3d $\mathcal{N}=2$ SO/USp adjoint SQCD: s-confinement and exact identites

We study 3d $\mathcal{N}=2$ SQCD with symplectic and orthogonal gauge groups and adjoint matter. For $USp(2n)$ with two fundamentals and $SO(N)$ with one vector these models have been recently shown to s-confine. Here we corroborate the validity of this proposal by relating it to the confinement of $USp(2n)$ with four fundamentals and an antisymmetric tensor, using exact mathematical results coming from the analysis of the partition function on the squashed three-sphere. Our analysis allows us to conjecture new s-confining theories for a higher number of fundamentals and vectors, in presence of linear monopole superpotentials. We then prove the new dualities through a chain of adjoint deconfinements and s-confining dualities.


Introduction
A crucial aspect underlining the study of gauge theories is that gauge invariance corresponds to a redundancy more than to a fundamental symmetry.This motivates the search of dual models, often described in terms of new gauge groups sharing the same IR properties of the original one.An interesting possibility is that the dual model is described in terms of the confined degrees of freedom of the original one.In this case the original model is referred as s-confining and it corresponds, in many cases, to a limiting case of a duality between two gauge theories.Examples of this behavior have been worked out in models preserving four supercharges in 4d and in 3d, namely with N = 1 and N = 2 supersymmetry respectively.
In the 4d N = 1 case with a single gauge group with W = 0 a systematic classification has been proposed by [2,3], and elaborating on that results many other examples have been found.Many examples of this phenomenon in the 3d N = 2 case can be obtained through the circle compactification of the 4d parent cases, along the lines of [4].
In 3d there is a new ingredient that makes the classification more intricate and offers new examples of gauge theories with confining dynamics, given by the possibility of turning on monopole superpotentials.Many examples of 3d s-confining gauge theories been studied in [4][5][6][7][8][9][10][11][12][13][14][15][16], where many checks of the new proposed dualities have been performed.In a recent paper [1] models with real gauge groups and adjoint matter have been studied and new confining dualities have been proposed.An interesting aspect of these cases is that the dualities can be proved by sequentially deconfining the adjoint (symmetric or antisymmetric tensors) in terms of other known dualities involving real gauge groups without any tensor.Such a deconfinement of two-index matter fields follows from the one originally worked out in 4d in [17] and then refined in [18] (see also the recent works [19,20] where such deconfinement technique has been reconsidered in the 4d case).In 3d the structure of confining gauge theories is richer because of the possibility of turning on monopole superpotentials.
In this paper we elaborate on these results, showing the matching of the threesphere partition function between the new dual phases proposed by [1].We find that there is a straightforward proof of the hyperbolic integral identity that corresponds to the matching of the squashed three-sphere partition functions between the dual phases.The result follows from the identity relating U Sp(2n) with the antisymmetric and four fundamentals without monopole superpotential and its description in terms of confined degrees of freedom.In this case by opportunely fixing the value of the mass parameters and by applying the duplication formula for the hyperbolic Gamma functions we observe that the identity can be manipulated into the expected ones for the new dualities proposed by [1].
This correspondence motivates us to make one step further, and to consider the case of U Sp(2n) with the antisymmetric and six fundamentals, in presence of a monopole superpotential (see [21][22][23][24][25][26][27][28][29][30][31][32] for recent examples of 3d N = 2 gauge theories and dualities with monopole superpotential turned on).This model is confining as well and it admits the same manipulation referred above on the integral identity matching the squashed three-sphere partition functions.Again we obtain identities relating, in this case, the partition function of models with U Sp(2n) or SO(N ) gauge groups with four fundamentals or three vectors and an adjoint matter field, and the partition function of models with (interacting) singlets.
We then analyze these models through sequentially deconfining the adjoint fields, obtaining a prove of the dualities.This last approach offers also an alternative derivation of the integral identities (obtained so far through the duplication formula), in terms of adjoint deconfinement.Indeed, as we will explicitly show below, each step discussed in the physical proof of the duality corresponds to the application of a known identity between hyperbolic integrals.
The paper is organized as follows.In section 2 we discuss some review material that will be necessary for our analysis.More concretely in sub-section 2.1 we review the dualities worked out in [1] while in sub-section 2.2 we focus on the hyperbolic integrals corresponding to the squashed three-sphere partition function that will play a relevant role in the rest of the paper.In section 3 we show how it is possible to reproduce the dualities of [1] by an application of the duplication formula on the partition function of U sp(2n) with four fundamentals and an antisymmetric.Section 4 is the main section of the paper and it contains the new results.Here we start our analysis by reverting the logic discussed so far in the derivation of the dualities.Indeed we first apply the duplication formula to the partition function of U sp(2n) with six fundamentals and an antisymmetric.This gives raise to three new integral identities that we interpret as examples of s-confining dualities for U Sp(2n) or SO(N ) gauge theories with four fundamentals or three vectors and an adjoint matter field.By flipping some singlets we propone also the structure of the superpotential for the confined phase in each case.Then in sub-section 4.1, as a consistency check, we engineer a real mass flow interpolating from our new dualities to the ones of [1].In sub-section 4.2 we prove the new dualities through deconfining the adjoint matter fields.As a bonus we show that this procedure can be followed step by step on the partition function, giving an independent proof of the integral identities we started with.In section 5 we summarize our analysis and discuss some further lines of research.In appendix A we discuss the physical derivation of the integral identities for the dualities of [1] by using the deconfining trick, corroborating the idea of proving exact mathematical identities from physical principles.In appendix B we derive the integral identities for SO(N ) gauge theories with N + 1 vectors and linear monopole superpotential, that have played a prominent role in our analysis.

