Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories

Journal of High Energy Physics, Jul 2015

We study the fermionic zero modes around 1 instanton operators for 5d supersymmetric gauge theories of type USp, SO and the exceptional groups. The major motivation is to try to understand the global symmetry enhancement pattern in these theories.

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Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories

Received: May Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge N 0 1 0 Haifa , 32000 , Israel 1 Department of Physics, Technion, Israel Institute of Technology We study the fermionic zero modes around 1 instanton operators for 5d supersymmetric gauge theories of type USp, SO and the exceptional groups. The major motivation is to try to understand the global symmetry enhancement pattern in these theories. Supersymmetric gauge theory; Solitons Monopoles and Instantons; Field The- 1 Introduction Review of the method The case of USp groups The case of SO groups With antisymmetric matter With symmetric matter Rank 3 antisymmetric tensor With antisymmetric matter The case of exceptional groups Gauge theories in 5d are non-renormalizable, and so naively are not good microscopic theories. However, at least for minimally supersymmetric 5d gauge theories, it is known that a fixed point can exist so these do in fact define a microscopic theory [1–3]. These theories in turn sometimes exhibit a peculiar phenomenon of symmetry enhancement in which the fixed point exhibit a larger global symmetry than the gauge theory. An important associated with every non-abelian gauge group. The particles, charged under this current, are instantons which are particles in 5d. In many cases these instantonic particles provide additional conserved currents so that the global symmetry of the fixed point is larger than that of the gauge theory. A classic example is an SU(2) gauge theory with Nf hypermultiplets in the doublet of SU(2). This theory is known to flow to a 5d fixed point provided Nf ≤ 7 [1]. Furthermore, the global symmetry at the fixed point is enhanced from U(1) × SO(2N ) to ENf +1 [1]. From the gauge theory viewpoint the enhancement is brought about by instantonic particles. 6d one [4]. This theory also has an enhanced symmetry but this time the Lorentz symmetry is enhanced. Again, the additional conserved currents are brought about by instanton A similar story is also believed to occur for the maximally supersymmetric theory, where the 6d theory is now the (2, 0) theory [5, 6]. A natural question then is how can we determine if there is an enhancement and if so what is the enhanced symmetry. One way is to infer this from the brane web presentation of the theory [7–9]. A more direct way is to find this from the superconformal index [10]. The superconformal index is a counting of the BPS operators of the theory where the counting is such that if two operators can merge to form a non-BPS multiplet they will sum to zero. Particularly for 5d SCFT’s, the theory is considered on S4 × S1. Then the representations of the superconformal group are labeled by the highest weight of its SOL(5) × SUR(2) subgroup. We will call the two weights of SOL(5) as j1, j2 and those of SUR(2) as R. Then following [11] the index is: I = Tr (−1)F x2 (j1+R) y2 j2 qQ . where x, y are the fugacities associated with the superconformal group, while the fugacities collectively denoted by q correspond to other commuting charges Q, generally flavor and More to the point, conserved currents are part of a BPS supermultiplet which contains calculate the index, and from the x2 terms infer the expected enhancement. There are two broad methods to calculate the index. One, directly from the gauge theory, relying on the localization calculation in [11]. For examples of calculations using these methods, see [11–18]. Alternatively, given a brane web presentation, it can be calculated using methods of topological strings, as first done in [19]. For examples of calculations using these methods, see [19–28]. While all of these methods are useful they have their shortcomings. First, they are very technical and in many cases of interest involve removing spurious contributions that need to be determined from additional input, for example using brane webs. Also, there are cases where these methods cannot be applied, for example, exceptional groups where both a brane web description and instanton counting are unknown. Recently, a simpler method for determining conserved currents coming from the 1 instanton was proposed in [29]. Besides being simpler, it also appear to be applicable to any gauge group and any matter content. The original article dealt with gauge group SU(N ). The purpose of this article is to explore the results of this method to other gauge groups. We consider the simplest possible case, being a simple gauge group with matter in some representation of that group. A first question is what sort of matter representations should we allow. We want to look only at cases that flow to 5d or 6d fixed point, but it is not clear under what conditions a 5d gauge theory flows to such a fixed point. We adopt the following rather broad condition. It is widely believed that the maximally supersymmetric gauge theory, as well as SU(2) + 8F , flow to a 6d fixed point. Also it is believed that adding additional flavors, of any non-trivial representation, leads to no fixed point. Thus, if a theory has a Higgs branch leading to such theories, it is reasonable that it has neither a 5d nor a 6d fixed point. We explore all cases where there is a Higgs branch leading to a pure gauge theory, maximally supersymmetric gauge theory or SU(2) + Nf F with Nf ≤ 8. This of course does not prove that these theories have a 6d or 5d fixed point. is the same as in 5d, one can largely borrow the classification of [30], adjusting the cases flowing to SU(2) + 4F to allow additional flavors. The structure of this paper is as follows. We start in section 2 with a review of the method. After that we move on to discus gauge groups USp(2N ) and SO(N ) in sections 3 and 4 respectively. Section 5 covers the exeptional groups. We end in section 6 with A word on notation: we will denote global symmetries associated with matter in the fundamental representation (“flavor”) by an F subscript, those associated with matter in the 2-index antisymmetric representation by an AS subscript, and those associated with matter in the 3-index antisymmetric representation by a T AS subscript. When discussing SO(N ) groups with spinor matter we will denote the spinor flavor symmetry by an S subscript. When N is dividable by 4, there are two different, self-conjugate, spinor representations and we use also a C subscript for flavor symmetry associated with the other type of spinors. In addition, we will use a T subscript for the topological (instanton) U(1) symmetries. If no subscript is written then this U(1) is a global one that is a combination of the various U(1)’s in the theory. Subscripts on gauge symmetries will sometimes be used to denote the When denoting the charges of states under the SUR(2) R-symmetry, we use R for the state in that multiplet. Review of the method We consider a 1 instanton of an SU(2) gauge theory. As the 1 instanton breaks part of the spacetime and gauge symmetry, there are zero modes associated with these broken symmetries. This builds an 8 dimensional moduli space. Besides these, when fermionic matter is present, there are also fermionic zero modes whose number depends on the matter representation. Specifically, a Weyl fermion in the doublet of SU(2) gives a single fermionic zero mode while a Weyl fermion in the Adjoint of SU(2) gives 4 zero modes. As we consider supersymmetric theories, we always have gauginos that are in the Adjoint of SU(2). These lead to 8 zero modes for the N = 1 case and 16 for the N = 2 case.1 = 1 case we can combine the 8 zero modes to form 4 raising operators whose application on the ground state leads to 16 distinct states. As analyzed in [29] these form a single supermultiplet dubbed broken current supermultiplet. This supermultiplet contains a conserved current, and it is these that lead to an enhancement of symmetry. If additional matter is present then there are additional zero modes coming from this matter. The application of these additional modes generically charges that current also under the = 2 case we can combine the 16 zero modes to form 8 raising operators whose application on the ground state leads to 256 distinct states. Again, as analyzed in [29] these form a single supermultiplet whose structure is identical to that expected is in accordance with the expectation that this theory lifts to 6d. For a general group G, the 1 instanton is in an SU(2) subgroup of G. Thus, to understand the behavior for general G it is sufficient to decompose all representations to those of SU(2) and use the preceding discussion to determine the possible states. Before moving to the actual calculation, a few words on the limitation of the method. First, this only tell us about conserved currents coming from the 1 instanton. There can also be conserved currents coming from higher instantons which may lead to additional enhancement of symmetry. In some cases the need to complete a simple group necessitates the existence of additional currents with higher instanton number. Another limitation is that we only look at fermionic zero modes while the 1 instanton spectrum also contains bosonic zero modes, and fermionic non-zero modes. With this in mind, we move to the The case of USp groups In this section we discuss the case of a USp(2N ) gauge theory. The 1-instanton is in an this breaking the Adjoint of USp(2N ) decomposes into the Adjoint of SU(2), the Adjoint of USp(2N − 2) and a bifundamental in the (2, 2N − 2). As mentioned in the previous section, the SU(2) Adjoint provides the fermionic zero modes to span the broken current 2N − 2 of the unbroken USp(2N − 2) gauge group, and in the 2 of SUR(2). The 4(N − 1) zero modes can be split to 2N − 2 raising operators, Bj , (with charge 1The instanton configuration is BPS, and breaks half of the superconformal symmetry. The resulting fermionic zero modes come from the broken supertranslations and special sperconformal transformations. with J ij BiBj operators leading to the N states: basic gauge invariant operator is given by J ij BiBj (where J ij is the invariant antisymmetric |0i , J ij BiBj |0i , (J ij BiBj )2 |0i , . . . , (J ij BiBj )N−1 |0i with r-charges − N2−1 , − Taking the product with the broken current supermultiplet gives a multiplet whose served current. From its charges, we expect this operator to contribute xN+1 to the index. This agrees with results found from index calculations, where the 1 instanton contribution for a pure USp(2N ) gauge theory indeed behaves as: ∼ xN+1 + O(xN+2) [11, 15]. Finally we need also consider the residual discrete gauge symmetry. This arises as an instanton of an SU(2) gauge theory does not completely break the SU(2) gauge symmetry, leaving the Z2 center unbroken. We must also demand gauge invariance under this symfrom the exact instanton counting [14]. From the results for SU(2) it is now straightforward to generalize to the case of USp(2N ). As the instanton is in an SU(2) subgroup, we again conclude that the ground operators Bi under this symmetry. As these arise from SU(2) doublets, each one transforms under this Z2 gauge symmetry. Thus, a state is even under Z2 if it is made from an It is straightforward to generalize this result by adding Nf hypers in the fundamental of USp(2N ). These provide 2Nf zero modes, which are gauge singlets, whose application on the previous states furnish