3d dualities from 4d dualities for orthogonal groups

We extend recent work on the relation of 4d and 3d IR dualities of supersymmetric gauge theories with four supercharges to the case of orthogonal gauge groups. The distinction between different SO(N) gauge theories in 4d plays an important role in this relation. We show that the 4d duality leads to a 3d duality between an SO(Nc) gauge theory with Nf flavors and an SO(Nf − Nc + 2) theory with Nf flavors and extra singlets, and we derive its generalization in the presence of Chern-Simons terms. There are two different O(N) theories in 3d, which we denote by O(N)±, and we also show that the O(Nc)− gauge theory is dual to a Spin(Nf − Nc + 2) theory, and derive from 4d the known duality between O(Nc)+ and O(Nf − Nc + 2)+. We verify the consistency of these 3d dualities by various methods, including index computations.


Introduction
A crucial role in gauge theory dynamics is played by various dualities. They give a weakly coupled description of some strong coupling phenomena (like confinement and chiral symmetry breaking), and may point to a deep structure underlying the theory. In different situations these dualities manifest themselves differently. Some 4d superconformal theories like N = 4 and certain N = 2 supersymmetric theories exhibit exact electric/magnetic duality, leading to several distinct descriptions of the same theory, with different values of the coupling constant and sometimes even different gauge groups. Many four-dimensional N = 1, 2, three-dimensional N = 2, 3, · · · , and certain two-dimensional theories have IR dualities, relating different theories with the same IR limit [1]. In some situations, including 4d N = 1 SO(N ) dualities, it is clear that these are also related to electric/magnetic JHEP08(2013)099 duality [1][2][3]; when the gauge group is broken to SO(2) they reduce to an Abelian electric/magnetic duality, and they exchange Wilson lines with 't Hooft-Wilson lines [4,5]. In 3d there are several known examples of such IR dualities, both with N = 2 and with higher supersymmetries. In a previous paper we argued that most and perhaps all such dualities in 3d originate from ancestor dualities in 4d [6] (see also [7]). The purpose of this note is to extend this discussion to theories with orthogonal gauge groups.
The discussion in [6] starts with any 4d N = 1 duality, and by carefully compactifying it on a circle, it leads to a clear prescription for how to generate from it a corresponding 3d duality. For example, we can start with the characteristic example of a 4d N = 1 duality. This is the duality between an SU(N c ) gauge theory with N f flavors Q i andQĩ, and its dual SU(N f − N c ) gauge theory with N f dual quarks q i andqĩ and elementary gauge neutral "mesons" M ĩ i and a superpotential W = M ĩ i q iqĩ [1]. It is common to refer to these theories as the electric and the magnetic theories, but we will refer to them as theory A and theory B. A naive dimensional reduction of any of these two dual theories to 3d leads to a theory with an additional "axial" U(1) global symmetry. This is the symmetry that is anomalous in 4d, but is preserved in 3d. The prescription of [6] is to modify the naive dimensionally reduced theory by adding to its Lagrangian a suitable operator, generated by non-perturbative effects in the theory on a circle, which explicitly breaks this anomalous U(1) symmetry. In theory A we add a superpotential where η = Λ b 0 is the instanton factor [3] of theory A, and Y is its monopole operator. In theory B, which already had a superpotential in 4d, we have whereη =Λb 0 = (−1) N f −Nc η −1 is the instanton factor of theory B, andỸ is its monopole operator. 1 The arguments of [6] imply that the two 3d theories (1.1), (1.2) are equivalent at low energies. Once such a 3d duality is established one can find many additional 3d dualities, which follow from it. First, we can turn on relevant operators in the two sides of the duality and flow to the IR. Second, we can gauge any of the global symmetries of the theories and generate new dual pairs. These two tools were used in [6] to reproduce all the known dualities between 3d N = 2 theories with SU(N c ), U(N c ) and USp(2N c ) gauge groups, and to generate many new dualities.
However, the application of this procedure to theories with orthogonal gauge groups turns out to need more care. In fact, already in 4d N = 1 theories the IR dualities for orthogonal groups are significantly more subtle than for unitary or symplectic gauge groups [1][2][3]8]. One underlying reason for this complexity was recently identified in [5]. It is known that if the Lie algebra of the gauge symmetry is so(N c ), the gauge group can be Spin(N c ) or SO(N c ) (and it could even have disconnected components, making it P in(N c ) or O(N c )). The main point of [5] is that even when the gauge group is SO(N c ), there are two distinct 4d gauge theories with that gauge group, denoted by SO(N c ) ± . In the Euclidean path integral they are distinguished by a new term in the Lagrangiana certain Z 2 -valued theta-like-angle, associated with the Pontryagin square P(w 2 ) of the Stiefel-Whitney class w 2 of the gauge bundle.
A simple physical way to distinguish between the three gauge theories SO(N c ) ± and Spin(N c ) already in R 4 is to study their line operators. The Spin(N c ) theory has a Wilson loop W in a spinor representation. 2 Its square W 2 can be screened by dynamical fields and we will view it as trivial. The two SO(N c ) theories do not have a Wilson loop in a spinor representation. Instead, they have 't Hooft loops carrying smaller magnetic charge than is allowed in Spin(N c ). The SO(N c ) + theory has a purely magnetic 't Hooft loop operator H, and the SO(N c ) − has the non-trivial loop operator HW . For a closely related earlier discussion, see [9].
The 4d N = 1 IR duality relates [1][2][3]8] an so(N c ) gauge theory with N f vectors Q i to an so(Ñ c = N f − N c + 4) gauge theory, with N f vectors q i and elementary gauge singlet mesons M ij , and with a superpotential W = 1 2 M ij q i q j . When either N c orÑ c are 2, 3, 4 a more careful discussion is needed. The analysis of [5], which took into account the global structure, 3 identified this duality as (1.5) Subtleties associated with the line operators in the 4d theory translate into subtleties with the local operators when the theory is compactified on a circle to 3d. In particular, JHEP08(2013)099 a 4d 't Hooft line operator H wrapping this circle turns into a local monopole operator Y in 3d. Hence, the choice of line operators in 4d becomes a choice of local operators in 3d, which has more dramatic consequences, as we will see in our discussion below.
An additional subtlety in the analysis of orthogonal groups is that the corresponding 4d supersymmetric QCD (SQCD) theories on S 1 have a Coulomb branch that is not lifted by quantum corrections. This did not occur in any of the cases analyzed in [6], and the mapping of the Coulomb branch across the duality turns out to be non-trivial.
In section 2 we discuss some classical and quantum properties of 3d N = 2 theories with orthogonal gauge groups. We identify the coordinates on their moduli space of vacua, paying particular attention to the global structure (the distinction between Spin(N ) and SO(N )). In section 3 we discuss the 4d gauge theories on R 3 × S 1 and their moduli space of vacua. Here the distinction between the three different theories with the same Lie algebra so(N c ) is crucial. In section 4 we follow [6] and consider two dual 4d theories compactified on a circle, and carefully identify their moduli spaces.
In section 5 we derive the main result of this paper. By taking an appropriate limit of the 4d theory on a circle we derive 3d dualities. In particular, the SO(N c ) SQCD theory with N f vectors Q is dual to an SO(N f − N c + 2) gauge theory with N f vectors q, with gauge singlet fields M and Y , and with an appropriate superpotential. Here the elementary fields M and Y are identified with the composite meson QQ and monopole operator of the original SO(N c ) theory. This is closely related to dualities of O(N c ) SQCD theories, previously found in [10][11][12]. We also find that the Spin(N c ) theory is dual to theory that we introduce in section 2.4. We perform various tests of these dualities, and deform them to obtain dualities for theories with Chern-Simons terms. Additional detailed tests are performed in section 6, where we discuss the S 2 × S 1 and the S 3 partition functions of all these theories.

Background
Much of the necessary background for this paper is found in the preceding paper [6], and in references therein. We will assume here familiarity with that paper, and discuss only the new issues which arise for orthogonal gauge groups. Some aspects of the theory depend on the precise choice of gauge group, while others depend only on the gauge algebra g = so(N c ), and we will try to distinguish the two in the following.

