N=1 QED in 2+1 dimensions: Dualities and enhanced symmetries

We consider three-dimensional sQED with 2 flavors and minimal supersymmetry. We discuss various models which are dual to Gross-Neveu-Yukawa theories. The $U(2)$ ultraviolet global symmetry is often enhanced in the infrared, for instance to $O(4)$ or $SU(3)$. This is analogous to the conjectured behaviour of non-supersymmetric QED with 2 flavors. A perturbative analysis of the Gross-Neveu-Yukawa models in the $D = 4 - \varepsilon$ expansion shows that the $U(2)$ preserving superpotential deformations of the sQED (modulo tuning mass terms to zero) are irrelevant, so the fixed points with enhanced symmetry are stable. We also construct an example of $\mathcal{N} = 2$ sQED with 4 flavors that exhibits enhanced $SO(6)$ symmetry.


Introduction and results
An interesting feature of gauge theories in 2 + 1 dimensions is infrared (IR) dualities that exchange standard mesonic operators with monopole operators.The first example was found long time ago: the bosonic particle-vortex duality [1,2] states that a U (1) gauge theory (QED) with one scalar flavor is dual to a theory without gauge fields: the O(2) vector model.
In this paper we study similar examples, in the case of QED with minimal N = 1 supersymmetry (SUSY).Recently, new dualities for N = 1 non-Abelian gauge theories have been investigated [3,4], suggesting that many interesting phenomena wait to be discovered.See also [5][6][7] for early, and [8,9] for recent, work on N = 1 theories in the string theory context.
Starting from well known N > 1 dualities known as mirror symmetries [10][11][12], we find dualities relating N = 1 sQED with 2 flavors to Gross-Neveu-Yukawa models.The latter theories have no gauge interactions, so they are much simpler than a QED to analyze.When they are supersymmetric they are usually called Wess-Zumino models.
Here SU (2) flav rotates the two flavors, while U (1) top is the "topological" symmetry that shifts the dual photon.The monopole operators are charged under both factors, while mesonic operators (polynomial in the elementary fields) have zero topological charge.We consider four different cases, distinguished by the amount of gauge-singlet fields.These gauge-singlet fields enter the superpotential multiplying the quadratic mesons and can be in a SU (2) flav -triplet or in a SU (2) flav -singlet. 1One model has enhanced N = 4 supersymmetry, and its well-known dual description in terms of free superfields is our starting point.Another model has N = 2 SUSY and displays SU (3) symmetry enhancement in the infrared, as recently pointed out in [13,14].The other two models are genuinely N = 1: one of them exhibits IR symmetry enhancement to O(4), while the other (with no gauge singlets and W = 0) shows no evidence of symmetry enhancement.
The enhanced symmetries are simple UV symmetries on the Wess-Zumino side, but they act non trivially on the sQED side: mesonic and monopoles operators combine into irreducible representations of the IR symmetry.
These features closely resemble the conjectured behavior of non-supersymmetric QED with 2 scalars or 2 fermions: using particle-vortex and bosonization dualities such theories can be argued to be self-dual, and it has been proposed that the SU (2) flav × U (1) top UV symmetry enhances in the IR to O(4) [15][16][17] or SO(5) [18].These fixed points are supposed to describe interesting quantum phase transitions.One important question is if they are stable, i.e. if there are deformations of the theories (on top of the mass terms, which are tuned to zero) which preserve the UV symmetry and are relevant in the IR.Such a deformation would destabilize the fixed point and rule out the possibility that the CFT with enhanced symmetry describes a physical quantum phase transition.
In our cases we can take advantage of the duality with the Wess-Zumino models, which are much simpler to analyze than a gauge theory.As we did in [4], we study the Wess-Zumino models in the D = 4 − ε expansion, which is not a particularly accurate technique, but it is good enough to learn qualitative features, like whether an operator is relevant in the IR or not.We find that in all four cases the SU (2) flav ×U (1) top preserving indeed include on one side of the duality gauge-singlets in the adjoint plus singlet of the SU (N flavor ) symmetry.The examples of this paper however lie outside the naive range of validity of the dualities of [4].
deformations (basically a quartic operator which is the square of the quadratic SU (2) flav singlet meson) are irrelevant.So these N = 1 fixed points are stable.