it with the Dirac spinor representation of SO(2Nf ). Enforcing N−1 and in a Weyl spinor representation of SO(2Nf ), where the chirality of the spinor is 2 Nf F . This is consistent with the expectations from instanton analysis and brane webs in the presence of orientifolds [31]. With antisymmetric matter We can add matter in the antisymmetric representation of USp(2N ). Under the breaking USp(2N − 2), that raise by 12 the charge under UAS (1) ⊂ SUAS (2). of USp(2N ) → SU(2) × USp(2N − 2) the antisymmetric of USp(2N ) decomposes to a singlet, the antisymmetric of USp(2N − 2), and a state in the (2, 2N − 2). Therefore, We can form a USp(2N − 2) invariant by J ij AiAj whose repeated application on (3.1) charges it in the N dimensional representation of SUAS (2). Alternatively, we can form a representation of (SUR(2), SUAS (2)). Additional applications of J ij AiBj generate new states until we apply N − 1 such operators.2 Tensoring this with the broken current under (SUR(2), SUAS (2)). In particular this includes a state in the (3, 1), given by: (J ij AiBj )N−1 |0i which corresponds to a conserved current. from an application of an even number of fermionic raising operators, they are kept only We thus find that USp0(2N ) + AS has a conserved current which is an SUAS (2) singlet at least up to the limitations of this method. This is in accordance with the identification It is now straightforward to generalize by adding Nf fundamental flavors. As previously stated this will just charge the instanton under the spinor representation of SO(2Nf ). This suggests an enhancement of UT (1) × SO(2Nf ) to ENf +1. This again is consistent with the identification of USp(2N ) + AS + Nf F with the rank N ENf +1 theory. of SO(16). However, there is no finite Lie group which contains U(1) × SO(16), and whose Lie group, E8(1), that has this spectrum.3 This suggests that this theory lifts to a 6d theory with E8 global symmetry in agreement with the results of [4, 33]. We do not expect either a 5d or a 6d fixed point to exist, for general N , when there are two or more antisymmetrics. This is because, when N > 6, there is a Higgs branch leading to an SU(2) gauge theory with more than 1 hyper in the Adjoint. However, for N ≤ 6 a fixed point might exist, and in the remainder of this subsection we will deal with these cases. We start with the case of USp(4) with 2 antisymmetrics and Nf flavors. This theory has a Higgs branch leading to SU(2) + 2Nf F , so as long as Nf < 4 a 5d fixed point First consider the case of Nf = 0. We still have the previous conserved current, appear, corresponding to applying JijAiBj N − 1 times. 2This hangs on the observation that the J matrix always connects a specific pair of indices. For example, the combination A1B2A3B4 is contained in JijAiBjJklAkBl, but not in JijAiAjJklBkBl. So the states 3We adopt the notation for affine Lie groups used in [32]. The physical interpretation of this is that the group written is the 6d global symmetry while the superscript denote whether the reduction is done with a twist in the outer automorphism of the group ((2) or (3)) or not (1). conserved current, now in the 2 1 representation. These two form the 4 of USpAS(4). of UT (1) × USpAS(4) → USp(6). Note that this enhancement also requires a USpAS(4) singlet conserved current with instanton charge ±2. global symmetry associated with the flavor is UF (1) and the conserved current acquires the U(1) × USpAS(4) → USp(6), just with the U(1) being a combination of UT (1) and UF (1). When Nf = 2 the global symmetry associated with the flavors is SU(2) × SU(2), and global symmetry consistent with this is USp(8). This also requires conserved currents with instanton number ±2 and in the (1, 3) of USpAS(4) × SU(2). Assuming these states are indeed present, the global symmetry of this theory is SU(2) × USp(8). When Nf = 3 the global symmetry associated with the flavors is SU(4), and the The smallest global symmetry consistent with this is USp(12). This also requires a conserved current, and it’s conjugate, with instanton number 2 and in the (1, 10) of USpAS(4) × SU(4). Assuming these states are indeed present, the global symmetry of this theory is USp(12). flavors is now SO(8), and the conserved currents are in the 8S or 8C , depending on the contains (4, 8)±1. However, this is contained in the affine Lie group A(121) So this theory There are two other theories in this family that may have a 6d fixed point. One is groups with vector matter. The second is USp(6) + 2AS where we find a conserved current With symmetric matter In this subsection we deal with adding a hypermultiplet in the symmetric representation. For U Sp groups, the symmetric representation is the Adjoint so this theory is the maximally supersymmetric U Sp theory and is expected to lift to 6d. Under the breaking USp(2N ) → USp(2N − 2), and a state in the (2, 2N − 2). The only difference between this case and the case with the antisymmetric is that we also have a state in the Adjoint of SU(2). As mentioned in the previous section, the SU(2) Adjoint hyper contributes additional zero modes generating a KK mode energy-momentum supermultiplet. We next need to take into account the effect of the bifundamental zero modes. As these are the same as in the previous subsection, we can just borrow the results so we get a tower of states in the (k, k) dimensional representations of (SUR(2), SUS(2)), for k = 1, 2 . . . , N . The groups SUR(2) × SUS(2) form a larger group SOR(5), which is the R-symmetry of the maximally supersymmetric theory, and these states form a single representation of the reader that 5d maximally supersymmetric USp0(2N ) theory is expected to lift to the 6d (2, 0) theory of type DN+1 where the compactification is done with a twist in the outer automorphism of SO(2N + 2) [32]. The (2, 0) theory of type DN+1 has a short multiplet for every SO(2N +2) invariant polynomial. This short multiplet contains a symmetric traceless D is the degree of the corresponding polynomial. For SO(2N + 2), there are N + 1 invariant polynomials of degrees 2, 4, . . . , 2N − 2, 2N When compactified, without a twist, on a circle we get the 5d maximally supersymmetric SO(2N + 2) theory. These short multiplets get expanded into KK modes that contribute to the 5d theory. The constant modes on the circle are masseless and gives the corresponding supermultiplets in the 5d theory. The first exited state are massive with mass R1 ∼ g52d which we expect to appear as instanton states. 