4D/3D reduction of dualities: mirrors on the circle

We engineer a brane picture for the reduction of Seiberg dualities from 4D to 3D, valid also in the presence of orientifold planes. We obtain effective 3D dualities on the circle by T-duality, geometrizing the non-perturbative superpotential which is an affine Toda potential. When reducing to pure 3D, we define a double-scaling limit which creates a confining sector, giving a unified mechanism for the brane reduction of dualities.


Introduction
We construct the brane representations for the reduction of gauge theory dualities from 4D to 3D including orientifold planes.This analysis was started in [1] where it was shown how to translate the dimensional reduction of dualities in terms of Dand NS-branes in type iia supergravity.The brane picture has certain advantages over a pure field theory description.It gives a unified treatment of various dualities involving different gauge groups and matter content, as opposed to an ad hoc case by case analysis.This allows an algebraic description of the superpotentials involved in the effective 3D dualities on the circle.It allows also to reduce further to pure 3D dualities, by a double scaling on the relative positions of some D-branes and the radius of the circle.In this process an extra, 3D-confining, sector is created in the magnetic theories, reproducing the gauge theory duality in the pure 3D limit.
The main motivation behind our analysis is the similarity between the electricmagnetic duality discovered by Seiberg in [2] for four-dimensional sqcd and the three-dimensional dualities studied in [3,4].The dualities can be connected by dimensional reduction, as discussed in [5] (see also [6]).In a necessary intermediate step of this reduction one finds another duality for effective 3D theories.The dual theories include the effect of the finite circle through the contributions to the nonperturbative superpotential of the Kaluza-Klein (kk) monopoles acting as instantons.The limit to pure 3D dualities differs depending on the details of the gauge and matter content [5,7].
In this paper we study the brane construction of this reduction, giving a physical origin for the differences in the pure 3D limit.We obtain the effective 3D dualities by T-duality, where Euclidean D1-branes reproduce the non-perturbative effects.Equivalently these effects are captured by an algebraic formulation in terms of Sdual F1-strings.Configurations creating the monopole superpotential are classified by affine Dynkin diagrams and this superpotential is an affine Toda potential.We take the pure 3D limit by moving some electric flavor branes to the mirror point x • 3 , when sending the radius to infinity, see Figure 1.The magnetic dual is obtained by a Hanany-Witten [8] (hw) transition, generating a confining gauge theory at x • 3 .Its effect survives the limit and reproduces the superpotential of 3D dualities.
This construction is quite general and can be applied to many other field theories that admit a type iia description in four dimensions.For example, the cases with adjoint matter and quivers with product groups have been discussed in [1].These cases are very similar to sqcd, and thus do not make evident the generality of our brane reduction.Indeed, as observed in [5,7], different treatments of the reduction become necessary in presence of real gauge groups or tensor matter.In brane language these theories are obtained including orientifold planes.Orientifolds are straightforwardly incorporated in our picture.Indeed, when the theory is put on the circle, a second orientifold plane is generated after T-duality at the mirror point x • 3 [9,10].In the hw transition the orientifold, carrying D−brane charge, modifies the rank of the gauge groups both at x 3 = 0 and at the location of the confining sector x 3 = x • 3 .By considering various brane realizations with orientifolds we obtain the pure 3D limit without modifications of the procedure explained above.
The plan of this article is as follows.In Section 2, we recap the reduction of N = 1 dualities from 4 to 3 dimensions.In Section 3, we outline the brane realization of the dimensional reduction.We discuss the generation of an affine Toda potential on the Coulomb branch variables when the theories are studied on a circle and its relation to the second orientifold appearing in the T-dual picture.Moreover, we explain the double-scaling limit and the reduction to pure three-dimensional dualities.In Section 4 we apply the brane picture on symplectic gauge theories with fundamental, antisymmetric and adjoint matter.In Section 5 we study the brane set-up of unitary gauge groups with antisymmetric flavor.In Section 6 we study orthogonal gauge groups, our analysis is at the level of the local properties and the gauge algebra.We conclude in Section 7 by outlining some open questions.
Before taking the limit r → 0 one has to consider an effective 3D theory on a finite circle.In addition to the usual 3D superpotential generated by bps monopoles acting as instantons, there is an extra superpotential generated by the kk monopoles.This extra superpotential breaks the global symmetry U(1) A , anomalous in 4D but welldefined in 3D.Starting with electric-magnetic dual theories in 4D, the corresponding effective 3D theories give rise to well defined dualities on the circle [5].
One can generate pure 3D dualities by integrating out some matter fields with suitable real masses turned on.This limit requires mapping the (non-trivial) vacua of the dual phases.Let us give few details for the example of unitary gauge groups.Real masses for a pair of fundamental and anti-fundamental matter fields in the electric theory map to a non-trivial vev for the scalar field in the vector multiplet.This Higgses the magnetic gauge group, which is important for correctly reproducing the pure 3D duality in the ir.Other gauge and matter content needs a separate discussion.For example in Sp(2N) sqcd the electric and magnetic vacua both have a trivial structure.In the orthogonal case the 3D limit can be taken without assigning any large mass at all, while it is required to Higgs the magnetic theory.In this paper we study the brane picture of the reduction, giving a unified description also for the pure 3D limits.

