Abelianization and sequential confinement in 2 + 1 dimensions

Journal of High Energy Physics, Oct 2017

We consider the lagrangian description of Argyres-Douglas theories of type A 2N −1, which is a SU(N) gauge theory with an adjoint and one fundamental flavor. An appropriate reformulation allows us to map the moduli space of vacua across the duality, and to dimensionally reduce. Going down to three dimensions, we find that the adjoint SQCD “abelianizes”: in the infrared it is equivalent to a \( \mathcal{N}=4 \) linear quiver theory. Moreover, we study the mirror dual: using a monopole duality to “sequentially confine” quivers tails with balanced nodes, we show that the mirror RG flow lands on \( \mathcal{N}=4 \) SQED with N flavors. These results make the supersymmetry enhancement explicit and provide a physical derivation of previous proposals for the three dimensional mirror of AD theories.

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Abelianization and sequential confinement in 2 + 1 dimensions

ArXiv ePrint: Abelianization and sequential con nement in 2 Sergio Benvenuti 0 1 2 3 5 Simone Giacomelli 0 1 2 3 4 Theory, Supersymmetry and Duality 0 Strada Costiera 11 , 34151 Trieste , Italy 1 Via Valerio 2 , 34127 Trieste , Italy 2 Via Bonomea 265 , 34136 Trieste , Italy 3 INFN , Sezione di Trieste 4 International Center for Theoretical Physics 5 International School of Advanced Studies , SISSA 6 From the equation @ We consider the lagrangian description of Argyres-Douglas theories of type A2N 1, which is a SU(N ) gauge theory with an adjoint and one fundamental avor. An appropriate reformulation allows us to map the moduli space of vacua across the duality, and to dimensionally reduce. Going down to three dimensions, we nd that the adjoint SQCD \abelianizes": in the infrared it is equivalent to a N Moreover, we study the mirror dual: using a monopole duality to \sequentially con ne" quivers tails with balanced nodes, we show that the mirror RG Extended Supersymmetry; Nonperturbative E ects; Supersymmetric Gauge - HJEP10(27)3 ow lands on N = 4 SQED with N avors. These results make the supersymmetry enhancement explicit and provide a physical derivation of previous proposals for the three dimensional mirror of AD theories. 1 Introduction and summary 1.1 Notation 2 Adjoint-SQCD with one avor in 3d: Abelianization 4d chiral ring: dressed baryons and dressed meson 2.1.1 N = 2 AD interpretation of the j multiplets Compacti cation to 3d: emergent symmetry Z-extremization: Abelianization 3d chiral ring: dressed monopoles S3 partition functions 3 Mirror RG ow to A2N 1 AD: sequential con nement 3.1 Basic ingredients 3.1.1 3.1.2 The mirror of U(N ) with 2N avors and the chiral rings map Con ning U(N ) with N + 1 avors and W = M+ The general picture of the mirror RG ow Mirror RG Mirror RG ow to A3 AD: the superpotential ow to A5 AD: the superpotential Comments about the higher N generalization The Maruyoshi-Song procedure in 3d A Nilpotent vevs 2.1 2.2 2.3 2.4 2.5 3.2 3.3 3.4 3.5 3.6 1 Introduction and summary Recently Maruyoshi and Song [1, 2] discovered `Lagrangians for Argyres-Douglas theories'. They coupled 4d N = 2 superconformal theories to a chiral eld A, transforming in the adjoint of the global symmetry group. Giving a nilpotent vacuum expectation value (vev) to A triggers an RG ow. Studying the infrared CFT, they found that sometimes the RG ow lands on N = 2 Argyres-Douglas theories [1{3]. For instance, in [2] it was shown that starting from SU(N ) gauge theory with 2N avors, TUV, a maximal nilpotent vev initiates an RG ow to the N = 1 gauge theory SU(N ) with an adjoint and one avor, plus some gauge-singlet elds. The IR theory, TIR (SU(N ) with an adjoint and one avor), turns out to be equivalent in the infrared to the so called A2N 1 Argyres-Douglas theory (see [4{7] for a detailed discussion about ArgyresDouglas theories) plus a free sector consisting of operators which violate the unitarity bound and decouple [8]. In this paper we provide two physical mechanisms for this duality going down to 3 dimensions, generalizing the case of SU(2) dual to A3 studied in [9]. { 1 { 1 singlet elds j which implement the decoupling of the operators that violate the 4d unitarity bound. This prescription provides a completion of the theory, allows all standard computations, and to preserve the 4d duality when going down to 3d. We would like to stress that this caveat is not related to the phenomenon observed in [10]. We will indeed see later that no monopole superpotential is generated in the compacti cation. More evidence that adding the elds j 's is necessary comes from the fact that the j 's map to a particular component of the Coulomb branch short multiplets of the N = 2 AD theory. As for the second modi cation, the superpotential written in [2] is incorrect, one term must be removed, in order to satisfy a criterion of chiral ring stability [9, 11]. The standard procedure of keeping all terms consistent with the symmetries in these cases must be improved. in the following diagram:1 We call the modi ed theories T40d;UV and T40d;IR. We study the dimensional reduction of the RG ow TUV ! TI0R, and its mirror dual T~UV ! T~IR. Our 3d results are summarized 0 mirror 1 N W = WN =4 (1.1) We analyze the left side of this diagram in section 2 and the right side in section 3. We exhibit strong evidence that TI0R;3d is equivalent in the IR to an N = 4 Abelian U(1)N 1 theory. Two di erent Lagrangian, UV free, theories are dual in the IR. The mechanism of this Abelianization duality is that in 3d there is an emergent U(1) global symmetry, and the result of Z-extremization [12, 13] is that the superconformal r-charge of the adjoint eld vanishes: r = 0. Using the input r = 0, we show that the integrand of ZS3 reduces to the integrand of the N = 4 U(1)N 1 theory. 1A circle is a U(n) gauge group, a double circle is a SU(n) gauge group, a square is a avor group. { 2 { We present and check numerically a map between the supersymmetric S3 partition functions of the non-Abelian and Abelian theories. We also show that the chiral ring of the SU(N ) gauge theory is isomoporphic to the chiral ring of the N = 4 abelian theory, using recent results about dressed monopole operators in 3d non-Abelian gauge theories [14]. The emergent U(1) symmetry enhances to an SU(N ) avor symmetry and the generators of the dressed monopoles of the SU(N ) gauge theory transform in the adjoint representation of the emergent SU(N ) avor symmetry. The Abelianization duality we propose is quite peculiar. For instance, in usual dualities, such as Seiberg duality [15] or 3d mirror symmetry [16] (see also [17]), at least the Cartan generators of the global symmetry group are visible in both descriptions. In our case ow T~UV ! T~IR, depicted on the right side. In this case we use very recent results for dualities of 3d N = 2 U(N ) gauge theories with linear monopole superpotentials [18]. Starting from the T~UV quiver, the monopole duality implies that all the gauge nodes in the lower row of T~UV con ne one after the other, starting from one U(1) node and ending with the opposite U(1) node. We call this phenomenon sequential con nement. It works for quiver tails with balanced nodes starting from an U(1) gauge group, and is the mirror counterpart of integrating out avors that get mass from the nilpotent vev. The left-over theory in the IR is N = 4 supersymmetric. This makes the supersymmetry enhancement explicit. In order to illustrate the procedure, we rst discuss the 3d mirror of A3 AD theory building on the results found in [9] and then proceed with the analysis of the general case. The mirror RG ow lands on SQED with N avors and enhanced N = 4 supersymmetry; the surviving U(1) is depicted in red. The latter theory is well known to be mirror of the linear quiver U(1)N 1, and was proposed to be the mirror of the 3d reduction of A2N 1 Argyres-Douglas [19], based on mathematical results [20]. The claim of [19] passes several nontrivial consistency checks and is perfectly consistent with the structure of the superconformal index [21, 22]. Our method clearly explains why the theory abelianizes in 3d. Using our 3d sequential con nement interpretation, it is possible to generalize the story, and nd a 4d Lagrangian for more general Argyres-Douglas models, like the ones arising from N M5's on a sphere with an irregular puncture [23]. Again, in the 3d mirror many nodes sequentially con ne, and in the IR the RG ow lands on the Abelian N = 4 theories of [19]. 