1 Now we want to take into account the effect of the outer automorphism twist on these results. Out of the N + 1 polynomials, the N polynomials of degrees 2, 4, . . . , 2N − 2, 2N are even while the one of degree N + 1 is odd. Thus, with the twist, we must enforce anti-periodic boundary conditions on the supermultiplet corresponding to the degree N + 1 invariant polynomial. As a result only those of degrees 2, 4, . . . , 2N − 2, 2N give massless states. These exactly match the degrees of the invariant polynomials of USp(2N ). The first massive state should now be the first term in the KK expansion of the degree N + 1 invariant polynomial, and is expected to appear in the gauge theory as a 1 instanton. It is also clear from the 6d reduction that, in 5d, this state should be in the representation traceless representation of SOR(5). This indeed matches the results we find from the 1 instanton. We expect the first KK modes of the degree 2, 4, . . . , 2N − 2, 2N polynomials to contribute in the 2 instanton. theory of type A2N with an outer automorphism twist along the circle. In general there are 2N invariant polynomials of SU(2N + 1) having degrees 2, 3 . . . , 2N, 2N + 1. Under the action of the Z2 outer automorphism of SU(2N + 1) the even degree polynomials are even while the odd degree ones are odd. Thus, we expect the first massive states to come from we find no states coming from the 1 instanton in this case. In this subsection we deal with other matter representations. The only other representations that seem to allow 5d fixed points is a rank 3 antisymmetric tensor. Going over the possible cases we seem to find 5 possibilities, 4 with gauge group USp(6) and one with gauge group USp(8). The rank 3 antisymmetric representation of USp(2N ) is pseudoreal so one can add a half-hyper. Specifically for USp(6), the representation is 14 dimensional and because of the anomaly of [34] adding a half-hyper is inconsistent unless one also adds a half-hyper in the fundamental. The possible candidates for a 5d or 6d fixed points are 12 , 1 and 32 rank 3 antisymmetric hypers with fundamental hypers and a rank 3 antisymmetric half-hyper with a rank 2 antisymmetric and fundamental hypers. Before discussing each one in turn we comment about the effect of half-hypers. Under the breaking USp(6) → SU(2) × USp(4), the 140 decomposes into states in the (1, 4) and (2, 5) of SU(2)×USp(4). Together with the additional half-hyper in the fundamental, which gives fermionic zero modes in the (2, 1), these give 3 fermionic raising operators whose Now we can discuss the possible cases. First if we have just a half-hyper in the 140 and the 6 then we essentially just need to repeat the original analysis, now with a ground state that is in the fundamental of USp(4). It is not difficult to see that there are just two USp(4) invariant states given by: These form a doublet of SUR(2) so there are no conserved currents. The generalization by adding Nf fundamental hypers is immediate. Like in previous cases, the additional zero modes charge this state with the Dirac spinor representation of SO(2Nf ) ⊂ SO(2Nf + 1). Enforcing Z2 gauge invariance reduce it to a Weyl spinor of SO(2Nf ). However, we now recall that there is an additional ground state that is not invariant under Z2. When flavors are added, this ground state also contributes a Weyl spinor of SO(2Nf ), but with opposite chirality. These combine to form an SO(2Nf + 1) spinor. So the end result is we find an SUR(2) doublet that is in the spinor representation of SO(2Nf + 1). This does not give a conserved current. If we add a rank 2 antisymmetric then we need to repeat the analysis of section 3.1 with a ground state in the fundamental of USp(4). It is not difficult to see that there are just two conserved currents, given by: which form a doublet of SUAS(2). The minimal enhancement consistent with this is UT (1)× We can further generalize by adding flavors in the fundamental of USp(4). The Higgs branch analysis suggests that we can add up to 2 before the theory is expected to lift to 6d. The addition of the flavors will still give the same conserved current, but now it will also be in the spinor of SO(2Nf + 1). Thus, if Nf = 1 we get a conserved current in the (2, 2) of SUAS(2) × SOF (3). The cannot fit in a finite Lie group, but can fit in the affine Lie group E6(2). minimal enhancement consistent with this is UT (1) × SUAS(2) × SOF (3) → SU(4). If enhancement consistent with this is UT (1) × SUAS(2) × SOF (5) → USp(8) which also requires conserved currents in the 3 of SUAS(2) with instanton number ±2. Finally, if Nf = 3 we get a conserved current in the (2, 8) of SUAS(2) × SOF (7). Such a spectrum Next we turn to the case of one full hyper in the 140 of USp(6). We find two conserved currents with charges ± 23 under UT AS(1). projected out while the other is kept. Thus, this suggests an enhancement of U(1) → SU(2). When flavors are added we still get the two conserved currents, but now both are in the Dirac spinor representation of SO(2Nf ). Half of these are projected out leaving two SO(2Nf ) spinors of opposite chirality and with charges ± 23 under UT AS(1). A Higgs branch analysis suggests that we can have at most 4 flavors and still have a SOF (2) and UT AS(1). The minimal enhancement consistent with this is U(1)2 one SUF (2) and the other in the 2 of the other. The minimal enhancement consistent with → SU(3)2. For Nf = 3 We get two conserved currents in the 4 and 4¯ of SUF (4). The minimal enhancement consistent with this is U(1)2 × SUF (4) → SU(6) which also requires two flavor singlet conserved currents with instanton number ±2. For Nf = 4 we get two conserved currents, one in the 8S and the other in the 8C of SOF (8). This suggests an enhancement of U(1)2 × SO(8) → E6 assuming we also get two conserved currents with instanton number ±2 and in the 80V of SO(8)UTAS(1). Finally, we consider the case of Nf = 5. In this case we get two conserved currents in the 16 and 1¯6 of SOF (10). This cannot fit in a finite Lie group, but can fit in the affine Lie group E6(1). Next, we consider the case of 32 hypers in the rank 3 antisymmetric. We do not find any conserved current in this case. The last case we consider is USp(8) with a rank 3 antisymmetric. The Higgs branch analysis suggests we can only put a single half-hyper. This theory does not suffer from the anomaly of [34] (see for example [30]). The effect of the half-hyper is to furnish the ground state in the 64 of the unbroken USp(6) gauge symmetry. We find no conserved current. The case of SO groups In this section we discus the case of SO(N ) gauge theory. The 1-instanton is in an SU(2) subgroup of the SO(N ) gauge group breaking SO(N ) to SU(2) × SU(2) × SO(N − 4). Under this breaking the Adjoint of SO(N ) decomposes into the Adjoints of both SU(2)’s, the Adjoint builds a broken current supermultiplet while the remaining states provide additional fermionic zero modes. In the case at hand these are 4(N − 4) states in the (2, N − 4) of the unbroken SU(2) × SO(N − 4) gauge group, and the 2 of SUR(2). These naturally form 2(N − 4) raising operators Baj (where a = 1, 2 ∈ SU(2), i = 1, . . . , N − 4 ∈ SO(N − 4)) from which we can form a gauge invariant by: abδij BaiBbj . the N − 3 states: |0i , abδij BaiBbj |0i , ( abδij BaiBbj )2 |0i , . . . , ( abδij BaiBbj )N−4 |0i whose r-charges are: − R-charge R = N2−4 . N2−4 , − Tensoring with the broken current supermultiplet we do not get index. Indeed, this matches the contribution one finds from the explicit index calculation. Next, we generalize by adding Nf hypers in the vector representation. These add 4Nf 2Nf of the USp(2Nf ) flavor symmetry. From these we can form 2Nf raising operators Ca in From these we can form the gauge invariant abCaCb. Applying this on (4.1) yields a new state which is in the rank 2 symmetric representation of SU(Nf ) ⊂ USp(2Nf ). representation should, on the one hand, be given by a symmetric product of the rank 2 symmetric representation of SU(Nf ), yet on the other hand, since the underlying operators are fermionic, it cannot be a completely symmetric product. The easiest way to determine the representation is to use the fact that the state given by applying ( abCaCb)l should be |Bi , abCaCb |Bi , ( abCaCb)2 |Bi , . . . , ( abCaCb)Nf |Bi USp(2Nf ) are −Nf , −Nf + 2, . . . , Nf while their SU(Nf ) ⊂ USp(2Nf ) representation is , . . . , where the last Young diagram has Nf rows. We next need to combine them into USp(2Nf ) representations. We claim that these states build the rank Nf irreducible antisymmetric representation of USp(2Nf ). Recall that under the U(1) × SU(Nf ) subgroup of USp(2Nf ), the 2Nf dimensional representation products of the fundamental representation. In order to get a state with lowest U(1) charge −Nf we must multiply Nf fundamentals. Furthermore, for that state to be an SU(Nf ) singlet the product must be completely antisymmetric. Also one can count the number of states, using Mathematica for example, and show that their number exactly match the dimension of the rank Nf irreducible antisymmetric representation of USp(2Nf ). In addition we may be able to form an invariant from both Ca and Baj . Since only the but, as this is a symmetric product and the B’s are fermionic, the gauge SU(2) indices must be contracted antisymmetrically forming a gauge invariant. This leads to the previously As this is an antisymmetric product, the gauge SU(2) indices must now be contracted symmetrically forming the N − 3 dimensional representation of SU(2). In order to get a different state we need to contract with something made from the C’s. To get the we must contract N − 4 C’s symmetrically. Yet, as the C’s are fermionic they must be antisymmetrized in the flavor index. Thus, we conclude that when Nf < N − 4 there are not enough C’s to form this irreducible antisymmetric representation of USp(2Nf ). When Nf = N − 4, in addition to the previously mentioned state, there is an invariant made from N − 4 B’s and N − 4 C’s given by: where we used f = 1, . . . Nf for the SU(Nf ) ⊂ USp(2Nf ) index. This is a singlet under USp(2Nf ) with R-charge zero and so leads to a conserved current multiplet when tensored with the basic broken current multiplet. This should lead to an enhancement of When Nf = N − 3 this state acquires charges in the fundamental of USp(2Nf ). The minimal symmetry containing U(1) × USp(2Nf ) with additional states in the 2Nf ±1 is USp(2Nf + 2). This requires also two USp(2Nf ) singlets with instanton charge ±2. The enhancement spectrum seen here is consistent with the results from the explicit index analysis and brane webs [31]. Finally when Nf = N − 2 this state acquires charges in the rank 2 irreducible antisymmetric representation of USp(2Nf ). There is no finite Lie group with this content, but Higgs branch analysis suggesting that this theory doesn’t possess a 5d fixed point, though it may possess a 6d one. With antisymmetric matter In this section we consider an SO(N ) gauge theory with a single hyper in the antisymmetric representation. Since for SO groups the antisymmetric is the