Review
2.1 3d confining models with real gauge groups and adjoint matter These dualities have been proved in [1] and they are the starting point of our analysis.Here we review the main properties of these dualities and briefly discuss their derivation.Then in appendix A we will provide the matching of the three-sphere partition function by reproducing the deconfinement of the adjoint matter fields.
The three classes of s-confining dualities with adjoint matter obtained in [1] are summarized in the following.
• In the first case the electric side of the duality involves an U Sp(2n) gauge theory with adjoint S and two fundamentals p and q with superpotential W = Tr(pSp).
The dual model corresponds of a WZ model with 4n chiral multiplets.These 4n gauge fields corresponds to gauge invariant singlets of the electric theory.There are 2n dressed monopole operators, Y j = Y U Sp Tr S j , j = 0, . . ., 2n − 1, where Y U Sp is the unit flux monopole of the U Sp(2n) gauge theory.Then there are n dressed mesons M = qS 2 +1 q with = 0, . . ., n − 1 and eventually there are n singlets σ k = Tr S 2k with k = 1, . . ., n.
• The second case involves an SO(2n) gauge theory with an adjoint A and a vector q, without superpotential.The dual theory is a WZ model with 4n chiral fields, corresponding to gauge invariant singlets of the electric theory.There are 2n − 1 dressed monopole operators, Y + j = Y + SO Tr A j , j = 0, . . ., 2n − 2, where Y + SO is the unit flux monopole of the SO(2n) gauge theory with positive charge with respect to the charge conjugation symmetry.Then there are n dressed mesons M = qA 2 q with = 0, . . ., n − 1 and n − 1 singlets σ k = Tr A 2k with k = 1, . . ., n − 1.The last two chiral fields correspond to the baryon B ≡ Pf A and to the baryon monopole Y − A n−1 , obtained from the unit flux monopole of the SO(2n) gauge theory with negative charge with respect to the charge conjugation symmetry.
• The third and last case involves an SO(2n + 1) gauge theory, again with an adjoint A, a vector q and vanishing superpotential.The dual theory is a WZ model with 4n + 2 chiral fields, corresponding to gauge invariant singlets of the electric theory.There are 2n dressed monopole operators, Y + j = Y + SO Tr A j , j = 0, . . ., 2n − 1, where Y + SO is the unit flux monopole of the SO(2n) gauge theory with positive charge with respect to the charge conjugation symmetry.Then there are n dressed mesons M = qA 2 q with = 0, . . ., n − 1 and n singlets σ k = Tr A 2k with k = 1, . . ., n.The last two chiral fields correspond to the baryon B = 2n+1 (qA n ) and to the baryon monopole Y − As stressed in [1] the superpotential of the dual models correspond to polynomials of the singlets and with complexity that rapidly grows when the ranks of the gauge groups increase.Nevertheless by flipping the singlets σ k , and the baryon and the baryon monopole in the orthogonal cases, these superpotentials are given by cubic combinations of the remaining singlets.
Let us briefly sketch the strategy for proving these dualities.The first step consists of deconfining the adjoint field.In the symplectic case the adjoint is in the symmetric representation and it can be deconfined in terms of an orthogonal gauge group.On the other hand in the orthogonal case the adjoint is in the antisymmetric representation and it can be deconfined in terms of a symplectic gauge group.In each case this step requires to find a confining duality that reduces to the original model.After deconfining the adjoint one is then left with a two gauge node quiver gauge theory and one can then proceed by dualizing the original gauge node, by using a known duality.In the cases at hand this duality corresponds to a limiting case of an Aharony duality or a modification of it, with monopole superpotentials.This gives raise to another model with a real gauge group and adjoint matter and generically a more sophisticated superpotential.By repeating the procedure of rank-two tensor deconfinement and duality one is left with the original gauge group but with rank of one unit less and it allows to iterate the procedure and arrive to the desired WZ model at the end of such a cascading process.
By inspection it has been shown in [1] that the adjoint of the U Sp(2n) case can be deconfined by an SO(2n+1) gauge group and a superpotential flipping the monopole.
After dualizing the U Sp(2n) gauge theory one ends up with an SO(2n + 1) gauge theory with an adjoint and a dynamically generated superpotentials flipping both the monopole and the baryon monopole.In this case the adjoint can be deconfined by an U Sp(2n − 2) gauge group and a more intricate flavor structure.Indeed the SO(2n + 1)/U Sp(2n − 2) gauge group have one extra vector/fundamental charged chiral fields and there is a superpotential interactions between these two fields and the SO(2n + 1) × U Sp(2n − 2) bifundamental.Furthermore there is a linear monopole superpotential for the U Sp(2n) gauge node.By dualizing the SO(2n + 1) gauge node with 2n vectors one ends up with an U Sp(2n − 2) gauge theory, with two fundamentals and a non trivial superpotential.By opportunely flipping some of the singlets of the original model one can recast that the original U Sp(2n), iterate the procedure and eventually prove the duality.Similar analysis have been used to prove the orthogonal dualities as well.In such cases after deconfining the antisymmetric in terms of U Sp(2n − 2) and dualizing the original orthogonal gauge group one is left with U Sp(2n − 2) and two fundamentals.Then the duality proven above for this case can be used to prove the duality for the orthogonal cases as well.

