes only if the endpoints agree as complete theories in the stated regime. Each vertical arrow carries more than a coupling: it selects vacua, integrates out masses, changes strong scales, can Higgs gauge groups, and can generate Chern–Simons, Wess–Zumino, or topological terms.

A useful record for each arrow is

$$(\mathcal O,\lambda,\text{vacuum},\text{scale hierarchy}, \text{induced terms},\text{massless sectors},\text{global data}).$$

If any component differs, the square may close only after adding a decoupled sector or narrowing the claim.

Worked mass flow in Seiberg duality

Take electric $SU(N_c)$ SQCD with $N_f$ flavors and add

$$W_{\mathrm{el}}=m Q^{N_f}\widetilde Q_{N_f}.$$

Below $|m|$, this flavor decouples. Holomorphic scale matching gives

$$\Lambda_{N_f-1}^{,3N_c-(N_f-1)} =m\,\Lambda_{N_f}^{,3N_c-N_f}$$

in a fixed holomorphic normalization.

The magnetic description has gauge group $SU(N_f-N_c)$, magnetic quarks $q,\widetilde q$, mesons $M$, and

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

The $M^{N_f}{}_{N_f}$ equation requires

$$q_{N_f}\widetilde q^{N_f}=-m\mu.$$

Thus magnetic quarks acquire a vacuum expectation value and Higgs

$$SU(N_f-N_c)\longrightarrow SU(N_f-N_c-1),$$

which is precisely the magnetic rank for $N_f-1$ flavors. Massive vector and chiral multiplets must be integrated out, and the magnetic scale must be matched with the same convention used in the original dual pair.

The square closes because “integrate out a flavor” on the electric side maps to “add a linear meson term, choose the F-flat Higgs vacuum, then integrate out the Higgsed sector” on the magnetic side. Treating the two vertical arrows as identical field operations would miss the mechanism. The deformation and scale-matching argument is given in Seiberg 1995, §4 and reviewed in Intriligator and Seiberg 1996, §5.5.

Vacuum choice and branch dependence

A deformation can have several supersymmetric vacua or trigger a runaway. A duality map acts on the vacuum space as well as on operators. For each chosen vacuum $v_A$, identify $v_B$ and compare:

• the unbroken ordinary and generalized symmetries;
• massless multiplets and topological sectors;
• order parameters and BPS charges;
• local counterterms after heavy fields are integrated out;
• domain walls connecting vacua.

If one side is expanded around the origin while the mapped vacuum on the other side lies on a Higgs branch, apparent rank and spectrum mismatches are expected. They are errors in the comparison, not failures of the duality.

Runaways require still more care. A formal F-term solution at infinite field value is not a normalizable vacuum at finite distance. A flow whose endpoint is a runaway cannot be compared to a gapped vacuum without specifying boundary conditions or additional stabilization.

Compactification carries towers and holonomies

Compactify a $d$-dimensional theory on a circle of radius $R$. The lower-dimensional data include:

• Kaluza–Klein modes with masses $|n|/R$;
• gauge holonomies around the circle, which become compact scalars;
• winding and wrapped defects;
• background holonomies that become real masses;
• induced parity-odd contact terms from massive fermion towers;
• monopole or instanton events involving the compact direction.

The naive zero-mode reduction is valid only when

$$E\ll R^{-1}$$

and every discarded mode remains heavy throughout the field region considered. On a Coulomb branch, some KK or winding states can become light and invalidate a uniform truncation.

For four-dimensional $\mathcal N=1$ gauge dynamics compactified to three dimensions, a KK monopole can generate a superpotential term schematically

$$W_{\mathrm{KK}}=\eta Y,$$

where $Y$ is a Coulomb-branch monopole coordinate and $\eta$ is related to the four-dimensional instanton factor. A genuine three-dimensional duality may require a further real-mass flow sending $\eta\to0$ while generating compensating contact terms. Simply deleting the KK term does not define the same limit. The Coulomb-branch coordinates, KK-monopole term, real-mass flows, and partition-function checks are worked through in Aharony, Razamat, Seiberg, and Willett 2013, §§2–5.

Noncommuting limits

Let $m$ be a mass and $R$ a compactification radius. Two dimensionless parameters are $mR$ and $ER$. The operations

$$\text{decouple }m\to\infty, \qquad \text{reduce }R\to0$$

can fail to commute because a massive fermion induces a lower-dimensional Chern–Simons term whose sign survives decoupling. The two paths can also differ by holonomy vacua or a topological theory.

To expose this, compute both endpoint functionals with background fields:

$$Z_{(m\to\infty)\circ(R\to0)}[B] \quad\text{and}\quad Z_{(R\to0)\circ(m\to\infty)}[B].$$

If their ratio is a quantized local counterterm, state it. If it is a nontrivial TQFT or changes genuine operators, the operations do not commute as complete theories.

Higgsing, confinement, and accidental sectors

Under a duality, an elementary Higgs field can map to a composite operator, a monopole, or a mass term. Consequently, Higgsing on one side may appear as confinement or a superpotential constraint on the other. Compare gauge-invariant spectra and topological responses rather than insisting on the same microscopic mechanism.

Near an endpoint fixed point, test every gauge-invariant chiral operator against the unitarity bound. If one becomes free, introduce its accidental symmetry and redo anomaly or R-charge extremization. A flow square that closes before this correction can fail afterward because it compared the wrong interacting sectors.

A six-step closure test

1. Map the deformation operator and its coupling normalization.
2. Choose corresponding vacua and state the scale hierarchy.
3. Integrate out or Higgs fields on each side, including induced local and topological terms.
4. Match strong scales, branches, global symmetries, anomalies, and genuine extended operators.
5. Add all decoupled free or topological sectors and accidental currents.
6. Compare the complete endpoints and repeat the check along alternative operation orders.

Only after step 6 does the flow provide new support for the original duality.

Common pitfalls

Mapping couplings but not vacua. The same deformation can have several branches; comparing different ones produces artificial contradictions.

Dropping induced topological terms. Heavy fermions and KK towers can leave quantized contact terms that remain visible in the infrared.

Assuming limits commute. Decoupling, compactification, gauging, and large-parameter limits should be composed in both orders when the claim uses them.

Exercises

In the Seiberg-duality mass flow above, explain why the magnetic endpoint cannot be obtained merely by deleting the $N_f$th magnetic quark and meson.

Solution

The linear term $mM^{N_f}{}_{N_f}$ makes the meson F-term require a nonzero product $q_{N_f}\widetilde q^{N_f}=-m\mu$. This selects a Higgs vacuum and reduces the magnetic rank by one. Deleting fields while keeping the original gauge group would give $SU(N_f-N_c)$ rather than the required $SU(N_f-N_c-1)$ and would miss the massive vector multiplets and scale matching. The deformation map is therefore a coupled F-term, Higgsing, and decoupling operation.

References

• Aharony, Ofer, Shmuel S. Razamat, Nathan Seiberg, and Brian Willett. “3d Dualities from 4d Dualities.” Journal of High Energy Physics 07 (2013): 149. arXiv:1305.3924.
• 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.