Monopole operators and Coulomb branch coordinates for
Three dimensional N = 2 gauge theories have classical Coulomb branches, where the adjoint scalar σ in the vector multiplet gets an expectation value, generically breaking the gauge group G to U(1) r G (where r G is the rank of G). On this branch we can dualize the r G photons to scalars a i , or supersymmetrically dualize the r G U(1) vector multiplets to chiral multiplets Y i . The expectation values of these chiral multiplets label the classical Coulomb branch of the theory. The chiral multiplets Y i are "monopole operators" in the effective lowenergy theory, creating a U(1) r G magnetic flux around them. In some cases they arise as low-energy limits of microscopic "monopole operators". The allowed spectrum of monopole JHEP08(2013)099 operators, and thus the appropriate coordinates on the Coulomb branch, depends on the choice of the gauge group; this choice determines the allowed Wilson line operators, and the monopole operators need to be mutually local with respect to these Wilson lines.
For theories based on the Lie algebra g = so(N c ), when N c is even, N c = 2r G and when N c is odd, N c = 2r G + 1. We write the adjoint matrix σ as a matrix in the vector representation of so(N c ), and we can always diagonalize it. For every non-zero eigenvalue, there is another eigenvalue of equal magnitude and opposite sign. For even values of N c we write the eigenvalues as {σ 1 , · · · , σ r G , −σ r G , · · · , −σ 1 }. By a Weyl transformation we can always choose N c even : If our gauge group includes reflections (namely, it is G = O(N c ) or G = P in(N c ) rather than G = SO(N c ) or G = Spin(N c )) then we can also set σ r G ≥ 0, while otherwise we cannot do this in general. For odd values of N c we can write the eigenvalues of σ as {σ 1 , · · · , σ r G , 0, −σ r G , · · · , −σ 1 }, and by a Weyl transformation we can always choose N c odd : The magnetic charges carried by the Coulomb branch coordinates should be thought of as charges in the magnetic-dual algebra to so(N c ). For even N c , this algebra is so(N c ), and for odd N c , it is usp(N c − 1). There are always operators carrying the charges of the roots of this algebra, and when the gauge group is G = Spin(N c ), these are the only allowed charges. We can then write the Coulomb branch coordinates semi-classically as Hereĝ 2 3 = g 2 3 /4π, where g 3 is the gauge coupling constant of the 3d gauge theory, normalized as in [6]. The dependence of these operators on the σ's that we wrote is valid far out on the Coulomb branch, and gets quantum corrections, while their dependence on the dual photons a i is exact. As usual, the global symmetry charges of these operators can be determined by summing over the charges of the fermions in chiral and vector multiplets, which are coupled to the corresponding σ's [13]. The SO(N c ) and Spin(N c ) theories have a global charge conjugation symmetry Z C 2 , which is gauged in the O(N c ) and P in(N c ) theories. In the theories with even N c , this symmetry exchanges the Coulomb branch coordinates Y r G −1 and Y r G .
As discussed in [6,14], some of the Coulomb branch coordinates are low-energy limits of microscopic monopole operators. These are defined so that their insertion at a point x generates some magnetic flux on the S 2 surrounding x, and takes the σ(y)'s pointing in the direction of the flux to +∞ as y → x. In the Spin(N c ) theory, the monopole operators all carry charges corresponding to roots of the dual magnetic group. The minimal monopole JHEP08(2013)099 operator Y Spin turns on one unit of flux, breaking so(N c ) → so(N c −2)×u(1), and takes one of the eigenvalues of σ to ∞. On the moduli space at low energies, using (2.1) and (2.2), this monopole Y Spin looks semi-classically like It is a combination of the Coulomb branch coordinates described in (2.3), Another monopole operator that will play a role in our discussion is the one that takes two eigenvalues of σ to infinity together, breaking so(N c ) → so(N c − 4) × u(2). This monopole semi-classically looks like and we will see that it will play an important role in the discussion of 4d so(N c ) theories on a circle. It obeys Y Spin = Y 1 Z, and For G = SO(N c ), Wilson lines carrying spinor charge are not allowed (we will always assume that Wilson lines in the vector representation are allowed, since we will be interested here in theories with matter fields in the vector representation). This means that extra Coulomb branch coordinates and monopole operators are allowed, carrying weights which are not roots of the dual magnetic group. For even N c they are allowed to carry weights in the vector representation of the dual so(N c ), and for odd N c in the fundamental representation of the dual usp(N c − 1). The basic monopole operator in this case behaves semi-classically as [10,12] Y ≈ exp and it obeys All other "new" monopole operators that exist in this case may be written as products of Y with the operators corresponding to roots of the dual gauge group. Note that while for the monopole operators corresponding to roots, there is a classical 't Hooft-Polyakov monopole solution (which is an instanton of the 3d theory) that is associated with them, there is no such solution for Y of (2.8). But this is not related to the definition of this operator, both microscopically and in the low-energy effective action. The fact that the quantum numbers of the Coulomb branch coordinates are determined by those of the matter fields implies that they change when some matter fields go to JHEP08(2013)099 infinite mass and decouple. In such cases we have a relation between the Coulomb branch coordinates in the high-energy and in the low-energy theories, which can usually be uniquely determined by matching their quantum numbers. For example, if we start from a Spin(N c ) theory with N f chiral superfields in the vector representation, and give a mass m to one of them in the superpotential, we have a relation of the form (2.10) Similarly, if we start from such a theory and break the gauge group to Spin(N c − 1) with (N f − 1) flavors, by giving an expectation value to one of the chiral superfields Q in the vector representation, we have a relation For low values of N c , N c < 5, some modifications are needed in our discussion. For N c = 2 the gauge group G = Spin(2) = U(1), and instead of the operator Y Spin we have the two standard U(1) Coulomb branch coordinates [13] parameterizing the parts of the Coulomb branch with σ positive and negative, respectively. Note that in Spin(2) we have particles of charge ±1/2 under the U(1) group, and hence the normalization of the monopole operators is twice the usual normalization. For G = SO(2) all particles have integer U(1) charge, and we have monopole operators carrying half the charge of (2.12), given by The charge conjugation symmetry Z C 2 exchangesV + andV − (or V + and V − ), and we will find it convenient to define the linear combinations that are even and odd under Z C 2 . For N c = 3, we have Spin(3) = SU (2), and the monopole Y Spin that we defined above is the standard Coulomb branch coordinate / monopole operator of the SU(2) theory (usually denoted by Y [13]). When the gauge group is SO(3) = SU(2)/Z 2 , there exists a monopole operator of lower charge, which we denoted by Y above.
For N c = 4, the group Spin(4) is equivalent to SU(2) × SU(2), and we can then have separate Coulomb branch coordinates and monopole operators in the two SU(2) factors. The Coulomb branch coordinates of the two SU(2)'s correspond to σ 1 ± σ 2 in our notations above. Thus, the Coulomb branch coordinates of the two SU(2)'s, which we will denote by Y (1) and Y (2) , look semi-classically like the operators Y 1 and Z discussed above. The operator Y Spin in this case is the product of these two SU(2) operators. When the gauge JHEP08(2013)099 group is G = SO(4) = (SU(2) × SU(2))/Z 2 , one does not allow Wilson loops carrying a charge under the center of each SU(2) separately, but only under both SU(2)'s. In this case the operator Y that we defined in (2.8) exists, and squares to the product of the two SU(2) monopole operators, Y 2 = Y (1) Y (2) . The charge conjugation symmetry Z C 2 exchanges the two SU(2) factors, exchanging Y (1) with Y (2) . As in (2.14), it is convenient to define the combinations which are even and odd under Z C 2 .

2.2
The quantum moduli space of 3d N = 2 theories with g = so(N c ) In the quantum theory, most of the Coulomb branch described above is lifted. Whenever two of the eigenvalues of σ come together at a non-zero value, the corresponding U(1) 2 symmetry is enhanced to U (2). As shown in [15], the corresponding 't Hooft Polyakov monopole-instanton solutions generate a superpotential in this case, which drives the eigenvalues apart. For even values of N c we have this effect whenever σ i approaches σ i+1 for i = 1, 2, · · · , r G −1, and also when σ r G −1 approaches −σ r G . Thus, in the 3d pure g = so(N c ) theory with N c even we have an effective quantum superpotential N c even : which completely lifts the Coulomb branch. For odd values of N c there is always one eigenvalue at σ = 0, and when σ r G approaches 0, the corresponding u(1) is enhanced to so(3) from 3 eigenvalues coming together at σ = 0. There is a similar superpotential arising here, with a different normalization [16], so that Again this completely lifts the Coulomb branch, so that the pure 3d supersymmetric Yang-Mills (SYM) theory has a runaway with no supersymmetric vacua. We can follow the reasoning used in [6] for SU(N ) gauge theories to show that (2.16) and (2.17) are in fact exact. More precisely, they are exact as functions of the chiral superfields Y i , but the Y i are complicated functions of σ i and a i . Similar superpotentials lift the Coulomb branch also for g = so(3) = su(2), and for g = so(4) = su(2) ⊕ su(2), where we have a separate superpotential of this type in each su(2) factor. For g = so(2) = u(1) there is no such effect, and the Coulomb branch of the (free) pure gauge theory is simply a cylinder, labeled byV + = 1/V − for G = Spin(2), and by V + = 1/V − for G = SO (2).
In theories with flavors in the vector representation that have no real mass, these flavors become massless whenever some eigenvalues of σ vanish. This gives extra zero modes to monopole-instantons corresponding to eigenvalues coming together at σ = 0, such that they no longer generate superpotentials; for odd N c this happens for the monopole Y r G , and for JHEP08(2013)099 even N c it happens either for Y r G −1 or for Y r G , depending on the sign of σ r G . In these theories the moduli space is not completely lifted, but a one (complex) dimensional branch remains, where only a single eigenvalue σ 1 is turned on (and σ 2 = σ 3 = · · · = σ r G = 0). In the Spin(N c ) theories, this quantum Coulomb branch may be parameterized by the operator Y Spin of (2.4), and for G = SO(N c ) theories it can be parameterized by Y of (2.8). Note that because of the quantum superpotentials (2.16), (2.17), the operator Z is not a chiral operator in the low-energy 3d theory, 4 but Y and Y Spin still are. As discussed in [10,12], for low numbers of flavors, N f < N c − 2, there are additional quantum effects that lift the Coulomb branch, while for N f ≥ N c − 2 the Coulomb branch is not lifted. For N f ≥ N c − 1, the quantum moduli space is the same as the classical moduli space (the discussion in [10,12] is just for G = O(N c ), but most of it can be generalized also to the other gauge groups with g = so(N c )).
The 3d SQCD theory with N f chiral multiplets Q i in the vector representation has a global SU(N f )×U(1) A ×U(1) R symmetry. We can choose the flavors Q i to transform in the fundamental of SU(N f ), with one unit of U(1) A charge and no U(1) R charge. In this case, the operator Y Spin has the same charges as Y 2 in the G = SO(N c ) theories.
Note that for N c = 2 our theories with flavors are the same as the U(1) theories with flavors discussed in [13]. However, for N c = 3 our so(3) theories have matter in the triplet (adjoint) representation, so they are not the same as su(2) theories with fundamental matter. In particular, for N c = 3 and N f = 1, the SQCD theory has enhanced supersymmetry, and it is the same as the 3d N = 4 SYM theory discussed in [17]. For N c = 4 the matter fields are charged under both SU(2)'s, and couple them together. Depending on where we are in the moduli space, the superpotential involving the Coulomb branch coordinate of one of the SU(2)'s is lifted by the matter zero modes, while the other one remains. For N c = 3, 4 we can still parameterize the remaining part of the Coulomb branch by Y Spin or Y , as for higher values of N c .