1.1 Notation Quiver diagrams. N a circle node denotes a U(N ) gauge group; N a double-circle node denotes a SU(N ) gauge group; { 3 { a square node denotes a U(N ) or SU(N ) avor group; sometimes we use an 8-supercharges notation N1 N2 , links are bifundamental hypers and adjoints in the vector multiplets are implicit; N1 N2 sometimes we use a 4-supercharges notation , arrows are bifundamental or adjoint chiral elds. through the term of ipping elds: Flips. A gauge singlet chiral eld ips an operator O when it enters the superpotential O. In this paper we consistently use di erent names for three classes r elds ip the dressed mesons operators, which are mapped to monopole operators M with topological charges (0; : : : ; 0; ; : : : ; ; 0; : : : ; 0) in the mirror quiver. j elds ip Tr( j ), which are mapped to length-j mesons in the mirror quiver. N elds are generated in the mirror quiver when gauge nodes con ne. They ip the N determinant of the dual Seiberg mesons. 2 Adjoint-SQCD with one avor in 3d: Abelianization The starting point is 4d N = 2 SU(N ) gauge theory, with 2N singlet eld A in the adjoint of the global symmetry SU(2N )F , coupled to the moment map H = Tr(q~iqj ). Notice that the latter coupling is marginally irrelevant and explicitly breaks N = 2 supersymmetry to N = 1. [1, 2] then gave a maximal 2N 2N nilpotent vev to A. We review the procedure of integrating out the massive avors due to the nilpotent vev [ 24, 25 ] in appendix A. The nilpotent vev breaks the SU(2N ) avor symmetry completely and leads to a N = 1 SU(N ) gauge theory with an adjoint eld and one avor q; q~: avors qi; q~i and an additional T4d;UV : N 2N W = WN =2 + P2N i;j=1 Aij Tr(q~iqj ) maximal nilpotent vev to A (2.1) T4d;IR : N q q~ 1 WIR = Tr(q~ 2N q) + PrN=02 rTr(q~ rq) Tr( j ); j = 2; 3; : : : N; are decoupled In T4d;IR the eld has R-charge R[ ] = 2 3(N+1) , as determined applying Amaximization. One important aspect of A-maximization is that the N 1 gauge invariant operators Tr( j ) with j = 2; 3; : : : ; N have R < 23 and must be decoupled. The N 1 singlet elds r are what is left-over from the 2N 2N matrix A. { 4 { Because of the singlets and the peculiar superpotential, the qualitative behavior of the TIR is quite di erent from the case of adjoint-SQCD with W = Tr( h) studied in the literature [26{31], both in 4d and in 3d. An important consequence of the decoupling of all the operators Tr( j ) with j = 2; 3; : : : ; N is that N , as N N matrix, is zero in the chiral ring. This implies a truncation in the spectrum of gauge invariant operators like mesons and baryons dressed by the adjoint elds. In particular, dressed mesons Tr(q~ rq) vanish if r N , so the rst term in the superpotential Tr(q~ 2N q) is zero in the chiral ring. This in turn implies that the chiral ring as de ned by the lower theory in (2.1) is unstable. Let us state in detail the criterion of chiral ring stability as in [9]. Starting from a theory T with superpotential WT = P needs, for each i, to: i Wi (where each term Wi is gauge invariant), one consider the modi ed theory Ti, where the term Wi is removed from W check if the operator Wi is in the chiral ring of Ti If one of the terms Wi does not pass the test, it must be discarded from the full superpotential WT . See [9] for a more detailed justi cation of this procedure and [11] for a geometric interpretation in terms of K-stability. If we drop Tr(q~ 2N q) from the superpotential, then Tr(q~ 2N q) is still zero in the modi ed chiral ring, so Tr(q~ 2N q) does not pass the test of chiral ring stability: the correct IR superpotential does not contain the term Tr(q~ 2N q).2 Moreover, in order to reduce to 3d, it is crucial that we do not simply reduce the N = 2 SU(N ) with 2N avors theory and then repeat the same procedure in 3 dimensions [9]: this strategy would lead to a di erent set of ipping r elds coupled to the 3d IR theory. For instance in the case of SU(N = 2), in [9] it was shown that, repeating the procedure of giving a maximal nilpotent vev to A in 3d, the IR theory contains also a ipping term 1Tr(q~ q), and instead of being dual to N = 4 U(1) with 2 avors, the IR theory contains two gauge singlets and is dual to N = 2 U(1) with 2 avors with both avors ipped. For general N , in appendix A, using the chiral ring stability criterion, we show that at most 1 r's remain in the IR, and as we discuss in more detail in section 3.6, the low energy theory is not N = 4 supersymmetric. We thus introduce in the UV precisely N j elds to ip the operators Tr( j ). In this way the UV description is complete and in the IR there in no unitarity violation. 2Notice that if one believes that the term Tr(q~ 2N q) can appear in the superpotential, then there would be an exactly marginal direction (this is because Tr(q~ 2N q) has R-charge 2 and does not break any nonanomalous global symmetry, for N > 2, so it generates an exactly marginal direction [32{35]), but in the A2N 1 AD model there are no marginal directions. For N = 2, the term Tr(q~ 4q) breaks the global SU(2) symmetry which must be present in the A3 AD model [9]. 1 r gauge singlets elds and also the N 1 { 5 { We call the modi ed theories T40d;UV and T40d;IR, and replace (2.1) with: discussed in more detail in [9], this operation has precisely the same e ect of stating that the operators Tr( j ) are decoupled as in [8]. The 3d and 4d superconformal indices, the S3 partition function and 4d a-maximization [36] are all the same. One advantage of this \completed" re-formulation is that now standard techniques can be used to compute the chiral ring and the moduli space of vacua. T40d can also be easily compacti ed on a circle. 2.1 4d chiral ring: dressed baryons and dressed meson Before compactifying to 3d, let us study the chiral ring of the 4d theory. The theory admits two non-anomalous global symmetries, acting on the elementary elds as U(1)4Rd 2 3(N+1) q; q~ 13 + 3(N+1) 2 j r 2 3(N+1) 2j 2r 4N 3(N+1) U(1)T 2 3(N+1) 2N 3(N+1) 2j 3(N+1) 4N 2r 3(N+1) U(1)B 1 0 0 0 (2.2) (2.3) where we normalized the non baryonic global symmetry U(1)T so that R[ ] = T [ ] and R[ r] = T [ r]. Notice that R[ r+2] = 63N(N+2+12)r = R[ r] + 23 . As pointed out in [2], the N 1 r's, r = 0; 1; : : : ; N 2, map to the Coulomb branch generators of A2N 1 AD. Let us study the rest of the chiral ring. First of all we claim that the operators j vanish in the chiral ring: they are Q-exact operators, where Q denotes the supercharges which emerge in the infrared. We postpone the discussion about this point to the end of this subsection; for the moment we just point { 6 { Since follows with with out, as a consistency check, that they cannot have an expectation value: such a vev would lead to a theory with no vacuum for quantum reasons. For instance, if 2 takes a vev, N = 1 SU(N ) with 1 avor and W = superpotential is dynamically generated. becomes massive, and the low energy theory is 0Tr(q~q), which has no vacuum because a ADS For a generic j N , giving vev to j brings us to a theory with W = Tr( j ). [26] showed that a SU(N ) gauge theory with Nf Nf jN1 . Since we have Nf = 1, giving a vev to j leads to a theory with no vacuum, for avors and W = Tr( j ) has a vacuum only if all j = 2; 3; : : : ; N . Assuming that all j 's vanish in the chiral ring, and using the powerful matrix relation N = 0, it is quite easy to discuss the full structure of the 4d chiral ring. The operators that are built using q; q~ and are generated by only three operators. N = 0, we can make only one dressed baryon, using N q elds and N 2 B = "i1;i2;:::;iN qi1 ( q)i2 ( 2q)i3 : : : ( N 1q)iN R[B] = There is a similarly de ned anti-baryon B~ using q~. See [37] for the Hilbert Series of adjoint SQCD with Nf avors. Because of the F -terms of r and the relation N = 0, there is only one non-vanishing dressed meson: B; B~ and M satisfy the chiral ring relation B B~ = "i1;i2;:::;iN "j1;j2;:::;jN qi1 ( q)i2 : : : ( N 1q)iN q~j1 (q~ )j2 : : : (q~ N 1 )jN = M N ; (2.8) where we used that Tr(q~ rq) = 0 in the chiral ring if r < N 1. The chiral ring relation B B~ = M N is precisely the de ning equation of C2=ZN , known to be the Higgs branch of A2N 1 Argyres-Douglas. The other generators of the chiral ring are the N 1 gauge singlets r, and map to the Coulomb branch of A2N 1 Argyres-Douglas. Let us study the chiral ring relations between the r's and B; M; B~. Contracting the F -terms of q~i with (q~ N 1 r)i we nd Contracting (2.9) with X s s( sq)i = 0 r M = 0 "i0;:::;ir 1;i;ir+1;:::;iN 1 qi0 ( q)i1 : : : ( r 1q)ir 1 ( r+1q)ir+1 : : : ( N 1q)iN 1 { 7 { elds as (2.4) (2.5) (2.6) (2.7) (2.9) (2.10) Similarly one can prove that So the 4d moduli space of vacua has two branches: one branch is CN 1, freely generated by the N 1 r's, the other branch is C2=ZN . The two branches intersect only at the origin of the moduli space. This is precisely the expected moduli space of vacua of the A2N 1 Argyres-Douglas theory. 