Brane reduction of dualities: the general strategy
The brane description of the dimensional reduction has been started in [1].The four dimensional gauge theory is engineered by a type iia brane system of D4-branes stretched between NS-and NS -branes.If one space-like dimension (say x 3 ) is compact, the reduction discussed above is reproduced by T-duality along that circle.
The compact Coulomb branch is parameterized by the scalars in the vector multiplet which correspond to the positions of the D3-branes in that direction.The repulsive interaction between these branes can be understood in terms of Euclidean D1-branes stretched between the NS and the D3-branes.The extra superpotential on the circle is reproduced by the extra D1-brane stretched between the first and the last D3 on compact x 3 .In the field theory language the Euclidean D1-branes correspond to bps monopoles, the one coming from the extra D1 is the kk monopole.The repulsive interaction of the branes corresponds to an Affleck-Harvey-Witten [22] (ahw) superpotential in the D3 theory.
There is an algebraic approach which allows an elegant classification of this superpotential.
• The fundamental (in the sense of [23,24]) bps monopoles are labeled by the simple co-roots of the Lie co-algebra.For unitary gauge groups G this corresponds to placing the i-th D1-brane between the i-th and the (i + 1)-th D3-brane (for i = 1, . . ., rank(G)).It is useful to study the S-dual configurations where D1-branes become F1-strings.In this picture the D3-branes are still distributed on the circle and connected by F1-strings, as depicted in the upper left corner of Figure 2.
The spectrum of the allowed bps F1-strings is given by the simple roots of the corresponding Lie algebra [10,25].For a unitary gauge group the simple roots of the A N series correspond to σ i − σ i+1 , i.e. to the difference between the positions of two consecutive D3-branes.We can include real gauge groups by adding orientifold planes (see Appendix B for details).The allowed spectrum of F1-strings is again given by the corresponding Dynkin diagrams, classified by the B N , C N and D N series.
• The picture incorporates very naturally the kk monopoles due to the compact direction.It turns out that the extra F1 string, which winds around the circle connecting, for unitary G, the 1st and the N-th D3 brane, can be accounted for by extending the Dynkin diagram to its affine version.Summing the contributions from the bps and the kk monopoles we obtain the superpotential on the compact Coulomb branch, finding an affine Toda potential for the associated affine algebra [26] where Σ = σ/e2 3 + iφ, α i are the simple roots and α * i are the associated co-roots.The extra simple root α 0 corresponds to the kk monopole and the corresponding contribution to (3.1) is commonly referred to as η-superpotential.
• In the unitary case, we have the affine algebra A N , where the extra simple root is associated to the combination σ N − σ 1 (see Figure 2).For SU(N) theories1 the superpotential associated to this diagram is where Y i = e (σ i −σ i+1 )/e2 3 +i(φ i −φ i+1 ) .The last term in (3.2) breaks explicitly the U(1) A symmetry in the three-dimensional field theory.This symmetry is associated to the rotation in the (4, 5) plane in the brane picture.The geometric realization of the breaking of this symmetry for compact x 3 has been discussed in [1].
• A similar analysis holds for real gauge groups, realized by including O3 or O5 planes.Let us first discuss the configurations with O3 planes on R 3 × S 1 .As reviewed in Appendix B the four differently charged orientifolds O3 + , O3 − , O3 + and O3 − project a unitary group to SP(2N), SO(2N), SP(2N) and SO(2N + 1) respectively.In presence of a compact direction orientifolds come in pairs and here 2 we have six different such pairs [10].Brane configurations with (O3 − , O3 − ), ( O3 − , O3 − ) and (O3 + , O3 + ) are associated to affine Dynkin diagrams.The other pairs (O3 + , O3 − ), (O3 + , O3 − ) and ( O3 − , O3 − ) correspond to twisted affine Dynkin diagrams.In this paper we are interested only in the "affine" pairs, since those are obtained by a T-duality from type iia configurations with O4-planes.These cases will be described in detail in the following sections.
A similar discussion holds with O6-planes, while the effect of the orientifold charge on the projection is exchanged.O5 + is associated to an SO(N) and O5 − to an Sp(2N) gauge group.The pairs (O5 ± , O5 ± ) are obtained by T-duality from a type iia configuration with O6 ± planes.
We can couple matter fields to the four dimensional gauge theory by adding stacks of F D6-branes in the type iia set-up.In the T-dual frame they become D5-branes.If we start in type iia with two Seiberg dual brane configurations, the T-duals correspond, at low energy, to the effectively 3D Seiberg-like dualities with η-superpotential.
Eventually we want to obtain pure 3D dualities from the brane picture.We take the double-scaling limit discussed in the introduction on the electric side.The magnetic theory is obtained by a hw transition swapping the NS5-branes in this configuration.
More precisely, in the electric theory, we move one stack of D5-branes clockwise and another one counterclockwise on the circle, until they reconnect at the mirror point x • 3 .When taking the limit r → 0 the D5-branes at x • 3 correspond to a set of massive singlets.In the magnetic theory, a D3 is created whenever a D5 crosses an NS in the hw transition, generating an extra gauge theory at x • 3 .This reproduces the Higgsing of the magnetic field theory discussed in Section 2. This sector is confining and can be effectively described by massless singlets, interacting through an ahw superpotential with the theory at x 3 = 0.The interaction survives the r → 0 limit and identifies the singlets of the confining sector with the electric monopoles.
In the rest of this paper we illustrate the generality of this picture, considering the effects of orientifold planes.Their D−brane charge modifies the standard brane creation effect in the hw transition.Nevertheless, the extra sector at x • 3 in the magnetic theory remains confining, allowing us to adopt the above picture to reduce dualities with orientifolds from 4D to 3D.