Confining theories and the three-sphere partition function
Here we review some known aspect of the 3d partition function for 3d N = 2 gauge theories on the squashed three-sphere preserving U (1) × U (1) isometry.
The real squashing parameter b can be associated to two imaginary parameters ω 1 = ib and ω 2 = i/b and their combination is usually referred as 2ω ≡ ω 1 + ω 2 .The matter and the vector multiplets contribute to the partition function through hyperbolic Gamma function, defined as The argument x represents a parameters associated to the real scalar in the (background) vector multiplet and it gives the informations about the representations and the global charges of the various fields.We refer the reader to [1] for further details.
Here we are interested in two confining gauge with U Sp(2n) gauge group and antisymmetric and six or four fundamentals.In the first case the theory has a monopole superpotential and it corresponds to the reduction of a 4d N = 1 confining gauge theory.In the second case the theory with four fundamenrtals can be obtained by a real mass flow, it is still confining but in this case the superpotential is vanishing.Details on these models have been discussed in [10,11].
In general the partition function of an U Sp(2n) gauge theory with 2n f funda-mentals and an antysimmetric tensor is Where the parameters τ and µ r are associated to the antisymmetric tensor and to the 2n f fundamentals respectively.The two confining dualities discussed above for 2n f = 6 and 2n f = 4 correspond to the following identities with the balancing condition signaling the presence of a linear monopole superpotential, and with unconstrained parameters, corresponding to the absence of any monopole superpotential.These identities are the starting point of our analysis, and they contain all the mathematical information on the models with real gauge groups and adjoint matter.
In order to transform symplectic gauge groups into unitary one we will use a well known trick, already used in the literature [33][34][35][36].It consists of using the duplication formula [37][38][39] to modify the partition function of the vector multiplet of U Sp(2n) into the partition function of the vector multiplet of SO(2n) or SO(2n + 1).This transformation requires to consider an U Sp(2n) gauge theory with fundamental matter fields and assign to some of the mass parameters some specific value as µ = ± ω i 2 or µ = ω or µ = 0. Then by applying the duplication formula (and the reflection equation Γ(x)Γ(2ω − x) = 1 when necessary) one can convert the contribution of U Sp(2n) with fundamentals in the one of SO(2n) or SO(2n + 1) with (few) vectors.Furthermore, by using the same mechanism, one can convert also the contribution of the U Sp(2n) antisymmetric field into the one of an adjoint (for both the symplectic and the orthogonal cases).
To simplify the reading of the various steps of the derivation we conclude this section by summarizing the integral identities for U Sp(2n) and SO(N ) s-confining SQCD, that we have used in the analysis below.These identities are indeed necessary for translating into the language of the squashed three-sphere partition function the chain of adjoint deconfiments and dualities introduced above.In the table we indicate the gauge group, the matter content, the superpotential and the reference to the integral identity equating the partition function of each gauge theory with the one of its confined description .

Gauge group Matter Superpotential Identity
U Sp(2n)

Proving known results
In this section we show how to obtain the integral identities for the three dualities reviewed in subsection (2.1) by applying the duplication formula (2.6) on the identity (2.5).Here and in the following section we will use three choice of masses, that are Here we did not specify the length n f of the vector µ.In the following we will have n f = 4 for the cases of [1] and n f = 6 for the new dualities discussed here.

Case I: U Sp(2n)
If we choose the masses µ r as µ , m and apply the duplication formula, the LHS of (2.5) becomes This corresponds to the partition function of U Sp(2n) with an adjoint S, a fundamental p and a fundamental q with superpotential W = T r(pSp), where the constraint imposed by the superpotential corresponds to the presence of the parameter ω − τ 2 in the argument of the last hyperbolic gamma function in the numerator of (3.1).
On the other hand the RHS of (2.5) requires more care.Let us separate first the contributions of the three terms.By substituting the parameters µ r and using the reflection equation we have where we used the shorthand notation Γ h (a, b) = Γ h (a)Γ h (b).By using the duplication formula it becomes This last formula can be reorganized as The three terms in the argument of these hyperbolic Gamma function correspond to the ones expected from the duality.Indeed if we associate a mass parameter τ to the adjoint and two mass parameters By using the constraint imposed by the superpotential on m 2 we then arrive at m 2 )τ − m, corresponding to the argument of the first hyperbolic Gamma function in (3.4).On the other hand the arguments of the second and of the third Gamma functions in (3.4) are straightforward and they correspond to the dressed mesons M = qS 2 +1 q and the to the singlets σ k = Tr S 2k .

Case II: SO(2n)
In this case we choose the parameters µ r as µ = ω 1 2 , ω 2 2 , 0, m and apply the duplication formula.On the LHS of (2.5) we obtain This corresponds to the partition function of SO(2n) with an adjoint A and a vector q with vanishing superpotential.Actually to correctly reproduce the expected partition function we need an extra factor of 2, in order to have 2 n−1 in the denominator, that correctly reproduces the Weyl factor.This extra 2 will be generated when looking at the RHS as are going to explain.The RHS of (2.5) can be studied as in the U Sp(2n) case above.In this case we obtain where we used the duplication formula, the reflection equation and the relations As anticipated above, the 1 2 term can be moved on the LHS reproducing the Weyl factor of SO(2n).The other contributions correspond to the 4n singlets of [1] .Let us discuss them in detail.Again we associate a mass parameter τ to the adjoint and a mass parameters m to the vector.The unit flux bare monopole In this case we choose the parameters µ r as µ = ω 1 2 , ω 2 2 , τ, m and apply the duplication formula.On the LHS of (2.5) we obtain This corresponds to the partition function of SO(2n + 1) with an adjoint A and a vector q with vanishing superpotential.Actually we are still missing a contribution Γ h (m) coming from the zero modes of the vector.As in the SO(2n) case discussed above, the extra term comes from the RHS, that in this case becomes As anticipated above the denominator can be moved on the LHS and it is necessary to reproduce the zero mode of the chiral fields in the vectorial representation of the SO(2n + 1) gauge group.The other 4n + 2 Gamma functions correspond to the singlets discussed in [1].Let us discuss them in detail.Again we associate a mass parameter τ to the adjoint and a mass parameters m to the vector.