2.1.1 N = 2 AD interpretation of the j multiplets The R-symmetry of any N = 2 SCFT is SU(2)R subalgebra is given by the combination U(1)RN=2 . The R-symmetry of an N = 1 RN =1 = 1 3 RN =2 + 4 3 I3 ; where I3 is the cartan generator of SU(2)R. The supercharges Q subalgebra (together with the corresponding Q _ ) are those with charge 12 under I3. In this way the scaling dimension of the N = 1 chiral primaries (de ned w.r.t. the above generating this N = 1 mentioned Q supercharges) satisfy = 32 R N =1. Instead, the only combination under which the gluinos in a Lagrangian N = 2 SCFT are uncharged is proportional to (uk) = 1 + k and Q _ supercharges are the only manifest supercharges in the lagrangian description of Argyres-Douglas theories. The AD theory of type A2N 1 contains Coulomb Branch operators, usually called uk, of dimension The uk operators transform in the trivial representation of SU(2)R, so they have charges Since AD theories have N = 2 supersymmetry, the uk operators are the lowest components of short N = 2 supermultiplets, which, in the Dolan-Osborn notation [38] are called E(RN=2;0;0). We denote the corresponding N = 2 multiplets as Uk. As we have already explained, the uk map to the lowest components of the chiral multiplets N 1 k in the nonabelian SU(N ) theory: where uk denotes the N = 1 chiral multiplet one gets acting with the supercharge Q on the chiral primary uk. The chiral multiplets N 1 k represent only half of the Uk N 1 k ! uk { 8 { CB multiplets and the remaining components are obtained by acting with the \hidden" supercharges, which have charge 12 under I3 (see also [39] for a discussion about this point). These extra components are organized into another N = 1 chiral multiplet (where again chirality refers to the Q supercharges described before) which we call vk: vk Z d2 ~Uk ; where ~ represent the IR emergent Grassmann variables of the N = 2 superspace (the notation is identical to that of [5]). The ~'s have charge 1 under RN =2 and -1/2 under I3, so HJEP10(27)3 The charge under the R-symmetry of the manifest N = 1 subalgebra is then This ts perfectly with the R-charge of the j elds given in (2.3), once we set j = N +1 k. Also the charges of the various elds under U(1)T in (2.3), which can be identi ed with the combination RN =2=3 claim that j and complete identi cation 2I3=3 in the N = 2 theory, and U(1)B are consistent with the j 2 are part of the same N = 2 multiplet. We therefore propose the r r+2 ! ! uN 1 r vN 1 r N = 2 supercharges In other words, r+2 is a supersymmetric partner of r, for an emergent supersymmetry. The triviality in the chiral ring of j 's now simply follows from the fact that in the N = 2 AD model they are Q-exact. 2.2 Compacti cation to 3d: emergent symmetry We now compactify on S1 the RG ow (2.2). First of all, can monopole superpotential be generated? Since the theory contains an adjoint eld , in order to possibly generate a monopole superpotential, two zero modes must be soaked up by the 4d superpotential [29]. Terms proportional to r cannot be generated because in a SU(N ) theory monopole operators MSU(N) cannot be dressed with fundamental elds q; q~ and because all dressed mesons Tr(q~ rq) vanish in the chiral ring if r < N 1. Terms proportional to j cannot be generated because j = 0 in the chiral ring,3 so terms like j fMSU(N) j 2g (we denote by fMSU(N) ig the monopole operators dressed by i factors of the adjoint eld) would lead to an unstable chiral ring. We conclude that no monopole superpotential is generated in the compacti cation. 3Here we are assuming that the 4d result j = 0 holds also in 3d, it would be nice to prove this statement. { 9 { (2.18) (2.19) (2.20) (2.21) So the 3d IR superpotential is the same as in 4d: This fact has the important consequence that in 3d there is an emergent symmetry, on top of the 4d symmetries. (2.23) Let us study the S3 partition function. The contribution of chiral eld with r-charge r is el(1 r). The function l(x) is de ned as follows: l(x) = xlog 1 e2 ix + i 2 1 x2 + Li2(e2 ix) i 12 and satis es the di erential equation @xl(x) = xcot( x). The S3 partition function for SU(N ) with an adjoint of r-charge r and a avor q; q~ of r-charge rq is e(N 1)l(1 r )Y el(1 r +i(zi zj))Y el(1 rq b izi) X z i dzi (2.27) T 0 is chosen so that the baryons B; B~ and the meson M are neutral. The basic monopole operator MSU(N) has GNO charges f+1; 0; : : : ; 0; 1g. In any 3d; N = 2 SU(N ) gauge theory with an adjoint eld , a fundamental q and an anti-fundamental q~, the monopole global symmetry charges can be computed in terms of the charges of the elementary fermionic elds in the lagrangian F [MSU(N)] = R[MSU(N)] = 1 In the rst line there is the contribution of the singlets r and j , the Haar measure and the N ! Weil-group factor. In the second line the contribution of the adjoint eld and the fundamental elds q~; q appear. b is the fugacity for the baryonic symmetry. Performing Z-extremization, we nd that ZSU(N) has a critical point at i6=j r = 0 ; rq = We checked this claim numerically for N = 2; 3. Since the baryonic symmetry doesn't mix with the R-symmetry, the critical point is obviously at b = 0. The following limit4 implies that in the limit r the Haar measure. The limit5 limr !0el(1 r ix)(2sinh( x))2 = 1 ! 0 the o -diagonal components of the adjoint cancel against limr !0el(1 (2 jr ))+l(1 r )) = j (2.29) (2.30) groups and 3N chiral elds i; Pi; P~i integrand of the partition function for an Abelian U(1)N 1 gauge theory. In the limit r ! 0 the integrand of the partition function for SU(N ) becomes the The Abelian gauge theory is the N = 4 supersymmetric linear quiver with N 1 gauge 1 1 1 1 W = WN =4 = PiN=11 i(PiP~i Pi+1P~i+1) (2.31) whose most general partition function depends on rP , a baryonic-like fugacity B and N 1 Fayet-Iliopoulos parameters j Notice that the reduction is at the level of the integrands, which is somehow stronger than the equality at the level of the integrals. The reduction holds on the two-dimensional j=1 (2.32) locus r = j = 0. 2.4 3d chiral ring: dressed monopoles desired result. asymptotics l(1 we conclude that In order to gain better understanding of the Abelianization, we study the complete 3d chiral ring, and map it to the chiral ring of the Abelian theory. Compared to 4d, in 3d there are also monopole and dressed monopole operators. MSU(N) can be dressed with the adjoint eld . Dressed monopoles were studied in order to compute the Coulomb branch of N = 4 gauge theories in [14], using Hilbert Series 4This can be proven as follows: using the explicit expression for l(z) and the identity l(z) + l( z) = 0 one can easily derive the equation l(1 + ix) + l(1 ix) = 2log(2sinh( x)); which immediately implies the x) = (x 1)cot( x), which implies the x) log(sin( x)) around x = 0. Using this result and the identity l(x) + l( x) = 0, el(1 (2 jr ))+l(1 r ) elog(sin(j r ))=elog(sin( r )) and the r.h.s. manifestly tends to j for r ! 0. Nc 1 X techniques [40]. Formula (5:11) of [14] gives the `Plethystic Logarithm' of the Coulomb Branch Hilbert Series for N = 4 SU(Nc) gauge theories with Nf avors: P Log[HSU(Nc);Nf (t)] = tj+1 +j(tNf j +tNf 2Nc+j+1) t2Nf 4Nc+6 +O(t2Nf 4Nc+7) The rst term represents the algebraic (linearly independent) generators of the Coulomb Branch chiral ring: (2.33) PjN=c1 1 tj+1 represents Nc These are the Tr( j ). 