Adjoint representation, this is the maximally supersymmetric case. The zero modes contributed by the antisymmetric gauge group. The effect of the 8 zero modes coming from the SU(2) Adjoint hypermultiplet are just to build the previously mentioned KK modes. The other 4(N − 4) zero modes can be combined to form 2(N − 4) raising operators, Aai, that are in the (2, N − 4) of the unbroken SU(2) × SO(N − 4) gauge group. These SUAS (2) is enhanced to SOR(5) of the maximally supersymmetric theory. In addition to its gauge charges, The operator Baαi is charged in the 21 of U(1) × SU(2) ⊂ SOR(5). Note that this SU(2) is neither SUR(2) nor SUAS(2), rather a different SU(2) subgroup of U(1) × SU(2) ⊂ SOR(5). We next need to apply these zero modes on the ground states, enforcing SU(2)×SO(N − 4) invariance, and compose the results into SOR(5) representations. As previously stated applying it on the ground states we generate the states: |0i , δij abBaαiBbβj |0i , (δij abBaαiBbβj )2 |0i . . . .(δij abBaαiBbβj )2(N−4) |0i . These states are in the 1−2(N−4), 3−2(N−5), ⊗s2ym3−2(N−6) , . . . , ⊗sNy−m53−2, ⊗sNy−m430, stands for the l symmetric product of the representation with itself). This in fact forms a single state in the rank N − 4 symmetric traceless representation of SOR(5). This can multiplication. After removing the trace, one can see that we get the spectrum of (4.4). Alternatively, we can contract the SU(2) ⊂ SOR(5) and SO(N − 4) indices antisymmetrically. This does not give an SO(N − 4) invariant, but we can form one by a symmetric product of two such operators. In term of the B operators, it is given by Acting with this operator on the ground state generates a new state, and repeated application of the operator δij abBaαiBbβj on it generates a single state in the rank N − 6 symmetric traceless representation of SOR(5). It is now clear that operating on the ground state with these two types of gauge invariants generates a series of states in the rank N − 4, N − 6, N − 8 . . . symmetric traceless representations of SOR(5). For N odd, these are the only gauge invariant states, but for N even, there is one more. We can form an SO(N − 4) invariant by contracting N − 4 B operators with an epsilon tensor, and if N is even, we can contract their SU(2) indices among themselves so as to form So, to recapitulate, for N odd we find a list of states in the rank N − 4, N − 6, N − 8 . . . , 3, 1 symmetric traceless representations of SOR(5). However, for N even, we get a list Next, we compare this against the expectations from the reduction of the corresponding (2, 0) theory. As stated in the previous section, in the N even case the theory is expected to lift to the 6d (2, 0) of type D N , and the short multiplets of this theory are expected to give KK modes that are precisely in the rank N − 4, N − 6, N − 8 . . . , 2, 0 and N−4 2 symmetric traceless representations of SOR(5). For the N odd case, the theory is expected to lift to the 6d (2, 0) of type AN−1 with a Z2 twist along the circle. Thus, the operators corresponding to the odd degree polynomials obey anti-periodic boundary conditions on the circle. The even ones contribute The first massive states correspond to the lowest KK mode of operators corresponding to the odd degree polynomials. This exactly matches our results for the 1 With spinor matter Next we consider the generalization by the addition of spinor matter. This is especially interesting as conventional instanton counting is unavailable in this case. two bispinors. So we are to determine whether, by applying all possible fermionic zero modes and limiting to gauge invariant states, there exists an R-charge singlet, and with what charges under the flavor symmetry. As this can be quite involved in general, we have resorted to numerical methods in many cases. The numerical strategy we use borrows significantly from index calculations. First, we pack all fermionic zero modes in a one particle index, defined by: IOP I = − where the sum goes over all types of fermionic zero modes, those coming from the Adjoint, vector matter and spinor matter. Here, xi stands for the fugacity of the global U(1) character of that type of zero modes under the non-abelian flavor and gauge symmetries. The minus sign is inserted as these are fermionic operators. Next, the one particle index is inserted into a plethystic exponent,4 which is expanded in a power series in all xi’s. This generates all possible products of these zero modes taking into account their fermionic nature. Next, we need to act with these operators on the ground state which in practice amounts to multiplying by the charges carried by the ground state. All that remains is to enforce gauge invariance, which is done by integrating over the gauge group with the appropriate Haar measure. From the final result we can identify whether there are SUR(2) singlets, and in what representation of the global symmetry. As the properties of spinors vary with N we concentrate on specific cases. Note, that with only spinor matter was covered in [29]. While, to our knowledge, the case with both vector and spinor matter was not addressed, we won’t discuss it here. Thus, the first case we discuss is SO(7). Also, a Higgs branch analysis suggests that for N > 14 a 5d fixed point is not possible, when spinor matter is present, so we won’t consider these cases. In this case the fermionic zero modes, provided by the spinors, are in the (1, 2) of the SU(2)2 unbroken gauge group. Next we state our results for the conserved currents. When 4The plethystic exponent is defined as P E[f (·)] = exp Pn∞=1 n1 f (·n) where the dot represents all the the 8 of USpF (8). The addition of spinors also gives it charges under the global symmetry SUS(2). We find no additional conserved currents, from the 1-instanton, besides this. The minimal global symmetry consistent with this is