New results
In this section we propose three new dualities, that generalize the ones reviewed above, in presence of two more fundamentals (or vectors) and of a monopole superpotential.
Here we propose such dualities by reversing the procedure adopted so far.We start from the integral identity (2.3) , that has a clear physical interpretation, because it gives the mathematical version of the confinement of U Sp(2n) with an antisymmetric, six fundamentals and the monopole superpotential.
Then we use the duplication formula and we obtain three new relations as discussed above in terms of U Sp(2n) (SO(N )) with an adjoint S (A), four (three) fundamentals (vectors) and W = pSp (W = 0).In each case the masses are constrained because the choice of parameters necessary to apply the duplication formula leaves us with a constraint, corresponding to the leftover of (2.4).
By applying the three choices of mass parameters discussed in Section 3 we arrive at the following three identities The first choice corresponds to choosing µ 3) it gives raise to the following identity Schematically this corresponds to: where = 0, . . .n − 1 and r, s = 1, 2, 3.The dual (confined) model corresponds to a set of singlets, σ k = TrS 2k , with k = 1, . . ., n, and dressed mesons.These are in the antisymmetric and in the symmetric representation of the flavor symmetry group that rotates q 1,2,3 and they can be defined as A (2 ) rs ≡ q r S (2 ) q s and S (2 +1) rs ≡ q r S 2 +1 q s respectively.By flipping the singlets σ k we modify the electric theory, adding the superpotential terms ∆ W ele = n k=1 ρ k Tr S 2k .In the dual theory we are left with the cubic superpotential On the identity (4.1) the effect of such a flip corresponds to moving the terms Γ h (2kτ ) on the LHS and taking them to the numerator by using the reflection equation, giving raise to the contribution Γ h (2ω − 2kτ ), corresponding to the singlets ρ k .
Case II: SO(2n) The second choice corresponds to choosing µ = ω 1 2 , ω 2 2 , 0, µ 1 , µ 2 , µ 3 .Substituting in (2.3) gives raise to the following identity with k = 1, . . ., n − 1, = 0, . . ., n − 1 and r, s = 1, 2, 3.The dual description consists of a set of chiral fields identified with mesons and baryons of the electric theory.The baryon B = Pf A is reproduced on the partition function by Γ h (nτ ) while the baryons B r = rst 2n (A n−1 q s q t ) are reproduced on the partition function by Γ h ((n − 1)τ + µ r + µ s ).There is also a tower of singlets σ k associated to the singlets Tr A 2k contributing to the partition function as n−1 k=1 Γ h (2kτ ).
The mesons are in the antisymmetric and in the symmetric representation of the flavor symmetry group that rotates the three vectors and they can be defined as A (2 +1) rs ≡ q r A 2 +1 q s and S (2 ) rs ≡ q r A 2 q s respectively.By flipping the singlets σ k and the baryons we are left, in the dual theory, with the cubic superpotential Again we can reproduce the effect of the flip on the partition function by moving the relative Gamma function on the LHS of (4.5) and using the reflection equation.

SO(2n + 1)
The third choice corresponds to choosing µ = ω 1 2 , ω 2 2 , τ, µ 1 , µ 2 , µ 3 .Substituting in (2.3) gives raise to the following identity with the condition (2n − 1)τ + with k = 1, . . ., n − 1, = 0, . . ., n and r, s = 1, 2, 3.The dual description consists of a set of chiral fields identified with symmetric and antisymmetric mesons as above, the baryons B ≡ 2n+1 A n−1 q 1 q 2 q 3 and B i ≡ 2n+1 A n q i and the singlets σ k = Tr A 2k .On the partition function such fields correspond to Γ h (ω − nτ ), Γ h (nτ + µ r ) and n k=1 Γ h (2kτ ) respectively.Again by flipping the singlets and leaving only the mesons on the dual side we are left with the superpotential (4.8).We can reproduce the effect of such flip on the partition function by moving the relative Gamma function on the LHS of (4.9) and using the reflection equation.
4.1 A consistency check: flowing to the cases of [1] Here we show that by giving large masses to two of the fundamentals (or two of the vectors in the theories with orthogonal group) the dualities (4.1), (4.5) and (4.9) reduce respectively to the dualities (5.1), (5.2) and (5.3) of [1].
We consider the real mass flow triggered by giving large real masses (of opposite signs) to two of the quarks, say q 1 and q 2 .On the electric side we are left with a U Sp(2n) theory with two quarks q = q 3 and p, one adjoint and W = pSp.The linear monopole superpotential is lifted in the mass flow.On the magnetic side the dressed mesons A with Y 2 +1 for = 0, . . ., n − 1.The leftover dressed mesons S 2 +1 33 correspond to M , for = 0, . . ., n − 1.The superpotential (4.4) reduces to the one of [1] when the singlets σ k are flipped.Indeed the only superpotential terms surviving the real mass flow are We can follow this real mass flow on the partition function in the following way.We parametrize the mass parameters as: and we take the limit s → ∞.The constraint from the monopole superpotential reads: On the RHS of (4.1) the Gamma functions with finite argument in the s → ∞ limit are: which correspond to the singlets M and Y j .On the LHS it corresponds to the partition function of U Sp(2n) with 2 fundamentals p, q, one adjoint S, n singlets ρ k and superpotential W = n k=1 ρ k Tr S 2k + pSp as expected.The Gamma functions with divergent argument can be written as an exponential using the formula: where ζ = exp 2πi The resulting phase on the LHS is then (we omit the prefactor iπ 2ω 1 ω 2 ): while on the RHS it is: Under the constraint (4.14) the divergent phases cancel between the RHS and the LHS.We are then left with an equation which corresponds to the identity between the partition functions of the theories of the duality (5.1) of [1].