1 generators, with scaling dimension 2; 3; : : : ; Nc t M+j 1), we see that the allowed values for k are monopoles made out of the basic monopole MSU(Nc); N =4 and k factors of the adjoint eld . We denote such operators fMSU(Nc) kg. The basic monopole MSU(Nc) has GNO charges (+1; 0; : : : ; 1) and in the N = 4 theory it has scaling dimension [MSU(Nc); N =4] = Nf 2Nc + 2. Rewriting the sum as PjN=c1 1 j(t M+2Nc 2 j + 0; 1; 1; 2; 2; 2; 3; 3; 3; 3; : : : ; 2Nc 4; 2Nc 4; 2Nc The second term in (2.33), with a minus sign, represents an algebraic non-linear relation satis ed by the generators, but it is valid only for N = 4 theories, in our N = 2 case it does not apply. independent dressed monopoles.6 The counting of the generators instead applies to our N = 2 case as well, even though the above results were derived for N = 4 gauge theories: there are N (N 1) linearly In our case of SU(N ) theory with one avor and superpotential WIR = N 2 X r=0 rTr(q~ rq) + X 1 Coulomb branch generators Tr( j ) are removed by the F -terms of j , but we can combine the dressed monopoles with the N 1 r's. All together there are N 2 1 operators with U(1)q-charge 1 and vanishing baryonic charge U(1)B. Using the input r = 0, rq = 12 , the scaling dimension of all these N 2 1 operators is = 1. They di er 6Giving a vev to MSU(N) breaks the gauge symmetry SU(N ) ! U(1) SU(N 2) U(1). The adjoint eld decomposes as diag( 1; ^; N ), where 1 and N are scalars and ^ is a traceless N 2 N 2 matrix. How many independent ways are there to dress MSU(N) with j factors of 's? We need to consider operators of the form fMSU(N) 1 a ^b cN g a + b + c = j (2.34) We need to impose that they cannot be written as a product of Tr( d) times some smaller dressed monopole of Tr( i<N ). Doing this type of analysis, one concludes that there are precisely N (N fMSU(N) d0 g and we need also to consider the constraints that Tr( j N ) can be expressed as a combination 1) dressed monopoles generators, see section 5 of [14]. by the U(1)T 0 charge, which goes from operators fMSU(N) N 1g having vanishing U(1)T 0 charge. 1 for M to 1 for 0, in steps of N1 1 , the N The dressed monopoles of the non-Abelian gauge theory SU(N ) with one avor map to the monopoles of the U(1)N 1 Abelian quiver. For instance for N = 4 0 fMSU(N) 3g1 fMSU(N) 4g1 fMSU(N) 5g B B fMSU(N) 2g1 fMSU(N) 3g2 fMSU(N) 4g2 1 C @ fMSU(N) g1 fMSU(N) 2g2 fMSU(N) 3g3 2 A M1;0;0 2 M 1;0;0 M1;1;0 M0;1;0 3 M1;1;1 1 M0;1;1 C M0;0;1 CA M 1; 1; 1 M0; 1; 1 M0;0; 1 where Ma;b;c are the monopoles of the U(1)3 quiver with topological charges (a; b; c).7 The r.h.s. of eq. (2.35) are the Coulomb branch generators of the linear quiver, with scaling (2.35) = 1. The latter operators in turn map to the mesons of the mirror theory, From eq. (2.35) we see that the global symmetry U(1)T 0 of the SU(N ) gauge theory descends to the sum of the N 1 topological symmetries of the linear quiver. The emergent symmetry which is generated compactifying to 3d is enhanced to SU(N ). Notice that we are not claiming a precise 1-to-1 map with eq. (2.35): the global symmetry analysis we made only implies, for instance, that dimensional space spanned by (fMSU(N) g 0 is mapped to M1;1;1, and the two 5 ; 1) is mapped to the two dimensional space spanned by (M1;1;0; M0;1;1). It would be interesting to derive the precise mapping of the dressed monopoles to the monopoles of the abelian quiver.8 On top of these N 2 1 generators, that map to the Coulomb branch of the Abelianized N = 4 theory, there are the three operators B; M; B~ discussed in section 2.1. These three operators in 3d have dimension [B; B~] = N2 and [M] = 1 and satisfy the same equation BB~ = M N . They generate the Higgs branch of the Abelianized N = 4 theory, indeed they 7Notice that fMSU(N) jg are not zero in the chiral ring even if j N , we are just using the symbol jg to denote the dressed monopole with j factors of , of the form (2.34). fM8STUh(Nis) mapping allows us to get one more check of the Abelianization duality. We focus on the SU(2) case. Adding to the superpotential term linear in 0, the meson tr(q~q) acquires a vev, breaking the SU(2) gauge symmetry completely. The IR description is a Wess-Zumino model W = 2Tr( 2) = 2 2 det( ): The Abelianized theory in this case is U(1) with 2 avors, using the mapping 0 $ 0-deformation corresponds to turning on W = MU+(1). Taking the mirror dual it becomes an o diagonal MU+(1), the linear mass term W = (p1p~1 + p2p~2) + p1p~2 We can now integrate out the massive elds, getting a U(1) gauge theory with one avor and W = Taking the mirror again (using that U(1) with one a WZ model with W = Z(XY 2), which is equivalent to (2.36). (Alternatively, we could have used the avor and W = 0 is dual to the XY Z model), we monopole duality discussed in the next section). It is interesting that after the linear 0 deformation, 2 is not forbidden anymore to acquire a vev. (2.36) (2.37) 2p2p~1. nd 0.6 0.5 map to the operators of the U(1)N 1 quiver as follows The N 1 r's have vanishing product with B; M; B~, for the same reasons explained in 4d. It would be nice to show that also the dressed monopoles have vanishing product with B; M; B~. Finding the chiral ring quantum relations satis ed by the dressed monopoles in our 3d N = 2 theory is an interesting problem that goes beyond the scope of this paper. 2.5 S3 partition functions At the level of S3 partition functions, we expect the equality of ZSU(N)[r ; rq; b] and ZU(1)N 1 as a function of 3 variables, for r > 0, which can be checked numerically for small values of N . As for the case of SU(2) studied in [9], the numerical evaluation of ZSU(N)[r ; rq; b] present a singularity at r = 0: the rst derivative with respect to r is discontinuos, as displayed in 1. We propose that ZSU(N)[r ; rq; b] should be continued analytically from the region r > 0. Using the mapping of the chiral ring generators found in the previous subsection it's possible to nd the mapping for the fugacities appearing in ZSU(N) and ZU(1)N 1 , which we recall are de ned by i6=j i N ! X zi dzi (2.41) From the mapping of the dressed baryons to the \long mesons" in the quiver we can infer where the baryonic and U(1)q fugacities map: N 2 1 r B = b : i = r : (2.42) (2.43) (2.44) (2.45) (2.46) and and From the mapping of the dressed monopoles to the monopoles in the quiver (2.35), we can guess that each Fayet-Iliopoulos fugacity maps to the r-charge of the SU(N ) adjoint : Combining these arguments, we arrive at the following equality among the ZS3 , as functions of three variables: ZSU(N)[r ; rq; b] = ZU(1)N 1 rP = rq + r ; B = b; i = r : (2.47) N 1 2 We checked this relation numerically for N = 2 and N = 3. It only holds if r ZSU(N)[r ; rq; b] in the region r < 0 should be analytically continued from the region r > 0. The previous equation provides an analytic continuation in terms of ZU(1)N 1 , which is perfectly regular around its minimum at rP = 12 ; B = i = 0. In particular the second derivatives, which are the two-point functions of the symmetry currents [41], are continuous and positive around the minimum. 3 ow to A2N 1 AD: sequential con nement In this section we provide a 3d interpretation of the results of [1, 2] and a further check of the claims of the previous section. The strategy is to reduce the 4d RG ow to 3 dimensions, use 3d N = 4 mirror symmetry, and analyze the mirror RG U(N ) gauge theories with monopole superpotential [18]. 3.1 Basic ingredients ow applying a duality for Reducing T40d;UV leads to 3d N = 4 theory SU(N ) with 2N avors with additional N = 2 superpotential terms and 2(N 1) additional chiral N = 2 singlets r and j . In the following we will refer to the mirror of T30d;UV as T~3d;UV. The crucial point is that N = 4 SQCD has a known mirror dual and we can study the RG ow T~3d;UV ! T~3d;IR induced by the additional superpotential terms. In order to proceed we now review the mirror of U(N ) with 2N avors, N = 4 susy. We will later adapt those results to the case of SU(N ) with 2N avors that we need. 3.1.1 avors is a U(ni) linear quiver gauge theory [42]: 3d mirror N 2N 1 2 2 N 2 1 (3.1) The SU(2N ) Higgs branch global symmetry on l.h.s. is mapped to the enhanced topological (or Coulomb branch) symmetry U(1)2N 1 ! SU(2N )C on the r.h.s. The U(1) Coulomb branch symmetry on the l.h.s. is also enhanced to SU(2)C , and is mapped to the SU(2) rotating the 2 avors of the central node on r.h.s. A proof of the equality of the re ned N = 4 S3 partition functions was given in [43]. The Higgs branch generators of the U(N ) with 2N avors Qi; Q~i theory have scaling dimension = 1, transform in the adjoint of SU(2N ) and map to the Coulomb branch generators of the r.h.s. as follows: B B B . . 