Aharony and Giveon–Kutasov-Type Dualities

Aharony and Giveon–Kutasov dualities are three-dimensional $\mathcal N=2$ analogues of Seiberg duality, but their complete dictionaries depend on data with no four-dimensional substitute: topological symmetry, monopole operators, quantized Chern–Simons levels, and parity-odd background contacts. The two families are related by real-mass flows, yet their magnetic spectra differ structurally because nonzero Chern–Simons level electrically charges a bare monopole.

Required background. We use parity anomalies and contact terms, monopole operators and dressing, and the standards for complete duality dictionaries. Helpful background. The flavor and meson assignments parallel four-dimensional Seiberg duality.

U(N) level conventions and parameter range

All theories on this page are oriented Lorentzian spin theories with signature $(+--)$; Euclidean partition-function claims use their specified supersymmetric contour. We write

$$U(N)_k\equiv \frac{SU(N)_k\times U(1)_{Nk}}{\mathbb Z_N}$$

for the single-trace Chern–Simons action $\frac{k}{4\pi}\int\operatorname{Tr}_{\boldsymbol N}\!\left(A\wedge dA-\frac{2i}{3}A\wedge A\wedge A\right)$. More general $U(N)$ theories have two independent non-Abelian and Abelian levels and require modified dictionaries; they are outside this representative family.

A “flavor” means a pair $(Q_i,\widetilde Q^i)$ in the fundamental and antifundamental. The axial symmetry assigns charge $+1$ to both, while $U(1)_J$ is normalized so a minimal positive monopole has charge $+1$. Background gauge, flavor, $R$, and gravitational counterterms are fixed in a common regulator scheme before the two theories are compared.

Aharony duality at level zero

For $N_c\ge1$ and $N_f\ge N_c+1$, the generic Aharony pair is Aharony 1997, pp. 71–76:

$$\begin{array}{c|c} \text{electric}&\text{magnetic}\\ \hline U(N_c)_0&U(N_f-N_c)_0\\ N_f\ (Q,\widetilde Q)&N_f\ (q,\widetilde q)\\ W_{\rm el}=0&M^i{}_j,\ v_+,v_-\ \text{singlets}\\ &W_{\rm mag}=M^i{}_j q_i\widetilde q^j +v_+\widehat V_-+v_-\widehat V_+ \end{array}$$

Here $\widehat V_\pm$ are the minimal monopoles of the magnetic gauge group, whereas $v_\pm$ are elementary singlets representing the electric monopoles. The cross-couplings lift the magnetic Coulomb coordinates that are not independent electric operators.

Let the electric quarks have trial scalar $R$-charge $r$. The essential global assignments are

$$\begin{array}{c|ccc} \text{operator}&U(1)_A&U(1)_J&U(1)_R\\ \hline Q,\widetilde Q&1&0&r\\ M=Q\widetilde Q&2&0&2r\\ q,\widetilde q&-1&0&1-r\\ v_\pm&-N_f&\pm1&N_f(1-r)-N_c+1\\ \widehat V_\pm&+N_f&\pm1&N_c+1-N_f(1-r) \end{array}$$

The magnetic quarks transform in conjugate representations of $SU(N_f)_L\times SU(N_f)_R$ so that $Mq\widetilde q$ is a singlet. Every term in $W_{\rm mag}$ has axial and topological charge zero and $R$-charge two. The monopole charges follow independently from the zero-mode formula; their sum in each cross-term is a useful arithmetic check.

The protected operator map is

$$Q_i\widetilde Q^j\longleftrightarrow M_i{}^j, \qquad V_\pm\longleftrightarrow v_\pm.$$

For $U(N)$, an ordinary determinant made only from fundamentals carries the central $U(1)$ gauge charge and is not a gauge-invariant baryon, unlike in $SU(N)$ SQCD. Dressed monopoles can replace baryonic-looking objects in related global forms; they must not be inserted into the present dictionary without recomputing charges.