Case II: SO(2n)
We can flow from the duality (4.5) to (5.2) of [1] by giving a large mass of opposite sign to two vectors.Indeed the only mesons that survive the projection are the ones labeled by A 12 and S 1,2 S (2 ) In order to follow the real mass flow on the partition function we parametrize the masses as: The constraint reads: 2(n − 1)τ + 2ν + m = ω (4.21) Taking the limit s → ∞ the LHS becomes the partition function for SO(2n) with one vector and one adjoint multiplied by a divergent phase.The singlets on the RHS of (4.5) that remain massless are: which correspond respectively to the singlets B, σ k , M , Y − A n−1 and Y + j discussed above.Along the lines of the computation done in the previous case one can show that the divergent phases cancel between the LHS and the RHS.The limit s → ∞ then gives the identity between the partition functions of the dual theories (5.2) of [1].

Case III: SO(2n + 1)
When we give large masses to two of the vectors this duality reduces to the duality (5.3) of [1].Analogously to the SO(2n) case the the superpotential reduces to the one of [1] when the singlets σ k , Y + j and Y − qA n−1 are flipped.
We parametrize the real masses as in (4.20).The constraint reads: The LHS becomes the partition function for a SO(2n + 1) gauge theory with one vector q and one adjoint A multiplied by a divergent phase.The singlets on the RHS of (4.9) that remain massless are: which correspond respectively to the singlets Y − qA n−1 , B, σ k , M and Y + j discussed above.The divergent phases cancel between the LHS and the RHS.The resulting identity corresponds to the duality (5.3) of [1].

Proving the new dualities through adjoint deconfinement
The dualities read above from the matching of the three-sphere partition functions can be proved along the lines of [1] by deconfining the adjoints as reviewed in sub-section 2.1.Even if the logic is very similar the presence of more fundamentals/vectors and the constraints imposed by the monopole superpotentials modify the analysis and it is worth to study explicitly the mechanism.Furthermore when translated to the three-sphere partition function this process offers an alternative derivation of the mathematical identities (4.1), (4.5) and (4.9) from a physical perspective.In Figure 1 we show schematically the confinement/deconfinement procedure we used to prove the confinement of the U Sp(2n) model with monopole superpotential.

