Phases of QCD$_3$ from Non-SUSY Seiberg Duality and Brane Dynamics

We consider a non-supersymmetric USp Yang-Mills Chern-Simons gauge theory coupled to fundamental flavours. The theory is realised in type IIB string theory via an embedding in a Hanany-Witten brane configuration which includes an orientifold and anti-branes. We argue that the theory admits a magnetic Seiberg dual. Using the magnetic dual we identify dynamics in field theory and brane physics that correspond to various phases, obtaining a better understanding of 3d bosonization and dynamical breaking of flavour symmetry in USp QCD$_3$ theory. In field theory both phases correspond to magnetic 'squark' condensation. In string theory they correspond to open string tachyon condensation and brane reconnection. We also discuss other phases where the magnetic 'squark' is massive. Finally, we briefly comment on the case of unitary gauge groups.


Introduction and conclusions
Quantum chromodynamics in three spacetime dimensions (QCD 3 ) is an interesting variant of the more familiar QCD in four dimensions. In a sense, the three dimensional theory is richer because three dimensions allow the addition of a topological Chern-Simons (CS) term which blocks the RG running of the Yang-Mills interaction towards strong coupling and alters the infra-red (IR) dynamics.
In the past decade or so there has been significant progress in understanding the dynamics of 3d supersymmetric gauge theories including the dynamics of a large class of super-symmetric Chern-Simons theories coupled to different types of matter. Steps forward have also been achieved in the study of non-supersymmetric Yang-Mills-Chern-Simons (YM-CS) theories like QCD 3 . For example, consider a CS theory coupled to N f Dirac fermions in the fundamental representation. A recent intriguing result is a conjectured duality between the following pairs of theories [1] SU (N c A variant of this duality postulates that 2 ) ⊕ N f scalars. (1.2) In these expressions, and in what follows, the notation SU(N) k quotes bare CS levels.
These are written in terms of the shifted level K defined as k bare − N f 2 . Similarly, in the notation U(N) k,ℓ the first level refers to the bare level of the SU part and the second to the bare level of the U(1) part. Since both of these dualities relate a theory with fermions to a theory with bosons they are frequently referred to as bosonization (in close analogy with the more familiar bosonization in two dimensions). The scalars on the bosonic side have quartic interactions tuned to a Wilson-Fisher fixed point. The duality holds for 1 2 N f ≤ K. Versions of this duality for SO(N c ) and USp(2N c ) gauge groups have also been formulated [2]. In this paper, we will mostly focus on results that are very closely related to the USp(2N c ) version of the duality Similar to the SU(N c ) case, the bosonic side requires that the scalars are at a USp(2N f ) invariant Wilson-Fisher fixed point. This duality is also expected to be valid for 1 2 N f ≤ K. More recently, Komargodski and Seiberg (KS) argued for a scenario [3] that describes the infra-red dynamics of the fermionic theory when N f > 2|K|. According to that scenario there is a window, for 2|K| < N f < N * , where the theory exhibits a phase of flavor symmetry breaking: U(N f ) → U(N f /2 − K) × U(N f /2 + K) in the unitary case and USp(2N f ) → USp(N f − 2K) × USp(N f + 2K) in the symplectic case. N * is a critical number of flavors whose precise value is currently unknown. The evidence in favour of this scenario includes the matching of anomalies and consistency under several RG flows [3].
For N f > N * it is conjectured [3,4] that the theory flows to some IR fixed point that does not exhibit flavor symmetry breaking.

