E 8 orbits of IR dualities

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Introduction
Different QFTs can flow to the same conformal field theory in the IR. This phenomenon is usually referred to as IR duality in high energy physics or universality in statistical physics. Considering supersymmetric theories in four dimensions following [1], one can discover various examples of seemingly very different models, for example possessing different gauge groups, which nevertheless reside in the same universality class. An understanding why two different looking theories in the UV flow to the same fixed point is rather lacking at 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 Q i , 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 E 7 . Here we discuss yet another duality transformation which relates theories with different rank. In particular a USp(2m) model with superpotential Q 2 Q 1 X n is in the same universality class as a USp(2n) model with superpotential Q 2 Q 1 X m 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 JHEP11(2017)115 of the eight quarks generates orbits of the Weyl group of SO (16). Then together with the dualities generating E 7 orbits the full duality web is that of orbits of the Weyl group of E 8 .
The note is organized as follows. We start the discussion with a review of dualities transforming USp(2m) models on an E 7 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 E 7 × U(1) symmetry. In the particular case of USp(4) this model sits on the same conformal manifold as the E 7 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 E 8 .

The E 7 orbits of dualities
The basic theory we consider is a USp(2n) gauge theory with four flavours, Q a with a ∈ {1, · · · , 8} in the fundamental representation, and one field in the antisymmetric representation, X. This model was studied by various authors, in particular see [12,13] and references below. The symmetry is SU(8) × U(1). We call this model T When n = 1 the field X does not exist and the model is SU(2) gauge theory with four fundamental flavors. The model above has multiple known dual descriptions. The dual description preserving 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], The charges of the fields are in table 2.3.
The map between the operators is, 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

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to the n = 1 case. The number of duals is 72 = W (E 7 )/W (A 7 ). 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.
We have thirty five choices to perform such splitting. One set of thirty five duals is then given by the following matter content, with superpotential given here, The mesons map to singlets and "baryons" to "baryons" as antisymmetric square of4 (4) is the real representation 6. This is the analogue of [1] Seiberg duality. Another thirty five duals have the superpotential given by the following, with the matter content given in table 2.9.
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 JHEP11(2017)115 supersymmetric index nomenclature. Remember that a chiral field of R charge r contributes to the index as Γ e ((qp) r 2 h) [17,18] with h being the fugacity of the U(1) symmetry under which the field transforms. We will then denote by u i = (qp) the weight for the ith quark. Here the fugacity t is for the U(1) symmetry, and fugacities h j for SU(8) (then 8 k=1 h k = 1). We can take the parameters t and u k to be general with one constraint coming from anomaly cancelation, t 2n−2 j u j = (qp) 2 . We define an ordered set of fugacities parametrizing the gauge sector of the USp(2n) theory to be (u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 , u 8 , n, t). Denote u 4 + = 4 j=1 u j and u 4 − = 8 j=5 u j . Then the three dualities we discussed imply the following transformations of this set, These transformations together with permutations of the first eight terms generate the Weyl group of E 7 . It has been observed by Dimofte and Gaiotto [11] that combining two copies of USp(2) theories by coupling the gauge invariant operators as Q j Q i q j q i , the theory has a point on the conformal manifold in the IR with, at least, E 7 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.
and assume m even. Turn on superpotential, We claim this model is self dual under the dualities we have considered. Note that under all the dualities it is either that Q i Q j X l−1 maps to itself or to M ij/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 E 7 and thus forms representations of E 7 . It is then plausible that on some point on the conformal manifold of this model the group enhances to U(1) × E 7 . Let us analyze one example in detail.

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Consider the USp(4) theory with the mesons and X 2 flipped. 2 That is the superpotential is Q l Q m M ml + X 2 x 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 E 7 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 E 7 surprise model if the mesons are flipped. We thus expect the USp(4) model to have E 7 symmetry somewhere on its conformal manifold. The model has U(1) × SU(8) symmetry visible in the Lagrangian and if we study the index we see that the symmetry is enhanced to E 7 × U(1). The index is given by, In particular we see explicitly the conserved current of E 7 appearing at order qp in the expansion of the index (see [22] Here λ is the gaugino. The fields ψ i are fermionic partners of Q i and the fields ψ L are fermionic partners of fields L with L being one of (M, X, x). The operatorψ i Q j forms the 63 + 1 of SU(8) × U(1) and to construct the adjoint of E 7 we also need the 70, fourth completely antisymmetric power of the 8, in addition to the 63. The operators Q i Q l M kl andψ M km M ij cancel each other in the index computation as a consequence of the chiral ring relation. The operator Q i Q j Q k Q n X 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 operatorψ M lk Q i Q j X is in the 28 × 28 = 336 + 378 + 70. The operator which gives us the required −70 is then obtained JHEP11(2017)115 from Q l Q m Xψ M ik . The relevant operators in the 56 are constructed from M ij which form the 28 of SU(8) and from XQ i Q j which forms the 28. Both operators have charge 1 2 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) × E 7 .