Case I: U Sp(2n)
The U Sp(2n) model with an adjoint S, four fundamentals {q 1,2,3 , p} and superpotential (4.3) is dual to the U Sp(2n) × SO(2n) quiver given in Figure 2. As discussed above the analysis is made easier by flipping the singlets Tr S 2k with k = 1, . . ., n.On the physical side this corresponds to adding singlets ρ k to the original U Sp theory with superpotential: while mathematically it corresponds to moving the tower Γ h (2kτ ) on the LHS of (4.1) and by using the reflection equation we are left with Γ h (2ω−2kτ ).The superpotential Indeed by confining the SO(2n) gauge node of this quiver we arrive at the original model.This can be proved thanks to a confining duality reviewed in the appendix Similarly Tr S n is a shorthand notation for Tr (S • J) n .In the rest of the paper we omit the matrix J, which is always understood whenever we contract the indices of a symplectic group.
where the duality map for the meson of SO(N ) is while the baryons are q b ≡ 2n (Q 2n ) and p ≡ 2n (Q 2n−1 b).The field S is in the adjoint of U Sp(2n) while the field p is the fourth fundamental of U Sp(2n) (the other three fundamentals q 1,2,3 are spectator when confining SO(2n)).By evaluating the F-terms of the massive fields we end up with the original U Sp(2n) gauge theory, with an adjoint, four fundamentals and superpotential We can expand the determinant of S in terms of traces 2 : det (S) ∝ Tr S 2n + multi-traces (4.30) By dropping the multi-trace terms and by comparing with the superpotential W = (4.3)+ (4.25) of the U Sp(2n) model we started with, we identify M bb = ρ n .On the partition function the mass parameters for the fields appearing in this U Sp(2n) × SO(2n) quiver are related to the ones of the original U Sp(2n) model (i.e.µ r and τ in formula (4.1)) by the following set of relations where µ r are the three mass parameter for the fields q 1,2,3 .Furthermore we can map these parameters to the ones in the confined SO(2n) model by imposing µ Q = τ 2 .In this way we arrive at the following identifications The duality between the original U Sp(2n) model and the quiver with the deconfined adjoint can be checked on the partition function by using the identity (B.3).This can be shown explicitly by considering the partition function of the quiver, i.e.
and then by using the relation (B.3).This is possible because the mass parameters 2 For a 2n × 2n symmetric matrix S: By applying (B.3) and by using the reflection equation we end up with the first line of (4.1), finding the expected result.
Next we can dualize the U Sp(2n) node with the linear monopole superpotential turned on.We are left with an SO(2n) SQCD with an adjoint A and superpotential In this case the fields are mapped to the ones in the U Sp(2n) × SO(2n) quiver as u r = Qq r , A = QQ, M rs = q r q s and v r = aq s .The fields u r are three vectors while A is in the adjoint of SO(2n).The fields M rs and v r are singlets.The term rst (. . . ) in the superpotential originates from the Pfaffian of the generalized meson, built up by contracting the fundamentals of the U Sp(2n) gauge node, after integrating out the massive component M Qa = Qa.
The partition function is obtained by the limiting case of the identity given in Proposition 5.3.4 of [40] and we report it in formula (B.1).It corresponds to the confining duality for U Sp(2n) with 2n + 4 fundamentals and linear monopole superpotential turned on.This identity was obtained also in [4] from the reduction of the integral identity relating the superconformal indices of the 4d duality of [41].The partition function obtained after confining the U Sp(2n) gauge node is As a consistency check we can now use formula (4.5) on the integral (4.38) because the mass parameters are constrained as in (4.6).After some rearranging we eventually checked that the integral reduces to the LHS of (4.1).This signals the consistency of the various steps done so far and motivated us to further deconfine the adjoint of SO(2n) in order to produce a new quiver with a symplectic and an orthogonal node.
The SO(2n) model with adjoint and three fundamentals is equivalent to the U Sp(2n − 2) × SO(2n) quiver given in Figure 3 The duality map reflects on the following relations between the mass parameters in the partition function Furthermore the superpotential imposes the following relations on the other parameters and the usual constraint We can see that this model reduces to the SO(2n) model discussed above when the U Sp(2n−2) node with 2n+2 fundamentals and a linear monopole superpotential confines.Again the confinement of the U Sp(2n − 2) symplectic gauge group gives raise to a superpotential term proportional to the Pfaffian of the generalized meson.By integrating out the massive fields and substituting in the Pfaffian we recover the superpotential (4.37).The partition function of the U Sp(2n − 2) × SO(2n) model is One can check that the partition functions for the SO(2n) model and that for the U Sp(2n − 2) × SO(2n) quiver are equal by applying the identity for the confining U Sp node with 2n + 2 fundamentals discussed above.The last step consists in performing a confining duality on the SO(2n) gauge node with 2n + 1 vectors and linear monopole superpotential turned on.This gives raise to an U Sp(2n − 2) gauge theory with an adjoint, four fundamentals and a series of singlets.The mesonic and baryonic operators associated to the SO(2n) gauge group are with l, m = 2, 3 and where S is in the adjoint of the U Sp(2n − 2) gauge group, while M Q,f , M Q,c 1 , M Q,c 2 and q are four fundamentals of U Sp(2n − 2).There are also two extra fundamentals of U Sp(2n − 2) corresponding to the fields d and e of the previous model, which are not modified by the duality on the SO(2n) gauge node.The superpotential of the dual U Sp(2n − 2) adjoint SQCD is then The determinant can be evaluated as We can then integrate out the massive fields {e, M Q,f , v, q f , M f, c } and we are left with U Sp(2n − 2) adjoint SQCD with four fundamentals.There is a rather rich structure of singlets that we do not report here but that can be read by computing the F -terms of (4.45).We can now iterate this procedure by alternating adjoint deconfinement and duality in order to arrive to the final step and eventually prove the duality.
As anticipated this procedure can be used on the mathematical side to prove the identity (4.1) from a physical perspective.In order to complete the proof we need to consider the partition function obtained so far after the final duality on the SO(2n) node (B.3).It is Where the masses μr are: Notice that the superpotential constraint reads: which is equivalent in form to the original superpotential constraint (4.2).
The contribution of the singlets can be written as: We can prove the confining duality for U Sp(2n) with four fundamental and linear monopole superpotential by iterating this procedure n times.In each step we obtain a new set of singlets as in (4.50), with the exception that the tower of Γ h (2ω − 2kτ ) reduces of one unit.Furthermore in each step the rank of the gauge group decreases by one and the real masses are redefined as in (4.48), so that the fundamentals of U Sp(2(n − h)) obtained after h steps are related to the original ones by: Thus iterating this procedure n times each term in (4.50) gives a tower of singlets of the final confined phase.Schematically: while the contribution of the tower n−1 k=1 Γ h (2ω − 2kτ ) reduces of one unit at each step, and eventually disappear.Together these reproduce the formula (4.1).