Benefits of an ultra-violet embedding
In this paper we provide new evidence in favour of the above scenario for QCD 3 (bosonization, symmetry breaking, CFT) by embedding the IR dynamics of QCD 3 in an ultra-violet (UV) YM-CS theory, which is a 3d cousin of a 4d non-supersymmetric orientifold QCD theory [6]. The main part of the paper will focus on the case of USp(2N c ) gauge group. In that case, the UV embedding involves a USp(2N c ) YM-CS theory at bare level 2k coupled to a real scalar field in the 2-index symmetric representation ('scalar gaugino'), a Dirac fermion in the 2-index antisymmetric representation ('gaugino'), 2N f scalars in the fundamental representation ('squarks') and 2N f fermions in the fundamental representation ('quarks'). We argue that there are regimes when k = 0 where the quantum effects lift the scalars and the IR behaviour is dominated by standard QCD 3 physics. The level K that appears in KS [3], e.g. in eq. (1.3), is related to k via the relation (1.4) The combination k + 1 − N c arises naturally when we integrate out the antisymmetric gaugino and will appear frequently in our formulae. It is therefore convenient to introduce the integer κ ≡ k + 1 − N c , (1.5) in terms of which K = |κ| − N f 2 . The great advantage of the additional degrees of freedom in the orientifold QCD theory is that they allow us to formulate a non-supersymmetric version of Seiberg duality. For N f = 0 this duality reduces to level-rank duality [5]. For general N f the dual 'magnetic' theory is a USp(2N f + 2κ) gauge theory whose details will be described in detail in section 2. 4d versions of the duality were studied in [6]. In this paper we will show that the finite N f duality passes the standard checks of consistency under RG flows, global symmetry and 't Hooft anomaly matching. Further favourable evidence is provided by string theory.
We will return to some of the salient features of the string theory embedding in a moment.
The electric-magnetic duality operates when the rank of the dual gauge group is positive, . For values of k, N c , N f outside this window there is no magnetic dual and one has to analyse separately the strong coupling dynamics of the electric theory. It is unclear what the infra-red physics of the theory are in this regime. In supersymmetric analogues, e.g. in 3d N = 2 SQCD theories, the supersymmetric vacuum is lifted by non-perturbative effects when the dual rank becomes negative. When the dual rank vanishes the dual theory is a theory of free chiral multiplets.
In this paper we will work exclusively in the regime where the dual magnetic description exists. The magnetic description will provide an illuminating perspective on the dynamics of the electric theory. In particular, we will find, under certain assumptions, that the magnetic theory leads to a rather natural universal description for all the phases of QCD 3 outlined above. We will show that when N f ≤ 2|K| bosonization emerges naturally in the IR via magnetic squark condensation; a mechanism that reminds of monopole condensation and the dual Meissner effect in four dimensional physics. We will not be able to prove conclusively the existence of magnetic squark condensation, but we will provide evidence indicating that it is a viable possibility and that it leads to a suggestive consistent synthesis of known results.
This mechanism provides a new explanation of 3d bosonization. Unlike previous explanations based on deformations of supersymmetric 3d Seiberg dualities and mirror symmetry (see e.g. [7] for beautiful work in these directions) in this mechanism bosonization arises dynamically in the infrared as a consequence of a non-supersymmetric version of Seiberg duality. These effects describe the part of the phase diagram denoted as I in the diagrams of Fig. 1.
Once magnetic squark condensation is assumed the phase of symmetry breaking for N f > 2|K| follows naturally. There are two regions in the phase diagram of orientifold QCD 3 , regions II and II ′ in diagram B of Fig. 1, that describe in the IR the physics of Interestingly, the magnetic description in regions II and II ′ is not identical. Consider first the situation in region II ′ where −N f < κ < 0. In this case squark condensation leads to a complete Higgsing of the gauge group leaving behind a sigma model of Goldstone bosons for the dynamically broken symmetry. The magnetic theory provides an explicit description of how the sigma model arises.
Region II of the phase diagram of orientifold QCD 3 refers to the parameter regime The dictionary (1.4) implies that this region describes the same IR physics as QCD 3 with N f > 2|K|. Naively, it looks like a natural continuation of the bosonization regime where squark condensation leads to a theory of 2N f bosons. It is obvious, however, that this theory cannot be the same as the bosonic dual of region I, because bosonization is inconsistent for N f > 2|K|, which is also clear in the behaviour of the electric and magnetic orientifold QCD theories under mass deformations. This suggests that N f = κ is a critical point above which the magnetic theory reverts to the symmetry breaking phase.
We propose a particular scenario for the mechanism behind this phase transition. This scenario is consistent with expectations about the massive deformations of the fermionic (electric) theory and predicts naturally a phase transition to a topological sector as the mass of the fermions is increased. A dual bosonic formulation appears at the transition point as anticipated by KS [3].
The symmetry pattern for sufficiently large N f persists as long as the magnetic squarks are tachyonic and condense. Competing effects in the one-loop mass normalisation of the squarks suggest the possibility that when N f is large enough, N f ≥ N * for some N * (k, N c ), the squarks become massive. Depending on the mass squared of the elementary magnetic meson fields, Seiberg duality suggests a specific description for the IR theory. We argue that the most likely scenario is one where the elementary magnetic mesons are tachyonic.
The condensation of the elementary magnetic mesons leads to an IR theory of free fermions and light mesons, a scenario that agrees with the large-N f results in [4].