Sp(2N) theories
In this section we discuss the reduction of the duality for Sp(2N) with 2F fundamentals.An Sp(2N) gauge theory with 2F fundamentals3 and global symmetry gauge theory with 2F dual fundamentals q and a meson M with superpotential W = Mqq.This duality was first presented in [27].The U(1) A symmetry is anomalous at the quantum level.We present the global charges associated to this symmetry because it is quantum realized in the three-dimensional case.The field content is given in Table 2.

Brane description
There are two ways to represent this theory, by either considering an O4 + -plane or an O6 − -plane.These two constructions give rise to similar theories, that differ for the representation of the matter fields under the global symmetries.When modifying the theory (by allowing larger numbers of NS-branes) the two constructions give rise to different two-index matter fields, adjoint or antisymmetric.We will study both possibilities.
• In the O4 + -plane case, the brane setup is summarized in Table 1 and Figure 3.In this case we consider a stack of 2N D4-branes and an O4 + -plane stretched between an NS and an NS -brane.We consider also 2F D6-branes on the NS -brane.At brane level this theory as an SO(2F) global symmetry, while on the field theory side there is the enhancement to SU(2F).This is similar to the usual doubling of the global symmetry for unitary gauge groups.The dual theory is obtained by an hw transition that exchanges the NS and the NS -branes.In the hw transition every time that a D6-brane crosses a non-parallel fivebrane, a D4-brane is generated.This brane creation mechanism in necessary to preserve the linking number [8].In presence of an orientifold the linking number is modified.For example an O4 + has to be treated as a stack of −4 D6-branes [28].After the transition we obtain the dual picture, in which the net number of D4-branes is 2(F − N − 2).
• A similar theory can be constructed by using an O6 − -plane.Consider two NS5branes, 2N D4s, 2F D6's and an O6 − -plane as in Figure 4(a).If all the NS5-branes are parallel, the system has N = 2 supersymmetry.The orientifold projects the SU(2N) gauge group to Sp(2N) (where, as usual, Sp(2) SU( 2)).The theory has an SU(F) global symmetry with F flavors.We expect this symmetry to be enhanced to SU(F) 2 .Here we rotate the NS5-branes and the D6'-branes by an angle θ as in Figure 4(a), and we have two stacks of NS ±θ and D6 ±θ .For generic angles the N = 2 adjoint is massive.If θ = π/2 the orientifold is parallel to the NS ±θ -branes and this field is massless and has to be considered in the low-energy spectrum.We will come back to this configuration later.This model (for θ = π/2) has a dual description as discussed above.In this case the O6 − behaves like a stack of −4 D6-branes in the hw transition.The brane picture becomes the one shown in Figure 4(b) where the dual gauge group is again Sp(2(F − N − 2)).

O3-planes
Let us begin with the reduction of the duality with an O4 + -plane.The threedimensional system is obtained by compactifying the x 3 -direction and T-dualizing.
The NS-branes remain invariant while the orientifold becomes an pair of (O3 + , O3 + )planes.We can study the properties of the Coulomb branch by looking at the spectrum of bps F1 strings as explained above.In this case the superpotential can be read off from the top half of Figure 5, where Y i = e (σ i −σ i+1 )/e 2 3 +i(φ i −φ i+1 ) and Y N = e 2(σ N /e 2 3 +iφ N ) .The extra root in the affine case is proportional to the variable Y 0 = e 2(σ 1 /e 2 3 +iφ 1 ) (shown in blue on the botton half of Figure 5) and it gives the η-superpotential The same result is obtained after the hw transition.
Now we want to flow to the Aharony duality.We start by considering 2(F + 1) D5-branes in the electric theory.We rotate two D5-branes on the circle and reconnect them on the other side of the circle.Since the D5s intersect the NS-brane in this configuration, there are no massless fields in this extra sector.If we take the r → 0 limit on this configuration, we obtain an Sp(2N) theory with 2F fundamentals.
Next we turn to the dual theory.In this case if we perform a hw transition, there are 2(F − N − 1) D3s at the origin.On the other side of the circle the two D3s created by the D5 crossing the NS-brane are destroyed by the extra orientifold plane located there.This example shows one of the general aspects of our analysis.In principle it is not necessary to reconnect the D5-branes at x • 3 .For example for unitary gauge groups the double scaling was realized by putting the D5-branes at 3 [1].Here avoiding the orientifold to create a negative number of D3-branes in the hw transition we have to reconnect the D5s at x • 3 .In the rest of the paper we will always follow this strategy.
The final configuration is represented in Figure 6.In this case, even if there is no i.e. the low-energy description of the η-superpotential.In the large-mass limit the effect of this interaction has to be considered.This reproduces the field theory expectation: the dual theory is an Sp(2(F − N − 1)) theory with 2F fundamentals, an antisymmetric meson M and superpotential where we identified the broken component of the electric singlet M that parametrizes a direction in the dual Higgs branch, with the electric monopole Y that parameterizes the Coulomb branch of the electric phase.This is commonly the case when dealing with mirror symmetry and in fact the electric singlet describes the Higgs branch of the dual phase, i.e. the Coulomb branch of the electric theory.