Baryon-monopole operators
In the g = so(N c ) SQCD theory with N f flavors Q i , the list of chiral multiplets includes the monopole operators discussed above, the mesons M ij = Q i Q j (symmetric in i, j) and (for N f ≥ N c , and for G = SO(N c ) or G = Spin(N c )) the baryons B = Q Nc , contracted with an epsilon symbol in so(N c ). The operator B 2 may be written as a combination of products of N c mesons M , but B is an independent operator, charged under Z C 2 . For G = O(N c ) and G = P in(N c ) the operator B does not exist, since the charge conjugation symmetry is gauged.
Consider now a monopole operator like Y in SO(N c ), which breaks

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Note that this includes transformations with determinant (−1) both in SO(N c − 2) and in SO (2). This means that the gauge-invariant operator Y must be charge conjugation even in SO (2), and thus it reduces to the operator W + (2.14) in this group. However, we can also build a "baryon-monopole" operator involving W − in SO(2) (2.14), by defining with the indices of the flavors contracted by an epsilon symbol in the SO(N c − 2) that is unbroken by the monopole. The product Q Nc−2 is invariant under the SO(N c − 2) × SO(2) subgroup of SO(N c ) that is left unbroken by the monopole, and its product with W − is invariant also under the extra Z 2 , so that (2.19) defines a gauge-invariant operator in SO(N c ). Note that the operators (2.19) exist for any N f ≥ N c − 2. The standard matching of quantum numbers for monopole operators, generalizing (2.11), implies that when we give a VEV to Q breaking SO(N c ) to SO(2), β reduces directly to W −low in the low-energy SO (2) theory (with no extra factors of Q ). As we discussed above, when the gauge group is Spin(N c ), the monopole operators Y and W ± do not exist. But we can still repeat the above discussion using the operator Y Spin (which reduces toŴ + ), and define a baryon-monopole as above. Note that in an SO(2) theory, The operator β satisfies an interesting chiral ring relation. Consider the 3d SO(N c ) gauge theory with N f = N c − 2. At a generic point on the moduli space of this theory, we break SO(N c ) → SO(2). The monopole operator Y reduces in the low-energy SO(2) theory to W +low ; the standard mapping of monopoles (2.11) implies that W +low = Y det(M ). The low-energy SO(2) theory has no massless flavors, and hence V +low V −low = 1. Therefore, in this vacuum This reflects an exact chiral ring relation which is valid in every vacuum of this theory. Classically this theory has a point at the origin of its moduli space where Y = M = β = 0, but we see that quantum mechanically the moduli space is deformed and obeys (2.22). This is similar to the deformation of the classical moduli space in some 4d theories [18] and in some 3d theories [13]. Similarly, we can use the extra monopole operators of SO(4), by having monopoles breaking SO(N c ) to SO(N c − 4) × U(2). More precisely, to define such monopoles we first choose some S(O(N c − 4) × O(4)) ⊂ SO(N c ), and then turn on a monopole like (2.6) in the SO(4) factor, breaking it to U(2). As in our discussion above, the monopole Z on its own reduces to the even operator Y + in SO(4) (2.15). But we can now consider instead the operator

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where the quarks are contracted with the epsilon symbol of SO(N c −4). As in our discussion above, this operator b is gauge-invariant in the G = SO(N c ) theory, including also the gauge transformations of determinant (−1) in the two factors. The same discussion applies also to Spin(N c ) theories. The baryon-monopole b exists for any N f ≥ N c − 4 (for N c = 4 it is simply Y − ). As in our discussion of Z above, due to quantum effects b is not really a chiral operator in 3d SQCD theories, but we will see that it still plays a role in our analysis. We cannot generalize this construction to breaking SO(N c ) → S O(N c − 2n) × O(2n) with n > 2, because there is no obvious monopole operator in SO(2n) that is odd under charge conjugation.

On 3d O(N c ) theories
We mentioned above that one can obtain O(N c ) theories by gauging the charge conjugation symmetry of SO(N c ), but in fact there are two different O(N c ) theories in 3d that will play a role in this paper. In one O(N c ) theory the minimal monopole operator Y of SO(N c ), which is charge-conjugation-even in the SO(N c ) theory, is gauge-invariant, and is the minimal monopole operator also for O(N c ). We will denote this theory by O(N c ) + ; this is the theory that was discussed in previous papers about 3d O(N c ) theories and their dualities. However, as in [5], one can also define a second O(N c ) − theory, in which the monopole operator Y is charge-conjugation-odd (it changes sign under gauge transformations whose determinant is (−1)). In this O(N c ) − theory, Y and B are both not gauge-invariant, but their product, as well as the operators Y Spin and β, are gauge-invariant (note that β is not gauge-invariant in the standard O(N c ) + theory).
In the Lagrangian language, the two theories differ by a discrete theta angle, analogous to the one that distinguishes the 4d SO(N c ) ± theories [5]. The relevant term in the Lagrangian is proportional to w 1 ∧ w 2 , where w i ∈ H i (X, Z 2 ) are Z 2 -valued characteristic classes of the O(N c ) bundle on a manifold X; w 1 is non-zero when the O(N c ) bundle cannot be written as an SO(N c ) bundle, while w 2 is non-zero when the O(N c ) bundle cannot be written as a P in(N c ) bundle. In particular, w 2 is non-zero on a two-sphere around an insertion of the operator Y . Note that the two options exist only for O(N c ) gauge groups, not for SO(N c ), Spin(N c ) or P in(N c ) (in which either w 1 , or w 2 , or both, are trivial). 5 3 The Coulomb branch of 4d so(N c ) theories on S 1 3.1 4d Spin(N c ) theories on S 1 As discussed in [6], when one compactifies a 4d gauge theory on S 1 and goes to low energies, naively one gets the same gauge theory in 3d, but there are two important differences. The first is that the Coulomb branch coordinates now come from holonomies of the gauge field on a circle, so the coordinates σ described above are periodic and the Coulomb branch is compact. The second is that there is an extra monopole-instanton in the theory on a circle, that gives an extra term in the effective superpotential. We will start by discussing these JHEP08(2013)099 aspects for the reduction of a 4d theory with G = Spin(N c ), and consider G = SO(N c ) theories in the next subsection. We begin with the case of N c ≥ 5.
In the 4d theory on a circle of radius r, the scalars σ described in the previous section originate from A 3 , but only the eigenvalues of U = P exp(i A 3 ) are gauge-invariant, so there are relations between different values of σ associated with large gauge transformations. In particular, the eigenvalues of U in the vector representation are exp(±2πirσ i ), so each σ i gives the same holonomy in this representation as σ i + 1/r. Note that the eigenvalues of U in the spinor representation are exp(2πir(±σ 1 /2 + · · · )), so the periodicity of each σ i for G = Spin(N c ) is actually 2/r (or one can shift two σ i 's together by 1/r). But the masses of W-bosons and matter fields in the vector representation are periodic in σ i with periodicity 1/r. In particular, whenever all the σ i are integer multiples of 1/r, the gauge group is unbroken and any matter fields in the vector representation are massless.
In the 4d theory on a circle, we can get an enhanced non-Abelian symmetry not just by having σ i → σ i+1 , but also by having eigenvalues meet the images of other eigenvalues. When σ 1 meets the image −σ 1 at σ 1 = 1/2r there is no enhanced non-Abelian symmetry, since we just have an enhancement of U(1) to SO (2) or Spin (2). However, when σ 1 meets the image of −σ 2 , when σ 1 = −σ 2 + 1/r, there is an enhancement of U(1) 2 to U(2) (if this happens at σ 1 = 1/2r then there is even an enhancement to SO(4) or Spin (4)). The same computation yielding the monopole-instanton contributions described above [15], thus gives in the theory on a circle an extra superpotential. The analogy with the 1/Y superpotential of [15] implies that semi-classically the extra superpotential looks like where Z was defined in (2.6), and η ≡ Λ b 0 = exp(−8π 2 /g 2 4 ) = exp(−4π/g 2 3 r) is the strong coupling scale of the 4d gauge theory (b 0 = 3(N c − 2) − N f is the 4d one-loop beta function coefficient, and we set the 4d theta angle to zero and the renormalization scale to one for simplicity). The precise form (3.1) follows by carefully analyzing all the instantons, as in [16,17,[19][20][21]. From the point of view of the effective 3d theory, (3.1) breaks precisely the global U(1) symmetry that is anomalous in the 4d theory. Note that in the 3d theory Z is not a chiral operator, but in the 4d theory on a circle, it can no longer be written as a descendant, due to the extra superpotential (3.1).
In the pure SYM theory, the extra term (3.1) stabilizes the runaway caused by (2.16) and (2.17), and leads to a finite number of supersymmetric vacua, obtained by solving the F-term equations for the Y i . One can check that, both for even and odd values of N c , this leads to (N c − 2) supersymmetric vacua, with This is the same number of vacua as in the 4d theory, as expected in this case [5], and the value of the superpotential also agrees with its 4d value. As discussed in [6], the 4d chiral operator S ∝ tr(W 2 α ) reduces in the theory on a circle to Z, with a chiral ring relation 3)