Lessons from string theory
It is useful to consider a further UV embedding of the orientifold QCD theory into a non-supersymmetric brane configuration in string theory. The immediate benefits of this embedding are: (a) String theory is a concrete guide to the tree-level Lagrangian of the magnetic dual and provides additional evidence in favour of the non-supersymmetric Seiberg duality that is the center-point of this paper.  [3]. Region I is the phase of bosonization, region II the phase of symmetry breaking and region III a phase with an IR CFT description. The curve separating the regions II and III occurs at some N * which is potentially a function of both κ and N c . The wiggly features of this curve are not a statement about its actual shape, but rather a symbolic depiction of our ignorance about its precise form. A constraint on the shape of this curve was determined in Ref. [3].
Diagram B represents the phases of orientifold QCD 3 that arise naturally (under a certain set of assumptions) from the magnetic description of the theory. With the exception of the region κ < −N f where no Seiberg duality is available the remaining regions I, II/II ′ and III, are expected to describe the same IR physics as the corresponding regions in diagram A. [9]. In order to break supersymmetry we consider a combination of an O3 plane, anti D5 branes and anti D3 branes suspended between two 5-branes, one of which is a bound state of an NS5 brane and anti D5 branes. 2 The mutual presence of anti-branes with an orientifold plane breaks the N = 2 supersymmetry completely. Supersymmetry is restored asymptotically when N c , N f and k are taken to infinity. This feature is useful. The large-N regime is a technically convenient regime where supersymmetry is softly broken by 1/N effects. 3 The broken supersymmetry leads to non-trivial potentials between different components of the brane configuration. On the electric side the potentials are attractive and lead to a stable brane configuration. On the magnetic side, however, the potentials can be repulsive, leading to open string tachyon condensation and brane reconnection between the flavor and colour branes. The rich phase diagram that arises in this manner is literally a web of phases driven by the presence or absence of open string instabilities. The result translates in gauge theory either as bosonization or dynamical symmetry breaking, or a CFT.

Outline of the paper
The paper is organised as follows. In section 2 we present the orientifold field theories of interest, describe how they are embedded in suitable brane configurations in ten-dimensional type IIB string theory, and formulate the non-supersymmetric electricmagnetic duality that they are conjectured to obey. The evidence in favour of the duality is summarised in a separate subsection.
In section 3 we discuss perturbative non-supersymmetric effects in both the electric and magnetic gauge theories. We discuss the expected behaviour of the electric theory in the IR and its relation to QCD 3 . A detailed description of the IR physics from the magnetic theory point of view is relegated to the subsequent sections.
In section 4.1 we consider the parameter regime where k + 1 − N c ≥ 0. In that case of the magnetic squarks can lead to 'full color-flavor recombination' (color-flavor locking) on the magnetic side. This is translated to 3d bosonization of the electric side. The magnetic description does not provide only the correct matter content for bosonization, but also the requisite USp(2N f )-invariant quartic interaction needed for the Wilson-Fischer fixed point.
It is argued that this picture is consistent for N f ≤ k + 1 − N c , but fails for larger values of N f , where the most natural scenario, in accordance with expectations from QCD 3 , is a scenario of a symmetry breaking phase.
The other regime is when −N f ≤ k + 1 − N c < 0. In that regime the squark condensation Higgses the magnetic gauge group completely and leads to 'partial color-flavor recombination'. Section 4.2 provides a detailed description of the effects that take place in this regime. On the electric field theory side these effects are translated as dynamical symmetry breaking. The Nambu-Goldstone bosons are identified as massless modes in the low-energy spectrum of the open string theory on the branes and as massless modes after squark condensation in the magnetic Lagrangian.
In section 4.3 we discuss the possibility of a phase where the magnetic squark becomes massive. This leads to a phase with a CFT description in the IR in accordance with field theory expectations in QCD 3 [4]. The magnetic dual provides a specific prediction for the IR CFT. This possibility requires the existence of a new critical number N * with the new phase being realised when N f ≥ N * .
In the above analysis it is always assumed that the bare CS level of the orientifold QCD theory k is non-zero. When k = 0 the IR physics is dominated by the YM interaction. The IR physics of this theory, which besides the effects of the fundamental fermions involves non-trivial dynamics from Dirac fermions in the 2-index anti-symmetric representation as well as additional scalar fields, is an interesting question that has not been explored in the past. This situation is discussed in section 5. A recent discussion of CS theories with matter in the adjoint representation but no fundamentals can be found in [10].