Case II: SO(2n)
Now we prove the confining duality for SO(2n) with one adjoint A, three vectors q 1,2,3 and monopole superpotential (4.5) by deconfining the adjoint.The mass parameters for the three vectors q r are referred as µ r with r = 1, 2, 3 and the one for the adjoint is referred as τ .The SO(2n) model is equivalent to the U Sp(2n − 2) × SO(2n) quiver in Figure 3, but this time the superpotential is The duality map is: The other parameters are fixed by the constraints given by the superpotential: with the constraint given by the monopole superpotential: The partition function of the quiver is: where M and Q are given by (4.44).Due to the rather complicated structure of such superpotential we decide to proceed by adding some interactions in the original theory.We turn on the extra superpotential term On the partition function this removes the contributions of Γ h (nτ ), r<s Γ h ((n − 1)τ + µ r + µ s ) and n k=1 Γ h (2kτ ) from the RHS of (4.5) giving raise to the contributions Γ h (2ω − nτ ), 3  r=1 Γ h (ω + (n − 1)τ + µ r ) and n k=1 Γ h (2ω − 2kτ ) on the LHS.Mathematically this is achieved by applying the reflection equation and the balancing condition (4.60) and it does not spoil the integral identity (4.5).Furthermore We can proceed by confining the SO(2n) node with 2n + 1 fundamentals and linear monopole superpotential after we have added the contributions of α 1,2,3 and β.The partition function for the U Sp(2n − 2) model is Where the masses are: If we now ignore the singlets we observe that the contribution of the U Sp(2n−2) gauge sector to this partition function corresponds to the LHS of the identity (4.1).The duality associated to such a sector was proven in the previous section.We can then use this duality to confine the U Sp(2n − 2) theory, resulting in a WZ model with partition function: which reproduces the RHS of (4.5) once the contributions of the baryons Pf A and 2n−2 (A n−1 q r q s ) and of the singlets Tr A 2k are removed.
Case III: SO(2n + 1) The SO(2n + 1) model with adjoint A and three fundamentals q 1,2,3 is equivalent to the U Sp(2n) × SO(2n + 1) quiver given in Figure 4.The superpotential of this The mass parameters in the partition function are with the constraint (4.10).The partition function of the SO(2n + 1) × U Sp(2n) quiver is given by Next we have to confine the SO(2n + 1) sector with 2n + 2 vectors and a linear monopole superpotential and we end up with U Sp(2n).The problem consists of understanding the interaction among the various singlets from the confining dynamics of SO(2n+1).Again we can simplify the problem by modifying the original SO(2n+ 1) model by considering the superpotential corresponding to remove the baryons and the singlets Tr A 2k from the confined phase and add the new singlets α 1,2,3 and β in the original model.On the partition function this removes the contributions of Γ h (ω − nτ ), 3 r=1 Γ h (nτ + µ r ) and n k=1 Γ h (2kτ ) from the RHS of (4.9) giving raise to the contributions Γ h (ω + nτ ), 3  r=1 Γ h (2ω − nτ − µ r ) and n k=1 Γ h (2ω − 2kτ ) on the LHS.Mathematically this is achieved by applying the reflection equation and it does not spoil the integral identity (4.9).By deconfining the adjoint A the superpotential (4.68) is modified as well.The new superpotential is We can proceed by confining the SO(2n + 1) node.By integrating out the massive fields we arrive to an U Sp(2n) gauge theory with an adjoint S, three fundamentals, identified by d and the two mesonic composites Qc 2 and Qc 3 , and a fourth fundamental corresponding to u = 2n+1 (Q 2n−1 bd) , interacting with the adjoint through a superpotential term W ∝ uSu.There is also a linear monopole superpotential and many more interactions with the singlets that we do not report here, but that can obtained by evaluating the determinant det S and the superpotential contraction of S with the baryons of the confined SO(2n + 1) node.The partition function of the model is and the constraints 4 =1 μ + 2nτ = 2ω and 2μ 4 + τ = 2ω.Also in this case we can borrow the results of the previous sections.Indeed if we ignore the singlets we observe that the contribution of the gauge U Sp(2n − 2) sector to this partition function corresponds to the LHS of the identity (4.1).The duality associated to such a sector was proven in the previous section.We can then use this duality to confine the U Sp(2n − 2) theory and prove the confining duality for the SO(2n + 1) model.