On the other hand, we find that there are relevant superpotential deformations that break the SU (2) flav × U (1) top symmetry.For instance in the model with W = 0 we can turn on a quartic superpotential in the isospin-2 representation of SU (2) flav .Studying these N = 1 deformations is certainly interesting, but goes beyond the scope of this paper [19].
We also construct a model of sQED with 4 flavors and N = 2 SUSY that displays enhanced symmetry in the infrared: the topological symmetry combine with an SO(4) flavor symmetry to form an SO (6).From this model it is possible to construct RG flows along which some symmetry enhancement is preserved, and for instance land on the sQED with 2 flavors and enhanced SU (3) symmetry.

Organization of the paper
In each Section from 2 to 5 we consider a different sQED with two flavors and its dual Wess-Zumino model.We describe the superpotential, the mapping of the basic operators (which are 4 mesonic operators and 4 monopoles) across the duality.We check that the dual pairs have the same massive deformations, and in one case show how to deform the duality to the duality for N = 1 sQED with just one flavor found in [4].
In Section 6 we study the theory with 4 flavors, argue for a self-duality, which is obtained combining Aharony duality [20] with mirror symmetry.We also check symmetry enhancement computing the superconformal index.
Note added.While this work was in preparation, [21] appeared, which has some overlap with our paper.

N = 4 mirror symmetry for sQED with flavors
We start setting out our notation and reviewing the well known mirror duality [10] for N = 4 sQED with a charge +1 hypermultiplet, that is two complex N = 1 scalar supermultiplets Q 1 and Q 2 with gauge charge +1.We will also consider some neutral matter fields, which we denote by Φ x .Each N = 1 real scalar supermultiplet A contains a real boson a and a Majorana fermion ψ A .
We use N = 1 superspace notation with anticommuting coordinates θ, and denote the matter superfields with capital letters: (2.1) In the theories we discuss, beyond the gauge interactions, there is also a cubic superpotential that in components leads to a quartic potential V ∼ (∂W )2 ∼ p 4 and cubic Yukawa terms of the form ∂ 2 W ψψ ∼ p ψ P ψ P .In other words, such theories can be considered as N = 1 supersymmetric Gross-Neveu-Yukawa models, possibly gauged.
Monopole operators.A crucial role is played by local gauge-invariant operators M that are usually called monopole operators [22,23].They carry non-zero charge under the topological symmetry associated to shifts of the dual photon.Since we consider a U (1) gauge theory with zero effective Chern-Simons term and two charge 1 fermions, the bare Chern-Simon term is 1.So the bare monopole M q bare has gauge charge q, where q ∈ Z is the charge of the monopole under U (1) top .In order to get a gauge-invariant operator, we need to dress the bare monopole M q bare with q charged elementary fields from Q or Q † , either the scalars q or the fermions ψ Q .Since the theory is supersymmetric, the monopoles form supermultiplets.We will mostly be concerned with the simplest monopoles M ±1 : they have minimal topological charge and are dressed by the lowest modes of the matter superfields Q.Since we have 2 flavors, the monopoles will transform in a representation of SU (2) flav , 2 which in this case is simply the two-dimensional representation.
Summarizing, the basic monopoles M ± of the various sQEDs with 2 flavors organize into 4 real supermultiplets, transforming as a complex doublet of SU (2) flav and with charge ±1 under U (1) top .Under the dualities we discuss, these two complex monopoles will always map to 2 complex superfields in the dual Wess-Zumino models we propose.Generically the dual WZ models will also contain additional superfields with U (1) top charge 0.

N = 4 SQED ↔ free hyper
The basic N = 4 Abelian duality states that the U (1) gauge theory with one hypermultiplet (i.e. two complex N = 1 multiplets) of charge 1 is dual in the IR to a free massless hypermultiplet [10,11].The gauge theory has R-symmetry SU (2) C × SU (2) H and a topological flavor symmetry U (1) top .
