E 8 orbits of IR dualities

Journal of High Energy Physics, Nov 2017

We discuss USp(2n) supersymmetric models with eight fundamental fields and a field in the antisymmetric representation. Turning on the most generic superpotentials, coupling pairs of fundamental fields to powers of the antisymmetric field while preserving an R symmetry, we give evidence for the statement that the models are connected by a large network of dualities which can be organized into orbits of the Weyl group of E 8. We make also several curious observations about such models. In particular, we argue that a USp(2m) model with the addition of singlet fields and even rank m flows in the IR to a CFT with E 7 × U(1) symmetry. We also discuss an infinite number of duals for the USp(2) theory with eight fundamentals and no superpotential.

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E 8 orbits of IR dualities

HJE Haifa 0 Israel 0 Theory, Supersymmetry and Duality 0 Kashiwa , Chiba 277-8583 , Japan 1 Kavli IPMU (WPI), UTIAS, the University of Tokyo 2 Physics Department , Technion We discuss USp(2n) supersymmetric models with eight fundamental fields and a field in the antisymmetric representation. Turning on the most generic superpotentials, coupling pairs of fundamental fields to powers of the antisymmetric field while preserving an R symmetry, we give evidence for the statement that the models are connected by a large network of dualities which can be organized into orbits of the Weyl group of E8. We make also several curious observations about such models. In particular, we argue that a USp(2m) model with the addition of singlet fields and even rank m flows in the IR to a CFT with E7 × U(1) symmetry. We also discuss an infinite number of duals for the USp(2) theory with eight fundamentals and no superpotential. Duality in Gauge Field Theories; Global Symmetries; Supersymmetric Gauge - E orbits of IR 1 Introduction 2 3 4 the moment.1 An interesting question one can ask to aid such an understanding is whether there is any structure relating different theories in a certain universality class. In this short note we will discuss such a structure in a very particular setup. We consider USp(2m) gauge theories with eight fundamental chiral fields Qi, a field in the antisymmetric representation X, and possibly a superpotential with gauge singlet fields. It so happens that these models are interrelated by a large network of dualities and this network has intriguing group theoretic structure. In particular there are dualities relating models with fixed rank but different superpotentials and a non trivial map of operators/symmetries between various sides of the duality. This duality web forms [9, 10] orbits of the Weyl group of E7. Here we discuss yet another duality transformation which relates theories with different rank. In particular a USp(2m) model with superpotential Q2Q1Xn is in the same universality class as a USp(2n) model with superpotential Q2Q1Xm when certain singlet fields and superpotentials involving them are added. This duality transformation was considered implicitly in [9] as a property of integrals which imply equality of the supersymmetric index of the two dual models. We will argue that turning on most general superpotentials of the form above, breaking all the flavor symmetry of the gauge models but preserving R symmetry, this duality transformation together with permutations 1There is though growing evidence that such dualities can be understood by constructing dual four dimensional theories as geometrically equivalent but different looking compactifications of a six dimensional model (see for examples [2–8]). – 1 – of the eight quarks generates orbits of the Weyl group of SO(16). Then together with the dualities generating E7 orbits the full duality web is that of orbits of the Weyl group of E8. The note is organized as follows. We start the discussion with a review of dualities transforming USp(2m) models on an E7 orbit. We also make a curious observation about a special property