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which is consistent with (3.2). Note that in the 4d theory on a circle, the monopole and baryon-monopole operators discussed in the previous section do not exist microscopically (due to the compactness of the Coulomb branch), but we can give a microscopic definition to Z using (3.3). Moving next to the theories with flavors, note that unlike in the cases of G = SU(N c ) and G = USp(2N c ) discussed in [6], here the extra superpotential (3.1) is not proportional to the coordinate Y Spin along the unlifted Coulomb branch of the 3d Spin(N c ) SQCD theory. Thus, this superpotential does not lift the Coulomb branch, but just adds extra interactions. The coordinate Y Spin on this Coulomb branch is uncharged under the continuous SU(N f ) × U(1) R global symmetry that is preserved by (3.1).
The global structure of the moduli space is interesting. First, σ 1 should be identified with (−σ 1 + 2/r) because they lead to the same holonomy, so we can take 0 ≤ σ 1 ≤ 1/r. In the quantum theory the moduli space is parameterized by Y Spin (2.4), and we identify the point Y Spin = 0 with σ 1 = 0, and the point Y Spin = ∞ with σ 1 = 1/r. Classically, at generic values of Y Spin the Spin(N c ) symmetry is broken, and at the two special points Y Spin = 0, ∞ the full Spin(N c ) symmetry is preserved.
Second, if our Spin(N c ) gauge theory does not couple to matter fields in a spinor representation, the compactified theory has a global Z 2 symmetry, acting on the moduli space by σ 1 → −σ 1 + 1/r. The point is that these two values of σ 1 represent two different holonomies in Spin(N c ), but this difference is not felt by any dynamical field. 6 The two special points on the moduli space Y Spin = 0, ∞ are not identified. Instead, they are exchanged by the global Z 2 symmetry, which acts on the moduli space as In the 4d G = Spin(N c ) theory there are several baryonic operators defined in [1,2], which all involve contractions with the epsilon symbol of Spin(N c ). The first operator obviously reduces to the same baryon operator in 3d. The second operator is useful when 6 More generally, whenever we have a Zp gauge theory in d dimensions that does not couple to charged fields, the theory compactified to d−1 dimensions has both a gauge Zp symmetry and a global Zp symmetry. One way to see that is to represent the Zp gauge theory by a one-form U(1) gauge field A (1) and a (d − 2)form gauge field A (d−2) with a Lagrangian given by p [22,23]. Reducing this system to d − 1 dimensions leads to four fields, with the Lagrangian p . The first term represents a Zp gauge theory, and the second term describes a Zp global symmetry (and a gauge symmetry for A (d−2) ). The global symmetry acts as If the original d dimensional theory includes matter fields charged under the Zp gauge symmetry, the global symmetry A (0) → A (0) + 2π/p in the d − 1 dimensional theory is explicitly broken, but the identification A (0) ∼ A (0) + 2π is preserved. A famous example of this phenomenon, due to Polyakov, is the compactification of a 4d SU(N ) gauge theory without matter on S 1 . The resulting 3d theory has a global ZN symmetry, which originates from the 4d ZN ⊂ SU(N ) gauge symmetry. The order parameter for its spontaneous breaking is the Polyakov loop e iA (0) ∝ 1 N tr(e i A ). In our case, the relevant Z2 gauge symmetry is the subgroup of the center of Spin(Nc) that acts on spinors.

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(N c − 2) quarks get expectation values breaking the gauge group to g = so (2), to label the remaining unbroken so(2) ⊂ so(N c ); in the effective 3d theory the same role is played by β or β Spin . Similarly, when the gauge group is broken to g = so(4), we have a relation between S 1 − S 2 (of SU(2) × SU(2) in 4d) and Y − of so(4) (2.15) that is analogous to (3.3), and that implies that the 4d baryon b 4d goes down in the low-energy effective theory to η times the baryon-monopole b defined in (2.23).
As in the 3d discussion, there are some modifications to this story for low values of N c . For N c = 2 there is no extra superpotential on the circle, and the moduli space for G = U(1) was discussed in section 4.2 of [6]. For N c = 3, there is only a single nontrivial eigenvalue of σ, and instead of (3.1) we get a superpotential from the fact that when σ 1 → 1/r the gauge group is enhanced again to so(3). The full superpotential of the 4d Spin(3) pure SYM theory on a circle takes the form W = 1/Y Spin + η 2 Y Spin , consistent with the global symmetry (3.4) that acts also in this case. Note that in this special case the instanton factor of Spin (3) is actually η Spin(3) = η 2 = Λ 2b 0 (keeping our general definition above for η), so this discussion is consistent with the standard discussion of 4d SU(2) theories on a circle [17]. The 4d Spin(3) SYM theory on a circle has two vacua at Y Spin = ±1/η = ±1/ √ η Spin (3) , which are fixed points of the Z 2 symmetry (3.4) (though they are related by a Z 4 global R-symmetry transformation).
For N c = 4 with Spin(4) = SU(2) × SU(2), we have in the pure 4d gauge theory on a circle two copies of the discussion of the previous paragraph, (consistent with the identification of Z discussed in section 2.3). Note that the two SU(2)'s have η 1 = η 2 = η. There are 4 supersymmetric vacua at In two of these vacua Y Spin = Y (1) Y (2) = 1/η, and in the other two Y Spin = −1/η; they are all fixed points of the global symmetry transformation (3.4).

4d SO(N c ) theories on S 1
We saw in the previous section that an important difference between G = Spin(N c ) and G = SO(N c ) is that in the latter case there is an extra monopole operator Y that can be used to label the Coulomb branch. The Coulomb branch of the SO(N c ) theory is a double cover of that of the Spin(N c ) theory, which is labeled by Y Spin = Y 2 . When we discuss the 4d theory on a circle, another difference is that the global symmetry transformation (3.4) becomes a large gauge transformation in SO(N c ), with the gauge transformation parameter periodic around the circle in SO(N c ) but not in Spin(N c ). Thus, the points on the Coulomb branch related by (3.4) are identified. This is related to the fact that we no longer have a spinor Wilson line, so the σ i have periodicity 1/r, and we can restrict the range of σ 1 to 0 ≤ σ 1 ≤ 1/2r. Let us see in more detail how the Z 2 large gauge transformation acts on Y . Given the relation Y 2 = Y Spin , and the action on Y Spin (3.4), there are two possible actions on Y : it could act either as Y ←→ 1/ηY , or as Y ←→ −1/ηY . These two options are related to the fact that there are two distinct SO(N c ) 4d gauge theories [5], SO(N c ) ± . As explained JHEP08(2013)099 in [5] and reviewed in the introduction, the difference between the three 4d gauge theories Spin(N c ) and SO(N c ) ± is in the choice of line operators. Y Spin has the same magnetic quantum numbers as the minimal 4d 't Hooft loop of the Spin(N c ) theory H 2 , wrapped on the circle. Y in the SO(N c ) + theory is related to the wrapped 4d 't Hooft loop H, and Y in the SO(N c ) − theory is related to the wrapped 4d 't Hooft-Wilson loop HW . (The wrapped 't Hooft loops are not BPS operators, but they reduce to chiral superfields in the low-energy effective action.) Correspondingly, the large gauge transformation in SO(N c ) ± acts on Y and leads to the identification (recalling that the wrapped spinor Wilson line W is odd under this transformation) In the 3d dimensionally reduced theory which arises as η → 0, we are left just with the region near Y = 0 so there is no longer any distinction between SO(N c ) + and SO(N c ) − , and there is just a single 3d SO(N c ) gauge theory.
In both cases, in the theories with flavors, the Coulomb branch of the 4d gauge theory on a circle is labeled by Y with the identification (3.7). These theories also have a Z 2 global symmetry taking Y → −Y , which acts on the Coulomb branch (this symmetry, acting on the non-trivial wrapped 't Hooft lines, is not present in the Spin(N c ) theory). Note that this symmetry remains also in the 3d limit, and that the baryon-monopole β of (2.19) is odd under it (while β Spin of (2.20) is even).
To summarize, in the SO(N c ) ± theories we have a gauge identification on the Coulomb branch given by (3.7), and in all 3 theories we have a Z 2 global symmetry changing the sign of the non-trivial line operator wrapped on the circle. We denote this symmetry by Z M 2 . It acts on the Coulomb branch as: We can now look at the pure so(N c ) SYM theory on a circle, and see how many vacua we have in the different so(N c ) theories [5]. Solving the F-term equations of the Spin(N c ) theory as above gives (N c − 2) different solutions for Z 4 The 4d duality on a circle

Dual theories on a circle
As discussed in [6], whenever we take two theories that are IR-dual in 4d, compactify them on a circle of radius r, and go to low energies (compared to the scales 1/r, Λ andΛ), the resulting low-energy 3d theories are IR-dual as well. We can start from the 4d duality The Z 2N f symmetry of theory A (which is a subgroup of the anomalous axial U(1) symmetry) is generated by g : Q → exp(2πi/2N f )Q, and that of theory B byg : q → exp(−2πi/2N f )q. They are mapped by the duality as g ↔gC [1,2].
When we compactify the two theories on a circle, we generate extra superpotentials where the 4d duality implies ηη = (−1) N f −Nc /256 [2,3]. We also get the compact Coulomb branches on both sides, described in the previous section. The general arguments of [6] imply that the low-energy theories with these extra superpotentials and compact Coulomb branches should be dual at low energies. The mapping between the chiral operators involving the flavors, and the associated flat directions, is the same in the theory on a circle as in 4d; M ij in theory B is identified with Q i Q j , and the baryonic operators B, W α and b 4d are identified withb 4d ,W α andB, JHEP08(2013)099 respectively [1,2]. But on a circle we have the extra Coulomb branch that we need to identify, which is not lifted by the extra superpotential (4.1), and we have the associated chiral operators that parameterize it. Note that the Coulomb branch coordinates are not charged under any continuous global symmetries, so these do not constrain their mapping.
However, there are discrete symmetries acting on the Coulomb branch (3.8), and these should map to each other under the duality (this follows from the mapping of the corresponding 4d non-trivial line operators [4,5]). In fact, the full mapping of the Coulomb branches is uniquely determined by requiring that we have a single-valued meromorphic transformation between them (after identifying by the large gauge transformations (3.7) for SO(N )), which correctly maps the global Z 2 symmetries (3.8), together with the extra requirement (for the SO(N ) + case) that the mapping is non-trivial (this follows from the analysis of the next subsection). The mapping between the Coulomb branch coordinate Y Spin of Spin(N c ), and the coordinateỸ of the dual SO(Ñ c ) − theory, takes the form (4. 2) The two choices of sign for √η are related by the global symmetryỸ → −Ỹ , which maps in the dual theory to the global symmetry Y Spin → 1/η 2 Y Spin . The two identified pointsỸ and (−1/ηỸ ) map to the same value of Y Spin , as they should. The map between the coordinate Y of the SO(N c ) − theory, and the coordinateỸ Spin of the dual Spin(Ñ c ) theory, is simply the inverse of this map (consistent with the fact that performing the duality twice should bring us back to the same point). The map between the two SO(N ) + theories is just the square root of (4.2), Again, one can check that it is consistent with the discrete gauge and global symmetries on both sides. For low numbers of flavors, N c = 3 orÑ c = 3, there are extra terms appearing in the dual superpotential [2], and the mapping of η toη is somewhat different because of the different instanton factor in so(3) theories. Both for N c = 3 and N c = 4 there are also extra "triality" relations between the 4d theories [2,5,8], because of the relations (1.3) and (1.4). In any case, the extra superpotential factors and dualities do not introduce any new issues when reduced on a circle, so we will not discuss them further here.
Note that we can get the dualities for the O(N ) and P in(N ) theories just by gauging the charge conjugation symmetries in the dualities for SO(N ) and Spin(N ). This is true both in 4d, and for the 4d theories on a circle. The main difference in the non-connected cases is that we do not have the baryon operators on both sides, so we have fewer distinguishable vacua and fewer chiral operators.