We can write the N = 4 sQED gauge theory in a way which is manifestly invariant under the antidiagonal subgroup SU (2) flav of the two R-symmetry factors, using the N = 1 notation. 3Then the duality can be stated as: and 3 real fields Φ I=1,2,3 Free theory of 2 complex fields M α=1,2 W = 0 . (2.4) Here σ I are the three Pauli matrices.With a slight abuse of notation, sometimes we use α = +, − interchangeably with α = 1, 2. The low-lying superconformal primary operators are mapped according to Here M α is the complex superfield whose lowest component is the scalar gauge-invariant monopole operator M ψ † Qα .The operators M α ↔ M α have charge 1 under U (1) top .In the last column we have indicated the dimensions of the operators.The last line charge +1 and −1, respectively, and a neutral chiral multiplet Φ c , with superpotential In this notation the Cartan subgroup of SU (2) C is visible as a U (1) R-symmetry under which Q, Q, Φ c have charges 0, 0, 2, respectively, while the Cartan subgroup of SU (2) H is visible as a U (1) R-symmetry under which they have charges 1, 1, 0, respectively.Their difference is an axial flavor symmetry.In the N = 1 notation there is one extra real singlet Φ r from the N = 2 vector multiplet, and the N = 1 superpotential is and grouping the three real gauge-singlet superfields Φ r , Φ c in a triplet Φ I , we get the presentation (2.4).In this presentation SU (2) flav is manifest, with Q α=1,2 forming a doublet and Φ I=1,2,3 forming a triplet.Its Cartan subgroup is the axial flavor symmetry mentioned above.
involves N = 1 SUSY descendants of φ I .Notice that the duality gives us the relation , which implies the quantum relation in the gauge theory.Notice also that since the RHS is free, we also know that the quartic operator Relevant N = 1 deformations include monopole superpotentials (up to order M 3 ) and terms linear in the Φ I 's.The relevant deformations break the UV SU (2) flav × U (1) top global symmetry. 3 From the duality (2.4) and the operator map (2.5) we perform an N = 1 flip of the three gauge-singlets Φ I ↔ M α (σ I ) αβ M † β , in other words we introduce three new fields µ I and add a superpotential This operation does not break the SU (2) flav × U (1) top flavor symmetry.On the sQED side the fields Φ I become massive and can be integrated out, while on the other side we obtain an interacting Wess-Zumino model: WZ model with a real triplet µ I and a complex doublet This duality appeared in [5], intepreting Hanany-Witten branes setups with N = 1 susy.In [5] it was checked that the moduli space of vacua (which is a 3-real dimensional cone) agrees on the two sides.The superpotential of the WZ model on the RHS can be written more explicitly as or as the determinant of a 3 × 3 matrix This is the most general cubic superpotential compatible with the SU (2) flav × U (1) top global symmetry.There are also Z 2 discrete symmetries: Z T 2 times reversal and . The mapping of the basic gauge-invariant operators across the duality is now: The approximate dimensions are computed at one-loop in the ε-expansion, as discussed below.In the last line, the particular linear combination of two flavor symmetry singlets on the RHS is dictated by the one-loop computation and it diagonalizes the dilation operator at that order.

Relevant and irrelevant deformations
In the WZ model, we have computed the scaling dimension of various operators in the D = 4 − ε expansion, as in [4], using the general formulas, which can be found for instance in the appendix of [24].The value of the coupling in W = λ det M U (2) at the RG fixed point at two-loops is The scaling dimension of the elementary fields are The 28 quadratic operators transform as where we denoted by N q an operator in the N -dimensional irrep of SU (2) flav with charge q under the U (1) top .The 3 0 is a SUSY descendant of µ I , while the 2 ±1 is a SUSY descendant of M α .
The two singlets 1 0 have one-loop scaling dimension , it is possible to flip it with a free N = 1 real superfield.We will study this deformation in Section 5. On the other hand, the singlet 2 µ 2 I + 3 |M α | 2 has ∆ > 2 and so it is an irrelevant deformation.The SU (2) flav invariant quartic term ( |Q α | 2 ) 2 is not a SUSY descendant in the sQED, and it maps to the primary singlet 2 From the duality we learn that the SU (2) flav invariant quartic superpotential is an irrelevant deformation.