O5-planes
Also in the case of the O6-plane realization one can reduce the duality to three dimensions by compactifying the x 3 -direction.After T-duality the type iib system contains a pair of O5-planes and describes a theory with the same η-superpotential as above.By considering F + 2 flavors and by integrating out of them we recover the usual Aharony duality.At the brane level this is obtained by introducing F + 2 D5 ±θ .We introduce real masses as in the construction with the O3-plane.The orientifold identification is however different: in this case we have a unitary symmetry.Moving a pair D5 ±θ along x 3 gives a mass to one flavor.One can flow to Aharony duality by taking the double scaling limit as above (see ).Let us explain the duality in this case.First we move the D5-branes in the x 3 -direction, assigning the real masses.We reconnect them on the other side of the circle.They reconnect at x 3 = x • 3 , where the second orientifold plane is located.The extra sector does not have massless degrees of freedom, and we can take the r → 0 limit in this case.We obtain a three-dimensional Sp(2N) theory with 2F flavors.Now we can turn to the dual picture, by exchanging the NS ±θ -branes.The D3s are created when the branes cross each other.While at the origin the orientifold cancels two D3s every time an NS5-brane crosses it, the net effect on the D3s at 3 is the absence of branes in the gauge theory.The final configuration is reproduced in Figure 7.
Like in the case with O3-planes, here we have an extra sector with massless singlets (coming from the original mesons).The r → 0 limit has to be taken by considering the effect of this sector on the Sp(2(F − N − 1)) theory.This is the same mechanism introduced above: the η-superpotential is absorbed in a scale matching, the meson couples with the magnetic monopoles, and in the final three-dimensional dual theory the extra interaction between the electric and magnetic monopoles takes place.

Sp(2N) with antisymmetric matter
An Sp(2N) gauge theory with F flavors Q and Q and an antisymmetric field A, with superpotential W = Tr A k+1 is dual to an Sp(2(k(F − 2) − N)) gauge theory with F flavors q and q, an antisymmetric a and superpotential W = Tr a k+1 + ∑ k−1 j=0 M k−j−1 qa j q, where M j = QA j Q is the generalized meson, with j = 0, . . ., k − 1.This duality was first presented in [29].
In this case we consider two stacks of k NS ±θ -branes.The gauge symmetry is broken by separating them along the directions 4 and 5 and leads to a polynomial superpotential for the antisymmetric field A. The electric theory is broken to x 6 x 4,5 k while the magnetic one becomes where r i = F − r i − 2 (see Figure 8).
In this case we can perform the reduction on each sector.The bare monopoles associated to each Sp(2r i ) factor recombine, through the scale matching relation, with the antisymmetric field when the superpotential deformations are turned off.This correspond to recombining the NS ±θ -branes.
The theory on the circle can be further reduced to an Aharony-like duality by integrating out some matter fields.In the three-dimensional case we refer to the antisymmetric representation discussed in [19], obtained by combining the irreducible antisymmetric with a singlet.
If we consider (F + 2) flavors and integrate out two of them in each sector we arrive in the dual at a ∏ k i=1 Sp(2(F − r i − 1)) gauge theory.After reconnecting the branes, the dual theory is Sp(2(k(F − 1) − N)).As a check we consider (F + 2K) flavors and flow to a known duality.Integrating out 2K flavors after assigning them the same large real mass generates a Chern-Simons (cs) term.We arrive at the duality of Kapustin, Kim and Park [19] between4 Sp(2N) 2K and Sp(2(k(F + |K| − 1) − N)) −2K .

Sp(2N) with adjoint matter
An Sp(2N) gauge theory with 2F fundamentals and an adjoint field X, with superpotential W = Tr(X) 2(k+1) is dual to an Sp(2((2k + 1)F − N − 2)) gauge theory with 2F fundamentals, an adjoint Y and superpotential x 6 x 4,5 NS5' where Y is in the adjoint of the dual group and M j = QX j Q.This duality was first presented in [30].
The electric theory is represented by 2N D4-branes and an O4 − -plane stretched between 2k + 1 NS-branes and one NS .In addition, there are 2F D6-branes on the NS-branes.By separating the NS-branes along the (45)-plane we have a polynomial deformation in the adjoint X.
In a generic vacuum the adjoint X acquires a vacuum expectation value.At matrix level there is a rank= 2r 0 sector at zero vev, and it gives rise to an SP(2r 0 ) gauge group.The other k rank= r i sectors, where the vev of the adjoint is non zero, give raise to a set of U(r i ) sectors.The ranks are chosen such that ∑ k i=0 r i = N.This Higgsing corresponds to separating the D4-branes along the directions 4 and 5 in the brane picture, as in Figure 9.Eventually, in a generic vacuum, the gauge group is broken as in the electric theory and in the magnetic theory.At the brane level this dual description is obtained by first separating the NS-branes, then performing the hw transition and eventually reconnecting the NS5s, see Figure 9.
In the case of Sp(2N) gauge theories with 2F fundamentals and adjoint matter with superpotential W = Tr X k+1 one can perform the reduction in the Sp(2r 0 ) and in the U(r i ) sectors separately.In each sector, an η-superpotential is generated.By reconnecting the branes and using the scale-matching relation one can identify the bare monopoles of the theory with the product Sp(2r 0 ) × U(r i ) with the dressed monopoles of the Sp(2N) theory.In the dual case the situation is similar.First one dualizes each sector, obtaining Sp(2(F − 2 − r 0 )) × ∏ U((F − r i )), then reconnects the branes and eventually uses the scale matching relation to recover the duality without the polynomial deformation in the adjoint field.We can flow to the Aharony-like duality.Let us consider 2F + 2 D5-branes in the electric phase.The dual gauge group is broken to where r 0 = F − r 0 − 1 and r i = F − r i .In the brane description, we move two D5-branes in each sector and perform the hw transition.The dual gauge group at This is given by imposing in the field theory description the correct vacuum structure preserving the duality.By joining the NS-branes back it becomes Sp(2((2k As a check we flow to a known duality.We can consider 2(F + K) fundamentals, integrating out 2K of them generating a cs term.One obtains the duality of Kapustin, Kim and Park [19], between an Sp(2N) 2K -and an Sp(2((2k + 1)(F + |K|) − N − 1) −2K gauge theory, with superpotential as in [19].