Conclusions
In this paper we have studied 3d N = 2 confining gauge theories with real USp/SO gauge groups, with fundamentals/vectors and adjoint matter.We have first shown that the symplectic and orthogonal cases recently studied in [1], with two fundamentals and one vector respectively, can be studied by the squashed three-sphere localization by applying the duplication formula for the hyperbolic Gamma function of another s-confining model, namely U Sp(2n) with an antisymmetric and four fundamentals.Motivated by this relation we then elaborated on the case of U Sp(2n) with an antisymmetric, six fundamentals and a monopole superpotential.By applying the same strategy we derived three new integral identities involving symplectic and orthogonal adjoint SQCD, with four fundamentals and three vector respectively and a monopole superpotential.We showed that the new confining cases reduce to the ones of [1] by a real mass flow and then we proved the dualities by sequentially deconfining the adjoint fields.This last step furnished an alternative proof of the identities and (4.1), (4.5) and (4.9) as we have explicitly shown.
This paper is the starting point of many further analysis.For example one can apply the duplication formula to the integral identities for U Sp(2n) theories with an antisymmetric and eight fundamentals, where the A 7 global symmetry enhances to E 7 .This case has been deeply investigated in the mathematical [40] and then in the physical literature [10,11] and it may be interesting to understand if similar enhancements or new dualities appear for models with adjoint matter as well.
Another interesting family of models that may deserve some further investigation are models with power law superpotential for the two index tensor.In this case the starting point of the analysis are the integral identities discussed in [42] for U Sp(2n) with antisymmetric and adjoint matter fields.Again applying the duplication formula in such cases could lead to new relations between these models and to new results for the orthogonal cases.
A deeper question that we have not addressed here consists of the physical interpretation, if any, of the duplication formula.As observed in the literature this formula allows to switch from the integral identities for the U Sp(2n) duality with fundamentals to the integral identities for the SO(n) dualities with vectors.This has been discussed in [35] for the superconformal index of 4d dualities and in [36] for the squashed three-sphere partition function of 3d dualities.In presence of monopole superpotential this issue is more delicate, because in some cases it can lead to a singular behavior that requires more care.In any case, when the procedure gives rise to a finite result, also in presence of monopole superpotential, the constraints imposed by anomalies (in 4d) and by monopole superpotential (in 3d) translate in a consistent way into the new identities, and the latter can be interpreted as new physical dualities (or in new examples of s-confining theories).It should be then important to have a physical interpretation of the duplication formula.
A last comment is related to the adjoint deconfinement and to a possible relation with another mathematical result, that consists of interpreting the various steps discussed when deconfining the adjoints as a manifestation of a generation of a chain or a tree of identities, along the lines of the Bailey's lemma.Such analysis has been first applied to the study of elliptic hypergeometric integrals (i.e. to the 4d superconformal index) in [43,44].Recently a 4d physical interpretation of such mechanism has been discussed in [45].It should be interesting to develop similar results in our 3d setup for the deconfinement of the adjoints in the hyperbolic hypergeometric integrals.we deconfine the adjoint with (A.2) and get to a quiver with gauge groups U Sp(2n − 2) × SO(2n): If we then assign the mass parameters as µ 1 = ω 1 2 and µ 2 = ω 1 2 , and we use the duplication formula on both sides of (B.1), we arrive at the identity with the constraint 2n+2 r=1 µ r = ω.This corresponds to the case of SO(2n + 1) with 2n + 2 fundamentals.The arguments of the singlets on the dual side correspond to the mesons and to the baryons of the electric theory.
The even case is obtained by considering also µ 3 = 0.In this case, by using the duplication formula on both sides of (B.1) we end up with with the constraint 2n+1 r=1 µ r = ω.This corresponds to the case of SO(2n) with 2n + 1 fundamentals.The arguments of the singlets on the dual side correspond to the mesons and to the baryons of the electric theory.
As a consistency check we can perform a real mass flow by giving large masses of opposite sign to two vectors and retrieve the limiting case of Aharony duality.In (B.2) we fix: and take the limit s → ∞.The constraint reads ω − 2n r=1 µ r = 2ν and the divergent phases cancel between the RHS and the LHS.We obtain: which corresponds to the limiting case of Aharony duality for SO(N ) = SO(2n + 1) and 2n vectors, with W = 0 [36].
corresponding to the last term in the second line of(3.6).The mass parameter associated to the baryon monopole is obtained by adding(n − 1)τ to m Y + SO .This gives m Y − A n−1 = ω − (n − 1)τ − m and it corresponds to the second term in the first line of (3.6).The first term of (3.6), with mass parameter nτ corresponds to the baryon B ≡ Pf A. The dressed mesons M = qA 2 q and the singlets σ k = Tr A 2k are associated to the combinations m M = 2 τ + 2m and m σ k = 2kτ respectively.Case III: SO(2n + 1) The unit flux bare monopole Y + SO has mass parameter m Y + SO = ω−(2n−1)τ −m.The dressed monopoles Y + j = Y + SO Tr A j have mass parameter m Y + j = ω − (2n − 1 − j)τ − m, corresponding to the term in the second line of (3.6).The baryon monopole Y − qA n−1 is obtained by adding (n − 1)τ + m to the contribution of m Y + SO .This gives m Y − qA n−1 = ω − (n − 1)τ , and this gives raise to the first term in the first line of (3.8).The second term in the first line of (3.8), with mass parameter nτ + m corresponds to the baryon 2n+1 (qA n ).The dressed mesons M = qA 2 q and the singlets σ k = Tr A 2k are associated to the combinations m M = 2 τ + 2m and m σ k = 2kτ respectively.

6 ) 3 W
This corresponds to the duality: SO(2n) w/ adjoint A and 3 vectors q 1,2,= Y + SO ⇐⇒ Wess-Zumino w/ 10n + 3 chirals rr with r = 1, 2 become massive and are integrated out in the IR.The dressed mesons A are identified with the dressed monopoles Y j of the electric theory.More precisely we identify A 0, . . ., n − 1.The leftover dressed mesons S2  33 correspond to M .After the real mass flow the superpotential (4.8) reduces to the one of[1] when the singlets σ k , Y − A n−1 and Y + j are flipped:

3 Figure 1 :
Figure 1: Schematic representation of one step of the deconfinement procedure used to prove the confinement of the U Sp(2n) model with monopole superpotential.The superpotential and three-sphere partition function of each model in Figure are:

Figure 2 :
Figure 2: Quiver description of the deconfinement of the adjoint S of the U Sp(2n) model with superpotential (4.3).In this and in the following quivers we decided to omit to represent the various singlets.

Figure 3 :
Figure 3: Quiver representation of the SO(2n) model after the adjoint field has been deconfined ±y a ± y b ) (4.61) Now we dualize the node with orthogonal group, this results in a U Sp(2n − 2) model with four fundamentals and superpotential: W = Y U Sp + h Tr(de) + Tr(M Qf e) + det M + T rQMQ (4.62)

Figure 4 :
Figure 4: Quiver representation of the SO(2n + 1) model after the adjoint field has been deconfined

Figure 5 : 3 Z 1 2 2
Figure 5: Schematic representation of one step of the deconfinement procedure of [1] for the U Sp(2n) model with adjoint.The partition functions of the model are: