A journey to 3d: exact relations for adjoint SQCD from dimensional reduction

In this note we elaborate on the reduction of four dimensional Seiberg duality with adjoint matter to three dimensions. We use the exact formulation of the superconformal index and of the partition function as instruments to test this reduction. We translate the identity between indices of the dual 4d theories to 3d. This produces various new identities between partition functions of 3d dual phases.


Introduction
A general program in the study of supersymmetric gauge theories is the connection between dualities in various dimensions. A systematic approach to the case with four supercharges in four and three dimensions has been recently proposed in [1].
The connection is somewhat subtle, it turns out that a plain dimensional reduction is too naive. This fact can be understood considering that in three dimensions there are extra, axial, symmetries, whose 4d parents would be anomalous. A careful analysis requires keeping the 4d theory on a circle with finite radius, whereby a superpotential breaking the axial symmetry is generated. This mechanism generates a well-defined, new, IR duality in three dimensions from a four dimensional parent. By adding real masses such 3d dual pairs flow to more canonical dualities. Along the flow the superpotential disappears and the axial symmetry emerges.
Exact results in supersymmetry, as from the recent progress in localisation, lead to efficient tests of dualities. For example, the superconformal index on S 3 × S 1 [14,15] has been matched between Seiberg dual phases [16][17][18]. A similar matching has been done in [6,19] for the partition functions on the (squashed) S 3 [20][21][22]. In both cases the identification is equivalent to non-trivial integral identities involving elliptic and hyperbolic Gamma functions respectively, recently analysed in the mathematical literature [23][24][25].
Moreover, the integral expressions for the index and the partition function provide a powerful tool for supporting the reduction set-up described above. It is possible to follow the reduction and the mass deformation on the integral expressions. This has been done in [1] for SQCD where the relations of [25] have been recovered from the ones of [23].
In this paper, we study the dimensional reduction of SQCD with an adjoint [11] at the level of the index and the partition function. Thereby we generate new integral identities involving hyperbolic Gamma functions, generalising some of the results in [25].
The paper is organised as follows. In section 2 we recap the electric-magnetic dualities discussed in the paper. We also give a brief review of the mechanism relating them by dimensional reduction. In section 3 we implement the reduction in the matrix integrals of the 4 and 3 dimensional theories, deriving the integral identities. In section 4 we conclude with some comments. We added an appendix A with few details about the partition function on the squashed three sphere.

Dualities and dimensional reduction
In sections 2.1 and 2.2 we review the four and three dimensional dualities with adjoint matter that are of interest in this paper. In section 2.3 we then review the dimensional reduction connecting them.
The electric phase is a SU(N c ) gauge theory with N f (anti-)fundamentals Q and Q and one adjoint X. There is a superpotential we denote the R charges of the (anti-)quarks as R Q . The R charge of the adjoint is fixed by (2.1) as R X = 2/(k + 1). The mesonic operators are The theory has stable vacua if N f > Nc k . The magnetic phase is an SU(kN f − N c ) gauge theory, with N f dual (anti-) fundamentals q andq and one adjoint Y . There is also a set of k gauge singlets, which correspond to the mesons (2.2) of the electric phase. The superpotential is The R charges of the dual (anti-)quarks are R q = R X − R Q .
By gauging the non-anomalous global U(1) B in the electric and in the magnetic phase one can extend the duality to the unitary gauge groups U(N c ) and U(kN f −N c ). This is the case considered in this paper.