of the USp(2m) with even m. We argue that this model with a particular superpotential flows to a SCFT with E7 × U( 1 ) symmetry. In the particular case of USp(4) this model sits on the same conformal manifold as the E7 surprise of Dimofte and Gaiotto [11]. In section three we discuss a duality which relates USp(2m) and USp(2n) models with superpotentials on both sides. In section four we finish by combining the two types of dualities and explain how they build orbits of the Weyl group of E8. are summarized in table 2.1. ing manifestly the most symmetry is given by the same gauge theory but with a collection of singlet fields coupled to gauge invariant operators through the superpotential [14], (2.1) (2.2) (2.3) (2.4) The charges of the fields are in table 2.3. The map between the operators is, W = This is a generalization of Intriligator-Pouliot duality [15] for n equals one case. We can build many other duality frames which will have less symmetry manifest in a similar manner – 2 – with superpotential given here, with the matter content given in table 2.9. to the n = 1 case. The number of duals is 72 = W (E7)/W (A7). To construct these dualities we split the eight fundamental fields into two groups of four. The matter content is then written in table 2.5. USp(2n) SU(4)1 SU(4)2 U( 1 )b U( 1 ) U( 1 )r We have thirty five choices to perform such splitting. One set of thirty five duals is then given by the following matter content, USp(2n) SU(4)1 SU(4)2 U( 1 )b U( 1 ) U( 1 )r (Mˆ i,j/lqiqj Xe l−1 + Mˆ i′,j/lqj+4qi+4Xe l−1) , USp(2n) SU(4)1 SU(4)2 U( 1 )b U( 1 ) U( 1 )r 1 2 1 2 0 1 2 1 2 0 1 1 2 1 2 0 1 1 The“baryons” map to the singlets and mesons to mesons. This is an analogue of the duality [16] discussed by Csaki, Schmaltz, Skiba, and Terning. The last two sets of dual descriptions were considered in [10]. It is convenient to encode the dualities as transformations on an ordered set of fugacities for the different symmetries. We will parametrize the symmetries as follows using (2.6) (2.5) (2.7) (2.8) (2.9) Q1,...,4 2n Q5,...,8 2n X – 3 – 1 −1 0 tributes to the index as Γe((qp) r2 h) [17, 18] with h being the fugacity of the U( 1 ) symmetry under which the field transforms. We will then denote by ui = (qp) 4 hit− n−41 1 the weight for the ith quark. Here the fugacity t is for the U( 1 ) symmetry, and fugacities hj for SU(8) (then Q8 with one constraint coming from anomaly cancelation, t2n−2 Qj uj = (qp)2. We define an ordered set of fugacities parametrizing the gauge sector of the USp(2n) theory to be (u1, u2, u3, u4, u5, u6, u7, u8, n, t). Denote u4+ = Q4 j=1 uj and u4− = Qj8=5 uj. Then the three dualities we discussed imply the following transformations of this set, k=1 hk = 1). We can take the parameters t and uk to be general HJEP1(207)5 → → (u1, u2, u3, u4, u5, u6, u7, u8, n, t) → (u1, u2, u3, u4, u5, u6, u7, u8, n, t) → (u1, u2, u3, u4, u5, u6, u7, u8, n, t) → u+u− , u+u− , u+u− , u+u− , u+u− , u+u− , u+u− , u+u− , n, t u1 u2 u3 u4 u5 u6 u7 u8 u+ , u2+ , u2+ , u+ , u2− , u− , u2− , u− , n, t 2 2 2 2 u1, uu−+ u2, uu−+ u3, uu+− u4, uu+− u5, uu+− u6, uu+− u7, uu+− u8, n, t (2.10) These transformations together with permutations of the first eight terms generate the Weyl group of E7. It has been observed by Dimofte and Gaiotto [11] that combining two copies of USp( 2 ) theories by coupling the gauge invariant operators as QjQiqjqi, the theory has a point on the conformal manifold in the IR with, at least, E7 symmetry. This fact was related in [8] to a statement that the two copies of USp( 2 ) with that superpotential can be obtained by compactification of the E string theory on a torus. We will now discuss here a generalizations of the former fact to higher rank. 2.1 E7 × U( 1 ) surprise Consider T(0m) and assume m even. Turn on superpotential, m/2 j=1 X QlQiXj−1Mil/j + X Xixi . m i=2 We claim this model is self dual under the dualities we have considered. Note that under all the dualities it is either that QiQjXl−1 maps to itself or to Mij/n−l. The powers of X map to the