Rank changing duality
We consider a deformation of T (n) 0 by a superpotential term, The theory then will be denoted by T (n) m . 5 We claim that if m is bigger than n T (n) m is dual to T (m) n with the additional superpotential and singlet fields, The symmetries of the theories are SU(6) × SU(2) × U(1) and the R-symmetry. When we consider n to be one, the symmetry should enhance to SU (8) On the USp(2m) side we obtain the charges in table 3.4.
The map of the operators is as follows, Note that because of the anomaly condition, the superpotential Q3Q4Q5Q6Q7Q8X 2n−2−m has R charge 4 minus the R charge of W n m 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.
A more symmetric way to think about the duality is to define the model T with the superpotential given by the following. (3.7) We denote this theory asĪ m to be I m n (u 1 , u 2 , · · · , u 8 , t) with the condition coming from anomalies t 2n−2 8 i=1 u i = p 2 q 2 . Here again u i 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 Q 1 Q 2 X m we identify u 1 u 2 t m = pq. If we define u 2 = pq t −n u 2 u 1 , the duality implies that the index satisfies the following identity, I m n (u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 , u 8 , t) = I n m u 1 u, u 2 u, This can be derived from the map of symmetries between the two dualities.

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 = q 2 q 1 X 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 q 2 q 1 X has R charge 1.15749 and thus is relevant. The operator X 2 violates the unitarity bound and needs to be decoupled by introducing a flip x appearing in the superpotential as x X 2 . After decoupling the X 2

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operator we get that the R charge of Xq 1 q 2 is 1.15331. We turn it on and flow to a new fixed point. At that point the operators (i, j = 1, 2) q i q j 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 X 2 and q i q j (i, j = 1, 2), and superpotential term q 2 q 1 X 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.

The duality orbit
We consider the more general superpotential for a USp(2a 9 ) theory, This superpotential breaks all the flavor symmetry but preserves the R symmetry. 6 The parameters a j are either all integer or all half integer. The R charges are, Some operators violate the unitarity bounds for general choices of a i and need to be decoupled. We define the parameters u i and t to be as in previous sections, Note also that now t = (qp) 1 2 r X . The index is given by The duality of the previous section then implies that this index is equal to, I a 9 a 1 +a 2 (pq) 1 2 t − 1 2 (a 1 −a 2 +a 9 ) , (pq) 1 2 t − 1 2 (a 2 −a 1 +a 9 ) , (pq) 1 2 t − 1 2 (2a 3 +a 2 +a 1 −a 9 ) , · · · , t (4.5) We parametrize again the theory by the nine numbers, which here are integers (or half integers), (a 9 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ). The duality transforms (a 9 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ) → (4.6) → a 1 + a 2 , 1 2 (a 1 − a 2 + a 9 ), 1 2 (a 2 − a 1 + a 9 ), 1 2 (2a 3 + a 1 + a 2 − a 9 ), · · · .
The transformations with permutations of the last eight numbers generate the Weyl group of SO (16). 6 Here we assume that a9 = 1 + 8 j=1 a j 2 . 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.

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We can also act with the duality generating the E 7 orbit. Denote a + = 1 . Then the three dualities generating the E 7 orbit imply the following transformations of this set, a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 a − a 1 , a − a 2 , a − a 3 , a − a 4 , a − a 5 , a − a 6 , a − a 7 , a − a 8 ) → → (a 9 , a ′ + a 1 , a ′ + a 2 , a ′ + a 3 , a ′ + a 4 , −a ′ + a 5 , −a ′ + a 6 , −a ′ + a 7 , −a ′ + a 8 ) . (4.7) These transformations and permutations of last eight elements generate the action of the Weyl group of E 7 as was observed previously. The SO(16) Weyl group with the E 7 transformations gives the Weyl group of E 8 . Note that the combination 2 − 2a 9 + 8 i=1 a i is invariant under all the transformations.
The transformations can generate a full orbit of the Weyl group of E 8 for general values of the parameters. For special values they might not act faithfully and generate smaller orbits. For example take all a i for i = 1 . . . 8 to be 1 2 a 9 . 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 a i 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 2−2a 9 + 8 i=1 a i , 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 E 8 implies that there is a model with E 8 symmetry, and in which particular way this fact is related to compactifications of six dimensional (1, 0) models.