Open problems
In this paper we focus on the case of (orientifold) QCD 3 theories with symplectic gauge group. This choice is dictated by the fact that this case is the most straightforward one from the viewpoint of the string theory construction that underlies part of this work. Since the analysis of unitary groups with even rank, U(2N), shares many similarities with the analysis of the USp(2N) case (and is related to it by a non-perturbative planar equivalence), it is rather natural to put forward a corresponding picture for U(2N) (orientifold) QCD 3 . Preliminary comments in this direction are summarised in appendix B.
The formulation of non-supersymmetric Seiberg duality for orientifold QCD 3 theories with general unitary gauge groups and orthogonal groups remains an open problem. We hope to return to these cases in a future publication.

Summary of conventions
In what follows when we mention the level of a UV gauge theory we will always refer, unless otherwise stated, to the bare level k (or 2k in the USp gauge theories). k is always an integer -it is the same integer (or related to the integer) that appears in the 5-brane bound states in our brane configurations. This convention is to be contrasted with other notation in the literature, e.g. [3] that is closely related to our discussion, where the quoted When we integrate out massive particles the CS level shifts. In the YM-CS regularisation (which we follow) only the fermions contribute to the shift. We will use conventions natural in string theory where the integration of a single Dirac fermion with positive mass m leads to the IR level Nf anti D3 branes Nc anti D3 branes O3 Another useful fact in our discussion is the fact that when we integrate out a gaugino (in the 2-index antisymmetric representation) in the USp(2N) theory we obtain the CS level shift

Non-supersymmetric USp Seiberg duality
The gauge theories of interest can be phrased independently of string theory. Nevertheless, since string theory provides a convenient organising principle for the non-supersymmetric duality of interest we will present in parallel the gauge theories in question and their string theory embedding. We consider the USp YM-CS theories that describe the IR physics on the Hanany-Witten brane configurations depicted in Figs. 2, 3.

Electric theory
Let us start with a description of the electric theory. The brane configuration that engineers the theory consists of N c anti D3 branes suspended between an NS5 brane and a tilted (1, 2k) fivebrane. In addition there is an orientifold O3. The orientifold plane changes from O3 + to O3 − when it crosses the fivebrane. This is identical to the brane configuration of [5]. In addition, we add N f anti D3 branes which are attached to the right of the NS5 brane and end on D5 branes. The resulting brane configuration is depicted in • a tilted (1, 2k) fivebrane along 012(37)89 directions. The latter fivebrane is a bound state of an NS5 brane and 2k anti-D5 branes, it is tilted in the (37) plane by an angle θ such that tan (θ) = −2g s k. With this choice of angle, we would preserve N = 2 supersymmetry, had we not had an orientifold. The minus sign reflects the presence of anti-branes in the construction.
• an O3 plane along the 0126 directions, which is O3 − at x 6 = ±∞, • N c (color) anti D3 branes (and their mirrors) along the 012|6| directions (|6| denotes that the branes have a finite extent in the 6-direction), The classical Lagrangian of the electric theory is The gauge part is Yang-Mills-Chern-Simons theory at level 2k D µ is the standard gauge covariant derivative and the contraction of spinor indices has been kept implicit. For the matter part The pseudo-reality condition imposes on bosons and fermions with Ω,Ω the USp(2N c ), USp(2N f ) symplectic tensors.

Magnetic theory
We define the proposed magnetic dual as the theory that describes the infra-red dynamics of the brane configuration in Fig. 3, which results from that in Fig. 2 by swapping the NS5 and (1, 2k) 5-branes across the x 6 direction. In the presence of the orientifold k + 1 additional D3 branes are created during this swap. As a result, we find a part of the Lagrangian includes the same terms for the magnetic squarks φ i , and quarks ψ i as the electric theory. In addition, it includes kinetic terms for the mesons M [ij] and mesinos χ (ij) and a set of interactions that constitute the non-supersymmetric orientifold version of the supersymmetric cubic superpotential between mesons/mesinos and squarks/quarks.
The complete matter Lagrangian is (2.11) and the total Lagrangian y is a coupling with mass dimension 1/2 that appears in front of the cubic superpotential interactions in the supersymmetric version of the theory. In the non-supersymmetric theory at hand the RG flow will not respect the relations between the couplings that appear in front of the terms in the last line of (2.11). The expression (2.11) is written here only as a specific bare Lagrangian that follows from its supersymmetric ancestor by appropriately modifying the representations of certain fields.