A test of the duality and of the Coulomb branch mapping
As a consistency check for our mappings (4.2), (4.3), let us analyze what happens far on the Higgs branch of theory A. In this theory we can turn on a vacuum expectation value JHEP08(2013)099 (VEV) for M ij = Q i Q j of rank N c , such that the gauge group is completely broken. For each such VEV there are two supersymmetric vacua, differing by the sign of the non-zero component of the baryon B = Q Nc (which squares to det Nc×Nc (M )). This is true both in 4d and in the theory on a circle. In the latter case, since the gauge group is completely broken by the quarks, we must be at a point on the Coulomb branch where classically the gauge group is unbroken, namely Y Spin = 0 or Y Spin = ∞ for Spin(N c ), and Y = 0 for SO(N c ).
We now discuss these vacua in theory B. The meson VEV gives a mass to N c quarks, leaving (N f − N c ) =Ñ c − 4 massless quarks. The low-energy theory is so(Ñ c ) withÑ c − 4 massless flavors q, with a scaleη low =η det Nc×Nc (M ). Let us first ignore the singlets and the superpotential. We then have classically a moduli space for the q's, where generically the gauge group is broken to so(4), with no light charged fields. We can think of the so(4) theory as an su(2) ⊕ su(2) gauge theory, where each su(2) factor has an instanton factorη su (2) related to that of the original so(Ñ c ) theory byη su(2) =η low /det(qq). In the 4d theory, gaugino condensation in each su(2) factor leads to an effective superpotential W = 2(±1 ± 1)η where at generic points on the moduli space thisW 2 α is the difference between the two gaugino bilinears of the two su(2) factors. It does not vanish in the SUSY vacua. This operator obeysb 2 4d = 4 det(qq)η su(2) = 4η low = 4η det Nc×Nc (M ). (4.5) Clearly, this relation is true for any value of the q's and is an exact ring relation. Hence, in the full 4d theory B we find thatb 4d is non-zero when M has rank N c , and obeys a similar relation to B of theory A, so that we can identifỹ b 4d = 2 ηB (4.6) (when we normalize the superpotential of theory B to be 1 2 M qq with no extra factors). We thus identify the two supersymmetric vacua discussed above for this VEV of M in theory B. Similarly, one can show that if rank(M ) > N c there is no supersymmetric vacuum in theory B.
Let us now repeat our discussion of theory B, when it is compactified on a circle. Most of the discussion is the same for Spin(N c ) and SO(N c ), so first we will not distinguish between them. Again we turn on a VEV of rank N c for M , leaving in theory B (Ñ c − 4) massless flavors. The matching between the high-energy and low-energy Coulomb branch coordinates implies thatZ low =Z/det Nc×Nc (M ) (as in (2.10)), so that the low-energy superpotential (4.1) includesηZ =η lowZlow . Again, let us ignore for a moment the extra 1 2 M qq superpotential in theory B, and imagine that we turn on generic VEVs for the remaining massless q's, breaking the gauge symmetry to so(4). Each of the su(2) factors JHEP08(2013)099 in so(4) now has a Coulomb branch coordinateỸ (j) , and, as in (2.11), the relation of the low-energy and high-energy coordinates is (4.7) The full low-energy superpotential, including the Affleck-Harvey-Witten superpotentials [15] of the two su(2)'s, is thus leading to four states withỸ (More precisely, in counting the physical states we should take into account the global aspects of whether our gauge group is Spin(Ñ c ) or SO(Ñ c ). We will do this momentarily.) Note that this is consistent with our discussion in the previous paragraph, and with the relation ηY = S for SU(2) theories on a circle [6]; in this case we have (in the chiral ring) η su(2)Ỹ (j) = S j . As in 4d, in order not to turn on a superpotential for the q's we need the expectation values to obeyỸ (1) = −Ỹ (2) . So, as in 4d, we find two supersymmetric vacua in theory B, in whichỸ For every value of M of rank N c in theory A there are two choices of B, and they are mapped under the duality to the two choices ofb in theory B. But we mentioned above that for fixed M of rank N c and fixed sign of B there could be either one or two vacua in theory A. Are they mapped correctly? For that we should be more careful about our gauge group.
When theory A is Spin(N c ), for each VEV of M of rank N c and each sign of B there are also two choices for the monopole operator: Y Spin = 0 or Y Spin = ∞ (related by Z M 2 ). In this case theory B is SO(Ñ c ) − , and we havẽ The two possible values ofỸ = ± i √η are fixed points of the identificationỸ ∼ − 1 ηỸ (3.7), and hence lead to two different vacua. They are related by the global ZM 2 symmetry whose JHEP08(2013)099 generator takesỸ → −Ỹ , and are the dual of the choice of Y Spin = 0, ∞ in theory A. This interpretation is consistent with our mappings (4.2).
When theory A is SO(N c ) − , for each value of M there are still two values of B, but there is a single choice, Y = 0 (which is a fixed point of Z M 2 ). Correspondingly, theory B is Spin(Ñ c ) in which there are two values ofb, as in (4.11), but no additional freedom in the VEV of the monopole operatorỸ Spin in (4.12).
When theory A is SO(N c ) + there is freedom only in the sign of B but not in Y = 0. In this case theory B is SO(Ñ c ) + , and there are again two possible values forỸ , which using (4.12) areỸ = ±i/ √η . But these two vacua are actually identified by (3.7), so we have a single vacuum, agreeing with the situation in theory A (and with (4.3)).
Thus, the mapping (1.5) leads to a consistent mapping of all these vacua far on the Higgs branch in the 4d theory on a circle, using (4.2) and (4.3).

Reduction of the SO(N ) + duality to 3d
The duality we found up to now is not purely a 3d duality, since it involves the compact moduli spaces that we get in the 4d theory on a circle. In this subsection we will see how we can turn it into a bona fide duality of 3d gauge theories.
Consider the duality between SO(N c ) + and SO(Ñ c ) + . The SO(N c ) + theory discussed above differs from the 3d SO(N c ) theory by the superpotential (4.1) and by the compactness of the moduli space. Let us focus on theory A near the origin of the moduli space Y = 0, and keep |ηY 2 | ≪ 1. In the dual theory B, this means that we are (using (4.3)) near Y = i/ √η (or equivalentlyỸ = −i/ √η ), with | √ηỸ − i| ≪ 1.
If we look at the low-energy superpotential in theory A we still have W A = ηZ, though the effects of this superpotential are very small in the region we are now discussing. In theory B we break the SO(Ñ c ) theory at this value ofỸ to SO(N f − N c + 2) × SO(2). 9 The operatorỸ maps at low energies to the Coulomb branch coordinate V + of the SO(2), and we can consider a newỸ low Coulomb branch coordinate for the low-energy SO(N f −N c +2) (defined as in (2.8)). In the low-energy superpotential of theory B we have contributions from the original W B of (4.1). The semi-classical forms of the monopole operators (2.6), (2.8), imply thatZ =ỸỸ low . In addition we have an Affleck-Harvey-Witten superpotential related to the breaking of the SO(Ñ c ) gauge group, which is proportional toỸ low /Ỹ . Thus, the full low-energy superpotential near this point is

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We can now use the mapping (4.3) betweenỸ and Y to rewrite this in terms of Y , which is now an elementary field in theory B: The choice of sign for the square root is arbitrary (the two choices are related by the global symmetry Y → −Y ). We can now simply take η → 0 on both sides (keeping ηη = (−1) N f −Nc /256 fixed); in theory A this is allowed since the effect of the superpotential smoothly goes to zero in the region we are keeping, and in theory B the same is also true (since |ηY 2 | ≪ 1). In this limit we find in theory A an SO(N c ) 3d theory with a noncompact Coulomb branch and with W A = 0, and in theory B an SO(N f − N c + 2) 3d theory, again with a non-compact Coulomb branch, and with where Y is now an elementary singlet in this low-energy theory, andỸ low is its standard Coulomb branch coordinate (2.8).
We can now lift this to a high-energy 3d duality between these two gauge theories, by replacingỸ low by the appropriate microscopic monopole operatorỹ of SO(N f − N c + 2), and the superpotential of theory B with Note that unlike in other cases discussed in [6], here we did not need to perform any real mass deformation in order to obtain the duality for the standard 3d SQCD theory from 4d, but just to take the 3d limit carefully. In the 3d limit we have an extra global U(1) A symmetry, that was anomalous in 4d. The quantum numbers of the various operators are consistent with the superpotential (5.4); using a specific choice for the 3d R-symmetry, they are