0 Tr(Q1Q~1) B Tr(Q2Q~1) Tr(Q1Q~2) : : : Tr(Q1Q~2N ) 1 Tr(Q2Q~2) : : : Tr(Q2Q~2N ) C . . . . . . . . . Tr(Q2N Q~1) Tr(Q2N Q~2) : : : Tr(Q2N Q~2N ) 0 Tr( LU(1)) M1;0;:::;0 M1;1;:::;0 B M 1;0;:::;0 Tr( LU(2)) M0;1;0;:::;0 B B B B C C$BBM 1; 1;:::;0 M0; 1;:::;0 . . . M 1;:::; 1 . . . : : : . . . . . . : : : : : : : : : . . . Tr( R U(2)) M0;:::;0; 1 Tr( UR(1)) M0;:::;0;1 CCA M1;1;:::;1 1 M0;1;:::;1 C . . . C C C C (3.2) Where Ma1;a2;:::;a2N 1 is the minimal monopole with topological charges (a1; a2; : : : ; a2N 1) in the r.h.s. quiver and the 's are the adjoint of the gauge nodes. See [44] for discussions and applications of such map. The Coulomb branch of the U(N ) with 2N avors theory is generated by 3N operators [14]. They transform as N triplets of the global SU(2)C symmetry, with = 1; 2; : : : ; N , and map to the Higgs branch of the r.h.s. quiver theory as follows: 0 B B B M fM g Tr( ) Tr( 2) : : : M+ fM+ g fM N 1 g Tr( N ) fM+ N 1 g 0 B ! BB Tr(qI q~J ) Tr(qI pp~q~J ) : : : Tr(qI pp : : : p~p~q~J ) 1 C C C A (3.3) where is the adjoint in the l.h.s. and fM g denote the basic monopole with GNO charges ( 1; 0; 0; : : : ; 0) dressed by j factors of the adjoint eld . On the r.h.s. qI ; q~J denote the 2 avors attached to the central U(N ) node, p; p~ generically denote the bifundamental elds of the lower row of the quiver. 3.1.2 Con ning U(N ) with N + 1 avors and W = M+ The RG ow on the mirror is triggered by linear monopole superpotentials. The analysis is accomplished using a recently found duality ([18], section 8) for 3d N = 2 U(Nc) with Nf avors (and Nf anti- avors) and W = M+. See [44, 45] for previous examples in the Abelian case and [46] for a brane interpretation. M are the basic monopoles of 1 C C C A j W = M+ + M [18]. model with Nf2 + 1 chiral elds: U(Nc) with GNO charges ( 1; 0; 0; : : : ; 0). The dual is a Aharony [47] magnetic description U(Nf Nc 1) gauge theory with Nf avors, Nf2 + 1 singlets and W = M + M+ + X MNf ij q~iqj The global symmetry on both sides is U(1)topological SU(Nf ) U(1)R. The duality can be obtained from a real mass deformation of a similar duality for U(Nc) with superpotential 2 The special case of interest to us is Nf = Nc + 1, in this case the dual is a Wess-Zumino U(Nc); Nc + 1 avorsfqi; q~ig; W = M+ WZ-model W = Nc+1 det(MNc+1) M Tr(qiq~j ) ! ! ! Nc+1 MNc+1 We also displayed the map of the chiral ring generators. In other words if Nf = Nc + 1, in the presence of a superpotential W = M+, the gauge We are able to apply the monopole duality since the relevant U(Nc) node loses additheory U(Nc) con nes. tional matter like the adjoint eld. 3.2 The general picture of the mirror RG ow Our set of theories of interest can be represented by the following diagram: where we are already anticipating the result: on the r.h.s. in the IR the manifestly N = 4 theory U(1) with N avors appears, as expected. The upper part of (3.6) is obtained from (3.1), where on the l.h.s. we gauged the U(1) SU(2) topological symmetry, so the gauge group, from U(N ), becomes SU(N ) and N 2N W = WN =4 + WN =2 RG ow: matter elds integrated out T30d;IR 3d mirror 3d mirror T~3d;UV 1 2 1 1 N W = WN =4 + ~ WN =2 2 1 RG ow: gauge nodes con ne 1 T~3d;IR: N the SU(2) topological symmetry is replaced by a U(1)baryonic symmetry. On the r.h.s. this maps to gauging one of the two avors (red node), breaking the global SU(2) symmetry and gaining an additional U(1) topological symmetry. WN =2 is given by the mirror of WN =2 = 2N 1 X i=1 N 2 X r=0 r i=0 WN =2 can be worked out adapting the maps (3.2) and (3.3) from U(N ) According to (3.2), the rst sum in (3.7) is mapped to a term linear in the 2N monopoles with precisely one positive topological charge: M1;0;:::;0 + M0;1;0:::;0 + : : : + M0;:::;0;1: (3.8) There are linear monopole superpotential only for the nodes in the lower row of the quiver T~3d;UV, the upper U(1) gauge node attached to the central U(N ) node will never have monopole potentials. Using (3.2) again, the second sum in (3.7) is mapped to ipping terms for monopoles with negative topological charges 0M 1;:::; 1 + 1(M 1;:::; 1;0 + M0; 1;:::; 1) + : : : + N 2(M 1;:::; 1;0;:::;0 + : : :) : (3.9) Finally, the third sum in (3.7), PjN=2 j Tr( j ), is mapped to ipping terms for mesonic operators appearing in the r.h.s. of the map (3.3), adapted from U(N ) to SU(N ) gauge symmetry. Sequential con nement. We start applying the monopole duality to the leftmost U(1) node in the upper-right quiver in (3.6). The U(1) node con nes and the Seiberg dual mesons give mass to the adjoint of the close-by U(2) node. At this point the U(2) node has no adjoint, 3 avors and a monopole superpotential M+, so we apply the monopole duality to the U(2) node. This pattern goes on until the left tail has disappeared and we reach the central node U(N ). When the central node con nes, some of the Seiberg dual mesons give mass to the adjoint of the U(N 1) node, some become bifundamental elds for the 3 groups U(1) U(1)F U(N 1). Going down along the right tail, at each dualization step we generate one more bifundamental avor between the two upper nodes. At the end we are left with just U(1) U(1)F with N bifundamental hypers, that is U(1) gauge theory with N hypermultiplet avors. This is the qualitative story, in the following we analyze in detail the process of sequential con nement including the superpotential, and con rm that the RG ow lands on U(1) avors with N = 4 supersymmetry. The only gauge-singlet, among the N 1 n's, that is massless in the IR is the N+1 singlet, generated when dualizing the central U(N ) node into a Wess-Zumino W = N+1det(MN+1). N+1 sits in the N = 4 vector multiplet of the U(1) gauge theory. The IR superpotential, modulo where Qi; Q~i is the fundamental hypermultiplet generated at the ith-step, dualizing down the second tail. Let us make a nal comment: if we had considered a non maximal Jordan block (and also non next-to-maximal), the sequential con nement would have stopped before, and the IR mirror theory would contain a non-Abelian node without the adjoint, so the mirror would clearly be only N = 2 supersymmetric. Since including the analysis of the superpotential leads to complicated expressions, we focus rst on the cases N = 2; 3. We will later comment about the generalization to N > 3. We start from N = 4 SU(2) SQCD with four avors, whose mirror is the N = 4 We numbered the abelian groups in the quiver and we call pi; p~i the U(2) U(1)i bifundamentals. The cartan subgroup of the SO(8) global symmetry of the theory is identi ed with the topological symmetries of the four gauge nodes. We are only interested in the SU(4) symmetry associated with the nodes U(1)1, U(1)3 and U(2). The singlets in the abelian vector multiplets will be denoted 'i (i = 1; 2; 3) whereas the trace and traceless parts of the U(2) adjoint are ^2 and 2 respectively. The operators Tr 2 and the monopole of SU(2) SQCD are mapped on the mirror side to p~4p3p~3p4 and p~4p3p~3p4 + p~2p3p~3p2 respectively. The SO(8) global symmetry of SQCD arises quantum mechanically in the mirror theory, due to the presence of monopole operators of scaling dimension 1, whose multiplets contain conserved currents [48]. We recall that the map between o -diagonal components of the SU(4) meson and monopoles is as follows: 0 BBq~2q1 q~4q1 q~4q2 q~4q3 q~1q2 q~1q3 q~1q41 0 M+00 M++0 M+++1 q~2q3 q~2q4C BM 00 q~3q4CCA$BB@M M The Cartan components of the meson matrix are mapped to '1, '3 and ^2. In (3.12) we have included only the charges under the topological symmetries related to U(1)1, U(1)3 and U(2), the others being trivial. Mapping the deformations of N = 4 SU(2) with 4 avors to the mirror theory T~U0 V, we nd that the mirror RG ow starts from WT~U0 V X 'ip~ipi i ^ 2 X p~ipi i 2 X pip~i i !! + + M+00 + M0+0 + M00+ + 0M + 2p~4p3p~3p4: (3.13) According to the monopole duality, the gauge group U(1)1 con nes 2 1 (3.14) HJEP10(27)3 2 det M2 + '1TrM2 2 TrM2 + X p~ipi ! Tr " 2 M2 + X pip~i i>1 !