Evidence for duality
We claim that the above electric and the magnetic theories form a Seiberg dual. Apart from the argument in string theory based on swapping fivebranes, in appendix A we provide standard evidence in the form of 't Hooft anomaly matching as well as a duality after deformations and RG flows.
Note that at large N c , N f and k the two theories become supersymmetric. The reason is that in this limit there is no difference between the two-index symmetric and the twoindex antisymmetric representations. In the brane picture this statement translates to the fact that the orientifold (the Möbius amplitude) is a 1/N effect. Therefore, in the large N limit the electric and magnetic theories are dual to each other as a supersymmetric pair similar to the one analysed in [9].
There is another argument, from string theory, in favour of the duality, due to Sugi- We obtained the non-supersymmetric duality between the electric USp(2N c ) theory and the magnetic USp(2(N f + k − N c + 1)) theory from the SUSY pair. It will be interesting to repeat this exercise in a pure field theory language.

Perturbative dynamics of the electric and magnetic theories
Given that the electric and magnetic theories are non-supersymmetric we anticipate potentials for the various scalars. 6 The potentials are due to non-planar effects (1/N effects) and they vanish in the large-N limit where both the electric and magnetic theories become a dual pair with N = 2 supersymmetry. In some cases the field theory potentials have an interpretation as potentials between branes in the brane configuration. The potentials depend on the UV cut-off of the theory. Within field theory we can remove this dependence by renormalisation. The embedding in string theory, however, provides a natural UV cutoff, Λ 2 = 1 α ′ , and a physical meaning to the potentials. In all cases the effects that we consider are due to the difference between the representations of bosons and fermions. Similar dynamics and considerations were involved in a proposal of a non-supersymmetric S-duality [11].
Consider a scalar propagator: in perturbation theory the difference between a bosonic loop and a fermionic loop will produce either a massive scalar or a tachyonic scalar, depending on whether there are more bosonic degrees of freedom or more fermionic degrees of freedom, as depicted schematically in Fig. 4 below.
If a scalar is massive it will not acquire a vev and the perturbative expansion near the origin (zero vev) is stable. If a scalar is tachyonic we anticipate a new minimum where a vev is acquired and symmetries (global or local) may be broken.

Bosonic loop
Fermionic loop − Fig. 4: Perturbative contributions to a scalar mass.

Electric theory
The electric theory contains two scalars, the squark Φ and the scalar gaugino σ.
The scalar gaugino couples to itself, to the gauge boson and to the fermionic gaugino λ. The generated mass due to the bosonic and fermionic loops is namely M 2 σ > 0. This suggests aht the point σ = 0 is stable, and moreover, that the field σ decouples from the low energy dynamics.
The potential V (σ) ∼ M 2 σ 2 has an interpretation in the brane picture: the N c colour branes are attracted to the orientifold plane. The brane configuration is hence stable (non-tachyonic).
A similar analysis can be performed for the squark Φ. The squark couples to the "gauge multiplet", namely to A µ , σ, λ. Since there are more bosonic than fermionic degrees of freedom perturbation theory suggests that M 2 Φ > 0. The squark therefore decouples from the low energy physics.
As a result, the low energy field theory contains a USp(2N c ) gauge field A µ , a gaugino λ and 2N f quarks Ψ.
When k = 0 there are Chern-Simons terms. The Chern-Simons terms provide a mass M CS = g 2 k to the gauge field and to the gaugino. We therefore anticipate that the IR dynamics will be dominated by the topological Chern-Simons theory coupled to the quarks. 7 In that sense for k = 0 the electric theory is very similar to QCD 3 .
The case k = 0 is special, as there is no Chern-Simons term. The IR theory involves 7 Integrating out the gaugino to obtain QCD 3 in the IR is most straightforward in the semiclassical regime k ≫ 1. Away from this regime the relation with QCD 3 in the IR is harder to establish, but the expectation is that the IR phases are the same even for finite k unless something drastic happens in the infrared behavior of OQCD 3 . The breakdown of the magnetic description outside the window of Seiberg duality could be such an effect. Inside the window of Seiberg duality with k = 0 there are currently no indications of drastic effects that would lead to significant deviations from QCD 3 dynamics. the strong coupling dynamics of the Yang-Mills interaction between the gauge field, the gaugino and the quarks.