3d dualities with adjoint matter
A similar duality exists in three dimensions. It generalises the duality studied in [3] by the inclusion of an adjoint and has been discussed by Kim and Park in [12].
The electric phase is a U(N c ) YM gauge theory with N f (anti-)fundamentals Q andQ and one adjoint X. The superpotential is (2.1), as in the 4d case. The global Recall that in three dimensions there is a topologically conserved current J = * dF , where F is the field strength of the U(1) inside the U(N) gauge group. The charges of the chiral fields under the global symmetries are There is a 2k dimensional unlifted Coulomb branch, parametrised by t j,± , where j = 0, . . . , k −1. These coordinates have an UV interpretation as monopole operators t 0,± , dressed with powers of the adjoint The operators t 0,± correspond to an excitation of the magnetic flux (±1, 0, . . . , 0).
The magnetic phase is a U(kN f − N c ) YM gauge theory with N f dual (anti-) fundamentals q andq and one adjoint Y . As in the 4d case there are k gauge singlets corresponding to the electric mesons M j . However in 3d we have additional 2k gauge singlets, t j,± , corresponding to the electric monopole operators (2.5). They couple to the magnetic theory through the superpotential where the 2k "monopole operators" t j,± are the Coulomb branch coordinates of the magnetic theory. In the UV theory they correspond to monopole operators dressed with powers of the adjoint,t j,± = tr (t 0,± Y j ). The chiral fields and their charges under the global symmetries are Theories with CS terms. There is another 3d duality with adjoint matter, having Chern-Simons (CS) interactions [26,27]. The electric phase is a U(N c ) K gauge theory (where K is the CS level). The magnetic phase has gauge group U(kN f −N c +|K|) −K . The Coulomb branch is lifted and no monopole operators appear in the description of the low energy theory.

From four to three dimensional dualities
In this section we review how one can obtain three dimensional from four dimensional dualities [1]. It turns out that a naive dimensional reduction on R 3 × S 1 does not lead to the right result. This can be understood as a consequence of Seiberg duality being an IR property, where the IR limit does not commute with the limit of the shrinking circle. More precisely, Seiberg duality is valid for energies much below the electric and the magnetic strong coupling scales, E ≪ Λ e , Λ m , which are each related to their coupling as Λ b ∼ e −1/g 2 4 . Here b is the coefficient of the 4d β-function. When dimensionally reducing the theory on the circle S 1 with radius r the coupling scales as g 2 4 ∼ rg 2 3 which implies that Λ → 0. Hence the IR limit E ≪ Λ is not well defined in three dimensions and in that sense the two limits do not commute. This argument mirrors the one presented in the introduction where we mentioned the existence of axial symmetries in 3d, whose 4d counterparts would be anomalous. We are hence looking for a reduction which commutes with the low-energy limit of Seiberg duality and forbids the existence of the axial symmetry.
In order to obtain well-defined three dimensional dualities from four dimensional parents we leave the theories on a circle S 1 with a finite radius r. At energies E ≪ 1/r that describes effectively three dimensional dynamics.
The finite radius implies that the Coulomb branch is compact. Recall that the adjoint scalar σ of the 3d vector multiplet comes from the forth component of the 4d gauge field, σ = A 4 . To make S 1 A 4 gauge invariant, one needs to require the periodicity σ ∼ σ + 1/r. As a consequence of this compactness, there is an additional Affleck-Harvey-Witten (AHW) superpotential for the unlifted coordinates of the Coulomb branch 2 , for U(N) theories generically looking like [29] Here η ∼ e −1/(rg 3 ) 2 and Y ± denote the unlifted directions of the Coulomb branch, with respective U(1) J charge ±1. From the perspective of the effective 3d field theory the Y 's are just the low-energy coordinates parametrising the Coulomb branch. One can though embed the η deformed theory in a purely 3d UV completion, where the high-energy operators Y ± typically correspond to monopole operators.
Let us point out that (2.8) breaks the axial symmetry. Since this was the symmetry without 4d counterpart the construction indeed overcomes the problem of the mismatching symmetries. In order to obtain conventional 3d theories with the axial symmetry, one can switch on a deformation by large real masses for some of the matter fields, generating a flow to theories without superpotential (2.8).

Reducing KSS to 3d
By generalising this analysis one can obtain the theories in section 2.2 from the ones in section 2.1. In the presence of an adjoint with superpotential (2.1) the Coulomb branch of a U(N) theory has 2k unlifted directions, parametrised by t i,± . The η superpotential in the electric theory is [11] With an analogue superpotential in the magnetic phase we end up with a pair of dual three dimensional theories.