U(N) groups and antisymmetric matter
For unitary groups with tensor matter there are two main cases: antisymmetric and symmetric tensors.We refer the reader to [31] where these dualities have been first presented.Here we focus on the antisymmetric case.In the antisymmetric case one has: • An SU(N) gauge theory with an antisymmetric tensor A, its conjugate A with W = Tr(A A) 2  (5.1) and F flavors is dual to an SU(3F − N − 4) with superpotential W = Tr(a a) 2 + M 1 q q + M 0 q aa q + Pq aq + P qa q , (5.2) where a, a are the dual antisymmetric fields, q, q the dual quarks and the mesons are • We can also consider the superpotential in the electric case.The SU(N) gauge theory is dual to an SU(2F − N − 4) gauge theory with superpotential W = Tr(a a) 2 + q a qa + (q q) 2 . ( This duality can be obtained from the previous one by a Higgs mechanism: a dual meson appears as a linear perturbation in the dual theory.After Higgsing we obtain (5.4) from (5.1) and the dual rank is modified accordingly.
• The discussion can be generalized to the superpotential W = Tr(A A) k+1 .In this case one can break the gauge group by adding a polynomial superpotential in (A A) j .By turning this superpotential off one then finds a generalized Kutasov-Schwimmer-Seiberg [32] (kss) duality with dual rank N = (2k + 1)F − N − 4 for the generalization of (5.1) and N = 2kF − N − 4 for the generalization of (5.4).

Brane description
The brane realization of these models has been done in [33,34].All cases in this family can be summarized in the brane cartoon in Figure 10.In order to understand the action of the orientifold we start by discussing a configuration with three NS5branes without the O6-plane.The theory is an N = 2 quiver with two unitary nodes, connected by a pair of bifundamentals and adjoints (see Figure 11).At each node there are F flavors.This configuration and its generalization to N = 1 where extensively studied in [35].Adding the orientifold plane the two nodes are identified and projected to a single U(N) gauge node.The matter fields are identified as well and there are two possibilities, corresponding to the different signs of the orientifold projection: the pair (A, A) or (S, S).Here we focus on the case with (A, A).Now we can break to N = 1 by rotating the external NS-branes: rotating the left and right NS5-brane by an angle θ (resp.−θ) corresponds to introducing a mass term µ(θ ± ) proportional to tan(θ ± ) for the adjoints in the N = 2 vector multiplet.Integrating out the massive adjoints we obtain the superpotential W = Tr A A. If the rotation angle is π/2, the adjoint is infinitely massive and the superpotential vanishes.More in general we can consider two stacks of k NS ±θ -branes, obtaining the superpotential W = (A A) k+1 .
The flavor branes can be added in two ways.In the first case one can add two stacks of D6s parallel to the orientifold and to the NS-brane, one on the left and one on the right.In the second case one can rotate the stack of D6s on the left (right) to a stack of D6 θ (D6 −θ ).In the first case we have to add the term Q A QA + (Q Q) 2 to the superpotential.In the second case, the flavor branes are parallel to the NS ±θ -branes and the quartic terms for the fundamentals are absent.The two configurations with D6 or D6 ±θ -branes have different ranks in the dual hw picture.
The Seiberg duality can be studied in terms of brane motions.It is convenient to describe the motion at first without the orientifold and then add the projection at the end.The starting theory has two unitary gauge groups connected by a pair of bifundamentals and extra flavors.The Seiberg-dual phase is obtained by a cascading process, first we dualize one gauge group, then the other, and finally we dualize again the first gauge group.In terms of branes it corresponds to exchanging the first two NS-branes, then the last two and then the first two again.Before this exchange, it is convenient to move the D6-branes.We have to distinguish the two situations, where we have either two stacks of D6 ±θ or two stacks of D6s parallel to the central brane.
• In the first case, the D6 θ crosses first the NS-brane and then the NS −θ -brane.Both times the crossing generates a stack of D4-branes.The same operation has to be performed on the second brane.In this case there are 2 D4-branes ending on each D6 ±θ .The S-rule is not violated because one stack of D4s is attached to an NS-brane and the other to an NS ∓θ .If we interchange the position of the NS5-branes we obtain the dual picture.The reduction of this duality has been studied in [1] from the brane perspective.At this point we can consider the effect of the O6 − orientifold on the central NS-brane.The following happens.
1. the gauge group is projected from SU(N) × SU(N) to SU(N); 2. the bifundamentals connecting the gauge groups become the tensor matter fields; 3. the two flavor groups are identified.
At the level of the duality the orientifold carries the charge of −4 D6-branes.By carefully considering the orientifold charge in each transition we end up with the SU(3F − N − 4) gauge theory as expected.
• In the second case the D6-branes are parallel to the NS-brane.We move the D6 on the left of the NS towards the NS θ and the other in the opposite direction.Once they cross the NS ±θ , each D6 generates a stack of F D4-branes.After this motion the duality works as in the case above.By carefully adding the orientifold charge, the dual SU(2F − N − 4) gauge theories are recovered.
One can also study the duality with k NS ±θ -branes.In this case one first separates these branes along the direction orthogonal to the plane that they occupy in (4589) and then studies the duality in each sector separately.By reconnecting the branes the expected dualities are recovered.