Magnetic theory
Similarly to the electric theory, the scalar gaugino of the magnetic theory acquires a mass and decouples. The color branes are therefore attracted to the orientifold plane.
The dynamics of the squarks and the mesons is more complicated. Let us focus on the squarks. At the one-loop level there are effects due to the coupling with the gauge multiplet and effects due to the coupling with the meson multiplet. Let us denote the magnetic gauge coupling by g m and the coupling to the meson multiplet by y.
There are more bosonic than fermionic degrees of freedom in the gauge multiplet and more fermionic than bosonic degrees of freedom in the meson multiplet. As a result, At the one-loop level the two effects compete and the squark may become massive or tachyonic. We will consider both possibilities and identify the various phases associated with each one of them. We note that at large-k the gauge field becomes very massive and decouples, therefore we anticipate that in this limit the dynamics is dominated by a tachyonic squark.
The magnetic theory includes a coupling of the form If the meson field acquires a vev of the form M ik M ij = v 2 δ j k the squark field becomes massive. If the squark field acquires a vev of the form φ a j = vδ a j , and if flavor symmetry is not broken (bosonized phase), the mesons become massive. We propose as the most likely scenario that in all phases In the following sections we will discuss several phases of the magnetic theory, their realisation in the brane picture and their relation to the electric theory.

Bosonization
We focus on the magnetic description of the orientifold QCD 3 theory. The first region of parameter space under consideration is the region where κ ≡ k + 1 − N c ≥ N f , namely region I in diagram B of Fig. 1. The rank of the magnetic gauge group,Ñ c = N f + κ is automatically positive in this regime. The one-loop computation of the previous section suggests that the 2N f squarks are tachyonic, at least for sufficiently small N f at fixed κ.
Henceforth, we will operate under this assumption for the whole region I in diagram B of After reconnection 2N f D3 branes are stretching between the NS5 brane and the 2N f D5 branes, and 2κ color D3 branes are stretching between the NS5 brane and the (1, 2k) fivebrane, see Fig.5.
Since the 2N f reconnected D3 branes have no common directions with the fivebranes on which they end their low energy spectrum does not have a gauge field, gaugino, mesons or mesinos. The only low lying modes arise from the strings on the 2κ color D3s and the strings stretching between the color D3s and the reconnected D3s. The theory on the color This is one of the two possible quartic scalar interactions that are USp(2N f )-invariant (see [2] for related comments; the interactionΩ ij φ aj Ω ab φ bkΩ kl φ cl Ω cd φ di is the second possibility).
The renormalization group will naturally induce this second interaction, as well as the USp(2N f )-invariant mass term a,i |φ ai | 2 . The match of massive deformations in the To summarise, after integrating out the gaugino, in the IR of the electric theory we have In the IR of the magnetic theory we have

Symmetry breaking
Next we discuss the regions II and II ′ in the phase diagram B in Fig. 1. We will argue in favour of a symmetry breaking scenario in both regions. Region II is characterised by the inequalities N f > κ > 0 and region II ′ by the inequalities −N f < κ < 0. It is convenient to consider first region II ′ .

Region II ′
κ < 0 means that there are more flavor D3 branes than color D3 branes. As a result, at most 2N f + 2κ D3 branes can reconnect. The color group is fully Higgsed and one is left with 2|κ| flavor D3s stretching between the (1, 2k) and the flavor D5s over an O3 − plane in addition to the reconnected D3s, see Fig.6.
the orientifold and their global symmetry remains SO(2|k|).
At the same time there is a global SO(2(N f +κ)) symmetry associated with the 2(N f +κ) D3s. We deduce that in string theory there is a breaking of the global SO(2N f ) symmetry of the form In the IR field theory the global symmetries enhance and it is not hard to show that the color-flavor locking vev of the squarks breaks the global USp(2N f ) in the following way This is exactly the same pattern of symmetry breaking anticipated in [3] with the identification (1.
This identification is consistent with massive deformations of the theory.
Consequently, the IR physics of this phase is described by the coset σ-model of the Nambu-Goldstone modes associated with the breaking (4.5). There are massless Nambu-Goldstone bosons, which arise as massless modes on the open strings stretching between the 2|κ| D3-branes and the 2(N f + κ) D3-branes, see Fig.6. As a check, notice that the result (4.8) (derived from (4.5)) agrees with the counting of Nambu-Goldstone bosons in QCD 3 (4.7).
The order parameter for the breaking is φ ai Ω ab φ bj .