Flowing to the Kim-Park duality
One can obtain Kim-Park duality by deforming the theories with η superpotential. As in [11] we consider N c colours and N f + 2 flavours, where in the electric theory we assign large real masses In the large M limit this theory flows to the electric phase of Kim-Park duality.
The magnetic phase has k(N f + 2) − N c colours and N f + 2 flavours. The real masses for the dual quarks follow from the electric ones, according to the global symmetries. In [11] the theory is perturbed by a polynomial superpotential in the adjoint. This breaks the gauge group to k sectors U(n i ) with i n i = k(N f +2)−N c . In order to preserve the duality there is a non trivial vacuum structure for the scalar σ in each U(n i ) vector multiplet, further breaking the gauge symmetry into U(n i − 2) × U(1) 2 . The 2k U(1) subsectors can be dualised to 2k XY Z models [30], made of gauge singlets. These singlets interact through a superpotential with the monopoles of the U(n i − 2) sectors. By turning off the polynomial deformation in the adjoint while taking the large M limit, one finds the magnetic phase of Kim-Park duality, with gauge group U(kN f − N c ) and superpotential (2.6). There the 2k surviving gauge singlets of the XY Z models are identified with theIn that case the exact result of interest here is electric monopoles.
In section 3 we reproduce this reduction mechanism on the index and the partition function. Actually, when restoring the dual gauge symmetry U( n i ) the k U(1) 2 sectors become enhanced to U(k) 2 , each with one flavour and one adjoint. In [11] it is assumed that this enhancement does not affect the duality. We support this assumption by using a duality discussed in [31].

Superconformal index and partition function
In four dimensions a powerful check of Seiberg duality comes from matching the index on S 3 × S 1 between the electric and the magnetic phases. The matching involves non-trivial integral identities as shown in [16][17][18]. A similar test exists for three dimensional N = 2 dualities [6,19], involving the partition function on the squashed S 3 [22]. It is known that one can obtain the partition function from the index in the limit of a shrinking circle [32][33][34][35][36].
In this section we will check the mechanism for obtaining 3d dualities with an adjoint from KSS duality at the level of these mathematical quantities. We start from the integral identity between the indices of the KSS pair in 4d. By dimensional reduction we obtain an identity for the partition functions of the 3d pair with η superpotential. The identity for Kim-Park duality is then obtain by implementing the real mass deformation (2.10). Eventually we flow to the duality of [26,27] with CS terms.

From KSS duality to 3d
Let us consider the duality reviewed in section 2.1. The electric phase is a U(N c ) gauge theory with one adjoint and N f + 2 (anti-)fundamental flavours. The partition function on S 3 × S 1 is The function Γ e is the elliptic Gamma function [23,24]. We also have defined Γ e ((z) ±1 ) ≡ Γ e (z)Γ e (1/z). The parameters p and q are the chemical potentials of the superconformal algebra, z i the ones of the U(N c ) gauge symmetry and s a and t a the ones of SU(N f ) L and SU(N f ) R respectively.
The magnetic phase is a U(k(N f + 2) − N c ) ≡ U(Ñ c + 2k) gauge theory with one adjoint, N f + 2 (anti-)fundamental flavours and the k electric mesons. The index is KSS duality predicts the integral identity I el = I mag . In [18] this identity has passed some partial nontrivial checks. Note that for the identity to hold, the chemical potentials in the index have to satisfy the balancing condition In order to reduce the four dimensional index to the three dimensional partition function, reviewed in appendix A, we redefine the chemical potentials as where σ, m a andm a parameterise the Cartan of the U(N c ) gauge and the SU(N f ) L × SU(N f ) R flavour group respectively. In the limit r → 0 the index reduces to the partition function. In particular, the elliptic Gamma function Γ e reduces to the hyperbolic Gamma function Γ h (A.2) as [1] lim r→0 Γ e e 2πirz = e − iπ The divergent term in the RHS of (3.6) is proportional to the contribution of each multiplet to the gravitational anomaly. Hence it will drop out when comparing the partition functions of two dual phases. As discussed in [1,34] also the four dimensional FI reduces to the three dimensional one. With this prescription the index (3.1) reduces to the partition function for the three dimensional electric theory with the η superpotential (2.9) where the function W Nc,k is defined in (A.1) and ∆ X = 2/(k + 1) denotes the R charge of the adjoint. From (3.5) we obtain where in three dimensions we call the R charge R Q = ∆ Q . The balancing condition (3.4) reduces to While in 4d the balancing condition reflects the anomaly cancellation for the R symmetry, in 3d it is imposed by the superpotential (2.9).
In the magnetic phase the index reduces to the partition function The four dimensional integral identity I el = I mag reduces to Z el = Z mag , with the constraint (3.9).

