Real-Mass, FI, and Compactification Flows

A deformation transports a duality only after its vacuum, massive determinants, and endpoint sectors have been transported with it. A real mass can change Chern–Simons levels; an FI parameter can select a Higgs or Coulomb vacuum; and a circle compactification creates an affine-monopole interaction that naive zero-mode truncation misses. The basic object on this page is therefore a flow record: signed masses, a chosen vacuum, a scale hierarchy, induced gauge and background terms, and the complete light theory on each side.

Required background. We use the regulator and normalization of parity anomalies and contact terms and the general logic of deformations, compactification, and duality flows. Helpful background. The endpoint theories are organized by Aharony and Giveon–Kutasov dualities.

A real mass is a signed matrix of thresholds

Let a chiral fermion have charges $Q_I$ under compact Abelian dynamical or background fields, and let its mass in a Coulomb background be

$$M_\rho=\rho(\sigma)+w(m).$$

Here $\rho$ is a gauge weight and $w$ is a flavor weight. In the chapter’s regulator convention, crossing no massless wall and integrating out this fermion gives

$$\Delta\kappa_{IJ} =\frac12Q_IQ_J\operatorname{sgn}(M_\rho).$$

The same determinant contributes $\operatorname{sgn}(M_\rho)T(R)$ to a simple non-Abelian level and one gravitational unit per complex two-component fermion. Its fermionic $R$-charge is $r-1$ when the chiral scalar has $R$-charge $r$. Gauge weights and all representation multiplicities must be summed before any field is deleted Closset et al. 2012, §§3–4.

One heavy flavor pair, with every multiplicity visible

Consider a $U(N_c)$ flavor pair $(Q,\widetilde Q)$ in the fundamental and antifundamental. Give both chirals the same axial mass $M$ and axial charge $+1$. In the single-trace $U(N_c)$ convention used in this chapter, the pair produces

induced term	$M>0$	$M<0$
dynamical $U(N_c)$ level $\Delta k$	$+1$	$-1$
axial contact $\Delta\kappa_{AA}$	$+N_c$	$-N_c$
axial–$R$ contact $\Delta\kappa_{AR}$	$+N_c(r-1)$	$-N_c(r-1)$
$R$ contact $\Delta\kappa_{RR}$	$+N_c(r-1)^2$	$-N_c(r-1)^2$
gravitational contact $\Delta\kappa_g$	$+2N_c$	$-2N_c$

The first row has no factor of $N_c$: a fundamental has Dynkin index $1/2$, so the two chirals shift the non-Abelian level by one. The background and gravitational rows do count all $N_c$ color components. Mixed central-gauge–axial and central-gauge–$R$ shifts cancel between the fundamental and antifundamental because their central gauge charges are opposite.

Opposite physical masses are different, not termwise trivial. Let a vector background $V$ give $Q$ and $\widetilde Q$ charges $+1$ and $-1$, so their masses are $+M$ and $-M$. Then $\Delta k$, the three displayed axial/$R$ contacts, and $\Delta\kappa_g$ cancel. Mixed entries can add: if the common axial background $A$ and the $R$ background are retained, the central $U(1)$ gauge field $a$ has

$$\Delta\kappa_{aA}=\Delta\kappa_{VA}=N_c\operatorname{sgn}(M), \qquad \Delta\kappa_{aR}=\Delta\kappa_{VR} =N_c(r-1)\operatorname{sgn}(M).$$

These mixed terms appear as effective FI or background-response terms when the corresponding multiplets are frozen. Thus a vector-mass flow can remove a flavor without generating a dynamical Chern–Simons level, but its full charge matrix must still be transported. This distinction separates it from the axial flow to a Chern–Simons theory.

Select a vacuum before taking the mass large

For a deformation scale $|M|$, the effective description is valid only at $E\ll |M_\rho|$ for every field declared heavy. A reproducible calculation proceeds in this order:

1. choose a candidate Coulomb/Higgs background and diagonalize every $M_\rho$ there;
2. sum the induced dynamical, mixed, flavor, $R$, and gravitational terms in that chamber;
3. include the induced FI shift and solve the corrected F- and D-term equations;
4. expand around every finite supersymmetric vacuum, including any block decomposition of the gauge group;
5. retain pure Chern–Simons, discrete-gauge, invertible-spin, and free-singlet sectors;
6. match each retained vacuum to a vacuum of the dual description.

For Abelian matter and the FI convention used here, the D equation has the schematic form

$$\sum_i q_i|\phi_i|^2-\zeta_{\rm eff} +\frac{k_{\rm eff}}{2\pi}\sigma=0.$$

Both $\zeta_{\rm eff}$ and $k_{\rm eff}$ are chamber dependent. If no solution remains at finite field distance as $|M|\to\infty$, the candidate vacuum runs away and is not an endpoint theory. Conversely, eigenvalues $\sigma_a\sim M$ can keep selected components light and break a magnetic group into several blocks. Such a vacuum can be the dual of an electric vacuum at $\sigma=0$; comparing only the two origins is generally wrong Aharony et al. 2013, §5.2.

A useful endpoint card contains the signed deformation, the limiting values of $\sigma$ and charged condensates, the light fields and superpotential, the residual global gauge group, all induced response terms, surviving monopoles or singlets, and the scale inequalities. A rank and a Chern–Simons level alone are not an endpoint card.

Axial flow from Aharony to Giveon–Kutasov

Set $N_F=N_f+p$ with $p>0$ and assume $N_F\ge N_c+1$, so the generic Aharony card applies. Begin with the level-zero pair

$$U(N_c)_0 \quad\longleftrightarrow\quad U(N_F-N_c)_0$$

with $N_F$ flavor pairs, magnetic mesons, magnetic monopoles $\widehat V_\pm$, and electric-monopole singlets $v_\pm$. Give the last $p$ electric pairs a common axial mass $sM$, where $M>0$ and $s=\pm1$, and select the electric vacuum at $\sigma=0$. The one-pair record above gives the light electric endpoint

$$U(N_c)_{sp} \quad\text{with}\quad N_f\ \text{flavors}.$$

Magnetic quarks have the opposite axial charge, so their corresponding masses have sign $-s$. At the paired supersymmetric vacuum the magnetic level is therefore $-sp$; mesons carrying a heavy flavor index and the Aharony monopole-singlet sector become massive, while the light block retains $M^i{}_j$, $q_i$, $\widetilde q^j$, and

$$W_{\rm mag}=M^i{}_j q_i\widetilde q^j.$$

The endpoint is the Giveon–Kutasov pair

$$\boxed{ U(N_c)_{sp} \quad\longleftrightarrow\quad U(N_f+p-N_c)_{-sp} }$$

with $N_f$ light flavors. The rank check works because $N_f+|sp|-N_c=N_f+p-N_c$. It does not replace the vacuum derivation, and other magnetic Coulomb vacua can retain additional topological blocks. At the boundary $N_F=N_c$, one must instead begin with the separate confining Aharony card; writing a fictitious $U(0)$ parent would discard its singlets and response. The same-sign mass flow and its background contacts are the field-theory origin of this relation Benini, Closset, and Cremonesi 2011, §§3–5; Giveon and Kutasov 2009, §§2–3.

Contact and sign check for the electric endpoint

For $p$ positive-mass pairs of common scalar $R$-charge $r$, the heavy electric fields give

$$\begin{aligned} \Delta k&=p, &\Delta\kappa_{AA}&=pN_c,\\ \Delta\kappa_{AR}&=pN_c(r-1), &\Delta\kappa_{RR}&=pN_c(r-1)^2,\\ \Delta\kappa_g&=2pN_c. \end{aligned}$$

All five signs reverse for negative axial mass. The magnetic result must be combined with the counterterms already present in the Aharony dictionary; counting only its heavy quarks is not a comparison of generating functionals. This full record is also why the notation $|k|$ in the magnetic rank does not erase the sign of $k$ from Hall and framing response.

FI flows exchange Higgs particles and Coulomb solitons

An FI parameter couples to the topological current. In an $\mathcal N=4$ mirror pair it maps to an ordinary real mass, so a Higgs vacuum on one side maps to a Coulomb chamber on the other. In $\mathcal N=2$ Aharony duality, an electric FI parameter maps through the topological-symmetry dictionary and can select a magnetic vacuum away from the origin.

As a normalization check, take a parity-anomaly-free $U(1)_0$ model with two charge-one chirals and $\zeta>0$. At $\sigma=0$ the D equation gives

$$|\phi_1|^2+|\phi_2|^2=\zeta,$$

so $U(1)$ is Higgsed. With FI action normalized as $-\zeta D$, a BPS vortex of flux $\int F=2\pi n$ has central-charge bound

$$M_{\rm vortex}=2\pi|\zeta n|.$$

If the duality maps the topological current $j_J=*F/(2\pi)$ to an ordinary flavor current, this vortex must map to a particle whose real-mass central charge has the same coefficient. The comparison independently fixes the BF normalization and the sign of the mass–FI map. A nonzero $k_{\rm eff}$ changes both the D equation and the electric charge of the flux sector, so the corresponding monopole or vortex may require matter dressing Intriligator and Seiberg 2013, §§2–4.

For non-Abelian Higgsing, state the unbroken group rather than only its Lie algebra. A quotient or discrete factor changes flux quantization, genuine lines, and the monopole lattice even when the local vector-multiplet spectrum looks unchanged.

Circle compactification retains an affine monopole

Compactify a four-dimensional $\mathcal N=1$ theory on $\mathbb R^{1,2}\times S^1_R$ with periodic fermions. The Wilson line supplies a periodic three-dimensional vector scalar. Ordinary BPS monopoles correspond to the simple roots; a further affine, or Kaluza–Klein, monopole winds the circle. Its amplitude is controlled by

$$\eta=\Lambda_4^{\,b} \sim \exp\!\left(-\frac{8\pi^2}{g_4^2(1/R)}+i\theta\right),$$

with powers of $R$ absorbed into the normalization of the monopole coordinate.

The resulting superpotential depends on the global group and on the chosen Coulomb coordinates. Representative unitary cases are

$$\begin{array}{c|c} SU(N_c)&W_{\rm KK}=\eta Y\\ U(N_c),\ N_c>1&W_{\rm KK}=\eta X_+X_- \end{array}$$

where $X_\pm$ are the two extremal $U(N_c)$ monopoles and $Y=X_+X_-$ is the corresponding $SU(N_c)$ coordinate. The $U(1)$ case is special and must not be obtained by blindly substituting $N_c=1$. These interactions break precisely the axial symmetry inherited as anomalous from four dimensions Aharony et al. 2013, §§1 and 4.1–4.2.

For a four-dimensional Seiberg pair, the finite-radius three-dimensional descriptions therefore have

$$\begin{aligned} W_{\rm el}&=\eta Y, &W_{\rm mag}&=M q\widetilde q+\widetilde\eta\widetilde Y,\\ \eta\widetilde\eta&=(-1)^{N_f-N_c}, \end{aligned}$$

in the $SU(N_c)$ presentation, with $Y$ replaced by $X_+X_-$ after gauging baryon number to obtain $U(N_c)$. Thus the finite-radius duality is similar to, but is not, the duality between the naive dimensionally reduced Lagrangians.

Why setting $\eta=0$ is not the Aharony derivation

At fixed three-dimensional coupling, $g_4^2=2\pi Rg_3^2$ and $\Lambda_4$ tends exponentially to zero as $R\to0$. But the four-dimensional duality window $E\ll\Lambda_4,\widetilde\Lambda_4$ then collapses at the same time. The operations “flow to the four-dimensional infrared” and “take the fixed-$g_3$ zero-radius limit” do not commute.

The controlled route to ordinary Aharony duality instead starts with the finite-$\eta$ pair with one extra flavor. An opposite-sign, or vector, real mass removes that pair without generating a Chern–Simons level. The electric vacuum remains near $\sigma=0$, while the paired magnetic vacuum lies at a correlated large value of $\widetilde\sigma$ and breaks the magnetic group. The light fields produced by that block decomposition include the two Aharony monopole singlets and their cross-couplings. The $\eta$ interaction disappears from the light variables through this vacuum flow; it is not deleted at the start Aharony et al. 2013, §§3.2 and 4.1.

One convenient separation of scales is

$$E\ll |m|\ll R^{-1},$$

with the finite-radius parent established before the flavor is integrated out and with the chosen vacuum kept at finite distance in shifted Coulomb coordinates. Other correlated limits are possible, but they must declare $mR$, $g_3^2R$, and which shifted eigenvalues stay finite. If $|m|$ crosses the KK scale, the tower contributes additional threshold terms and the purely three-dimensional determinant is no longer the complete calculation.

Partition functions remember the discarded fields

The hyperbolic limit of a four-dimensional supersymmetric index reduces elliptic gamma functions to three-dimensional hyperbolic gamma functions. Its divergent polynomial phase is not disposable: its quadratic terms encode gauge and background Chern–Simons contacts, while its metric-dependent part carries gravitational response. Likewise, a large-mass limit of a localized three-dimensional integral can have distinct saddles associated with distinct Coulomb vacua.

Consequently $R\to0$, $|m|\to\infty$, and $E\to0$ need not commute. A partition-function identity supports a particular endpoint only after the same saddle, counterterm representative, and asymptotic phase have been retained on both sides. Equality after dropping those phases is weaker than equality of the generating functionals Aharony et al. 2013, §§5.1–5.3 and Appendix A.

Common pitfalls

Deleting a field before evaluating its determinant. Its gauge, flavor, $R$, and gravitational terms remain below the threshold. Representation dimension multiplies background contacts even when it does not multiply the non-Abelian level.

Using “vector” and “axial” mass interchangeably. Same-sign masses for $Q$ and $\widetilde Q$ generate a level; opposite-sign masses cancel it. The two operations lead to different endpoint dualities.

Matching only the origins of two Coulomb branches. A vacuum at $\sigma=0$ can map to one with $\widetilde\sigma\sim m$. Shift the eigenvalues before taking the limit and retain every light block.