same powers on the dual side. This implies that the superpotential maps to itself under the three dualities. The only effect of the duality is the non trivial identification of symmetries. These imply that for example the protected spectrum is invariant under th Weyl group of E7 and thus forms representations of E7. It is then plausible that on some point on the conformal manifold of this model the group enhances to U( 1 ) × E7. Let us analyze one example in detail. – 4 – Consider the USp(4) theory with the mesons and X2 flipped.2 That is the superpotential is QlQmMml + X2x with M and x gauge singlet fields. The superconformal R symmetry derived by a maximization [21] assigns R charge zero to X and R charge half to the quarks. The superconformal (a, c) anomalies coincide with two copies of SU( 2 ) theories glued together with a superpotential, the E7 surprise model. This suggests that the USp(4) model sits on the same conformal manifold. In particular as the superconformal R charge of X is zero,3 giving it a vacuum expectation value takes us on the conformal manifold of that model.4 Giving such an expectation value Higgses the USp(4) gauge group to SU( 2 )2. This generates the E7 surprise model if the mesons are flipped. We thus expect the USp(4) model to have E7 symmetry somewhere on its conformal manifold. The model Here λ is the gaugino. The fields ψi are fermionic partners of Qi and the fields ψL are fermionic partners of fields L with L being one of (M, X, x). The operator ψ¯iQj forms the 63 + 1 of SU(8) × U( 1 ) and to construct the adjoint of E7 we also need the 70, fourth completely antisymmetric power of the 8, in addition to the 63. The operators QiQlMkl and ψ¯kMmMij cancel each other in the index computation as a consequence of the chiral ring relation. The operator QiQj QkQnX is in the representation 378 + 336 of SU(8). In particular this lacks the rank 4 antisymmetric representation of SU(8). This arises as to get it one has to contract the Q’s antisymmetrically in the flavor index, but, since these are bosonic fields, they must then be contracted antisymmetrically also in the gauge indices. However the fourth totally antisymmetric product of the 4 of USp(4) is a singlet, and so the product with X then cannot be made gauge invariant. The operator ψ¯lMk QiQj X is in the 28 × 28 = 336 + 378 + 70. The operator which gives us the required −70 is then obtained 2Flipping, that is introducing chiral fields φO and coupling them to a theory as φOO with O being an operator to be removed in the IR, is a standard technique in CFT. For recent discussion of some aspects of this procedure see [19, 20]. 3 This will cease to be the case for higher rank, and in particular there will be no conformal manifold. 4More specifically, an expectation value for X is forbidden by both the F-term and D-term conditions. from QlQmXψ¯iMk . The relevant operators in the 56 are constructed from Mij which form the 28 of SU(8) and from XQiQj which forms the 28. Both operators have charge 21 under the U( 1 ). We note that the fact that at order qp we see −133 − 1, assuming that the theory flows to interacting SCFT in the IR, is a proof, following from the superconformal representation theory [22], that the symmetry of the fixed point enhances to at least U( 1 ) × E7. 3 Rank changing duality We consider a deformation of T(0n) by a superpotential term, Wmn = Q1Q2Xm . (3.1) to T(m) with the additional superpotential and singlet fields, n The theory then will be denoted by T(mn).5 We claim that if m is bigger than n T(mn) is dual ΔWnm = xkXe k + m X k=n+1 m−n X X the same across the duality. The charges on the USp(2n) side are detailed in table 3.3. The map of the operators is as follows, Xj → Xe j , Q1,2Ql>2Xj−1 → q1,2ql>2Xe j−1 , j ≤ m − n Q1Q2Xj−1 → xm−j+1 , j ≤ n − 1 (Q4)ckXj−1 → Mck/n−j . 