Let us look at the SU (2) flav isospin-2 operators 5 0 .They have scaling dimension therefore they are relevant.They map to the quartic SU (2) flav isospin-2 operators in the sQED (the SU (2) flav isospin-1 operators are descendants of the SU (2) flav isospin-1 quadratic mesons).This means that it is possible to use these 5 operators to deform the sQED with W = 0.

Massive phases
We can give mass to both flavors of the sQED with W = 0 in two different ways.One way is SU (2) flav invariant, and uses a ∆ ∼ 1 operator: On the sQED side, depending on the sign of m, we are left with the pure N = 1 CS gauge theory U (1) ±1 , which has a trivial gapped vacuum (the gaugino is massive because of the non-zero Chern-Simons term).On the WZ side, the F-terms equations are satisfied only at a single point, where all 7 fields have vanishing VEV. 4 At such a point all scalars and fermions are massive, and we are left with a trivial gapped vacuum for both signs of m.
Another way to give mass to the two flavors breaks SU (2) flav and uses a ∆ ∼ 0.66 operator: On the sQED side, for both signs of m we are left with the pure N = 1 gauge theory U (1) 0 .Dualizing the photon, we get an S 1 worth of vacua plus a free Majorana fermion (the gaugino).On the WZ side, for m > 0 the complex scalar M + acquires a VEV with |M + | 2 = m.This spontaneously breaks U (1) top and gives an S 1 of vacua, over which one real boson and one Majorana fermion are massless.For m < 0 the story is the same, except that M − instead of M + takes a VEV.

Other deformations
We can flip the operator by means of a real scalar superfield Ψ; this operation explicitly breaks SU (2) flav to U (1) flav .On the sQED side this corresponds to the superpotential The resulting theory is, in fact, N = 2 sQED with two chiral multiplets Q = Q 1 and Q = Q † 2 of gauge charge +1, −1, respectively.On the WZ side the superfields µ 3 and Ψ become massive and can be integrated out.One is left with the superpotential − , we obtain an N = 2 Wess-Zumino model of three chiral multiplets with superpotential W N =2 = XY Z. Succinctly: We have obtained the well-known N = 2 Abelian mirror duality [12].Another interesting case is to turn on both massive deformations in (3.11) and (3.12), with a tuning that keeps one massless flavor Q 1 = Q in the sQED.The resulting theory has gauge group U (1) 1/2 and is not parity invariant in the UV, therefore we expect a superpotential term W = −|Q| 4 to be generated along the RG flow.On the WZ side the massive deformation corresponds to The solution to the F-term equations is µ 3 = −m 2 /4m 1 while all other fields have vanishing VEV.With a tuning m 2 = 4m 2 1 we can arrange such that, around the vacuum, only M − and µ 1,2 are massive, while M + is not.We are left with a WZ model of a real superfield µ 3 = H and a complex superfield M + = P , with a superpotential coupling W ⊃ H|P | 2 .It has been shown in [4] that in such a WZ model the term −H 3 is generated by the RG flow, while H 2 is irrelevant.We are led to the duality WZ model with a real field H and a complex field This is the duality proposed in [4].
) where the dots stand for quantum corrections that we describe below.The operator map is As we are going to describe, this duality implies an IR symmetry enhancement in the sQED: On the WZ side, the four real superfields contained in M + , M − transform in the vector representation of O(4), while H is a singlet.From the duality we infer how the operators organize on the sQED side.For instance, on the WZ side there are nine quadratic operators in the symmetric traceless representation (3, 3) of O(4).The three of them with charge 2 under U (1) top , namely M 2 + , M + M − , M 2 − , are mapped to monopole operators M +2 on the sQED side, while the three of them with charge 0, namely M α (σ I ) αβ M † β , are mapped to the singlets Φ I .We conclude that the 9 operators M +2 , Φ I , M −2 must be degenerate in the sQED.