Dimensional reduction
Now we compactly x 3 and T-dualize along this direction.We consider the U(N) case, where the baryonic symmetry is gauged.On the T-dual circle, the theory develops a superpotential of the form where Y + = e σ 1 /e 2 3 +iφ i and Y − = e −(σ N /e 2 3 +iφ N ) .This can be understood from the brane picture as follows: there are two sets of D3s, one connecting the NS θ and NS-branes and the other connecting the NS −θ and NS-branes.On each stack an η-superpotential is generated by the Euclidean D1-branes.The two superpotentials are identical and identified by the orientifold.Finally, one has (5.6).Now we want to investigate the dual phase.As discussed above there are two possible situations: the D5-branes are parallel to the NS-branes or to the NS ±θ -branes.In the first case N = 3F − N − 4, while in the second case we have N = 2F − N − 4.
On the circle, the extra superpotential is generated.Here y + = e σ 1 / e 2 3 +i φ i and y − = e −( σ N / e 2 3 +i φ N ) .Figures 12 (a) and 12 (b) show the brane cartoon of the electric theory in the case without the extra superpotential.The NS-branes are drawn in black, the D5s in green, the orientifold plane is orange and the D3s are red.In Figures 12 (c) and 12 (d) we represent the case with the superpotential turned on.Now we want to flow to the theory without the η-superpotential.We consider the case with F + 2 green branes in each sector, and assign a positive large real mass to one flavor and one negative large real mass to a second one.We rotate one pair of D5 ±θ clockwise on the circle and another pair counterclockwise.Finally, we reconnect the pairs at x 3 = x • 3 , where the second orientifold is placed.Now we can proceed as above, we interchange the NS-branes and arrive at the dual configuration.Finally, we obtain the setup in Figure 13.These pictures represent the Aharony-like duality for the models with antisymmetric matter.The extra sectors are dualized to singlets, as done in [1] for the U(N) sqcd.The extra singlets that are generated interact with the monopoles of the magnetic theory, and they are identified with the Coulomb branch variable of the electric theory.This can be explicitly verified on the field theory side.The duality now involves a U(3F − N − 2) gauge group in the case where the superpotential is W = (A A) 2 .One can add the extra deformation (Q Q) 2 + A Q AQ (corresponding to rotating the branes as in Figure (c)).In the dual phase this deformation enforces a Higgs flow to the theory with U(2F − N − 2), and it exactly corresponds to the expected dual, after dualizing the extra sectors and considering the ahw superpotential.This confirms the validity of our rules and of our picture.We can reproduce the same story by considering k external NS ± θ-branes.In this case we can break the NS ±θ -branes, e.g.generating a power superpotential W ∑ i λ i (A A) j .By breaking the gauge group in the decoupled sectors we can use the same rules used above and reconstruct the dual theory.Finally, we obtain the dual ranks U((2k + 1)F − N − 2k) and U(2kF − N − 2k).As a final check, we can flow to the case with cs terms.In this case we reproduce the duality between the U(N) K theory with F flavors and the dual U((2k + 1)(F + K) − N − 2k) −K studied in [19].

Orthogonal gauge groups
In this section we discuss orthogonal gauge groups.An SO(N) gauge theory with 2F fundamental vectors and global symmetry SU(2F) × U(1) A × U(1) R without superpotential is dual to an SO(2F − N − 4) gauge theory with 2F fundamental vectors q and a meson in the (conjugate) symmetric representation of the global SU(2F) with superpotential W = Mqq.This duality was first presented in [36].The field content is given in Table 3.

Aspects of field theory
On can associate three distinct gauge groups to the Lie algebra so(N), as discussed in [37] where they were called SO(N) ± and Spin(N).In four dimensions the different choices depend on the spectrum of line defects, while in three dimensions they depend on the monopole charges in the dual algebra.
The Coulomb branch variables associated to the so(N) algebra are and At finite radius there is also an η superpotential W = ηZ from the kk monopoles [7,26,38], where in the odd case.The two expressions finally boil down to Z = e (σ 1 +σ 2 )/e 2 3 +i(φ 1 +φ 2 ) .In presence of matter fields this superpotential still contributes to the theory, but there is a difference with the symplectic and unitary cases: the η-superpotential does not completely lift the Coulomb branch, parameterized by Y Spin = e 2σ 1 /e 2 3 +2iφ 1 in the Spin(N) case and Y = e σ 1 /e 2 3 +iφ 1 in the SO(N) case.There are three possible dualities: Spin(N) ↔ SO( N) − , SO(N) − ↔ Spin( N) or SO(N) + ↔ SO( N) + , where in each case It is possible to reduce the 4D dualities to 3D dualities by considering the limit r → 0, i.e. η → 0, without adding real masses.This is possible because of the presence of a Coulomb branch.A region near the origin of the moduli space on the electric side of SO(N) + corresponds in the dual to the region Y = i/ η.This breaks the gauge symmetry to SO(F − N + 2) × SO(2).This last sector in the ir is described by its Coulomb branch variable interacting with the monopole of the unbroken sector through an ahw superpotential.Similarly, one obtains a duality between SO(N) and Spin(N) theories in pure 3D.At the local level, the O(N) duality studied in [18,38] is recovered.