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in theory A, and and (N f ) Nc A denote totally antisymmetric products. Z C 2 is the charge conjugation symmetry, generated by C, and Z M 2 and ZM 2 are the global symmetries (3.8), generated by M andM, respectively. We included their action only on the gauge singlets. The compositesB andβ in theory B are defined as in theory A (see (2.19)), and their identification in theory A will be discussed below.
The three symmetries U(1) A , Z C 2 and Z M 2 are actually not independent. In theory A with gauge group SO(N c ), the action of e πiA (which is in SU(N f ) for even values of N f ) on Q is part of the gauge group for even values of N c , and is the same as C for odd values of N c . The action of e πiA on Y is the same as M N f . Thus, on gauge-invariant operators we have and In the dual theory we have e πiA C N f −Nc+2MN f = 1, implying that for odd values of N f , M ←→MC. (5.8) We will see below that this must be true for even values of N f as well.
The duality we find is very similar to the one discussed for O(N c ) theories (more precisely, O(N c ) + theories) in [10][11][12]. Indeed, if we now gauge the charge conjugation symmetry Z C 2 on both sides, we obtain precisely that duality, so our discussion is a derivation of this duality from 4d. But we obtain a duality also for SO(N c ) groups, meaning that there should be a consistent mapping of the charge-conjugation-odd baryons between the two sides. We can follow what happens to the 4d baryon mapping by our reduction procedure. In 4d the baryon B = Q Nc mapped tob 4d /2 √η (4.11). In the reduction on the circle we say that this first becomes equal to √ηb /2 (4.11), where the latter operator (2.23) involves a monopole operator in so(4). When we go onto the Coulomb branch as above, this monopole operatorỸ − becomes i/ √η (from the VEV ofỸ ) times the odd monopole operatorW − of SO(2), so we find that B maps to iβ/2, withβ defined as in (2.19). This is consistent with their global symmetry quantum numbers as in (5.5), (5.6). The 4d operator b 4d goes to zero in the η → 0 limit that we described, as does its 4d dualB 4d = q N f −Nc+4 (since only (N f −N c +2) components of the quarks remain massless in JHEP08(2013)099 the limit we took in theory B). However, we now get a new relation (required by consistency of the duality), mapping β to the 3d baryonB = q N f −Nc+2 . We cannot derive this duality directly from 4d, but on the part of the moduli space where we break both gauge groups to SO (2), it follows by dualizing the vector multiplets (3.5) in the 4d relation W α ↔W α into chiral multiplets (taking into account again the VEV ofỸ in theory B). It is also consistent with the global symmetries as in (5.5), (5.6). We conclude that the baryons map in the 3d duality between SO(N c ) and SO(N f − N c + 2) by Note that this mapping requires that the Z M 2 symmetry (3.8), which takes B → B and β → −β, maps under the duality by (5.8) for all values of N f . As we mentioned above, in the 3d theory b and Z are not chiral, so they do not have a simple mapping under the duality.
We can perform many tests of this duality, comparing moduli spaces, chiral operators, deformations, and so on, but most of these tests are identical to tests of the O(N c ) + duality that were already performed in [12]. We can find new tests by involving also the baryon operators. For instance, suppose that we turn on a VEV for M of rank N c , as in our discussion of the previous section. In theory A we still have two vacua for every such M , with B 2 = det Nc×Nc (M ). In theory B we now give a mass to N c quarks, so that we are left at low energies with N f − N c =Ñ c − 2 massless quarks, and with a low-energy Coulomb branch coordinateỸ N f −Nc =ỹ/ det Nc×Nc (M ). Our discussion around (2.22) implies that in this low-energy theory, in which the superpotential sets its meson qq = 0, there is a relationβ 2 N f −Nc = −4. Translating this into the high-energy theory (using (2.10)) we find β 2 = −4 det Nc×Nc (M ), so that we can indeed identifyβ with (−2iB).
For low values of N c and of N f − N c there are slight modifications of this discussion, as in the 4d duality [2] and in the analysis of the pure 3d theory [12], but these do not raise any new issues so we will not discuss them in detail here.

Reduction of the
We can similarly obtain the dual of the 3d Spin(N c ) theory, by starting from the 4d duality between Spin(N c ) and SO(Ñ c ) − . We can again focus on the same points Y Spin = 0 and Y = −i/ √η in the moduli space, and obtain the low-energy superpotential (5.1). However, now we are at a fixed point of (3.7), so the discussion in footnote 9 implies that the unbroken gauge symmetry in theory B is (2)). The moduli space coordinateỸ is the monopole operator of the SO(2) factor; when expressed in terms of Y , the SO(2) ⊂ O(2) is not visible. Focusing on the region aroundỸ = −i/ √η , it is convenient to change variables from the approximate free fieldỸ to y defined by Now we expand around y = 0, the mapping from theory A to theory B (4.2) identifies Y Spin = y 2 , and the extra Z 2 gauge identification acts as y → −y. This Z 2 also acts on the JHEP08(2013)099 SO(N f −N c +2) theory as charge conjugation, and it also changes the sign of the monopole operatorỹ =Ỹ low of SO(N f − N c + 2). Thus, we recognize the gauge theory we get as an O(N f − N c + 2) − theory, with the elementary field y odd under ZC 2 ⊂ O(N f − N c + 2). The low energy superpotential after taking η → 0 is The bottom line is that the dual of the Spin(N c ) SQCD theory is similar to (5.4), but the dual gauge group is O(N f − N c + 2) − , and y andỹ are odd under its ZC 2 subgroup. The Z C 2 global symmetry of theory A is mapped under this duality to ZM 2 . In the Spin(N c ) theory B is a Z C 2 -odd gauge-invariant operator, while Y and β do not exist. In the dual O(N f − N c + 2) − theoryβ is a ZM 2 -odd gauge-invariant operator (mapped to B), while y,ỹ andB are not gauge-invariant. The operator β Spin of (2.20) is present in theory A, and maps to the gauge-invariant operatorBy in theory B.
In the discussion above we expanded around the point Y Spin = 0, which is mapped tõ Y = −i/ √η . Instead we could expand aroundỸ = 0, which corresponds to Y Spin = −1/η.
Here the SO(N f − N c + 4) gauge group of theory B is unbroken, while the gauge group of theory A is broken as Spin(N c ) → (Spin(N c − 2) × Spin (2))/Z 2 , where the Z 2 acts on the spinors in both groups. Next we analyze the low energy dynamics of theory A, as we did in theory B above, focusing on the Spin(2) dynamics. It is important that there are no massless fields charged under this group. Hence, its Wilson loops become trivial at long distances. Further, we can map each (Spin(N c − 2) × Spin(2))/Z 2 bundle to an SO(N c − 2) bundle by simply ignoring the Spin(2) transition functions. Hence, when we integrate out the Spin(2) dynamics we are left with an SO(N c − 2) gauge theory. The dual of the Spin(2) gauge field, Y Spin , is an almost free field, which we can replace using (4.2) by an elementary fieldỸ , identified with the monopole operator of theory B. Finally, we can identify the gauge-invariant operator (iY low,Spin / √ ηZ) in theory A (where Y low,Spin is the Coulomb branch coordinate of Spin(N c − 2)) with Y low of SO(N c − 2), and, as in our analysis above, the various monopole-instantons coupleỸ to Y low . Forη → 0 we find a superpotential term proportional to Y lowỸ , as in (5.3). The duality we derive this way is precisely the inverse of the SO(N ) duality that we derived in section 5.1. This is a non-trivial consistency check on our web of dualities, because in section 5.1 we derived this duality from the compactification of a different 4d duality. We can also obtain a dual for P in(N c ), by gauging the global symmetry Z M 2 in the duality for O(N c ) + groups. The fact that in the O(N c ) + duality the symmetry Z M 2 maps to itself implies that the P in(N c ) theory is dual to a P in(N f − N c + 2) theory.

Dualities with Chern-Simons terms
As in [6], we can flow from the duality above to a duality with Chern-Simons terms. We can obtain an SO(N c ) theory with N f flavors and a Chern-Simons term at level k > 0 by starting from the theory with N f + k flavors and giving k flavors a positive real mass, by turning on a background field for the U(N f + k) global symmetry. In the dual SO(N f + k − N c + 2) theory, this maps to giving k flavors a negative real mass, giving the JHEP08(2013)099 mesons they couple to a positive real mass, and also giving a negative real mass to the singlet Y . Integrating out the massive fields we find an SO(N f + k − N c + 2) theory with N f flavors, level (−k), and a W B = 1 2 M qq superpotential. This is precisely the duality conjectured in [24] for the O(N ) theories (more precisely, O(N ) + theories), and here we see that it is true also for SO(N ).
The difference between O(N ) + and SO(N ) is that now we need to understand also how to map the baryon operators in the two sides, and this is more complicated (as in the discussion of SU(N c ) Chern-Simons-matter theories in [6]) since they involve monopoles. In theory A we still have the baryon operator B = Q Nc , while the baryon-monopole β = Q Nc−2 W − is no longer gauge-invariant in the presence of the Chern-Simons term, and similarly in theory B. We claim, similar to what we found for SU(N c ) Chern-Simons-matter theories, that the dual of B is now given by a monopole operatorβ ′ = q N f −NcW − (W α ) k , which is gauge-invariant. The quarks are contracted with an epsilon symbol, and break SO(N f + k − N c + 2) to SO(k + 2). The monopole operatorW − breaks SO(k + 2) to SO(k) × U(1), and because of the Chern-Simons term it carries a charge (−k) under the U(1). The gluinosW α are off-diagonal gluinos in SO(k) × U(1) which cancel this charge, and carry k different vector indices of SO(k) that are contracted by an epsilon symbol, such that the total operator is gauge-invariant. One can verify that the global symmetry charges of thisβ ′ exactly match with those of B. Similarly, we can construct an operator β ′ in theory A that matches withB = q N f +k−Nc+2 .
In theory A we have the relation B 2 = det Nc×Nc (M ). To see this in theory B we turn on a VEV of rank N c for M , leaving N f − N c massless flavors q, and we then ignore for a moment the superpotential and imagine giving an expectation value to the remaining massless flavors. This breaks the gauge group to SO(k + 2) with level (−k) and no massless flavors. At low energies this is a purely topological theory, in which we can construct a singlet operatorβ ′ =W −W k α as above, which is charged under the charge conjugation symmetry of this theory, and argue that it squares to one (similar to our discussion of the SU(k) theory at level (−k) in [6]). Lifting this to the high-energy theory using (2.10) we get precisely the expected relation (which turns out to be independent of the VEVs of the q's, so it is valid even for q = 0).
For N f = 0 our duality reduces to a duality of pure supersymmetric Chern-Simons theories, SO(N c ) k being identified with SO(|k| − N c + 2) −k . This is just the standard level-rank duality of SO(N ) Chern-Simons theories. At low energies we can integrate out the gauginos, shifting the SO(N c ) level to k → k − (N c − 2)sign(k). We then obtain the standard level-rank relation [25][26][27] for n, m > 0, that can be proven by studying nm real fermions in two dimensions. Similarly, we can flow from our Spin(N c ) duality to find a duality between Spin(N c ) k and (O(N f +|k|−N c +2) − ) −k (and a corresponding non-supersymmetric level-rank duality taking Spin(n) m to (O(m) − ) −n ), and a duality between the supersymmetric P in(N c ) k and P in(N f + |k| − N c + 2) −k theories (with a level-rank duality taking P in(n) m to P in(m) −n ).