The full quantum superpotential for the sQED has the general form On the other hand, the full quantum superpotential for the Wess-Zumino model has the general form This WZ model was studied in [4]: it was found that the sign of the quantum-generated cubic superpotential is negative (i.e.λ > 0).Also, the one-loop scaling dimensions reported in eqn.(4.2) were computed.
The flavor-singlet operators.On the sQED side, the first parity-even flavor-singlet scalar operators are A linear combination of them is a descendant of H. Similarly, on the WZ side the first parity-even flavor-singlet operators are A linear combination of them is a descendant of H.The other linear combination was found in [4] to be irrelevant: It is also likely that all higher operators, such as and so on, have ∆ > 2. If that is true, all parity-even O(4) invariant operators have ∆ > 2 in the WZ model.Using the duality, it follows that all parity-even SU (2) flav × U (1) top invariant deformations of the sQED are irrelevant.

Massive deformations
The most natural deformation is O(4) invariant and parity-odd: δW = m H . (4.9) Recall that parity-odd deformations of the superpotential preserve parity.
Consider first the sQED side, for m > 0. At least assuming that the signs in the superpotential at the fixed point are as in (4.4), the triplet Φ I acquires a VEV and breaks the SU (2) flav symmetry to U (1).The vacua sit on a S 2 : At each point of S 2 , the two flavors Q α get opposite mass and leave a free N = 1 gauge theory U (1) 0 in the IR.The bosonic part of the theory describes a S 1 fibered over S 2 , namely a NLSM with target S 3 .For m < 0 we find two vacua related by spontaneously-broken parity symmetry: In each vacuum the two flavors get a mass of the same sign, leaving a pure N = 1 CS gauge theory U (1) ±1 with a trivial gapped vacuum.
Let us now look at the WZ side.For m < 0 there is an S 3 worth of vacua, |M α | 2 = −m and H = 0, from the spontaneous breaking O(4) → O(3).For m > 0 there are two vacua sitting at M α = 0 and H = ± √ m, where parity is spontaneously broken.The vacua match upon mass deformations (up to an uninfluential sign redefinition of H).

N = 2 sQED ↔ N = 1 8-field SU (3) WZ model
Starting from the dualities in the previous sections, we can obtain a duality for the N = 2 theory U (1) with 2 chiral multiplets of charge +1. 5 In N = 1 language, this theory reads U (1) with 2 flavors Q 1 , Q 2 and a (real) singlet Φ (5.1) The continuos UV global symmetry is SU (2 Enhanced global symmetry.This N = 2 sQED is expected to have enhanced SU (3) global symmetry in the IR, as pointed out in [13,14], using geometrical features of the 3d/3d correspondence. 5In 3d QED's, the notion of sign of gauge charges is meaningful only for N = 2 SUSY.For N > 2 the flavors always come in pairs, that in N = 2 language have opposite charge.For N = 0, 1 it is always possible to make a field redefinition (exchange a field with its complex conjugate) that changes the sign of the charge.In the N = 2 case, instead, if we write a theory with N + flavors of charge +1 and N − flavors of charge −1 in N = 1 language, there are N = 1 superpotential interactions that break SU (N + + N − ) to SU (N + ) × SU (N − ).
On the LHS we obtain the N = 2 gauge theory U (1) with two flavors of charge +1 , while on the RHS we obtain a cubic WZ model with a total of eight real superfields: On the sQED side no additional superpotential terms can be generated, due to the enhanced N = 2 SUSY.On the WZ side, instead, we expect a cubic superpotential term Φ 3 to be generated quantum mechanically: this is the only other cubic SU (2) flav × U (1) top singlet which is also parity-odd.The most general cubic superpotential that respects the SU (2) flav × U (1) top symmetry is then (5.6) The reason we chose such normalization of the four couplings is that if we organize the 8 real superfields in a 3 × 3 matrix as The operator map is very similar to the ones for the other dualities: At the SU (3) invariant point, the scaling dimension of all 8 fields in M SU (3) is expected to be ∆ = 1 (as imposed by N = 2 supersymmetry).