Flowing to Kim-Park duality
Next we turn on the real masses in the electric theory as in (2.10). The parameters µ and ν become Observe that the global SU(N f + 2) 2 symmetry is broken to SU(N f ) 2 × U(1) A in the large M limit. Indeed in this limit the η superpotential that prevents the U(1) A symmetry disappears [1]. At large M (3.7) becomes In the magnetic case the situation is more involved. The real masses can be read from the electric theory. Furthermore there is a non trivial vacuum structure for the scalar σ in the vector multiplet Note that in absence of the R symmetry breaking polynomial in the adjoint the gauge symmetry U(Ñ c + 2k) is broken to U(Ñ c ) × U(k) 2 . In the large M limit (3.10) becomes × W kN f −Nc,0 (ω∆ X − ν a ; ω∆ X − µ a ; ω∆ X ; −λ) Z + Z − (3.14) The additional terms Z ± are the partition functions of the two U(k) sectors and effective FI terms Recall that the U(k) 2 sector is the enhancement of the k U(1) 2 sectors discussed in [11]. As we mentioned at the end of section 2.3 the U(k) 2 sector can be dualised to a set of 6k singlets. We can see this on the partition function exploiting the integral identity [25] W Nc,0 (µ; ν; ω∆ X ; λ) = 4k singlets acquire mass from a superpotential and are integrated out. On the partition function this is reflected in the relation Γ h (z)Γ h (2ω − z) = 1. In this process we used the constraint on the R charges coming from (3.9) We are left with This is the contribution to the partition function from the 2k singlets which remain light. They have exactly the right global charges for coupling through the superpotential interaction (2.6) with the magnetic monopolest i,± . It is hence natural to identify them with the electric monopoles t i,± . Eventually we arrive at the identity Note that as expected this identity holds without any balancing condition. In the limiting case N f = 1 and k = N c it corresponds to the identity (3.18) from [25]. This fact is as a consistency check of the procedure.

Flowing to the case with CS terms
In [26,27] another three dimensional duality with adjoint fields has been discussed. It has a CS gauge interaction and extends the duality [5]. The electric phase is a U(N c ) K gauge theory at CS level K with N f (anti-)fundamentals and one adjoint, with superpotential (2.1). The magnetic phase is a U(k(N f + |K|) − N c ) −K gauge theory at CS level −K with N f (anti-)fundamentals, one adjoint and k singlets, with superpotential (2.3).
The duality can be derived from Kim-Park duality by an RG flow. In the electric phase we consider a U(N c ) 0 gauge group and N f + K (anti-)fundamentals, giving a large real mass to K of them. In the dual U(k(N f + K) − N c ) theory the fields acquire their masses accordingly. More explicitly, K magnetic flavours (q,q), the electric monopoles t j,± and K 2 +2KN f components of each meson M j become heavy. In the large mass limit these fields are integrated out. This procedure generates the CS levels and one recovers the duality of [26,27].
One can study this RG flow at the level of the partition function. Assigning the masses as described above and taking the large mass limit, the identity (3.21) becomes 3 where ζ = exp πi(ω 2 1 +ω 2 2 ) 24ω 1 ω 2 and the extra phase is (3.22) is the identity between the electric and the magnetic partition functions of the duality in [26,27]. Note that the limiting case N f = 0, K = 1 and k = N c is the third integral identity in theorem 5.6.8 of [25]. Similar dualities appeared in the physics literature [36][37][38][39].