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5.4 The special case of SO(2) = U(1) with N f = 2: a triality of dualities We now have two different dualities for SO(2) = U(1) gauge theories with N f flavors, which we refer to as theory A. First, we can view the gauge group as U(1) and find a dual theory based on U(N f − 1) [28]. We will refer to this dual theory as B1. Alternatively, as in this paper, we can view it as SO(2) and find a dual theory based on SO(N f ). We will refer to it as B2. The B1 dual exhibits the full SU(N f ) × SU(N f ) × U(1) A × U(1) J × U(1) R global symmetry, while the B2 dual exhibits explicitly only SU(N f ) × U(1) A × U(1) R , and the other symmetries arise as accidental symmetries at low energies.
The SO (2) theory with N f = 2 deserves special attention. In this case the gauge groups of theories A, B1 and B2 are all U(1), and they all have two flavors. Furthermore, in this case there is also a mirror theory, that also has gauge group U(1) and two flavors [29,30]- [13,31]. We will refer to this theory as B3.
Let us understand the relation between these dual descriptions (see also [12]). We begin with theory A. We can think of it either as a U(1) theory with two flavors Q a ,Qã (a,ã = 1, 2), or as an SO(2) theory with two doublets P i (i = 1, 2). Let us work out the translation between these two languages. In the U(1) description of this theory, the chiral operators are the magnetic monopoles V ± (2.13) and four mesonsM ab = Q aQb . The translation to SO(2) variables, if we keep the standard normalization for the kinetic terms, is by Defining the standard SO(2) mesons M ij = P i · P j , the symmetric part ofM is related to M byM ij +M ji = M ij . The anti-symmetric part ofM is related to the SO(2) baryon B ≡ P 1 1 P 2 2 − P 2 1 P 1 2 byM 12 −M 21 = −iB. The natural monopole-related operators in the SO(2) language are 14) The former is the basic monopole in the SO(2) language, and the latter is the baryon operator β (2.19) in this special case. The charges of the different objects under the global U(1) A × U(1) R symmetry that is visible in all descriptions are (using our standard conventions): The dual description which has all the symmetries of theory A manifest is B1. This is a U(1) theory with two flavors q (1) a ,q (1) a (a,ã = 1, 2) and with additional singlet fieldsM ab (1) and V (1) ± . The superpotential is [28] W B1 = q (1)M (1)q(1) + V

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For comparison with other duals it will be useful to translate this to the SO(2) language, as we did above. We define as abovẽ Translating the quarks q,q to SO(2) quarks p as above, the superpotential (5.16) becomes with the singlets B (1) and M (1) related toM (1) , and the singlets β (1) and Y (1) related to V (1) ± , as in theory A. These singlets are identified with the corresponding operators in theory A. The U(1) A × U(1) R charges of the different objects are: The normalization of the first term in (5.18) is half of the normalization in the standard SO(N c ) duality (5.4), and it will be easier to compare them if we have the same normalization in both cases. Since the p's do not appear in any gauge-invariant chiral operator, we can simply rescale them to new variablesp This also rescales the baryonB (1) toB (1) =B (1) /2, and because of the relation of the quantum numbers of the monopoles to those of the quarks, the latter are also rescaled toỸ (1) = 2Ỹ (1) ,β (1) = 2β (1) . We can now write (5.18) as The dual description B2, with gauge group SO(2), is quite similar to B1. The difference is that we do not have the singlet fields B and β, and the superpotential is (5.4) The map between the chiral operators here is Note that in description B2 the U(1) J symmetry is not present in the UV gauge theory, as the singlet Y mixes with β under this symmetry. Moreover only an SU(2) ⊂ SU(2) × SU(2) flavor symmetry is visible in the UV. The symmetries broken in this UV description should appear as accidental symmetries of the IR physics. Finally, in description B3 we do not have M 12 and Y , and the superpotential is

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Using the fact that mirror symmetry exchanges the monopoles V ± with the off-diagonal mesonsM 12 ,M 21 with coefficient one, the map between the chiral operators here is (5.24) Here U(1) J is present in the UV description and is identified with part of the dual flavor group, but only U(1) × U(1) ⊂ SU(2) × SU(2) is a symmetry of the UV theory. As in the B1 theory, it is convenient to rescale the p's by √ 2, and in the rescaled variables defined as above we have (3) 1p with the new mapping .
(5.26) Note that the mapping between the B's and β's is now the same as in (5.22).
Let us now relate these theories, by understanding their deformations. We claim that the IR superconformal field theory that all these theories flow to has eight marginal deformations, and that two of them are exactly marginal. In the B1 description the eight marginal operators areM aã (1) V (1) ± . Denote their coupling constants by λ ± aã . The space of exactly marginal deformations is generally given by the (complexified) quotient of the space of marginal deformations by the global symmetries, which act on it non-trivially [32]. This can be found by noting the invariant combinations constructed out of λ ± aã . All these deformations preserve U(1) A , but they are charged under SU(2) × SU(2) × U(1) J . There are two non-trivial invariants of this group: ǫ ab ǫãbλ + aã λ − bb and ǫ ab ǫãbλ + aã λ + bb ǫ cd ǫcdλ − cc λ − dd . This shows that there are two exactly marginal deformations. Equivalently, the global symmetry SU(2) × SU(2) × U(1) J is not completely broken on the space of couplings, but a U(1) always remains (which is a subgroup of the diagonal SU(2)).
By a global symmetry transformation, we can choose the two exactly marginal deformations to be given by (in the SO(2) language) δW = γ B β + ρ M 12 Y . (5.27) Note that these deformations are invariant under Z C 2 , but they break Z M 2 (3.8). Suppose we first add the term with γ to theory A. In the B1 dual description, the superpotential becomes after integrating out the massive singlet fields (1)B (1) .

(5.28)
This is exactly the same as the superpotential W B2 , deformed by −1

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Next we add the term with ρ to theory A. The superpotential in the dual B2 description becomes, after integrating out massive singlet fields, This is exactly the same as the superpotential W B3 , deformed by − 1 4ρỸ (3)p (3) 1p (3) 2 ; as we mentioned, the baryon mappings are also consistent. On the other hand, adding ρ M 12 Y to theory A translates in theory B3 to adding ρỸ 2 . Thus we conclude that our IR superconformal field theory also possesses an exact duality ρ ←→ (−1/4ρ).
Assuming the three dualities, we deduced that the exactly marginal couplings enjoy dualities taking γ ←→ (−1/4γ) and ρ ←→ (−1/4ρ). Alternatively, if one could prove the duality of the marginal deformations, one could deduce all three duals of U(1) with two flavors from knowing any one of them.

Partition functions and indices for so(N ) dualities
A set of useful checks of dualities is given by comparisons of supersymmetric partition functions of the putative dual pair: if the two 3d UV theories describe the same IR physics, their S 3 and S 2 × S 1 supersymmetric partition functions should agree. In this section we will discuss these checks for the dualities of the previous section.

The partition function on S 2 × S 1
Let us start by discussing the matching of the partition function on S 2 × S 1 , also known as the supersymmetric index. The indices for the O(N c ) + versions of the dualities discussed in this paper were checked to match in [11,33]. The index is sensitive to the global structure of the gauge group, and thus the matching of the indices for SO(N c ) does not directly follow from these computations. 10 We will check here that the supersymmetric indices match also for the SO(N c ) dualities. In the process of doing this, we will see that, since the index contains information about local operators, it can test the proposed mapping of the baryon operators to the baryon-monopole operators discussed in the preceding sections.
First, let us briefly review the definition of the 3d supersymmetric index. It is defined by the following trace over states on S 2 × R (see [34][35][36][37]- [6,38] for details): (6.1) Here ∆ is the energy in units of the S 2 radius (related to the conformal dimension for superconformal field theories), J 3 is the Cartan generator of the Lorentz SO(3) isometry of S 2 , and e a are charges under U(1) global symmetries (which could be subgroups of non-Abelian global symmetries). The states that contribute to this index satisfy ∆−R−J 3 = 0, where R is the R-charge (that is used in the compactification on S 2 ).