# leaving behind the U(2) adjoint chiral M2, which enters in the superpotential with terms M2 and 2 become massive and can be integrated out, the equations of motion impose the constraint M2 = Pi>1 pip~i. At this stage the U(2) gauge group has three avors and no adjoint matter, so according to the monopole duality it con nes and is traded for a 3 3 chiral multiplet M3, which is nothing but the dual of p~ipj (i; j = 2; 3; 4). This also generates the superpotential term 3 det M3. The constraint M2 = Pi>1 pip~i allows to express det M2 in terms of traces of M3: detM2 = (TrM2) 2 2 TrM22 = (p~ipi)2 Tr((p~ipj )2) 2 = (TrM3) 2 2 TrM32 : (3.15) In theory (3.11) the cartan subgroup of the U(3) symmetry under which p~ipj (i; j = 2; 3; 4) transforms in the adjoint representation is gauged: the U(1)2;3;4 symmetries are generated respectively by the 3 3 matrices diag(1; 0; 0), diag(0; 0; 1) and diag(0; 1; 0). Our convention is that these groups act in the same way on the matrix M3 after con nement of the U(2) gauge group. As a result, the o -diagonal components of M3 become bifundamental hypermultiplets charged under the leftover U(1)i symmetries and we relabel the elds as follows: (M3)12; (M3)2 $ Q1; Q~1; (M3)13; (M3)3 $ v; v~; (M3)23; (M3)3 $ w; w~: 1 1 2 After con nement of the U(2) gauge group the theory (3.11) becomes: 2 i>1 ! 1 1 Tr 4 3 2 1 v; v~ 1 w; w~ 4 (3.16) The elds 'i now appear only in the superpotential terms W = ('2 ^2)(M3)11 + ^2(M3)22 + ('3 ^2)(M3)33 : : : (3.17) As a consequence they become massive and their F-terms set to zero the diagonal components of M3. The remaining elds are 0, 2, 2;3 and the three bifundamental hypermultiplets with superpotential W = 2 2 (Q~1Q1 + v~v + w~w) + 2(w~w) + 3(Q~1v~w + w~vQ1) + M+ + 0M ; (3.18) where the monopoles are charged under the topological symmetry of U(1)3. Finally, the gauge group U(1)3 con nes and its meson components v~v, w~w and v~w, w~v become elementary elds of the theory. The rst two are singlets, which we call x and y, whereas the other two are charged under the surviving gauge group U(1)2 and we call them Q2, Q~2. 2 1 1 4 After con nement of U(1)3 (3.18) becomes 6 i=1 5 i=1 p1; p~1 1 W = 2 out we are left with in [19]. and all the elds except 3 and the two U(1)2 avors become massive. Integrating them WT~I0R = 3(Q~1Q2 + Q~2Q1); which is equivalent to the standard superpotential of N = 4 SQED with two avors after a change of variable and this is precisely the mirror of A3 Argyres-Douglas theory proposed 3.4 ow to A5 AD: the superpotential We now focus on the case N = 3. The prescription to obtain the A5 AD theory is to start from SU(3) SQCD with six avors, turn on ve o -diagonal mass terms and ip the operators Tr 2 and Tr 3 . We also introduce the two ipping elds ( 0 and 1) which do not decouple in the IR. The superpotential is WT30d;UV = X Tr(q~i qi) + 2Tr( 2) + 3Tr( 3) + X Tr(q~iqi+1) + 0Tr(q~6q1) + 1Tr(q~5q1 + q~6q2): We refer to this model as the T30d;UV theory. Its mirror is the quiver [48, 49] (3.19) (3.21) (3.22) (3.23) Every unitary gauge group gives rise to a topological U(1) symmetry and the U(1)5 global symmetry arising from the nodes in the lower row enhances to SU(6). We denote the gauged U(1) in the upper-row (depicted in red) U(1)red, and the avor U(1) as U(1)F . These two nodes will survive in the IR. We denote the bifundamental matter elds in the quiver as explained in (3.23).9 Since the theory is N = 4, every vector multiplet includes a chiral multiplet transforming in the adjoint representation. We denote the trace part of the adjoint chirals in the two tails (from left to right) as 'i (i = 1; : : : ; 5), the singlet of the U(1)red as '6 and the traceless part for the non abelian nodes as 2;L, 2;R and 3 . In order to study the mirror RG ow we need to map in the mirror theory (3.23) all the superpotential terms appearing in (3.22). Adapting the mapping (3.3) from U(N ) to SU(N ), we claim that the Casimirs Tr( 2) and Tr( 3) of the UV N = 4 SU(3) SQCD are mapped in the mirror (3.23) to Tr( 2) $ q2p3p~3q~2; Tr( 3) $ q2p3p4p~4p~3q~2: Using (3.24) and the observations of the section 3.2 the complete UV mirror superpotential reads 5 i=1 WT~3d;UV = WN =4 + X Mi+ + 1(M where Mi are the 5 monopoles with just one lower-row topological charge turned on. The rest of this section is devoted to the study of the RG ow. As in section 3.3, our basic tool is the monopole duality for N = 2 U(N ) SQCD with N +1 avors (and no adjoint matter) reviewed in section 3.1. Using this duality, the nal result will be that all the gauge nodes at which we have turned on the monopole superpotential term M+ con ne and our strategy is to follow the evolution of theory (3.25) step-by-step, sequentially dualizing one node at each step. This is essentially the mirror counterpart of integrating out massive avors one by one. At the end of this process, once we have dualized all nodes with the monopole term, the two monopoles multiplying 1 in (3.25) become the same operator, in analogy with the mirror theory (3.22), where both Tr(q~5q1) and Tr(q~6q2) become Tr(q~ q) in the IR. The rst step is to apply the monopole duality to the abelian node on the left, the relevant superpotential terms are W = M1+ + '1p~1p1 + '2(Trp~2p2 p~1p1) + Tr 2;L(p~2p2 p1p~1) + M2+ + : : : (3.26) 9We slightly change notation with respect to the SU(2) case since the four tails are not on equal footing for N > 2. The theory becomes 2 1 2 chiral M2 appears, and the above superpotential terms become: W = 2 det M2 + '1TrM2 + '2(Trp~2p2 TrM2) + Tr 2;L(p~2p2 M2) + M2+ + : : : (3.28) HJEP10(27)3 The elds M2, '2 and 2;L are now massive and can be integrated out. From the above formula one can easily see that the equations of motion identify M2 with the 2 matrix p2p~2. At this stage the neighbouring U(2) node has three avors and no adjoint multiplets ( 2;L has become massive), so the left U(2) node con nes and gets replaced by a 3 chiral multiplet M3 and singlet 3: 1 X Trq~iqi + Trp~3p3 i=1;2 1 Trp2p~2A + Integrating out all massive elds we get W = 2 det(p2p~2) + 3 det M3 + '2(Trp2p~2) + '3@ 0 p2p~2A + : : : As before, the multiplets M3, '3 and 3 become massive and can be integrated out, implying that the U(3) gauge group now has four avors and no adjoint matter. Let us now pause to explain how to treat the determinants which arise dynamically at each dualization step, like 2 det(p2p~2) + 3 det M3 in (3.30). We need to rewrite these determinants in terms of the elds that survive the various dualization steps as in the SU(2) case discussed before. Considering for instance the term 2 det M2 generated at the rst step, as already explained F-terms identify the multiplet M2 with p~2p2 and then when the U(2)L node con nes p2p~2 is identi ed with M3. So we need to rewrite det M2 in terms of the surviving eld M3. This is accomplished by rst rewriting the determinants in terms of traces.10 Using (3.31) and (3.35) the relation is 1 2 det M2 = ((Trp~2p2)2 Tr(p~2p2)2) = ((TrM3) 2 TrM32): More in general, if we had considered the mirror of SU(N ) SQCD with 2N avors, by turning on monopole superpotential terms at all the nodes along a tail the various nodes con ne and at the k-th step the U(k) gauge group disappears and is replaced by a (k + 1) (k + 1) chiral multiplet Mk+1. The superpotential term 2 det M2 generated at the rst step can be rewritten as 2 2 ((TrMk+1) 2 TrMk2+1): A similar observation applies to the terms generated at the subsequent dualization steps, using (3.32), (3.33) and generalizations thereof. In this way it is possible to keep track of all the terms generated along the process of sequential con nement and write all the superpotential terms as functions of the surviving fundamental elds of the theory. Going back to the analysis of our RG ow, we can now dualize the U(3) node of (3.29), generating the superpotential term 4 det M4. Now the operators q~1q1 and q~2q2 become diagonal elements of M4, which are elementary elds of the theory. The chirals '3 and '6 become massive and their F-terms set to zero the two diagonal elements of M4 they couple to. Another important fact is that, since a U(2) U(1) subgroup of SU(4) is gauged in the quiver, the massless components of M4 decompose as a U(2) adjoint (which we call ), a U(2) U(1)red bifundamental, a U(2) U(1)F bifundamental (we denote them as v; v~ and w; w~ respectively) and a U(1)red U(1)F bifundamental which we call Q1; Q~1: 1 2 A 3 M4 = B Q~1 0 w~ C w TrMkn = TrMkn+1 8n: (3.36) HJEP10(27)3 (3.37) (3.31) (3.32) (3.33) (3.34) (3.35) 10We will use the following identities for k k matrices Mk: det M2 = det M3 = TrM33 + (TrM2)2 det M4 = 3 2 4 TrM22 ; 2 (TrM3)3 6 24 2 TrM3 TrM32; 8 TrM44 + (TrM4)4 + (TrM42)2 + TrM4 TrM43 (TrM4)2 TrM42: 4 These are special cases of the formula det Mk = Yk ( 1)nl+1 X n1;:::;nk l=1 lnl nl! (TrMkl )nl ; where the sum is taken over the set of all integers nl 0 satisfying the relation