Parity can be taken to act as M SU (3) → −M SU (3) , while charge conjugation acts as 3) .Notice that the U (1) R-symmetry and the N = 2 supersymmetry are not visible in the UV in the WZ model.
The one-loop beta-functions for the four couplings in (5.6) are: (5.10) We look for critical points where the beta-functions vanish.Notice that there is no solution with δ = 0 and β = 0 or γ = 0: as expected the term Φ 3 is generated.Modulo α → −α and δ → −δ (which can be obtained by field redefinitions), the solutions of the beta-function equations (with all couplings turned on) for the four rescaled couplings are: (5.12) It can be checked that the first fixed point-with enhanced SU (3) symmetry-and the last one are stable.At the other two fixed points the direction parametrized by γ is unstable.This implies that the SU (2) flav × U (1) top global symmetry enhances in the IR to SU (3).We have computed the scaling dimension of the 8 elementary fields at the SU (3) invariant point at two-loops, 8 finding Notice that this result is quite far from the exact scaling dimension ∆ = 1 that is implied by the duality.This means that the ε expansion has poor accuracy and can only be used to infer qualitative features of the IR CFT.The 36 quadratic operators transform under SU (3) as The 8 is a SUSY descendant of the elementary fields.
The other two representations have one-loop scaling dimension Also in this theory the flavor singlet is irrelevant, so the quartic SU (2) flav invariant superpotential on sQED side cannot be turned on.

Massive phases
We consider the SU (3) → SU (2) × U (1) breaking deformation δW = m Φ . (5.18) The discussion is parallel to the case of the previous section, since the superpotentials are the same: what changes is on which side the gauge field sits.On the sQED side, for m > 0 the quarks Q α acquire a VEV which sits on an S 3 : |Q α | 2 = m and Φ = 0. Quotienting by the gauge action, we obtain an S 2 worth of vacua.For m < 0 we need to include one-loop quantum effects, since we are going to move along the Coulomb branch of an N = 2 gauge theory.When Φ acquires a VEV, the quarks get masses of the same sign and an effective Chern-Simons term with level sign(Φ) as well as its supersymmetric partner δW = sign(Φ)Φ 2 are generated.This happens both for Φ = m and Φ = −m.Concluding, for m < 0 there are two gapped vacua.
dimensions are (5.13) Notice that the beta-function equation should be solved perturbatively in ε.
On the WZ side, for m > 0 the triplet µ I acquires a VEV: the vacua sit on a S 2 parametrized by µ 2 I = m and M α = Φ = 0.For m < 0 there are two vacua related by spontaneously-broken parity symmetry: Φ = ±m and µ I = M α = 0.The vacua match upon mass deformations.
It is conceivable that similar self-dualities exist for QED with 4 flavors and less supersymmetry.Such phenomena might also have interesting implication for quantum phases transitions.Here we present an example with N = 2 SUSY, that to the best of our knowledge has not been discussed in the literature. 9In this section we use N = 2 notation and all superpotentials are complex N = 2 superpotentials.
We consider the N = 2 sQED theory with 2 flavors Q α of charge +1, 2 flavors Q β of charge −1 and 4 gauge-singlet chiral fields η αβ . 10The complex N = 2 superpotential is manifestly SU (2) L × SU (2) R invariant.The continuos UV global symmetry of the theory is Here SU (2) L and SU (2) R rotate the quarks Q α and Q β , respectively, while η αβ transform as a bifundamental of SU (2 The chiral ring is generated by the four gauge singlets η αβ and the two SUSY chiral monopoles M ± (which are singlets under the SU (2) 2 ).The holomorphic mesons Q α Q β are set to zero by the F-terms of η αβ .We will see that these operators satisfy a single quadratic quantum relation.