USp(12). This also requires conserved currents with instanton number ±2 and in the (1, 3) of USpF (8) × SUS(2). Assuming these states are indeed present, the global symmetry of this theory is USp(12). When Ns = 2, we get a conserved current in the (5, 8) of USpS(4) × USpF (8). This cannot fit in a finite Lie group, but can form an affine one, A(122). For Nf = 3, the maximal number of spinors is Ns = 3 (if Ns = 4 the theory has a there is a conserved current which is a USpF (6) singlet. The addition of spinors gives it the conserved current is now in the 5 of USpS(4). The minimal global symmetry consistent The minimal global symmetry consistent with this is F4, which also requires two extra conserved currents with instanton number ±2. Finally, for Ns = 4 the conserved current is now in the 42 of USpS(8). There is no finite Lie group that can accommodate this structure, but there is an affine group, E6(2), that can. Also it turns out that there is an additional conserved current in the (1, 14) of USpS(8) × USpF (6). Again this appears to suggest an enhancement to an affine group A(52). Thus, in this case, both the global symmetries appear to be affinized which is consistent with this theory lifting to 6d. For Nf = 2 we find no conserved currents unless Ns ≥ 4. When Ns = 4, this current is in the (1, 4) of USpS(8) × USpF (4). This appears to suggest an enhancement of UT (1) × USpF (4) to USp(6). This also requires two states with instanton number ±2. This is the leading to SU(2) + 8F so one might expect this theory to lift to 6d. Indeed, in this case, we find the conserved current to be in the (10, 4) of USpS(10) × USpF (4) which is consistent For Nf = 1 we find no conserved currents unless Ns ≥ 4. When Ns = 4, this current is a singlet under the flavor symmetry. This should lead to an enhancement of UT (1) → SU(2). to USp(12) (again this also requires states with instanton number ±2). Finally, in the case This cannot be accommodated in a finite Lie group, but rather in the affine A(121). This is consistent with a Higgs branch analysis, as for this case the theory can be reduced to SU(2) + 8F . Strangely, SUF (2) does not appear to be affinized at this level. Finally we can consider the case with no vectors. The maximal number of spinors is 6, as for 7 spinors there is a Higgs branch where the theory breaks to SU(2) + 8F . To SU(2) × SUF (2) × USpS(8) SUF (2) × USp(12)∗ USpS(8) × USp(6)∗ USpF (6) × F4∗ symmetry is UT (1) × USp(2Nf ) × USp(2Ns). Written are the found conserved currents with their representation under the USp(2Nf )×USp(2Ns) global symmetry, and minimal enhanced symmetry consistent with these currents. Only cases where conserved currents were found, and where a 5d currents that are in the (1, 3) with instanton number ±2. three B operators. However, it is not possible to form an SU(2) gauge invariant from three doublets, so one cannot form a state that is both a gauge and R-charge singlet. Therefore, there are no conserved currents in this case, for any number of spinors. The important element here is the need to apply an odd number of B operators in order to get an R-charge singlet. This is true for any SO(M ) with spinor matter only, for M odd, so in all these cases there is no symmetry enhancement coming from the 1-instanton. We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 1. Next we move to the case of SO(8). There are two different, self-conjugated, spinor representations. Under the unbroken SU(2)3 gauge group, the fermionic zero modes, provided by the spinors, are in the (1, 2, 1) or (1, 1, 2) depending on the chirality.5 Next we state our results for the conserved currents. To save space, henceforward we shall state only those cases where a conserved current was found. We also present only cases unrelated by triality, in this case, or, in other cases, the duality exchanging the two spinor representations. For Nf = 5 we can have at most a single spinor of any chirality as for Ns = 2 or there is a single conserved current in the 10 of USpF (10). The addition of spinor matter SUS(2) so we find a single conserved current in the (10, 2) of USpF (10) × SUS(2). This should lead to an enhancement of UT (1) × SUS(2) × USpF (10) → USp(14). This also requires conserved currents with instanton number ±2 and charges (1, 3). 5In this notation the fermionic zero modes provided by the vector are in the (2, 1, 1), and triality of SO(8) is correctly implemented by permutating the three SU(2) groups. The cases of Ns = 2 and Ns = Nc = 1 are also interesting as they may possess a 6d not fit in a finite Lie group, but rather in the affine A(123). find a conserved current in the (10, 2, 2) of USpF (10) × SUS(2) × SUC (2). This again does ralities, is three, as the case of four spinors has a Higgs branch leading to two copies of USpF (8) singlet. The addition of spinor matter charges this current under the spinor flaconserved current in the (1, 2) of USpF (8) × SUS(2). This should lead to an enhancement of UT (1)×SUS(2) → SU(3). For Ns = 2, Nc = 0, it acquires the 5 of USpS(4) and we find a conserved current in the (1, 5) of USpF (8) × USpS(4). This should lead to an enhancement of UT (1) × USpS(4) → SO(7). Alternatively, for Ns = Nc = 1, we find a conserved current in the (1, 2, 2) of USpF (8)× SUS(2) × SUC (2). This should lead to an enhancement of UT (1) × SUS(2) × SUC (2) → USpS(6). This should lead to an enhancement of UT (1) × USpS(6) → F4, which also requires conserved currents with instanton number ±2 that are flavor symmetry singlets. charge the current in the (5, 2) of USpS(4) × SUC (2). So we find a conserved current in the (1, 5, 2) of USpF (8) × USpS(4) × SUC (2). This should lead to an enhancement of UT (1) × USpS(4) × SUC (2) → SO(9), which also requires conserved currents with instanton number ±2 that are flavor symmetry singlets. The cases with four spinors are also