Plk=1 lnl = k: These identities will be used to handle the superpotential terms which arise dynamically. We will also need the following simple observation: in the mirror quiver there are bifundamental hyperwhereas Mk+1 = bik~bjk transforms in the adjoint of U(k + 1) and the following identity holds multiplets bij; ~bij charged under U(k) U(k + 1). The operator Mk = ~bikbjk transforms in the adjoint of U(k) All in all, we get the theory 1 v; v~ with the following, complete, superpotential W = 2(: : :)+ 3(: : :)+ 4 det M4 +'4(Tr M4+ + M5+ + 1(M4 + M 0) + 0M p~4p4)+'5p~4p4 +Tr 2;R(p4p~4 + 2w~w + 3Tr(wp4p~4w~); where M4;5 are the monopole operators charged under one of the topological symmetries of the U(2) and U(1) nodes of the lower row in (3.38). M is the monopole with charge -1 under both topological symmetries and M 0 is the operator to which M0 (appearing in (3.25)) is mapped under these dualities. We will discuss it in more in detail later. , '4 and 2;R become massive, leaving 0 0 Q1 v~ w~ C 1 A v w p4p~4 (3.38) (3.39) (3.40) (3.41) (3.42) (3.43) (3.44) (3.45) HJEP10(27)3 and rst three terms in (3.39) W = 2(: : :) + 3(: : :) + 4 det M4 + '5p~4p4 + M4+ + M5++ + 0M + 1(M4 + M 0) + 2Tr(w~w) + 3Tr(wp4p~4w~); From the explicit form of M4 (3.40), using (3.31){(3.33), one can write explicitly the 2(Q~1Q1 + v~v + w~w) + : : : 3(Q1w~v + Q~1v~w + v~p4p~4v + w~p4p~4w) + : : : 4(v~vw~w The dots stand for all terms proportional to the trace of M4, which is just equal to p~4p4. As will become clear shortly, they don't play any role in our analysis, so we do not write them explicitly. This is essentially due to the F-term of '5 (the singlet in the vectormultiplet of the rightmost U(1) node in the lower row in (3.38)), which implies that p4p~4 squares to zero. The U(2) node now con nes and is traded for a 3 3 chiral multiplet Y3 0 Y11 Q2 Y13 1 Y31 Y32 Y33 which provides one extra U(1)red U(1)F bifundamental (we named those components Q2 and Q~2). We have the usual superpotential term det Y3 and according to the monopole duality M4 is identi ed with . The operator p~4p4 is now replaced by Y33. At this stage we are left with the theory 1 1 Q1;2;Q~1;2 1 (3.46) (3.47) (3.48) HJEP10(27)3 and in terms of the matrix Y3 the superpotential reads W = 3(Q1Q~2 + Q~1Q2 + Y13Y31 + Y23Y32) + 4(Y11Y22 Q2Q~2 Q~1Y13Y32 Q1Y23Y31) 2(Q~1Q1 + Y11 + Y22) + '5Y33 + M+ + 0M + 2Y22 + 3Y23Y32 det Y3 + 1( + M 0) + Y33(: : : ): The last term Y33(: : : ) denotes all the terms in (3.42){(3.44) we did not write explicitly, which are all proportional to TrM4 = Y33. The monopoles M are charged under the topological symmetry of the node with two avors. The diagonal elds Yii (i = 1; 2; 3) and the singlets '5, 2 and 2 are now massive and can be integrated out. The F-term for '5 sets Y33 to zero, hence also the last term in the superpotential vanishes: W = 3(Q1Q~2 + Q~1Q2 + Y13Y31 + Y23Y32) 4(Q2Q~2 + Q~1Y13Y32 + Q1Y23Y31) det Y3 + 1( + M 0) + M+ + 0M + 3Y23Y32 : Finally, when the abelian node with two avors con nes the superpotential term Y2 det Y2 is generated, with the 2 2 matrix Y2, whose o -diagonal entries combine into a hypermultiplet charged under the leftover U(1)red gauge group: Y2 = Y101 Q3 ~ Q3 Y202 ! As we mentioned before, the eld M 0 is now identi ed with which becomes massive and the terms in the third line of (3.46) can be dropped from the superpotential because of F-terms. The superpotential term proportional to det Y is also set to zero by the F-term for 0 ( Y2 = 0) and we are left with W = 3(Q1Q~2 + Q~1Q2 + Y11 + Y22) 4(Q2Q~2 + Q~1Q3 + Q1Q~3) + 3Y202: (3.49) The diagonal components of Y2 are massive and can be integrated out. The elds Q1, Q~1, Q2, Q~2, Q3 and Q~3 survive in the IR, they transform in the bifundamental under the IR U(1)red U(1)F theory: Q1;2;3;Q~1;2;3 T~IR : 1 1 1 3 The superpotential for T~IR 4(Q2Q~2 + Q~1Q3 + Q1Q~3); (3.50) is just (modulo a eld rede nition) the superpotential of N = 4 SQED with three avors, as we wanted to show. Notice also that the equations of motion set to zero 2 and 3 , which is consistent with our ndings in section 2.1 that the j 's vanish in the chiral ring. Comments about the higher N generalization The analysis in the general case proceeds in the same way, although the detailed computation quickly gets involved. In this section we will give the answer for some higher rank cases, namely A7 and A9 AD theories. We will just state the result without providing all the details of the derivation. SU(4) SQCD and A7 AD theories. The UV theory which in 4d ows in the IR to A7 AD and constitutes our starting point is SU(4) adjoint SQCD with eight avors and superpotential 8 W = X q~i qi + X q~iqi+1 + X 4 i=2 7 iTr i + 0q~8q1 + 1(q~7q1 + q~8q2) + 2(q~6q1 + q~7q2 + q~8q3): In the mirror quiver, after the dualization of all the gauge groups of one tail and the central node, we are left with the theory (3.51) (3.52) (3.53) 1 2 p; p~ 3 v; v~ w; w~ 1 1 Q1; Q~1 0 0 Q1 v~ 1 M5 = B Q~1 0 w~ C w pp~ A In this quiver the operator pp~, which is a U(3) adjoint, satis es the chiral ring relation (pp~)3 = 0. This is due to the F-term relations of the linear tail. When the gauge group U(4) con nes we are left with a 5 5 chiral M5, which in terms of the elds appearing in (3.52) takes the form Along the way we generate the superpotential terms i det Mi, whose form can be derived using (3.34). The terms proportional to 2 and 3 are exactly as in (3.42), (3.43) (with p4p~4 replaced by pp~) so we don't write them again. The term involving 4 is as in (3.44) except for two extra terms: 4(v~vw~w The term involving 5, namely the determinant of M5, reads 5(Q~1v~(pp~)2w + Q1w~(pp~)2v + w~vv~pp~w + v~ww~pp~v v~vw~pp~w w~wv~pp~v): (3.55) Dualizing the remaining three gauge nodes with monopole superpotential we generate the surviving elds Q2; Q3; Q4 and land on the theory namely SQED with four avors and superpotential 1 1 W = 5(Q~1Q4 + Q~2Q3 + Q~3Q4 + Q~4Q1): SU(5) SQCD and A9 theory. In order to engineer A9 AD theory we start from N = 4 SU(5) SQCD with 10 avors and modify the superpotential according to the procedure discussed above. After the dualization of all the gauge nodes in one tail and the central node, we nd the model 1 2 3 p; p~ Q1; Q~1 Again the central node is con ned and can be traded for a 6 6 chiral multiplet M6 which reads 0 0 Q1 v~ 1 v w pp~ The U(4) adjoint pp~ now satis es the constraint (pp~)4 = 0. In terms of these elds the superpotential terms proportional to 2 and 3 are as in (3.42) and (3.43) respectively. The term proportional to 4 is as in (3.54) and the one proportional to 5 is 5(v~(pp~)3v + w~(pp~)3w + : : : ); where the dots stand for all the terms appearing in (3.55). The determinant of M6 reads w~wv~(pp~)2v + v~vw~(pp~)2w v~ww~(pp~)2v w~vv~(pp~)2w + (v~pp~v)(w~pp~w) (w~pp~v)(v~pp~w) Q1w~(pp~)3v Q~1v~(pp~)3w: When the gauge nodes in the linear tail in (3.57) con ne, only the superpotential term proportional to 6 survives and we are left with SQED with ve avors Qi; Q~i i = 1; 2; : : : ; 5. In terms of these elds the superpotential reads W = 6 Q~1Q5 + Q~2Q4 + Q~3Q3 + Q~4Q2 + Q~5Q1 : 1 4 v; v~ w; w~ 4 1 1 (3.56) (3.57) (3.58) (3.59) (3.60) (3.61) If we repeat the procedure of Maruyoshi and Song in 3d, we nd that more r elds remain coupled to the SU(N ) gauge theory. In order to identify them, we can use algebraic arguments instead of performing Z-extremizations. Moreover, the conclusions are valid both in 3d and 4d. Let us start from the 8-supercharges theory SU(N ) with 2N avors and couple a 2N 2N matrix to the Higgs Branch moment map. After giving a maximal nilpotent vev to A, the superpotential is given by eq. (A.4). It contains many terms, not necessarily linear in r. As we explain in A, chiral ring stability [9] arguments, analogous to those given in section 2, imply that all the terms containing Tr(q~ rq) drop out in the IR if r N . The remaining superpotential contains N 1 r's (the others decouple) and is simply W = N 1 X r=0 rTr(q~ rq): (3.62) Performing a-maximization in 4d, it turns out that N 1 decouples from the theory. We can also think of a theory with the superpotential (3.62) as the naive compacti cation to 3d of the 4d theory: we start in 4d from SU(N ) with 2N avor coupled to A which takes a nilpotent vev, and compactify to 3d before owing to the IR. The arguments given in section 2.2 imply that also in this case no monopole superpotential terms