Region II
The region II, characterised by the parameter regime N f > κ > 0, is more intriguing.
On one hand, this is a case where, at first sight, all 2N f flavor D3s can reconnect with color D3s leading to an IR USp(2κ) theory coupled to 2N f bosons implying bosonization and no global symmetry breaking. On the other hand, the dictionary (1.4) implies that this phase should still be describing the symmetry breaking phase of QCD 3 with N f > 2|K| where bosonization is inconsistent. It is hard to believe that the association of the orientifold QCD 3 with QCD 3 breaks down in this region because we can go to a limit of large k where the gaugino becomes arbitrarily massive and can be integrated out safely. The only logical conclusion seems to be that something critical is happening to the brane configuration as one crosses the threshold N f = 2κ.
A natural guess is the presence of additional instabilities in the brane setup that describes the magnetic theory. Previously, in region I, it was sensible to expect that the brane reconnection leads to a stable vacuum. We would like to propose that the vacuum after brane reconnection is unstable, namely the 2N f scalars in the resulting IR description have a negative mass squared. In that case, it seems likely that open string condensation could proceed by annihilating the 2κ color D3s against another set of 2κ color D3s eventually leaving behind 2(N f − κ) reconnected D3s (between 2(N f − κ) D5s and the NS5) and 2κ flavor D3s (between 2κ D5s and the (1, 2k) bound state). This is identical to the symmetry breaking configuration in region II ′ .
As a check of this scenario let us consider a deformation of the configuration with  Fig. 6 of Ref. [3]. Similar statements can be made for the mass deformation represented by the brane move 2 in Fig. 7 in appendix A.
The above-mentioned checks imply an overall picture which is consistent with the proposed scenario of a transition in brane physics at N f = κ. It would be interesting to find further evidence in favour of this scenario with a more explicit computation in string theory.

Scenario of symmetry restoration
In QCD 3 it is expected [4] that there is some N * (k, N c ) such that when N f ≥ N * the IR CFT is interacting and there is no symmetry breaking. Can this possibility arise naturally in our framework? We would like to argue that this could be described in the magnetic brane setup by a regime where the magnetic squarks are massive.
Assume that quantum effects make the squarks massive (namely, their mass squared is positive). Integrating out the massive squarks we obtain

The special case of Yang-Mills theory without a Chern-Simons term
Consider the special case where k = 0. In the absence of a bare CS term the IR of the field theory is dominated by the Yang-Mills interaction. The theory is parity invariant classically. Due to the Vafa-Witten theorem [12] that states that parity cannot be broken spontaneously, the theory is also parity invariant quantum mechanically.
The brane dynamics is also expected to be different with respect to the case k = 0. Instead of having configurations with an NS5 brane and a tilted fivebrane we have configurations with parallel NS5 branes.
The magnetic theory is USp (2N hence brane dynamics implies that the bosonization phase cannot occur.
According to the philosophy advocated in this paper, whether flavour symmetry is broken or not depends on whether the magnetic squarks condense or become massive.
Brane dynamics suggests that if the squarks condense the flavour symmetry is broken . This pattern, however, is not consistent with the Vafa-Witten theorem, because such a breaking pattern does not preserve parity.
According to eq. (3.2) the squarks may become tachyonic or massive. For large enough k they become tachyonic. We propose that for k = 0 the squarks are massive.
When the squarks become massive flavour symmetry is not broken. The dynamics is the same as that of the case N f ≥ N * , which was described in section (4.3). This result is different from the one for QCD 3 with K = 0, as described in the literature [4]. For sufficiently small N f QCD 3 without a Chern-Simons term is expected to break the flavour symmetry in a pattern consistent with parity Integrating out the gaugino on the dynamical USp(2N c ) and USp(2Ñ c ) factors we recover level-rank duality in the IR. The match of the CS levels of USp(2N f ) on both sides requires The global symmetry that acts faithfully on local operators is USp(2N f )/Z 2 . This puts restrictions on the CS counterterms. Consistency with the Z 2 quotient on the electric side On the magnetic side it requires We notice that Since (N f + k + 1)(N f − k) ∈ 2Z identically, we deduce that the conditions (A.2) and (A.3) are automatically satisfied together, which is a good consistency check.