Brane description
Now we turn to the brane picture.We will limit the discussion to the study of the local properties, without focusing on the difference between the (S)pin(N) and the (S)O ± cases.We will comment on the possibility of extending the analysis to the global property of the gauge group in the conclusions.At the brane level these theories are obtained in two different ways.In one case, we put the O4 − ( O4 − ) on N = 2n (N = 2n + 1) D4-branes stretched between the NS and NS -brane.This theory has an Sp(2F) global symmetry and we expect that this symmetry is enhanced to SU(2F).In the second case we consider two NS ±θ -branes connected by a stack of D4-branes intersecting the O6 + -plane symmetrically with respect to the NS ±θ -branes.We can distinguish between the even N = 2n case and the odd N = 2n + 1 case, essentially this corresponds to the number of D4-branes.In this case the global symmetry is SU(F) and we expect this enhances to SU(F) 2 .
In the T-dual type iib description there is an O3-plane between the two NS ±θ and the 2F D5 ±θ -branes.The gauge theory lives on the N D3s extended along x 6 .First we study the generation of the superpotential in the Coulomb branch in the case of a pure gauge theory.Then we discuss the new duality obtained on the circle and finally we reproduce the 3D limit studied in [7].The three-dimensional theory on the circle has two possible orientifolds O3 − or O3 − .In the first case we have to consider an even number of D3-branes while in the second case they have to be odd.We can study in both cases the generation of the superpotential in terms of the Coulomb branch variables.
The superpotential on the Coulomb branch is obtained in terms of the spectrum  of the allowed bps F1-strings in presence of the orientifold, as discussed in Section 3.
In the orthogonal case we can represent the two different possibilities for the O3 − or O3 − with the B N and the D N series.(see Figure 14 and Figure 15).The extra superpotential corresponds in both cases to the extra term Z = e (σ 1 +σ 2 )/e 2 3 +i(φ 1 +φ 2 ) .Since it identifies two eigenvalues after we cross from one half to the other of the circle it involves the identification and add the η-term.Finally, we have When we consider the D6-branes there is an unlifted direction in the Coulomb branch, corresponding to the term e 2σ 1 /e 2 3 .We consider F + 2 D5 on each NS-brane and take the pure 3D limit.In the electric theory we are left with a pure 3D so(N) theory with 2F flavors.In the dual theory the situation is more intricate.At x 3 = 0 there is an so(F − N − 2) theory with superpotential W = Mqq.At x 3 = x • 3 the confining sector corresponds to an so(4) theory with two fundamentals.It can be dualized to a singlet Y, interacting with the so(F − N − 2) through an ahw superpotential.This interaction is W = yY where y is the magnetic monopole.By interpreting Y as the electric monopole acting as a singlet in the magnetic theory we arrive to the expected duality.

An alternative limit
Differently from the unitary and symplectic cases, here the pure 3D limit can be obtained without any real mass flow [7].The reason is that the region x 3 0 of the Coulomb branch in the electric theory corresponds to the region x 3 x • 3 in the magnetic one.A confining so( 2) is created at x • 3 in the magnetic theory, and the pure 3D limit can be taken directly, preserving the duality.
In the brane description we consider the D3s in the electric theory at the origin, while in the magnetic theory the orientifold generates automatically a pair of D3branes at x • 3 .The dual gauge theory becomes so(2F − N − 2) × so(2).The final configuration is in Figure 16.The so(2) sector is dual to a singlet Y, that interacts with the so(2F − N − 2) sector through an ahw superpotential.Again, in the pure 3D case, Y has the same quantum numbers as the electric monopole.
It is possible to study the case with O6-planes as well.In this case the discussion follows the one of the symplectic case, and we do not report the whole analysis.
The mirror orientifold is created on the circle and the extra sectors can be studied with the usual brane techniques.One can also study the cases with tensor matter, by adding k NS -branes in the case with an O4-plane and k NS ±θ for the cases with the O6-planes.Moreover, one can consider the cases with O6-planes and an extra NS-brane, this leads to unitary theories with symmetric matter and the discussion follows the one in Section 5.In all the cases new examples of threedimensional dualities can be worked out.We conclude by observing that the known three-dimensional case studied in [20] can be recovered from these dualities.