Comments on the phases and CS terms
In this section we studied RG flows by turning on real masses. These flows generate not only CS terms for the gauge but also for the global symmetries. The real masses arise as vevs after weakly gauging the global symmetries and the global CS give rise to a phase, quadratic in ω and in the mass parameters [6]. When flowing from the duality with η superpotential to Kim-Park duality, we checked that the generated CS all cancel. In fact, in the relation 3.21 there is no extra phase. However in the flow to the duality of [26,27] we found a global CS action. We checked that it precisely corresponds to the phase (3.23).

Conclusions and further directions
In this paper we studied the dimensional reduction of four dimensional dualities to three dimensions. We considered the reduction of KSS duality discussed in [11]. This yields a new 3d duality with the superpotential (2.9). Other previously studied 3d dualities can be deduced from this by an RG flow. We studied how the identity of the 4d superconformal index leads to identities between 3d partition functions. As a partial check of these identities we observed their agreement with theorem 5.6.8 of [25]. Furthermore we matched the phases in the integral identities with the global CS terms generated along the RG flows.
Let us comment on the validity of the mathematical identities of section 3. The parent relation in four dimensions I el = I mag has not been rigorously proven in the mathematical literature, though several partial checks have been performed (see [18] for references). In that sense the derivation of the identities between hyperbolic hypergeometric integrals Z el = Z mag in section 3 is not a proper mathematical derivation. However we can think of this as an instance where physical intuition may be helpful in making predictions about unknown mathematical results.
We conclude with an outlook of some possible lines of future research. Note that the relation between the partition functions of the SU(N) duality [40] can be found by the "ungauging" procedure of [1]. Alternatively one can apply the reduction discussed in [11] to the 4d superconformal index. It would be desirable to extend the dualities studied in this paper by considering a chiral flavour sector, as done for the case without adjoint in [6]. This would also be useful in studying the inverse flow from the dualities with CS terms in [26,27] to Kim-Park duality, along the lines of [41][42][43]. On the partition function this corresponds to deriving the identity (3.21) from (3.22). Moreover, one might want to reduce the 4d dualities with multiple adjoint matter [44] to 3d, obtaining additional integral identities. Another interesting extension of the programme is the analysis of dualities with real gauge groups and tensor matter.

A The three dimensional partition function
The partition function on the squashed S 3 for a gauge group U(N c ) K is a matrix integral over the Cartan of the gauge group, parametrised by the scalar σ in the N = 2 vector multiplet [20][21][22]. For N f (anti-)fundamental flavours and one adjoint the matrix integral is where ω 1 and ω 2 are related to the real squashing parameter b of S 3 as ω 1 = ib and ω 2 = i/b, and ω ≡ (ω 1 + ω 2 )/2. The function Γ h is the hyperbolic Gamma function, it can be written as [25] Γ h (z; ω 1 , ω 2 ) ≡ Γ h (z) ≡ ∞ n,m=1 (n + 1)ω 1 + (m + 1)ω 2 − z nω 1 + mω 2 + z (A.2) We also introduced the shorthand notation Γ h (±x) ≡ Γ h (x)Γ h (−x). Let us comment on the different terms in the partition function and their physical interpretation. The exponential is the contribution from the CS action and the FI term ξ = 2λ. The hyperbolic Gamma functions are the 1-loop contributions of the various multiplets, their arguments reflect the charges under the local and global symmetries. More precisely, ±(σ i − σ j ) are the weights of the adjoint representation while ±σ i the ones of the (anti-)fundamental. Similarly τ, µ and ν are collective parameters for the weights under the global symmetries for the adjoint, the fundamental and the anti-fundamental respectively.