Normalizing the U (1) a × U (1) R charges of the fundamental flavors as (1, r), all six chiral ring generators have U (1) a × U (1) R charges (−2, 2 − 2r).The superconformal R-charge r of the fundamental flavors can be determined to great accuracy numerically using Z S 3 -extremization: r = 0.6696 . . . .(6.3) 6.1 Self-duality and symmetry enhancement We want to argue for a self-duality of the theory in (6.1).We focus on the chiral ring generators.Let us recall that the 3d N = 2 theory U (1) with 2 + 2 flavors and W = 0 satisfies two different IR dualities.The first one is Aharony duality [20]: ) with 2 flavors P i , P i W = i,j µ ij P i P j + ± µ ± M ± (6.4) The six chiral ring generators map as The second duality is an N = 2 version [12] of mirror symmetry [10]: The six chiral ring generators map as We can apply the dualities above to the theory in (6.1).Basically, we start from either one of the above dualities and flip on both sides the right operators, using the mapping of chiral ring generators.Starting from (6.1), applying Aharony duality horizontally and mirror symmetry vertically, we obtain the following duality web: Looking at the top-left and bottom-right theories, we recognize that-composing Aharony duality and mirror symmetry-we get a self-duality of the theory in (6.1).We can think of this as a quantum Z dual 2 symmetry emerging in the infrared. 11Under self-duality, the six chiral ring generators map as From the final diagram we learn that Z dual 2 acts on the six chiral ring generators as is part of the Weyl group of the enhanced symmetry, which in this case must be SO(6) en .Thus, the combination of Aharony duality and mirror symmetry implies that the IR symmetry of the theory is enhanced to SO(6) en × U (1) a × U (1) R .
The adjoint of SO( 6) en decomposes into irreps of SU (2) L × SU (2) R × U (1) top as follows: 15 The last term represents eight emergent current multiplets, which are monopoles with topological charge ±1.They are N = 2 real multiplets, whose bottom component is a scalar with ∆ = 1 and zero R-charge. 12assive deformation to the SU (3) sQED with 2 flavors.The N = 2 "real mas" operator in the Cartan of SU (2) R , Here z, a, b, t a are the fugacities for U (1) gauge × SU (2) L × SU (2) R × U (1) a , while r is the R-charge of the flavors, whose superconformal value is given in (6.3).
modes, as required by gauge invariance (recall that the bare Chern-Simon level is −N f /2).Since the two fermions must be antisymmetrized, we have in total 6 + 6 states.4 of them are chiral/antichiral BPS monopoles, the other 8 are in the 8 conserved supermultiplets which emerge in the IR.
The superconformal index that includes the fugacity t for U (1) top [25] is a sum over all monopole sectors: It can be checked that it indeed admits an expansion at small x in terms of characters χ so(6) of SO( 6) en : SC-I = 1 + x 2−2r t −2 a χ so (6) [6] + x 4−4r t −4 a χ so (6) [20 ] + x 3−3r t −6 a χ so (6) [50] − x 2 χ so(6) [15]  The first three contributions in (6.17) are chiral ring operators (linear, quadratic, cubic in the generators), while the term proportional to −x 2 is the contribution of fermionic operators sitting in the IR conserved current multiplets.We indeed see that the adjoint of SO(6) en appears there.
Let us close with a description of the chiral ring and the moduli space of vacua.From the expansion (6.17The moduli space of vacua of the gauge theory is the 5-complex dimensional cone defined by one SO(6) invariant quadratic equation in 6 variables.So in particular the moduli space of vacua is a complete intersection.This implies that the chiral ring Hilbert Series [26] is a simple Plethystic Exponential: HS = P E χ so(6) [6]t − t 2 = 1 + χ so(6) [6]t + χ so (6) [20 ]t 2 + χ so (6) [50]t 3 + χ so (6) [105]t 4 + . . .(6.20) 13 The Plethystic Exponential P E of a function f (t) such that f (0) = 0 is defined as .15)where t = x 2−2r is the scaling dimension and the R-charge of the chiral ring generators.
In terms of Dinkyn labels of SU (4) ∼ SO(6), we get an all-order expansion as sum over all representations with Dinkyn labels proportional to the Dinkyn label of the 6:14 [0, n, 0] su(4) t n .(6.21) Notice that only SU (4) representations of even quadrality appear, consistent with the fact that the enhanced symmetry is SO(6) = SU (4)/Z 2 .