5Note that because of the anomaly condition, the superpotential Q3Q4Q5Q6Q7Q8X2n−2−m has R charge 4 minus the R charge of Wmn and opposite charges under other symmetries. This implies that at least one of these operators is relevant if m < 2n − 2 and that the two terms are marginal in IR. − 1 n 2 − 16 (2m − n − 2) 1 −y − 31 (n − 2m + 3y − 1) i, l 6= 1, 2 QiQlXj → qiqlXe m−n+j , j > m − n Q1Q2Xj−1 → q1q2Xe j−1−m+n , 1 6 15 1 1 6 2 1 1 1 1 (q1, q2) 2m q3,...,8 X e M/y xy 2m 1 1 USp(2m) SU(6) SU( 2 ) U( 1 ) m(2m − 1) − 1 1 On the USp(2m) side we obtain the charges in table 3.4. – 6 – The supersymmetric index of the two sides of the duality agrees as was shown by Rains [9]. The fact that the index agrees guarantees in particular that the anomalies agree and that the protected operators map to each other. Nevertheles, let us detail the anomalies here. We can encode anomalies involving abelian symmetries in the trial c and a anomalies. Defining R = R′ + sq with R′ and q the R symmetry and the U( 1 ) charge in the tables above, with s a parameter, the conformal anomalies are, a(s) = c(s) = 1 32 1 32 s3 −3 m2 +4m−2 n2 +6(m+2)n3 −(m−1)2(4m+5)n−4n4 −9 (3.6) −3 s2 2m2 +8m−1 n+(2−8m)n2 +8n3 −9 +12s 2mn+n2 +n−2 +4n+6 HJEP1(207)5 s3 −3 m2 +4m−2 n2 +6(m+2)n3 −(m−1)2(4m+5)n−4n4 −9 −3 s2 2m2 +8m−1 n+(2−8m)n2 +8n3 −9 +2s 3(4m+1)n+8n2 −11 +16n+4 . A more symmetric way to think about the duality is to define the model T(0n) with the superpotential given by the following. n k=2 n We denote this theory as I¯(0n) and the model with Q1Q2Xm as I¯(mn). We then claim that Let us define the index of a theory I¯(n) to be Inm(u1, u2, · · · , u8, t) with the condition m coming from anomalies t2n−2 Q8 i=1 ui = p2q2. Here again ui are the weights of the quarks as defined in the previous section and t is the fugacity for the U( 1 ) under which the antisymmetric chiral field is charged. Turning on the superpotential Q1Q2Xm we identify 2 = pq ut2−un1 , the duality implies that the index satisfies the Inm(u1, u2, u3, u4, u5, u6, u7, u8, t) = Imn u1u, u2u, u3 , u4 , u5 , u6 , u7 , u8 , t . (3.8) u u u u u u This can be derived from the map of symmetries between the two dualities. 3.1 Duals of SU(2) SQCD with four flavors Consider an example of the rank changing duality with n = 1 and general m. The theory on one side is always USp( 2 ) = SU( 2 ) SQCD with eight fundamental chiral fields and no superpotential. On the dual side we have a USp(2m) model with superpotential W = q2q1Xe and other terms involving singlet fields we have discussed. We then have an infinite number of duals for the SU( 2 ) theory with eight fundamental chiral fields. The fact that this model has an infinite number of duals is not surprising [4], but the surprising point is that the duals are rather simple. Let us work out a simple example. We consider m to be two. At the fixed point of the gauge theory with no superpotential the operator q2q1Xe has R charge 1.15749 and thus is relevant. The operator Xe2 violates the unitarity bound and needs to be decoupled by introducing a flip x appearing in the superpotential as xXe2. After decoupling the Xe2 – 7 – operator we get that the R charge of Xe q1q2 is 1.15331. We turn it on and flow to a new fixed point. At that point the operators (i, j 6= 1, 2) qiqj violate the unitarity bound and need to be decoupled by introducing flippers. After this there are no operators violating unitarity bounds and we get precisely the superpotential we obtain from the duality. Thus USp(4) theory with the superpotenial flipping Xe 2 and qiqj (i, j 6= 1, 2), and superpotential term q2q1Xe flows to USp( 2 ) with no superpotential. For low values of m we can repeat such an analysis though for higher values it becomes rather intricate. 