A.2. RG flows
In the tree-level orientifold QCD 3 theory we can consider four types of mass deformations. In this appendix we mostly ignore the potential quantum-induced instabilities.
When such instabilities exist we implement the deformations at the unstable vacuum. It is convenient to see first how the deformations of interest map between the electric and magnetic descriptions in the brane setup. See [9] for analogous deformations in a d = 3 N = 2 setup. The brane deformations are summarised pictorially in Fig. 7.
The first deformation, number 1 in Fig. 7, moves an image pair of D5s in the (45) plane. On the electric side this reduces N f → N f − 1 while N c → N c and k → k. On the The corresponding deformation gives a vev to φ a1 so that The vev of φ a1 breaks the magnetic gauge group,Ñ c →Ñ c − 1, removes one flavor (N f → N f − 1) and leaves k invariant as anticipated.
The second deformation, number 2 in Fig. 7, moves an image pair of D5s in the 3 direction dragging along a pair of flavor D3s stretching between a D5 and the NS5. On the electric side this reduces N f → N f − 1 while N c → N c and k → k. On the field theory side this involves equal and opposite sign masses for one flavor The scalars are not involved in this deformation. Indeed, (A.7) removes the fermions of one flavor and after integrating out all auxiliary fields we find a potential for the scalars φ a1 with a quartic and sextic term. There are two potential vacua for this potential. One at the origin and one away from the origin. The deformation number 2 in Fig. 7  This operation reduces N f → N f − 1 but N c → N c and k → k + 1. On the field theory side this operation involves giving same sign real masses to one flavor δL electric = mψ a1 ψ a1 + mψ a,N f +1 ψ a,N f +1 = 2mψ a1 ψ a1 . (A.8) On the magnetic side the magnetic quarks acquire corresponding masses and as these masses are sent to infinity N f → N f − 1,Ñ c →Ñ c and k → k + 1 consistently with the duality.

Appendix B. Unitary groups
In this section we propose a duality for a U(2N c ) orientifold QCD 3 theory. In order to have a string theory realisation of a non-supersymmetric theory and a Seiberg duality, we follow [5] where the N f = 0 case was considered (see also [14,15] for the analogous 4d theory).
The field theory lives on a type 0B brane configuration. The electric and magnetic brane configurations are identical to those we considered in section 2. The orientifold projection is special to type 0 strings: it decouples the bulk tachyon from the brane and projects out half of the RR fields [16]. 9 The matter content of the electric theory on the brane configuration is given in table 3. In the planar (large-N) limit the theory is equivalent to the corresponding N = 2 U(2N) gauge theory that lives on type IIB branes.
Note also that the U(2N f ) global symmetry is the symmetry of the IR theory. By swapping the fivebranes we arrive at the magnetic theory. Its matter content is given in table 4.
The dynamics of the theory based on unitary groups is similar to the dynamics of the theory based on USp groups. This is not surprising: there exists non-perturbative planar equivalence between USp(2N) and U(2N) gauge theories.
On the electric side the scalars (squark and scalar gaugino) acquire mass and decouple.
The CS level is shifted due to the antisymmetric gluino.
The IR levels of the electric theory 2K SU , 2K U(1) are given by thanks to the fundamental fermions and the antisymmetric fermion that shift the bare level 2k. Note that in the bosonized phase K U(1) = K SU + N c .
In the preliminary discussion of this appendix we single out the case with even color, 2N c and even flavor, 2N f , which proceeds as in the USp case.
When N f ≤ 2K SU the squarks condense, the Higgs mechanism takes place, and we find a dual bosonized theory of the form U(2K SU + N f ) with shifted levels When 2K SU ≤ N f ≤ N * the reconnected branes suggest that U(2N f ) → U(N f + 2K SU ) × U(N f − 2K SU ), in agreement with the conjecture by KS [3]. There are massless Nambu-Goldstone bosons that corresponds to massless modes of the oriented open strings stretched between the 2|κ| branes and 2(N f + κ) branes The case k = 0 (or K SU =0) is special. As in the USp theory we propose that there is no symmetry breaking and the theory is described by the magnetic theory of N f ≥ N * .
For N f ≥ N * the magnetic theory consists of a massless mesino and a light meson of mass square 1/N f .