Sporadic dualities from tensor deconfinement

In this paper we give a field theory explanation of two confining dualities that have been proposed in the literature based on exact results from supersymmetric localization. The first confining model under investigation is 4d $SU(N_c+1)$ SQCD with a conjugate rank-$2$ anti-symmetric tensor, $N_c+3$ anti-fundamentals, $2N_c$ fundamentals and a superpotential that couples the anti-symmetric tensor and the fundamentals. The second confining model studied here is $3d$ $\mathcal{N}=2$ $USp(4)$ gauge SQCD with two fundamentals, two rank-$2$ anti-symmetric tensors and vanishing superpotential. Here we prove that these models are confining by using the technique of deconfining the anti-symmetric tensors and then by flowing to the IR description by sequential dualities. As a bonus the analysis provides (alternative) proofs of the identities obtained from supersymmetric localization.


Introduction
The low energy dynamics of UV free strongly coupled supersymmetric gauge theories can be often be simplified by the existence of infrared dualities.The dual descriptions are in general associated to (more) weakly coupled QFTs, described by a different set of fields and interactions that share in the IR the same correlation functions for the physically observable conserved currents of the original description.The prototypical example of such dualities is the electromagnetic duality and for this reason the two dual models are usually referred to as the electric and the magnetic phase.
Restricting to cases with four supercharges the basic example of these dualities was discovered by Seiberg in [1] for SU(N c ) 4d SQCD with N f > N c + 1 flavors and vanishing superpotential.This duality has also a limiting case, where the magnetic description does not correspond to any gauge theory but to a WZ model consisting in a collection of mesons and baryons of the electric description, in addition to a (classical) constraint among them.In this case, corresponding to the choice N f = N c + 1, the electric gauge theory confines without breaking the chiral symmetry (i.e.s-confines [2]), and the magnetic theory describes the dynamics of the confined degrees of freedom, with a superpotential imposing the classical constraint on the moduli space.There is also another confining case, corresponding to SU(N c ) 4d SQCD with N f = N c flavors, where the low energy dynamics described by the mesons and the baryons requires a quantum constraint on the moduli space.Such constraint breaks the chiral symmetry and for this reason this case is referred to as confinement with chiral symmetry breaking.This idea of confinement as a limiting case of a supersymmetric duality was then extended to various generalizations of Seiberg duality.Furthermore a full classification of s-confining gauge theories with vanishing superpotential was worked out in [3] for theories with a single gauge group.In this classification there are many models that do not correspond to any limiting case of any known duality.Such models are characterized usually by the presence of matter fields in a rank-two tensor representation of the gauge group.Despite the fact that gauge theories of this type do not have in general a Seiberg-like dual description, it has been shown in [4] that the s-confining dualities can be derived using only Seiberg-(like) dualities thanks to the rank-2 tensor deconfining technique originally proposed in [5] and subsequently generalized in [6].The technique consists of substituting a rank-2 tensor matter field with a bifundamental field charged also under another (auxiliary) confining gauge group, such to recover the original description once this new gauge group confines.After deconfining the rank-2 tensors it has been possible to apply sequences of Seiberg dualities (see [7] for a general construction) and than to recover the confined phase proposed in [3], using only the s-confining dualities of SU(N c ) with N c + 1 flavors of [1] and USp(2N c ) SQCD with 2N c + 4 fundamentals of [8].This construction may require further refinements for models with a superpotential deformation, due to the possible presence of an Higgsing that breaks partially or completely the gauge group (see [9] for a general discussion).Recently new confining gauge theories have been obtained for 4d models with rank-2 tensors and non-vanishing superpotential [10].Furthermore the deconfinement techniques have been applied to 3d N = 2 gauge theories [11], where the zoo of confining gauge theories is richer, because of the presence of a dual photon and of a Coulomb branch.New confining dualities in this direction have been obtained in [12,13].
In this paper we apply these techniques to a 4d and a 3d model that have been claimed to be confining because of integral identities in supersymmetric localization.We find a physical origin of these integral identities that allowed to state the new confining dualities, finding a field theoretical explanation for them.The 4d duality under inspection corresponds to SU(N c + 1) SQCD, with a rank-2 conjugate antisymmetric tensor, N c + 3 anti-fundamentals and 2N c fundamentals.This theory is claimed to be confining if a cubic superpotential between the anti-symmetric and the fundamentals is turned on.Such claim was originally proposed in [14] based on the fact that the supersymmetric index1 on S 3 × S 1 [15,16] of this theory was computed exactly in [17].The final result has a field theory interpretation representing the low energy description of the baryons and mesons of the SU(N c + 1) gauge theory with the expected constraints from the truncation of the chiral ring and the moduli space.
This duality has been referred to as Spiridonov-Warnaar-Vartanov (SWV) duality in [18], where it has been used in the study of 4d compactification of the 6d minimal (D,D) conformal matter theories on a punctured Riemann surface (see also [19]).
Here we provide a physical derivation of the duality from the field theoretical perspective, by deconfining the rank-2 anti-symmetric tensor and sequentially dualizing the gauge groups.In the process we find that one of the steps requires a partial Higgsing, analogously to the analysis recently performed in [10] for similar 4d confining dualities.The partial Higgsing is triggered in our case by an USp(2N c ) gauge group with 2N c + 2 fundamentals, that confines breaking the chiral symmetry.Furthermore, following the various steps on the supersymmetric index we provide an alternative derivation of the identity of [17].
In the second part of the paper we study a 3d confining duality recently obtained in [20], corresponding to USp(4) with two rank-2 anti-symmetric tensors and two fundamentals.The existence of such a duality has been claimed by extending to the 3d bulk a boundary duality constructed from N = (0, 2) half-BPS boundary conditions in 3d N = 2. Again we deconfine the two rank-2 anti-symmetric tensors and then provide the sequential dualities leading to the final description in terms of the gauge singlets of the original model.

The Spiridonov-Warnaar-Vartanov 4d duality
In this section we derive the SWV duality from a physical approach, by deconfining a rank-2 conjugate anti-symmetric tensor with an auxiliary symplectic gauge group and then by sequentially applying IR dualities.Actually referring to the last step in such a sequence as a duality is improper, because, as we will see in the following, it corresponds to the case of a symplectic gauge theory that confines with a quantum constraint on the moduli space.The crucial aspect of this constraint is that it forces a Higgs mechanism on the leftover unitary gauge group, breaking it to a symplectic one, and assigning a superpotential mass term to some of the fields in the spectrum.This leads to the final step of the construction, where one is left with an s-confining gauge theory (namely USp(2M ) with 2M + 4 fundamentals).After confining this theory we eventually find the expected WZ model, describing the magnetic phase of the SWV duality.
The analysis is supported at each step by the relative (integral) identities matching the 4d supersymmetric index.On one hand this corroborates the validity of the results and on the other hand it provides an alternative derivation of the integral identity discovered in [17].
Let us start the analysis discussing the gauge theory that can be read from the Spiridonov-Warnaar identity [17].It consists in SU(N c + 1) SQCD with N c + 3 anti-fundamentals Q 1 and 2N c fundamentals Q 2 .In addition there is a rank-2 antisymmetric conjugate tensor A. This is a non-anomalous asymptotically free theory and it becomes confining if the superpotential deformation2 is turned on.This deformation is relevant for any value of N c [18] and it breaks the SU(2N c ) flavor symmetry group into USp(2N c ).The representations of the fields and their charges under the gauge and the flavor groups are summarized in the following The S 3 × S 1 supersymmetric index of this model has been explicitly computed [17].
The identity is I E = I M with and with the constraint S = Nc+3 m=1 s m imposed on the charges.It is possible to read from this identity that there are two contributions arising from the meson M = Q 1 Q 2 and the baryon B = Q Nc+1 1 .The field content allows a non-vanishing superpotential of the form [18] W mag = M 2 B (2.5) In the rest of this section we provide the physical derivation of this confining duality.We start by adding a singlet α in the electric theory, flipping the meson M through a superpotential In this way the dual superpotential becomes that vanishes once we compute the F-terms of the massive fields α and M .The next step consists in deconfining the field A. We distinguish two cases, depending on the parity of N c .Let us study them separately.

Deconfinement with odd N c = 2k + 1
In this case we deconfine the rank-2 conjugate anti-symmetric tensor A of SU(2k + 2) We depicted the model in Figure 1 in terms of a quiver gauge theory.
Quiver description of the model after the deconfining of the SU(2k + 2) rank-2 conjugate anti-symmetric tensor A. Gauge groups are represented as circles while flavor nodes are represented with squares.Symplectic groups are depicted in blue and unitary groups are depicted in red.
The superpotential for the deconfined model is schematically where the field β corresponds to the 2 × 2 antisymmetric matrix, i.e. it is a singlet.The anti-symmetric tensor A is recovered by confining the USp(2k) gauge node in terms of the field B, i.e.A ∼ B 2 , where the contraction is done on the USp(2k) indices.
The next step consists of Seiberg duality on SU(2k + 2).This gauge group is self dual and the quiver is represented in Figure 2. The superpotential of the dual theory is where b, c, q 1 , q 2 are the dual quarks and the mesons M 1,2,3,4 are associated to the quarks of the previous phase through the dictionary Quiver obtained after Seiberg duality on SU(2k + 2).The rank of the dual gauge group is the same as above, but there are new mesonic degresse of freedom that modify the superpotential.
By integrating out the massive fields this superpotential becomes Then we consider the USp(2k) gauge group with 2k+2 flavors.This gauge theory confines with a quantum constraint enforced on the moduli space.By considering the low energy dynamics we are left with a single gauge group SU(2k + 2) and the field content can be read from the quiver in Figure 3.The superpotential becomes 12) The last term in (2.12) enforces the quantum constraint on the moduli space through the Lagrange multiplier λ.
The constraint breaks the gauge symmetry to USp(2(k + 1)) and this Higgsing gives mass to Q 2 as well.The leftover superpotential is This model s-confines and the final superpotential is where L, U and γ correspond to the USp(2k + 2) contractions q 1 c, q 2 1 and c 2 respectively.Integrating out the massive fields we are left with just the meson L in the anti-symmetric representation of the flavor symmetry group SU(N c +3) = SU(2k+4).This field corresponds indeed to the baryon B expected in the SWV duality.
In order to connect with the WZ superpotential of the SWV duality (2.5) we have to flip the field α in (2.5).This turns off the field α in the derivation and keeps the meson M 1 massless in (2.9).The superpotential (2.11) then becomes The other steps in the derivation are straightforward and in the final superpotential (2.14) there is a further contribution ∆W ∝ M 2 1 U .This terms survives after the massive fields are integrated out and by the identification U ↔ B and M 1 ↔ M we obtain exactly (2.5) as expected.
This concludes the proof of the duality from the field theory analysis in the case with N c = 2k + 1.Before moving to N c = 2k it is instructive to reproduce the analysis using the S 3 × S 1 supersymmetric index.
In order to have a better physical intuition of the duality from localization we start by rewriting I E and I M by modifying the USp(2N c ) fugacities as t → t pq/S.This gives and again with the balancing condition S = Nc+3 m=1 s m .Then we proceed to deconfine the rank-2 conjugate anti-symmetric tensors, to dualize the SU(2k + 4) node and to integrate out the massive fields.These steps are done by using the integral identities collected in [14] (which are reproduced in the appendix A) and the reflection equation for the elliptic gamma functions Γ e (pq/x)Γ e (x) = 1.We skip these standard elementary steps and focus on the quiver described in Figure 4, where we also highlighted in blue the fugacity of each field in the S 3 × S 1 supersymmetric index.The S 3 × S 1 supersymmetric index for these models is given by formula We then consider the change of variables z i = e 2πiφ i , where φ i are real and the balancing condition is 2k+2 i=1 φ i = 0.With such a change of variables we can substitute in the index the following terms Γ(e 2πi(± φi ± φj ) ; p, q) where we used the identity (A.6).This identity was derived in [21] and it represents the evaluation of the superconformal index for USp(2M ) SQCD with 2M + 2 fundamentals.The fact that the models confines with a quantum superpotential that breaks the chiral symmetry is reflected in the structure of the δ-functions in (A.6).

Deconfinement with even N c = 2k
In this case we deconfine the rank-2 conjugate anti-symmetric tensor A of SU(2k + 1) We depicted the model in Figure 5 in terms of a quiver gauge theory.The superpotential for the deconfined model is given again by formula (2.8).The next step consists of Seiberg duality on SU(2k + 1).The dual gauge group is SU(2k + 2) and the quiver is represented in Figure 6.The superpotential of the dual theory is again given by (2.9).The fields b, c, q 1 , q 2 are the dual quarks and the mesons M 1,2,3,4 are associated to the quarks of the previous phase through the dictionary spelled out in (2.10).By integrating out the massive fields the superpotential becomes the one in (2.11).Then we observe that USp(2k) with 2k + 2 flavors confines with a quantum moduli space and after such confinement we are left with a SU(2k + 2) gauge group with superpotential (2.12).The partial Higgsing triggered by the quantum constraint enforced by the Lagrange multiplier reduces the theory to USp(2k + 2) with 2k + 6 fundamentals and superpotential (2.14).Integrating out the massive fields we are left with a single field L that correspond to the baryon B of the original theory.
The analysis for even N c is then almost identical to the case of odd N c .For this reason we skip the derivation of the duality from the superconformal index.The interested reader can reproduce it by following the stepwise procedure that we described for odd N c .

The Okazaki-Smith 3d duality
In this section we study a 3d N = 2 confining duality recently proposed in [20].The electric model is USp(4) SQCD with two fundamentals and two rank-2 antisymmetric tensors.The model has an U(2) 2 × U(1) R global symmetry and the charges of the fields under these symmetries are summarized in (3.1).
The model has vanishing superpotential and its low energy dynamics is described by the gauge invariant combinations These fields interact through a superpotential with a singlet T 4 that corresponds to the minimal monopole of USp(4).The charges of the fields with respect to the global U(2) 2 × U(1) R symmetry are: In the following we will derive this confining duality by deconfining the antisymmetric tensors and then by sequentially dualizing the gauge groups.We found that in order to proceed it is very useful to add to the electric theory an SU(2) A vector s = (s 1 , s 2 ) interacting with PfA 1 and PfA 2 through the superpotential where the trace of an anti-symmetric matrix is defined as

Field theory analysis
In the following we will derive the duality using the field theory approach.We proceed by representing the model in terms of a quiver gauge theory, using the same conventions of the previous section: the blue circles refer to symplectic gauge groups while the red squares identify the special unitary flavor groups.We start by considering the model with the flip in formula (3.3) We then deconfine the two rank-2 anti-symmetric fields A I with two auxiliary USp(2) nodes with the assignment where the i-index refers to the USp(4) node and (α 1,2 , β 1,2 ) are indices of the two USp(2) 1,2 gauge groups.Therefore the deconfined theory is Observe that the superpotential is vanishing because the singlets s 1,2 have flipped the monopoles of the USp(2) 1,2 gauge groups.The central USp(4) node in this theory has then 6 fundamentals and therefore it confines [22].The IR description has then two USp(2) 1 and USp(2) 2 gauge groups connected by a bifundamental field.There is still a manifest SU(2) flavor symmetry associated to a node in the quiver and there are further fundamental fields for both the USp(2) 1,2 gauge factors.There is also a singlet Y 4 identified with the monopole of the USp(4) gauge group for the model in (3.6), that interacts through a superpotential with the generalized meson of USp( 4) itself.The quiver and the superpotential for this dual theory are 2 2 2 Observe that the components X 11 , X 22 and X 33 of the meson X correspond to 2 × 2 anti-symmetric matrix, i.e. they are singlets.The two USp(2) nodes have each 4 fundamentals and are therefore confining [22].Here we choose to confine the USp(2) 1 group.The other choice is completely equivalent because of the SU(2) A global symmetry that rotates the two anti-symmetric in the original description of the model (we will further comment on this equivalence below).After confining the USp(2) 1 gauge group we are left with a USp(2) 2 SQCD with four fundamentals and a non-trivial superpotential.The quiver and the operator mapping are reported below 2 2 while the superpotential is The field Y (1) 2 is the monopole of the USp(2) 1 gauge group acting as a singlet in the confined phase.
We conclude the sequence by confining the USp(2) 2 gauge group, that has indeed four fundamentals.This leads to the final, confined, theory where the new mesons are mapped to the fundamentals of USp(2) 2 as Furthermore there are two singlets of USp(2) 2 that we redefine as where the field Y (2) 2 is the monopole of the USp(2) 2 gauge group acting as a singlet in the confined phase.The expression (3.11) needs some massage in order to simplify its interpretation.For example some fields appear quadratically in the superpotential and they can be integrated out.By writing we see that the v 3 field is massive.The singlet field αβ T αβ also acquires a mass and it can be integrated out in the IR.By considering the various F-term conditions, we get the final superpotential We can identify the fields here with the ones in formula (3.2) by first flipping the singlets s I in the original USp(4) gauge theory.This can be done by adding two fields, denoted as r I through the superpotential ∆W = r I s I .These fields can be integrated out in the electric description and F -terms of s I leave us with the identification r I ∝ Tr A 2 I .In the dual description the fields r I are crucial in order to reconstruct the correct field content of the duality.
Looking at the global symmetry structure the explicit mapping is then Substituting this mapping into the superpotential (3.13) we obtain where we absorbed the numerical coefficients into the fields.Actually we could have reversed the order of the last two confining dualities on USp(2) 1 and USp(2) 2 , arriving to a different results, with the role of B I and φ I exchanged.However the two WZ models must be equivalent, and this equivalence corresponds to the following more symmetric formulation of the superpotential The last step of the derivation of the superpotential of the WZ model consists of flipping s I in the electric model.This gives rise to the interactions involving the fields φ 11 and φ 22 .By a symmetry argument we claim that the interactions allowed by the global symmetry are generated at quantum level and that the flipped fields reconstruct the SU(2) A adjoint field φ IJ .In the next subsection we will confirm this expectation from the analysis of the partition function.

3d partition function
We complete our analysis by reproducing the derivation of the duality from supersymmetric localization on the squashed three sphere.Such procedure gives rise to the identity between the partition function of USp( 4) with with two anti-symmetric and two fundamentals and the partition function of the WZ model for the gauge singlets B α,β , B I , φ I , φ IJ , M and T 4 .The global symmetry enters in these identities in terms of real masses, that from the field theory side are associated to vevs of the reals scalars in the vector multiplets of the weakly gauged background flavor symmetries.Before studying the deconfinement of two rank-2 anti-symmetric tensors from the three sphere partition function we briefly review the necessary definitions.The partition function on the squashed three sphere S 3 b , obtained from localization in [23] (see also [24][25][26] for the round case) is a matrix integral over the reals scalar in the vector multiplet in the Cartan of the gauge group.There is a classical term corresponding to the CS action (global and local) and the matter and the gauge multiplet contribute with their one loop determinant.These last can be associated to hyperbolic Gamma functions, formally defined as The argument of such Gamma functions is physically interpreted as a holomorphic combination between the real masses for the gauge and the global symmetries and the R-charges (or mass dimensions).The purely imaginary parameters ω 1 = ib and ω 2 = i/b are related to the squashing parameter of the three sphere S 3 b .Here we will only focus on the case of symplectic gauge group.Let us consider the partition function of an USp(2N c ) gauge theory with 2N f fundamentals.It is given by In our analysis we will use an identity involving this partition function and its dual Aharony phase [22].The identity is (see Theorem 5.5.9 of [27]) Observe that the identity (3.19) remains valid for N f = N c + 1, that corresponds to the confining case of Aharony duality [22], where only the meson M and the minimal USp(2N c ) monopole Y survive in the WZ model and they interact through the superpotential W = Y PfM .We start considering the original model, adding also the flippers s I arising from the superpotential (3.3).The partition function is In this formula m 1,2 are the real masses of the two fundamental fields and n 1,2 are the real masses of the two anti-symmetric fields.We can also use a different basis giving an explicit parameterization it terms of the Cartan of the U(2) 2 flavor symmetry.Indeed in this way σ and ν parameterize the Cartan of SU(2) a and SU(2) A respectively.
We then proceed by deconfining the two rank-2 anti-symmetric tensors.This step produces two USp(2) gauge nodes, two bifundamentals, each connecting one of these USp(2) gauge groups to the original USp(4).The partition function of the model becomes The partition function (3.23) corresponds to the one for the model in represented in (3.7).
The next step consists of a confining limit of Aharony duality on one of the USp(2) factor.Choosing one of the two USp(2) nodes has the effect of making the SU(2) A × SU(2) a global symmetry not manifest in the integrand of the partition function.Following the discussion on the field theory side here we choose to dualize the USp(2) 1 gauge group, such that the partition function becomes The partition function (3.24) corresponds to the one for the model in represented in (3.8).The last step of the procedure requires a confining limit of Aharony duality on the leftover USp(2) 2 gauge group.This gives the final partition function This expression still needs some massage.First we can integrate out the massive fields, as done on the field theory approach.Here this integration corresponds to take advantage of the formula Γ h (2ω − x)Γ h (x) = 1.After this step we can also write down (3.25) in a manifestly SU(2) A × SU(2) a invariant form.We arrive to the expression This is the final expression that matches with (3.20).We can also flip the singlets s I in the electric side, and on the magnetic side two new singlets appear with their contribution to the Γ h (2n A ) = Γ h (2µ ± 2ν) to the partition function.In this way we can see that all the fields B I , φ I , φ IJ , B α,β , M and the monopole T 4 appear in the partition function with the expected real masses.Explicitly we can associate these Gamma functions to the singlets of the confined phase using the mapping B αβ ↔ Γ h (2ρ ± 2σ + 2µ, 2ρ + 2µ) φ IJ ↔ Γ h (2µ ± 2ν, 2µ) Indeed the arguments of hyperbolic Gamma functions correspond to the real masses that can be read from the charges in formula (3.2).

Conclusions
In this paper we have derived, using field theory arguments, two confining dualities that have been proposed in the literature from supersymmetric localization.Here the dualities have been derived by combining the technique of rank-2 tensor deconfinement of [5] together with the sequential application of ordinary dualities and/or confining dualities.There are many interesting directions that would be worth to explore.For example the Higgsing triggered, in the 4d model, by the quantum correction imposed on the moduli space, corresponds, on the superconformal index, to the pole pinching [28] vastly used in [4] for the derivation of 4d confining dualities in presence of a superpotential.The confining theories obtained in this way were not discussed in [2], because of the absence of a superpotential.On the other hand confining gauge theories of this type have been discussed in [29], at least for the limiting cases of ordinary dualities with rank-2 tensor matter fields.Therefore we expect many more 4d confining gauge theories with a simple gauge group not discovered yet.
Another interesting question regards the 3d duality for USp(4) with two rank-2 anti-symmetric tensors and two fundamentals.As discussed in [20] the field content in this case corresponds to the one of the dualities studied in [30,31] with D-type superpotential.Nevertheless as observed in [20] there are differences in the operator mapping and in the charge spectrum.Furthermore the USp(4) duality discussed here appears sporadic and its generalization to USp(2N c ) does not seem straightforward.For example we did not find any confining duality by increasing the rank of the gauge group and keeping fixed the field content (i.e.keeping two rank-2 antisymmetric tensors and possibly increasing the number of fundamentals).It is nevertheless possible that further fields and interactions should be considered in order to have an USp(2N c ) confining theory with two rank-2 anti-symmetric tensors.
A last, related, question regards the existence of 4d confining dualities with two rank-2 tensors.Beyond the case of USp(2N c ) with two rank-2 anti-symmetric tensors, one can imagine also cases with unitary or orthogonal gauge groups or cases with more general rank-2 tensor matter fields.

Figure 3 .
Figure 3. Quiver obtained after confining the USp(2k) gauge node.The anti-symmetric field a gets a vev from the quantum constraint on the moduli space.

Figure 5 .
Figure 5. Quiver description of the model after the deconfining of the SU(2k + 1) rank-2 conjugate anti-symmetric tensor A
.22)As a check we can see that(3.20)isobtained by applying(3.19)to the two USp(2) gauge groups in(3.22).The partition function(3.22)corresponds to the one for the model in represented in(3.6).The next step consists of Aharony duality on USp(4).At the level of the partition function it corresponds to use the identity(3.19)on the gauge theory identified by the variables z 1,2 .The partition function becomes