4 The duality orbit We consider the more general superpotential for a USp(2a9) theory, HJEP1(207)5 Wa9;a1,··· ,a8 = X QiQj Xai+aj . i6=j This superpotential breaks all the flavor symmetry but preserves the R symmetry.6 The parameters aj are either all integer or all half integer. The R charges are, rQi = 1 − airX , rX = 4 2 − 2a9 + P8 j=1 aj . Some operators violate the unitarity bounds for general choices of ai and need to be decoupled. We define the parameters ui and t to be as in previous sections, – 8 – 1 Note also that now t = (qp) 2 rX . The index is given by ui = (pq) 12 t−ai . Iaa91+a2 ((pq) 12 t−a1 , · · · , (pq) 12 t−a8 , t) . The duality of the previous section then implies that this index is equal to, Iaa19+a2 (pq) 12 t− 21 (a1−a2+a9), (pq) 12 t− 21 (a2−a1+a9), (pq) 12 t− 21 (2a3+a2+a1−a9), · · · , t We parametrize again the theory by the nine numbers, which here are integers (or half integers), (a9, a1, a2, a3, a4, a5, a6, a7, a8). The duality transforms (a9, a1, a2, a3, a4, a5, a6, a7, a8) → → a1 + a2, 21 (a1 − a2 + a9), 1 (a2 − a1 + a9), (2a3 + a1 + a2 − a9), · · · . 1 2 2 The transformations with permutations of the last eight numbers generate the Weyl group of SO(16). 8 a 6Here we assume that a9 6= 1 + Pj=1 2j . If this is not true then there is no R symmetry, but instead there is an anomaly free U( 1 ) global symmetry. We shall not discuss this case in any detail. (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) We can also act with the duality generating the E7 orbit. Denote a i=5 ai, a = 12 (a− + a+), a′ = 21 (a− − a+). Then the three dualities generating the E7 orbit imply the following transformations of this set, (a9, a1, a2, a3, a4, a5, a6, a7, a8) → → (a9, a+ − a1, a+ − a2, a+ − a3, a+ − a4, a− − a5, a− − a6, a− − a7, a− − a8) → → (a9, a − a1, a − a2, a − a3, a − a4, a − a5, a − a6, a − a7, a − a8) → → (a9, a′ + a1, a′ + a2, a′ + a3, a′ + a4, −a′ + a5, −a′ + a6, −a′ + a7, −a′ + a8) . (4.7) These transformations and permutations of last eight elements generate the action of the invariant under all the transformations. The transformations can generate a full orbit of the Weyl group of E8 for general values of the parameters. For special values they might not act faithfully and generate smaller orbits. For example take all ai for i = 1 . . . 8 to be 21 a9. Then the SO(16) transformations become self dualities. There is an issue of whether or not the theories defined via the superpotential (4.1) are indeed all unique and define a non-trivial SCFT. The basic problem is that the superpotentials seem irrelevant in the IR so one may fear that the RG flow they generate is trivial. A related issue is that for generic values of ai there will be operators with R charge below the unitarity bound. To deal with the latter problem we can add, to all sides of the duality, flipping fields that remove these operators. The exact number of operators required may depend on the value of 2−2a9 +P8 i=1 ai, but one can still perform this action so as to result in an orbit where all operators are above the unitarity bound. Here it is important that the R symmetry is fixed so this does not lead to any flow that may cause more operators to go below the unitarity bound. Therefore this statement can be phrased as an identity between a collection of theories where all operators are above the unitarity bound, and so there is no contradiction with them flowing to an SCFT. Of course we cannot rule out the possibility that in some cases the flow may be trivial and the superpotential are truly irrelevant rather then dangerously irrelevant. It will be interesting to understand whether the fact that there are theories residing on an orbit of the Weyl group of E8 implies that there is a model with E8 symmetry, and in which particular way this fact is related to compactifications of six dimensional ( 1, 0 ) models. Acknowledgments We would like to thank O. Aharony, D. Gaiotto, K. Intriligator, H. C. Kim, Z. Komargodski, and C. Vafa for useful comments and discussions. GZ is supported in part by World Premier International Research Center Initiative (WPI), MEXT, Japan. SSR is a Jacques Lewiner Career Advancement Chair fellow. The research of SSR was also supported by Israel Science Foundation under grant no. 1696/15 and by I-CORE Program of the Planning and Budgeting Committee. – 9 – Open Access. 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Shlomo S. Razamat, Gabi Zafrir. E 8 orbits of IR dualities, Journal of High Energy Physics, 2017, 115, DOI: 10.1007/JHEP11(2017)115