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This index can be computed by a partition function on S 2 ×S 1 , and localization dictates that the index gets contributions only from BPS configurations. For example, for a U(1) gauge multiplet, we can take the gauge field to have a holonomy z ∈ U(1) around the S 1 and magnetic flux m ∈ Z on the S 2 , which then determines the configurations of the other fields in the gauge multiplet. The 1-loop determinant of a chiral multiplet of R-charge R coupled with unit charge to this gauge multiplet is: For a general gauge theory with gauge group G of rank r G , one introduces fugacities z i (i = 1, · · · , r G ) parameterizing the maximal torus of G, with corresponding GNO magnetic fluxes m i on S 2 . One can similarly introduce fugacities u a and fluxes n a for background gauge multiplets coupled to global symmetries. The 1-loop determinant in such a configuration is given by taking the product of the contributions (6.2) of the chiral multiplets, along with a contribution from the vector multiplet: where the product is over the roots of the gauge group. One can also include Chern-Simons terms for background or dynamical gauge multiplets, whose contribution, for instance, for a level k term for a U(1) gauge multiplet, is z km . Finally, the partition function is given by integrating over the gauge parameters z i and summing over the gauge fluxes m i . We will be interested in the SO(N c ) gauge theory with N f chiral multiplets Q a of R-charge R in the vector representation of SO(N c ), and with a Chern-Simons term at level k. We include fugacities and fluxes, µ a and n a , a = 1, · · · , N f , for the U(N f ) flavor symmetry, as well as fugacities ζ = ±1 for the global symmetry Z M 2 , and χ = ±1 for the charge conjugation symmetry Z C 2 . Let us write down explicitly the relevant indices. The index with χ = +1 is given by: Here N c = 2r G + ǫ with ǫ = 0, 1. The integers m i run over the Weyl-inequivalent GNO charges, and |W {m} | is the order of the residual Weyl group [35]. In the first term on the JHEP08(2013)099 first line we have introduced background Chern-Simons terms for the global symmetries. k F is the level of a background Chern-Simons term for the U(N f ) global symmetry [39,40], 11 and k ζ (obeying k ζ ∼ k ζ + 2) is a similar term mixing the discrete Z M 2 symmetry with U(1) A . 12 We are free to choose the values of these terms, as long as parity anomalies are canceled so that the index is well defined, namely, it has an expansion in fugacities with integer powers. 13 This requires where the second requirement is the standard parity anomaly [41,42].
Next we want to compute the index with χ = −1, where we should sum over holonomies of O(N c ) that have determinant (−1). The computation is different for the cases of odd and even N c (see [11,33] and also [43]). First, let us discuss the odd N c case. A general O(2r G + 1) holonomy of determinant χ can be brought to the form Thus, the indices with χ = −1 are given by 14 We introduced here a background Chern-Simons term with coefficient k χ (k χ ∼ k χ + 2) mixing the charge conjugation symmetry Z C 2 with U(1) A , and for the partition function to be well-defined we must have (6.8) 11 We could also introduce different levels for U(1)A and SU(N f ). 12 If we describe the background Z M 2 gauge symmetry by two U(1) gauge fields A1,2 with an action given by an off-diagonal Chern-Simons term at level two [22,23], we can write k ζ as the coefficient of an ordinary Chern-Simons term that mixes A1 with the background U(1)A gauge field. 13 Note that, in the building blocks defining the index, there appear half-integer powers of the fugacities (see, e.g. (6.2)), and so these factors are not well-defined individually. However, when we expand the index as a series and include the appropriate background Chern-Simons terms, this expansion has only integer powers of the fugacities, and thus is well-defined. 14 In the expression of the index here and in the even Nc case below, we write the last eigenvalue of the holonomy as χ in the contribution of the chiral multiplets, in order to keep track of the fractional powers of χ appearing in the intermediate expressions in a consistent way.

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where againÑ c = N f + |k| − N c + 2 = 2r G +ǫ. Note that the parameter ζ now also appears in the contribution of the elementary field Y , since the ZM 2 symmetry in theory B also acts on this singlet. The factors on the first and the second line represent the contribution of background Chern-Simons terms. The background Chern-Simons terms on the second line are the relative ones, which must be included when k = 0 [10] (here we defined sign(k) = 0 for k = 0) for the duality to work. The expressions for I B SO(Ñc) with χ = −1 are obtained in an analogous way to our discussion of theory A above.
The dualities discussed in this paper imply the following equality for the indices, I A SO(Nc) (x; µ a ; n a ; ζ, χ) = I B SO(N f +|k|−Nc+2) (x; µ a ; n a ; ζ, ζ χ) . (6.12) We have checked this equality for various values of the discrete parameters k, n a , ζ and χ, by expanding both sides in a power series in x and comparing the leading coefficients. We also can write the indices for other orthogonal gauge groups. In the SO(N c ) index computation we introduced a fugacity χ = ±1 for the global charge conjugation symmetry Z C 2 . Similarly, we can introduce in the computation for an O(N c ) gauge group a discrete theta-like parameter χ ′ = ±1, determining whether we project on even or odd states under Z C 2 . The O(N c ) + result for χ ′ = 1 is half of the sum of the SO(N c ) results with χ = 1, −1, and the O(N c ) + result for χ ′ = −1 is half of their difference. Thus, allowing for arbitrary χ and χ ′ one can relate the SO(N c ) and the O(N c ) + expressions. 15 In the O(N c ) − case we need to change the sign of the projection for states charged under Z M 2 . Similarly, the Spin(N c ) and P in(N c ) indices are given by summing over the sectors with different ζ, and we can define for them an index with ζ ′ = 1 that projects on the Z M 2 -even states (which make up the standard Spin(N c ) and P in(N c ) theories), and an index with ζ ′ = −1 that projects on the odd states. We then have: I Spin(Nc) (x; µ a ; n a ; ζ ′ , χ) = 1 2 I SO(Nc) (x; µ a ; n a ; ζ = +1, χ) + ζ ′ I SO(Nc) (x; µ a ; n a ; ζ = −1, χ) , I O(Nc) + (x; µ a ; n a ; ζ, χ ′ ) = 1 2 I SO(Nc) (x; µ a ; n a ; ζ, χ = +1) + χ ′ I SO(Nc) (x; µ a ; n a ; ζ, χ = −1) , I O(Nc) − (x; µ a ; n a ; ζ, χ ′ ) = 1 2 I SO(Nc) (x; µ a ; n a ; ζ, χ = +1) + χ ′ I SO(Nc) (x; µ a ; n a ; −ζ, χ = −1) , I P in(Nc) (x; µ a ; n a ; ζ ′ , χ ′ ) = 1 2 I Spin(Nc) (x; µ a ; n a ; ζ ′ , χ = +1) + χ ′ I Spin(Nc) (x; µ a ; n a ; ζ ′ , χ = −1) . (6.13) Our tests of the SO(N c ) duality (6.12) with general ζ and χ provide tests also for the dualities between two O(N c ) + theories, between Spin(N c ) and O(Ñ c ) − theories, and between two P in(N c ) theories. 16

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When the indices are expanded as a series in the various fugacities, the terms in the series represent the contributions of BPS operators with the corresponding charges, as follows from the definition of the index (6.1). Thus we can use the expressions above to attempt to trace how the baryons map between the two dual SO descriptions (6.12). 17 On the electric side the usual baryons are Q Nc , with an anti-symmetric product of some choice of N c flavors. For example, let us define B to be the operator constructed out of the flavors a = 1, · · · , N c . In the dual side, this maps to a monopole-baryon operator, β ′ , as described in section 5.3. Specifically, in this case: (6.14) Here eσ 1 /ĝ 2 3 +iã 1 is the basic SO(Ñ c ) monopole with GNO charges (1, 0, 0, . . . , 0),λ αβ are the gluinos, and q a are the dual quarks, with their color indices contracted antisymmetrically in the subgroup SO(N f − N c ) ⊂ SO(Ñ c ) that is left unbroken by the monopole and the gluinos. To test this mapping in the index, note that each chiral multiplet Q a,α (α = 1, · · · , N c , a = 1, · · · , N f ) contributes a factor 18 x R w α µ a , so the operator B contributes a term x RNc Nc a=1 µ a to the index. On the dual side, q a,β contributes x 1−R w β µ a −1 , each gluino contributes −x w α /w 1 , and the monopole background Putting this together we find that the contribution ofβ ′ matches that of B (there is also a factor of χ in theory A, and a factor of ζχ in theory B). The baryons of theory B qÑ c are mapped in a similar way to β ′ = e σ 1 /ĝ 2 3 +ia 1 k+2 α=3 λ 1α Q Nc−k−2 in theory A.

The partition functions on S 3
Let us now comment on the S 3 partition functions. The partition functions on S 3 for N = 1 SQCD with O(N c ) + gauge group were computed and found to agree for the O(N c ) + dualities discussed in [10,11,24,33]. In fact, the equality of the partition functions of the theories with SO(N c ) (and Spin(N c )) gauge groups discussed in the preceding sections follows directly from the equality of partition functions for theories with O(N c ) + gauge groups. The S 3 partition functions are computed by a matrix integral over the Lie algebra, which is the same in all these cases, and thus the partition functions differ only by overall factors of 2 due to the different volumes of the gauge groups. Hence, the results of [10,11,24,33] straightforwardly imply that the SO(N c ) dual pairs discussed in this paper have the same S 3 partition functions. In certain cases, e.g. the dualities discussed in [6], the equality of the partition functions on S 3 of the 3d theories follows in a simple way from the equality of the 4d partition JHEP08(2013)099 functions on S 3 × S 1 (the supersymmetric index) of the 4d theories from which these 3d theories descend. However, this is more subtle in the case of dualities with orthogonal groups, as we will now explain.
First, let us briefly outline how the 3d partition functions are obtained from the 4d indices: for more details see [6,7,[46][47][48]. The partition function of a 4d theory on S 3 ×S 1 can be thought of explicitly as an S 3 partition function of the dimensionally reduced theory with all the KK modes included. 19 The (inverse) radius of the S 1 appears in the S 3 partition function as a real mass for the U(1) symmetry associated with the rotation around the circle. Taking the small radius limit corresponds to taking this real mass to be large, and thus decoupling the massive KK modes. The fugacities for the 4d global symmetries become real mass parameters in 3d. Some of the classical symmetries of the 4d gauge theories are anomalous, but the 3d theories obtained by dimensional reduction of the matter content of the 4d ones do have these symmetries at the full quantum level. The 4d index cannot be refined with fugacities for the anomalous symmetries, and thus the 3d partition functions obtained by this reduction procedure are not refined with the corresponding real mass parameters. This is an indication that the 3d theory obtained by the reduction has a superpotential breaking the symmetry that is anomalous in 4d [6].
The above discussion presumes that the dimensional reduction produces a well-defined and finite S 3 partition function. This presumption is true for the cases discussed in [6], but it is not true for the SO(N c ) theories discussed in this paper: the reduction of the 4d index 20 for SO(N c ) SQCD produces a divergent 3d partition function. The divergence can be explained physically by the fact that not all of the Coulomb branch is lifted when putting the theory on the circle, as discussed in the previous sections. In particular, in the 4d theory on S 1 , the operators Y or Y Spin , parameterizing the Coulomb branch, have no continuous global symmetry charges and no R-charge, and the presence of such a field leads the 3d partition function to diverge. 21 We have seen in the previous sections that due to the intricate moduli space on the circle, the 3d SO(N c ) dualities are obtained by focusing on certain regions of the Coulomb branch. It is possible that this more intricate procedure can also be mimicked at the level of the index, 22 and we leave this question to future investigations.