are generated in the compacti cation. In 3d, the di erence between (3.62) and the theory T30d;IR, analyzed in detail in section 2, is the presence of the superpotential term N 1Tr(q~ N 1q). The crucial point is that the singlet N 1 does not decouple from the rest of the theory. In analogy with [9], the 3d theory with superpotential (3.62) abelianizes to U(1)N 1 linear quiver and N singlet elds that ip each meson. Instead of (2.31), T3d;IR is dual to the N = 2 quiver 1 1 1 1 W = PiN=1 iPiP~i (3.63) ring relation (2.8) BB~ = M in T30d;IR becomes BB~ = 0 in T3d;IR. Here the `long mesons' Q Pi and Q P~i have vanishing product, due to the F -terms of i's. This is consistent with the properties of the SU(N ) model, in which the F -terms of set to zero Tr(q~ N 1q), the operator we called M in section 2.1. As a result, the chiral N 1 We can at this point de ne a T3d;UV involving the elds r r = 0; : : : ; N 1, analogous to T30d;UV: this is a SU(N ) theory with 2N avors, an adjoint and superpotential 2N i=1 W = X Tr(q~i qi) + 2N 1 X i=1 N 1 r X X r=0 i=0 rTr(q~2N+i rqi+1); (3.64) which reduces precisely to (3.62) upon integrating out massive avors (see also appendix A). We will now discuss the mirror dual of the RG ow T3d;UV ! T3d;IR. In the case N = 3, by repeating the analysis of section 3.4 for T3d;UV, we nd that the mirror theory still reduces to SQED with 3 avors: the mirror of T3d;UV has superpotential (3.65) (3.66) (using the same notation as in section 3.4) 00 + M0 2(M 0 + M00 ): W = WN =4 + X Mi+ + 0M + 1(M We should now repeat the procedure explained in section 3.4, replacing (3.25) with the above equation. All the gauge groups in the lower row of (3.23) con ne as before and the monopole operators appearing in the second row of (3.65) are identi ed with 4 . As a result (3.50) is replaced by 4(Q2Q~2 + Q~1Q3 + Q1Q~3) + 3 2 4: Since the 2;3 terms in this case are absent, the singlets Y22 and Y202 appearing in section 3.4 decouple and become free instead of acquiring mass. The crucial di erence with respect to the analysis of section 3.4 is that 2 makes the singlet 4 massive and the superpotential vanishes. All other singlets r and i still become massive. The same conclusion holds for arbitrary N : the singlet N 1 makes N+1 massive and we are left with N = 2 SQED with N avors and no superpotential. In conclusion, we nd that T3d;UV ows in the IR to the mirror of N = 2 SQED, which is precisely the abelian linear quiver discussed around (3.63) [17]. Acknowledgments We are grateful to Francesco Benini, Matthew Buican, Amihay Hanany and Alberto Za aroni for useful discussions and comments. S.B. is partly supported by the INFN Research Projects GAST and ST&FI and by PRIN `Geometria delle varieta algebriche'. The research of S.G. is partly supported by the INFN Research Project ST&FI. A Nilpotent vevs In this section we discuss, following [25], the superpotential generated by the MaruyoshiSong procedure for SU(N ) with 2N avors. When we turn on a nilpotent vev for the matrix of ipping elds A, in the form of a single Jordan block of size 2N , we break the SU(2N ) symmetry completely, leaving just the baryon number unbroken. In the resulting RG ow some chiral multiplets decouple and in the IR we are left with a free sector consisting of decoupled chiral multiplets plus an interacting theory which turns out to be equivalent to (A1; A2N 1). As is well-known, SU(N ) nilpotent orbits are in one-to-one correspondence with SU(2) embeddings into SU(N ). For every such embedding ( +) is nilpotent and we can assume it is in Jordan form, with blocks of size ni. Under the above mentioned embedding, the fundamental representation of SU(N ) decomposes into irreducible representations of SU(2) as N ! Pli=1 ni. We can easily derive from this formula the decomposition of the we break spontaneously the global symmetry down to the commutant of SU(2) inside SU(N ). By expanding the superpotential around the vev we nd The rst term is the source of global symmetry breaking and as a result several components i of the SU(N ) moment map (in our case the meson) will combine with the current multiplets into long multiplets. The components of the ipping eld A coupled to i's will now decouple and become free. These are the Goldstone multiplets associated with the spontaneous symmetry breaking. How can we determine which components i decouple? Given the form of the superpotential, we can observe that under an in nitesimal complexi ed SU(N ) transformation we can obtain all the components of except those which commute with ( +). On the other hand, since ( +) is the SU(2) raising operator, we immediately conclude that the only components of the moment map which commute with it are the highest weight states in each Vs appearing in (A.1). Accordingly, the only components of A which remain coupled to the theory are the lowest states of each SU(2) irreducible representation. In writing the superpotential as in (A.2), we should keep only the components of the ipping eld which do not decouple. In the case relevant for AD theories, a single Jordan block of size 2N , A (actually its vev plus uctuations around it) takes the form: hAi = ( +); W = Tr ( + ) + TrA : 0 B B 0 . . . 0 B B 2N 2 A = BBB 2N 3 2N 2 1 0 . . . : : : . . 0 1 . . . . : : : : : : . . . 0 2N 3 2N 2 0 0 1 C C C C l ni 1 M M Vs i=1 s=1 adj: = (l 1)V0 2 nj M V ni+n2j 2k 5 3 where Vs is the spin s representation of SU(2). When we turn on a nilpotent vev of the form The vev of the ipping eld indeed breaks the UV R-symmetry, which is now mixed with ( 3). After the vev, the trial R-symmetry should then be rede ned by subtracting (1 + ) ( 3). The value of can be found performing a-maximization. In order to complete the analysis, we take into account the fact that the vev for A gives mass to all the SU(N ) fundamentals except one and we should integrate out all the massive multiplets. This can be done following the procedure described in [25]: the superpotential becomes W = ZZe + AZeZ + X (ZABnZe + ZBnZe ); 2N 1 n=1 (A.1) (A.2) (A.3) (A.4) where we suppressed the color indices, and the matrix B is More explicitly, the cubic and quartic terms have the following form: W = Q2N 0Qe1 Q2N 2 1 Z = (0 : : : ; 0; Q2N ); . 0 . . 2N 2 2N 2 X k=2 : : : 0 1 : : : 0 C . : : : 0 CC : . . ... CC C A 0 Qe1 + : : : This argument is valid in any spacetime dimension. Of course the set of operators which violate the unitarity bound and decouple is dimension dependent: in 4d N 1 decouples and the corresponding superpotential term drops out, whereas in 3d all the singlets are above the unitarity bound. At the cubic level only the singlet 0 appears. The other 2N 2 chiral multiplets r (r = 1; : : : ; 2N 2) appear only in quartic or higher terms. Using the chiral ring stability criterion of [9], we can signi cantly simplify (A.6): rst of all we notice that the N N matrix satis es the charachteristic polynomial equation, so Qe1 j N Q2N can be written as a polynomial in Qe1 j<N Q2N and the Casimirs of . Then we can notice that 0 and 1 appear only in the terms respectively, so their F-terms set to zero Qe1Q2N and Qe1 Q2N . 0Qe1Q2N and 1Qe1 Q2N This implies that all other terms of the form Qe1Q2N (: : : ) and Qe1 Q2N (: : : ) such as the last term in (A.6) can be dropped. At this stage it is straightforward to check that the only surviving term containing 2 is 2Qe1 2Q2N . Combining the F-terms for 0 and 2, which reads Qe1 2Q2N = 2N 2Qe1Q2N ; we conclude that Qe1 2Q2N is zero in the chiral ring, hence all terms proportional to this operator can be dropped. Proceeding recursively in this way, we nd that the F-terms for r with r < N set to zero all dressed mesons of the form Qe1 j<N Q2N . Consequently, operators of the form Qe1 j N Q2N automatically vanish in the chiral ring because of the characteristic polynomial constraint. The conclusion is that the term Qe1 2N Q2N can be removed and all the singlets r with r N disappear from the superpotential and decouple. This observation tells us that the superpotential reduces to the simpler form N 1 X r=0 W = rQe1 rQ2N : 0 Qe1 B 0 C Ze = BBB ... CCC ; 1 A B = B 0 . . . 1 0 . . . : : : (A.5) (A.6) (A.7) (A.8) Open Access. 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Sergio Benvenuti, Simone Giacomelli. Abelianization and sequential confinement in 2 + 1 dimensions, Journal of High Energy Physics, 2017, 173, DOI: 10.1007/JHEP10(2017)173