The stated $N_f\ge N_c+1$ range is exactly the range in which the displayed magnetic gauge group has positive rank and its monopoles $\widehat V_\pm$ exist. At $N_f=N_c$, the gauge sector disappears and the correct infrared variables are only $M$, $v_+$, and $v_-$. Up to a holomorphic normalization and an absorbable overall sign, their superpotential is

$$W_{N_f=N_c}=-v_+v_-\det M.$$

Its axial charge vanishes and its $R$-charge is two: $R(v_+v_-)=2-2N_cr$ while $R(\det M)=2N_cr$. This is a separate confining description, not the generic card with a fictitious $U(0)$ monopole Aharony 1997, §3, pp. 75–76. Lower-flavor theories require their separately derived quantum constraints or generated superpotentials; depending on the range, a finite supersymmetric vacuum may be absent.

Giveon–Kutasov duality at nonzero level

For $N_c\ge1$, $N_f\ge0$, nonzero integer $k$, and the supersymmetric range $N_c\le N_f+|k|$, the single-trace Giveon–Kutasov pair is

$$\begin{array}{c|c} \text{electric}&\text{magnetic}\\ \hline U(N_c)_k&U(N_f+|k|-N_c)_{-k}\\ N_f\ (Q,\widetilde Q)&N_f\ (q,\widetilde q),\quad M^i{}_j\\ W_{\rm el}=0&W_{\rm mag}=M^i{}_j q_i\widetilde q^j. \end{array}$$

This is an infrared strong–weak duality, not an equality of ultraviolet couplings Giveon and Kutasov 2009, §§2–3. The magnetic rank is sometimes called the dual color,

$$\widetilde N_c=N_f+|k|-N_c.$$

If $N_c>N_f+|k|$, the standard brane and field-theory analysis indicates supersymmetry breaking rather than a magnetic gauge group of negative rank. At the boundary $\widetilde N_c=0$, the magnetic gauge sector is absent, but the $N_f^2$ meson singlets remain. The endpoint is therefore not generically gapped. Only when $N_f=0$ are there no meson chirals; even then, the residual topological and gravitational response fixed by the regulator and supersymmetric level/rank relation must not be silently discarded.

There are no elementary Aharony-type monopole singlets in the generic nonzero-$k$ dictionary. With the chapter’s $+\frac{k}{4\pi}\int a\wedge da$ and positive-flux conventions, a flux $m$ carries classical electric charge $+km$, so a bare minimal monopole is not gauge invariant. Dressed monopoles may exist, but their matter dressing, flavor representations, and spin depend on $k$, $m$, and the chosen chamber. The absence of $v_\pm$ is thus a consequence of Gauss law, not a declaration that magnetic sectors cease to exist.

What survives when a rank becomes small

Situation	Gauge sector	Local fields that still remain	Required qualification
Aharony, $N_f=N_c$	none on the magnetic side	$M,v_+,v_-$	use $W=-v_+v_-\det M$; there are no $q$ or $\widehat V_\pm$
Giveon–Kutasov, $\widetilde N_c=0$, $N_f>0$	none on the magnetic side	the meson matrix $M$	the theory is not gapped; retain background and any decoupled invertible response
Giveon–Kutasov, $\widetilde N_c=0$, $N_f=0$	none on the magnetic side	no local chiral singlets	the statement is entirely about the residual topological/gravitational response
Either family with rank one	Abelian	the fields in the relevant card	root contributions vanish, but compact flux sectors and contact terms do not

The notation $U(0)$ is therefore only a reminder that a gauge factor has disappeared. It is never permission to delete elementary singlets or parity-odd response data.

Contact terms complete both dictionaries

The tables above give local fields and continuous charges, but a complete equality of generating functionals also specifies parity-odd contact terms. In a common regulator convention one couples background fields for

$$SU(N_f)_L\times SU(N_f)_R\times U(1)_A\times U(1)_J\times U(1)_R$$

and compares every flavor–flavor, flavor–$R$, topological, and gravitational Chern–Simons term. Their fractional parts must agree; an integer mismatch must be supplied as an explicit local counterterm. The precise integer matrix changes under convention changes, so a table of bare numbers without its regulator is not universal.

A concrete prescription makes the statement computable. Choose Pauli–Villars regulator masses so that every massless complex fermion contributes $+\frac12$ times the quadratic form of its charges to the local representative, and let $u_X=(Q_A,Q_J,Q_R)$ be its Abelian background-charge vector. For Aharony theory, the fermion charge data are

$$\begin{aligned} u_{Q,\widetilde Q}&=(1,0,r-1), &n_{Q,\widetilde Q}&=2N_cN_f,\\ u_{q,\widetilde q}&=(-1,0,-r), &n_{q,\widetilde q}&=2(N_f-N_c)N_f,\\ u_M&=(2,0,2r-1), &n_M&=N_f^2,\\ u_{v_\pm}&=(-N_f,\pm1,N_f(1-r)-N_c), &n_{v_\pm}&=1. \end{aligned}$$

The electric and magnetic gauginos add $(0,0,1)$ with multiplicities $N_c^2$ and $(N_f-N_c)^2$, respectively. Define

$$K_{\rm reg}=\frac12\sum_X n_X\,u_Xu_X^{\mathsf T}.$$

If the electric background counterterm is defined to vanish in this regulator scheme, choosing

$$K_{\rm ct}^{\rm mag}=K_{\rm reg}^{\rm el}-K_{\rm reg}^{\rm mag}$$

fixes the magnetic Abelian contact representative; non-Abelian flavor entries are obtained by replacing charge squares with the relevant Dynkin indices. The same counting with one gravitational unit per complex fermion fixes the relative gravitational term. This formula fixes a background local-counterterm representative after the symmetry basis is identified; it does not by itself determine dynamical level/rank equivalence, genuine lines, or a decoupled topological sector. Changing all regulator signs shifts both representatives by quantized counterterms. For Giveon–Kutasov duality, the analogous $A$-, $R$-, and non-Abelian flavor entries follow from the same rule, while contacts involving $U(1)_J$ and residual invertible sectors are most safely fixed by the level-changing real-mass flow Benini, Closset, and Cremonesi 2011, §§3–5.

Real-mass deformations provide a robust way to derive the required relative terms. Give equal-sign large masses to a vectorlike pair: its two fermions shift the non-Abelian level by one unit in the fundamental-trace convention. Repeating this operation $|k|$ times flows from an Aharony parent with $N_f+|k|$ flavors to a level-$k$ theory. When the parent has positive magnetic rank, the corresponding magnetic vacuum, induced Abelian levels, and gravitational response determine the Giveon–Kutasov counterterms. A boundary parent with $N_f+|k|=N_c$ must instead start from the separate $N_f=N_c$ Aharony card above. Simply deleting the heavy flavor from both tables misses this information.

Worked dictionary check

Take Aharony duality with $N_c=2$ and $N_f=3$. The magnetic gauge group is $U(1)_0$. Electric minimal monopoles have

$$R(v_\pm)=3(1-r)-1=2-3r, \qquad Q_A(v_\pm)=-3.$$

The magnetic quarks have $R=1-r$ and axial charge $-1$. A $U(1)$ magnetic monopole has no non-Abelian root contribution, so

$$R(\widehat V_\pm)=3r, \qquad Q_A(\widehat V_\pm)=+3.$$

Therefore $R(v_+\widehat V_-)=2$ and its axial/topological charges vanish, exactly as required. This check uses the rank-dependent gaugino contribution on the electric side; omitting it would give the wrong superpotential charge.

For a complementary endpoint check, take Giveon–Kutasov data $(N_c,N_f,k)=(2,1,1)$. Then

$$\widetilde N_c=1+1-2=0.$$

The magnetic gauge field, magnetic quarks, and magnetic monopoles are absent, but the single meson $M=Q\widetilde Q$ remains as an elementary chiral. There is no cubic term because there are no $q,\widetilde q$. Thus the local magnetic spectrum is not empty, and the relative background/gravitational contact terms still complete the duality. This one-line example catches the common but incorrect inference “zero dual rank implies a gapped theory.”