Conclusions
In this paper we completed the analysis started in [1] of the reduction of fourdimensional dualities to three dimensions via brane constructions.We have shown that this picture captures the relevant properties of the reduction of the duality on R 3 × S 1 .By T-duality the Coulomb branch on the circle is correctly described, after separating the D3-branes in the compact direction, by an affine Toda potential for the F1-strings in an S-dual frame.When considering real groups or tensor matter fields, a crucial role is played by the behavior of the orientifold under T-duality.A second orientifold plane is generated at an opposite point on the T-dual circle.We have shown that it is necessary to consider the physics at this mirror point when taking the three-dimensional limit.This limit is a double scaling on the real masses and the radius.The masses correspond to the positions of certain D5-branes (and in the magnetic phases also D3-branes).By reconnecting the branes at the mirror point, a new unified scenario to study the reduction of four-dimensional dualities admitting a type iia description in four dimensions emerges.The construction presents an algorithmic way to obtain many new three-dimensional dual pairs from their four-dimensional parents which we have discussed in this article.
The construction presented here is generic for 4D dualities that can be described by type iia brane systems and several extensions are possible.E.g. one could apply the reduction to type iia setups involving chiral matter and orientifolds like the ones studied in [39][40][41].
It would be interesting to study the spectrum of line defects and their connection to dualities in the brane picture.The relations of line defects to global properties of the gauge groups has been pointed out [37] and there are various implications for the duality involving the orthogonal algebras so(n) [7].It should be possible to distinguish between "Spin(N)" and the "SO(N) ± " (in the language of [7]) also in the brane set-up, e.g.following the discussion in [42].More explicitly, in the type iib description, one can separate the D3-branes, studying configurations of semi-infinite (electric) D1-branes and (magnetic) F1-strings with endpoints on the D3-branes.In the presence of an orientifold, this analysis should give rise to the distinction between Spin(N) and SO(N) ± .We leave this problem for future investigation.
Another interesting extension of our analysis involves the pairs of orientifolds associated to twisted affine Dynkin diagrams.These cases do not descend from a compactification of a type iia background, and they do not represent a fourdimensional theory.However, they do correspond to well-defined theories on R 3 × S 1 .One might expect obtaining new Seiberg-like dualities corresponding to these configurations.By assigning suitable real masses one may even expect to obtain new purely three-dimensional dualities, without four-dimensional parents.It would be interesting to further investigate in this direction.
Let us comment on the generation of the monopole charges.The axial U(1) A , anomalous in four dimensions, is broken by the kk monopole superpotential at finite radius.However, rotating the D5s on the circle partially breaks the non-abelian flavor symmetry and generates the axial symmetry.The massless singlets in the confining sector are charged under this symmetry and survive the pure 3D limit.They correspond to the monopoles of the electric theory and, at the same time, their U(1) A charge is imposed by the original global symmetry.This observation explains the relation pointed out in [43], between the equations governing the cancellation of the anomalies in 4D and those governing the monopole charges in 3D.
It is possible to reproduce our results when reducing the four-dimensional superconformal index [44,45] to the three-dimensional partition function [46,47].That was done for the case of sqcd in [5] and in presence of adjoint matter in [48] 5 .One should consider the identities summarized in [50] and obtain new identities for the three dimensional dualities.A possible strategy for this calculation is the kk reduction of the one-loop determinants while shifting some fugacities of the global and local symmetries.This reproduces the double-scaling limit discussed in this paper.One should check that the surviving zero modes remove the possible divergent contributions found in [7].To simplify the notation in the paper we work with the coupling e 2 3 = g 2 3 /(4π).

B Orientifolds
Orientifold planes played a special role in our discussion, therefore we briefly review here some of their basic aspects [52][53][54].A p-dimensional orientifold (Op-plane) is defined in string theory by its perturbative action.It corresponds to the projection σ • Ω • (−1) F L , where σ is the parity inversion of the coordinates transverse to the plane, Ω is the world-sheet parity and F L is the left-moving fermion number.The orientifold acts both on the NS [55] and on the R sector [56,57], by two distinct Z 2 parities.The action on the NS sector is perturbative in string theory and we denote it with a + or a −.The Z 2 acting on the R sector is non-perturbative in string theory and we denote it with the presence or the absence of a tilde (∼) on the orientifold [9,41].These charges identify the action of the orientifold on the gauge symmetry.We summarize the different possibilities in table 4.
In this paper we have been mostly interested in the O3 and O5 cases, coming from a T-duality from type iia.After compactification an O(p + 1) plane becomes a pair of orientifolds which turn, after T-duality, into a pair of Op-planes (see Figure 1) [9,41], and this fact has been crucial in our analysis.There are in principle

Figure 1 :
Figure 1: Geometry of the compact direction.The possible orientifolds are depicted in red color.LHS: type IIA circle of radius r.RHS: T-dual circle of radius α /r.The black arrowheads indicate the motion of the D-branes to the mirror point x 3 = x • 3 ≡ πα /r.

Figure 2 :
Figure 2: Branes and Dynkin diagrams for A N and A N .The left column shows the S-dual configuration of F1-strings stretched between D3-branes.In the right column we depict the corresponding A N and A N Dynkin diagrams.After compactification a new string appears between σ 1 and σ N and corresponds to the affine node (in blue in the brane cartoon and in the Dynkin diagram).

Figure 3 :Figure 4 :
Figure 3: Brane cartoon for the realization of an Sp(2N) theory in the electric phase with an O4-plane.

Figure 5 :
Figure 5: Dynkin and affine Dynkin diagrams and spectrum of BPS F1-strings associated to the fundamental monopoles for Sp(2N) theories in the linear case and on the circle.The affine root is represented in blue on the affine Dynkin diagram and in the brane cartoon.

Figure 6 :
Figure 6: Dual Aharony flow Sp in the O3 configuration

Figure 8 :
Figure 8: Electric and magnetic sides of the duality for Sp(2N) gauge theories with antisymmetric matter.

Figure 9 :
Figure 9: Electric and magnetic sides of the duality for Sp(2N) gauge theories with adjoint matter.

Figure 10 :
Figure 10: Brane cartoon summarizing all the constructions of unitary gauge theories with tensor matter

Figure 11 :
Figure 11: Orbifold projection of the A 2 quiver realizing a U(N) theory with tensor matter.

Figure 12 :
Figure 12: U(N) gauge theory with antisymmetric matter, electric theory.(a) and (b) show the case without superpotential (D5-branes parallel to NS ±θ ), (c) and (d) show the case with superpotential (D5-branes are parallel to the NS-branes).

Figure 14 : 3 .
Figure 14: Dynkin and affine Dynkin diagrams and spectrum of BPS F1-strings associated to the fundamental monopoles for SO(2N + 1) theories (B N algebra).The orientifold at the mirror point x 3 = x • 3 is an O − 3 , while the one at x 3 = 0 is an O − 3 .For this reason the Dynkin

Figure 15 :Figure 16 :
Figure 15: Dynkin and affine Dynkin diagrams and spectrum of BPS F1-strings associated to the fundamental monopoles for SO(2N) theories (D N algebra).The affine root is in blue, the root due to the orientifold in x 3 = 0 in teal.Both orientifolds are O −

Table 1 :
Brane setup for the realization of the gauge theories of interest in this paper.