Dropping an apparently trivial topological factor. A pure level-one or level/rank-dual block can be invertible yet still carry a framing phase or transparent spin sector.

Putting $W_{\rm KK}$ to zero before compactifying. At finite radius it controls the anomalous axial symmetry and the Coulomb branch. Remove it only through a specified vacuum and scaling flow.

Exercises

1. A $U(N_c)$ flavor pair has axial charge $+1$, scalar $R$-charge $r$, and common negative mass. Compute its dynamical, axial, axial–$R$, $R$, and gravitational shifts.

Solution

The fundamental and antifundamental each contribute non-Abelian index $1/2$, so $\Delta k=-1$. There are $2N_c$ fermion components. Their axial charge is $1$ and their $R$-charge is $r-1$, hence

$$\Delta\kappa_{AA}=-N_c, \quad \Delta\kappa_{AR}=-N_c(r-1), \quad \Delta\kappa_{RR}=-N_c(r-1)^2, \quad \Delta\kappa_g=-2N_c.$$

The mixed central-gauge–axial and central-gauge–$R$ shifts cancel between the opposite central gauge charges.

2. Start with $U(N_c)_0$ and $N_f+3$ flavors in an Aharony pair. Give the last three pairs a common negative axial mass. State both Giveon–Kutasov endpoints and the electric heavy-field response.

Solution

Each pair shifts the electric level by $-1$, so the electric endpoint is $U(N_c)_{-3}$ with $N_f$ flavors. The paired magnetic endpoint is $U(N_f+3-N_c)_{+3}$ with $N_f$ flavors, mesons, and $W=Mq\widetilde q$. For common scalar $R$-charge $r$, the electric heavy fields contribute

$$(\Delta\kappa_{AA},\Delta\kappa_{AR},\Delta\kappa_{RR},\Delta\kappa_g) =(-3N_c,-3N_c(r-1),-3N_c(r-1)^2,-6N_c).$$

The starting Aharony counterterms must also be transported before comparing the magnetic response.

3. Why does $W_{\rm KK}=\eta X_+X_-$ obstruct an independent continuous axial symmetry at finite radius, and why does simply writing $\eta=0$ not derive Aharony duality?

Solution

The monopole product carries the axial charge generated by its matter zero modes, while the four-dimensional anomaly fixes the compensating transformation of $\eta$. Holding the microscopic coupling fixed leaves only the nonanomalous subgroup. Writing $\eta=0$ enlarges the symmetry but does not identify the magnetic vacuum or produce the monopole singlets. In the controlled derivation, an extra flavor receives an opposite-sign real mass and the dual vacuum moves on the Coulomb branch; the light block decomposition produces the Aharony fields and makes the $\eta$ term disappear in the appropriate low-energy coordinates.

4. In a limit with $E\ll |m|$ but $|m|R\gg1$, why is the three-dimensional shift $\frac12Q_IQ_J\operatorname{sgn}(m)$ by itself insufficient?

Solution

The mass lies above the Kaluza–Klein scale, so modes with masses $m+n/R$ cross thresholds as well. Their regulated sum changes local Chern–Simons and gravitational terms and can depend on the Wilson-line chamber. One must evaluate the compactified determinant, or first choose a hierarchy with $|m|R\ll1$, rather than retaining only the zero-mode contribution.

References

• Aharony, O., Razamat, S. S., Seiberg, N., and Willett, B. (2013), “3d Dualities from 4d Dualities,” Journal of High Energy Physics 2013(07), 149. doi:10.1007/JHEP07(2013)149. Open PDF
• Benini, F., Closset, C., and Cremonesi, S. (2011), “Comments on 3d Seiberg-Like Dualities,” Journal of High Energy Physics 2011(10), 075. doi:10.1007/JHEP10(2011)075. Open PDF
• Closset, C., Dumitrescu, T. T., Festuccia, G., Komargodski, Z., and Seiberg, N. (2012), “Comments on Chern–Simons Contact Terms in Three Dimensions,” Journal of High Energy Physics 2012(09), 091. doi:10.1007/JHEP09(2012)091. Open PDF
• Giveon, A., and Kutasov, D. (2009), “Seiberg Duality in Chern–Simons Theory,” Nuclear Physics B 812, 1–11. doi:10.1016/j.nuclphysb.2008.09.045. Open PDF
• Intriligator, K., and Seiberg, N. (2013), “Aspects of 3d $\mathcal N=2$ Chern–Simons–Matter Theories,” Journal of High Energy Physics 2013(07), 079. doi:10.1007/JHEP07(2013)079. Open PDF

Next steps

These protected flow records supply the parents and endpoint checks for the less-protected mirror, particle–vortex, and bosonization web.