Evidence and failure modes

The dualities are supported by complementary checks: moduli spaces and chiral rings, real-mass flows, monopole quantum numbers, supersymmetric indices, localized $S^3$ partition functions, and matching of gapped Chern–Simons phases. Their logical roles differ. A partition-function identity is powerful but is evaluated with particular global forms and counterterms; a chiral-ring match does not see every transparent topological sector.

Before applying either dictionary, test for:

• accidental $U(1)$ symmetries and monopoles or mesons at the unitarity bound;
• special ranks $N_c=0,1$ or $\widetilde N_c=0,1$;
• superpotential deformations that remove a flavor or lift a monopole;
• alternative $U(N)$ level pairs, quotient groups, or gauged topological symmetries;
• boundary conditions, where bulk Chern–Simons response requires anomaly inflow data.

These are not cosmetic qualifications. Any one can change the local operator spectrum or the residual topological theory.

The chapter’s monopole–contact–duality map displays the Aharony card and one controlled mass-flow edge together with the data that must travel along it.

Common pitfalls

Confusing $v_\pm$ with $\widehat V_\pm$. The former are elementary magnetic singlets representing electric monopoles; the latter are disorder operators of the magnetic gauge group and are lifted by the cross-couplings.

Replacing $k$ by $|k|$ everywhere. The magnetic rank depends on $|k|$, but the magnetic level is $-k$ and response terms retain the sign.

Applying the generic formula after a gauge rank vanishes. When the magnetic rank is zero, its monopoles and gauge dynamics cannot remain in the superpotential, but elementary singlets and response terms can. Use the separately derived endpoint description.

Exercises

1. Verify that every term in the Aharony magnetic superpotential is neutral under $U(1)_A\times U(1)_J$ and has $R=2$ using the table above.

Solution

$Mq\widetilde q$ has axial charge $2-1-1=0$ and $R$-charge $2r+(1-r)+(1-r)=2$. For $v_+\widehat V_-$, axial charges $-N_f$ and $+N_f$ cancel, topological charges $+1$ and $-1$ cancel, and the R charges sum to $N_f(1-r)-N_c+1+N_c+1-N_f(1-r)=2$. The other monopole term is identical with signs reversed only for $U(1)_J$.

2. Starting with $U(N_c)_0$ and $N_f+p$ flavors, give $p$ flavor pairs a large positive axial real mass. What non-Abelian level and candidate Giveon–Kutasov magnetic rank remain?

Solution

Each pair contains a fundamental and antifundamental Dirac fermion, each shifting the level by $+1/2$, so the electric level becomes $k=p$. The light theory has $N_f$ flavors. Its Giveon–Kutasov dual rank is $N_f+p-N_c$. Deriving the full magnetic theory additionally requires selecting the appropriate magnetic Coulomb vacuum and retaining all induced Abelian, background, and gravitational terms.

3. Apply the Giveon–Kutasov rank formula to $(N_c,N_f,k)=(3,1,-2)$. What remains on the magnetic side, and why is the answer not an empty gapped theory?

Solution

The dual rank is

$$\widetilde N_c=N_f+|k|-N_c=1+2-3=0.$$

There is no magnetic gauge factor, and hence no magnetic quarks or gauge monopoles. The $1\times1$ meson matrix is nevertheless one elementary chiral $M$, inherited from the generic operator map $Q\widetilde Q\leftrightarrow M$. Its cubic superpotential vanishes because $q$ and $\widetilde q$ are absent. In addition, the sign $k=-2$ fixes parity-odd background and gravitational response through the chosen real-mass flow. The local chiral and that response prevent the endpoint from being described as an empty gapped theory.

References

• Aharony, O. (1997), “IR Duality in $d=3$ $N=2$ Supersymmetric $USp(2N_c)$ and $U(N_c)$ Gauge Theories,” Physics Letters B 404, 71–76. doi:10.1016/S0370-2693(97)00530-3. 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
• 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

Next steps

The controlled derivation of level-changing dictionaries is developed in real-mass, FI, and compactification flows. Their use as parents of less-protected conjectures is qualified in the mirror, particle–vortex, and bosonization web.