Deformations, Compactification, and Duality Flows

A duality that survives a deformation should relate not only the endpoint theories but also the two composed flow maps. Establishing that square requires the operator map, scale matching, vacuum choice, induced interactions, global sectors, order of limits, and an explicit comparison of composition. Merely adding terms with the same name on both sides can hide Higgsing, runaways, accidental symmetries, endpoint automorphisms, 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

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

For a supersymmetric F- or D-term, replace $d^dx$ by the appropriate superspace measure.

The proposed square is

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

Let $F_A:\mathcal T_A\to\mathcal T'_A$ and $F_B:\mathcal T_B\to\mathcal T'_B$ denote the complete flow maps. Isomorphic endpoint theory cards are necessary but not sufficient. The square commutes only when there is a declared equivalence of composed maps,

$$\eta:\ D'\circ F_A\Longrightarrow F_B\circ D,$$

including their induced maps of operators, backgrounds, extended sectors, and any equivalence modulo a predeclared local-counterterm lattice. 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

$$(\text{theory version},\text{deformation and measure},\mathcal O,\lambda, \text{vacuum},\text{hierarchy and limit order},\text{induced terms}, \text{massless and decoupled sectors},\text{global data}, \text{equivalence convention}).$$

If any component differs, the square may close only after an explicit endpoint automorphism, a decoupled sector, or a narrower claim is incorporated into $\eta$.

The chapter’s typed dictionary-and-operation map gives a visual key for this distinction: vertical arrows are operations, the lower relation is conditional, and graph connectivity alone is not a composition witness.

Worked mass flow in Seiberg duality

Take electric $SU(N_c)$ SQCD with $N_c\geq3$ in the standard left–right flavor basis and add

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

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

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

in a fixed holomorphic normalization.

For the generic residual nonabelian description, take $N_f\geq N_c+3$ and set $\widetilde N_c=N_f-N_c$. The magnetic description has gauge group $SU(\widetilde N_c)$, 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

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

A D-flat representative is

$$q_{N_f}^{\,\widetilde N_c}=v, \qquad \widetilde q_{\widetilde N_c}^{\,N_f}=\widetilde v, \qquad v\widetilde v=-m\mu, \qquad |v|=|\widetilde v|.$$

Thus magnetic quarks acquire vacuum expectation values and Higgs

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

whose number of colors, $\widetilde N_c-1$, is precisely the value for $N_f-1$ flavors. Massive vector and chiral multiplets must be integrated out. The daughter comparison applies below both heavy thresholds,

$$E\ll |m|, \qquad E\ll m_{\mathrm{Higgs}}, \qquad m_{\mathrm{Higgs}}\sim\sqrt{|m\mu|},$$

where the vector-multiplet mass also contains the magnetic gauge coupling evaluated near the Higgs scale. Writing

$$b=3N_c-N_f, \qquad \widetilde b=3\widetilde N_c-N_f,$$

the same holomorphic convention gives

$$\Lambda_{\mathrm e,L}^{\,b+1}=m\Lambda_{\mathrm e}^{\,b}, \qquad \widetilde\Lambda_L^{\,\widetilde b-2} =\frac{\widetilde\Lambda^{\,\widetilde b}}{-m\mu}.$$

Together with $\Lambda_{\mathrm e}^{b}\widetilde\Lambda^{\widetilde b}=(-1)^{\widetilde N_c}\mu^{N_f}$, these imply the correct $N_f-1$ scale relation. At $N_f=N_c+2$, the magnetic group is completely broken; a broken-group instanton generates an additional term, so the generic residual-gauge-theory argument must not be extrapolated to that endpoint.

Thus the generic square passes a nontrivial number-of-colors, operator-map, vacuum, and scale check: “integrate out a flavor” on the electric side maps to “add a linear meson term, choose the D- and 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. Seiberg gives this analysis in Seiberg 1995, §3.2, “Mass terms in the magnetic theory,” arXiv PDF, including the $N_f=N_c+2$ exception; Intriligator and Seiberg 1996, §5.5, arXiv PDF gives the scale relations. The later mass-flow page develops the full threshold derivation.

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. A mismatch caused solely by comparing noncorresponding vacua is not evidence against the duality; failure to find any corresponding branch after an exhaustive branch analysis is.

Runaways require still more care. A runaway means that no finite-field stationary vacuum realizes the endpoint while the potential approaches its infimum only asymptotically along field space. Such an endpoint cannot be compared to a gapped vacuum without specifying additional stabilization or boundary data.

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 $\mathcal N=1$ $SU(N_c)$ gauge dynamics compactified to three dimensions, in the single-coordinate convention used here 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. For $U(N_c)$, common conventions instead give $\eta X_+X_-$. 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 and KK term appear in Aharony, Razamat, Seiberg, and Willett 2013, §2.2, eq. (2.10), arXiv PDF; §§3.1 and 5.2–5.4 treat the reduction and real-mass flows.

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, relative to a fixed regulator, a three-dimensional charge-$q$ fermion shifts a parity-odd level by $\tfrac12q^2\operatorname{sgn}(m)$. The completed level for a dynamical gauge field must obey its quantization condition, while scheme-independent fractional background contact terms remain physical Closset et al. 2012, §§2–3, arXiv PDF. 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 lies in the predeclared lattice of allowed local counterterms for the specified backgrounds and tangential structure, state that equivalence convention. 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 can the flow provide new support for the original duality—and only if the flow was not used to construct the dictionary and supplies materially independent information.

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 number of magnetic colors 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.

Now verify the scale part of the square. Given

$$\Lambda_{\mathrm e}^{b}\widetilde\Lambda^{\widetilde b} =(-1)^{\widetilde N_c}\mu^{N_f}, \qquad \Lambda_{\mathrm e,L}^{b+1}=m\Lambda_{\mathrm e}^{b}, \qquad \widetilde\Lambda_L^{\widetilde b-2} =\frac{\widetilde\Lambda^{\widetilde b}}{-m\mu},$$

derive the matching relation for the daughter pair with $N_f-1$ flavors.

Solution

Multiplying the two threshold relations and then using the parent-pair relation gives

$$\begin{aligned} \Lambda_{\mathrm e,L}^{b+1} \widetilde\Lambda_L^{\widetilde b-2} &=m\Lambda_{\mathrm e}^{b} \frac{\widetilde\Lambda^{\widetilde b}}{-m\mu}\\ &=(-1)^{\widetilde N_c-1}\mu^{N_f-1}. \end{aligned}$$

The daughter has $\widetilde N_c-1$ magnetic colors, so this is precisely $(-1)^{\widetilde N_c-1}\mu^{N_f-1}$ in the same convention. The check uses the scale map in addition to the reduction in the number of colors.

References

• Aharony, Ofer, Shmuel S. Razamat, Nathan Seiberg, and Brian Willett. “3d Dualities from 4d Dualities.” Journal of High Energy Physics 07 (2013): 149. DOI. Open PDF.
• Closset, Cyril, Thomas T. Dumitrescu, Guido Festuccia, Zohar Komargodski, and Nathan Seiberg. “Comments on Chern–Simons Contact Terms in Three Dimensions.” Journal of High Energy Physics 09 (2012): 091. DOI. Open PDF.
• Intriligator, Kenneth, and Nathan Seiberg. “Lectures on Supersymmetric Gauge Theories and Electric–Magnetic Duality.” Nuclear Physics B Proceedings Supplements 45BC (1996): 1–28. DOI. Open PDF.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. DOI. Open PDF.