interesting as they may have a 6d fixed point. one in the (42, 1), and the other in the (1, 42) of USpF (8) × USpS(8). These appear the (1, 140, 2) of USpF (8) × USpS(6) × SUC (2). This should affinize USpS(6) × SUC (2) to F4(1). Incidentally, USpF (8) is not affinized, at least not at this level. Finally for (27, 1, 1) of USpF (8) × USpS(4) × USpC (4). These appear to affinize USpF (8) to A(72) and There are just two other cases, not related by triality to the previous cases, where we current which is a flavor singlet. This should lead to an enhancement of UT (1) → SU(2). USpF (6)×USpS(4)×USpC (4). This should lead to an enhancement of UT (1)×USpF (6) → USp(8), which also requires conserved currents with instanton number ±2 that are flavor The case of Nf = Ns = 3, Nc = 2 is also interesting as in that case there is a Higgs branch leading to SU(2)+8F so while we do not expect a 5d fixed point, a 6d one is possible. Indeed, we find a conserved current in the (6, 6, 1) of USpF (6) × USpS(6) × USpC (4). This SU(2) × USpF (4) × USpS(4) × USpC (4) USpS(4) × USpC (4) × USp(8)∗ USpF (8) × SO(9)∗ USpF (8) × F4∗ classical global symmetry is UT (1) × USp(2Nf ) × USp(2Ns) × USp(2Nc). Written are the found conserved currents with their representation under the USp(2Nf ) × USp(2Ns) × USp(2Nc) global symmetry, and minimal enhanced symmetry consistent with these currents. conserved currents were found, and where a 5d fixed point is not ruled out, are shown. ∗ This enhancement also requires two conserved currents that are flavor singlets with instanton number should affinize USpF (6) × USpS(6) to C6(1). Incidentally, USpC (4) is not affinized, at least We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 2. For SO(9), there is a single spinor representation. The fermionic zero modes, provided by the spinors, are in the (1, 4) of the unbroken SU(2) × SO(5) gauge group. Next, we state our results for conserved currents. The additional fermionic zero modes charge it in the 3 of SUS(2) so we find a conserved current in the (12, 3) of USpF (12) × SUS(2). This cannot be contained in a finite Lie For Nf = 5, we can have at most one spinor, as for Ns = 2 there is a Higgs branch conserved current which is a USpF (10) singlet. The additional spinor zero modes charge it under SUS(2), and we find a conserved current in the (1, 3) of USpF (10) × SUS(2). This suggests an enhancement of UT (1) × SUS(2) → USp(4). For Ns = 2, it is now charged in the (1, 14) of USpF (10) × USpS(4). In addition we also find a conserved current in the (44, 1). These cannot be contained in a finite Lie group, rather suggesting both groups lift USpF (6) × USpS(4) × SU(2) USpS(4) × USp(10)∗ USpF (10) × USp(4) global symmetry is UT (1) × USp(2Nf ) × USp(2Ns). Written are the found conserved currents with their representation under the USp(2Nf ) × USp(2Ns) global symmetry, and the minimal enhanced symmetry consistent with these currents. Only cases where conserved currents were found, and currents that are flavor singlets with instanton number ±2. For Nf = 4 we find a conserved current only if Ns ≥ 2. Specifically for Ns = 2 we find a conserved current in the (8, 1) of USpF (8) × USpS(4). This should lead to an enhancement of UT (1) × USpF (8) → USp(10) which should also require conserved currents with instanton number ±2 that are flavor symmetry singlets. This is the maximal allowed number of spinors as for Ns = 3 there is a Higgs branch leading to SU(2) + 10F . When Nf = 3 we again find a conserved current only when Ns ≥ 2. This current is a flavor singlet, and should lead to an enhancement of UT (1) → SU(2). This is the maximal Indeed, in that case, we find a conserved current in the (1, 21) of USpF (6) × USpS(6). Thus, USpS(6) appear to be affinized to C3(1). Interestingly, USpF (6) is not affinized at The only other relevant case where we find a conserved current is Nf = 1, Ns = 4. Note that this theory has a Higgs branch leading to two copies of SU(2) + 8F so we do not expect a 5d fixed point, but a 6d one is possible. Indeed, we find a conserved current in the (1, 42) of SUF (2) × USpS(8) suggesting an enhanced affine Lie group E6(2). The vector flavor symmetry is not affinized at this level. We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 3. The spinor representation of SO(10) is complex and the two chiralities are complex conjugates. The fermionic zero modes, provided by the spinors, are in the (1, 4) of the unbroken SU(2) × SO(6) gauge group. Next, we state our results for conserved currents. For Nf = 7, Ns = 1 there is a Higgs branch leading to SU(2) + 8F so this theory may in the 14 of USpF (14). This suggests an enhancement to the affine Lie group A(125). have a 6d origin. Indeed, we find two conserved currents with charges ±2 under US(1) and For Nf = 6, the maximal allowed number of spinors is Ns = 1. In that case, we find two conserved currents of charges ±2 under US(1). This suggests an enhancement USpF (12) × SUS(2) × US(1). The last two suggest an enhancement of USpF (12) to the SUS(4) × SU(4)∗ USpF (8) × SUS(2) × SU(2) × US(1) US(1) × SUS(2) × USp(12)∗ global symmetry is USp(2Nf )×SUS(Ns)×US(1)×UT (1). Written are the found conserved currents with their representation under the USp(2Nf ) × SUS(Ns) × US(1) global symmetry, and minimal enhanced symmetry consistent with these currents. Only cases where conserved currents were found, and where a 5d fixed point is not ruled out, are shown. ∗ This enhancement also requires two conserved currents that are flavor singlets with instanton number ±2. affine A(121) and of SUS(2) to A(22). In that light, we expect the first two currents to affinize US(1), though understanding exactly what it leads to requires additional input. For Nf = 5 we find a conserved current only if Ns ≥ 2. A Higgs branch analysis reveals 10 of USpF (10). This suggests an enhancement of USpF (10) × UT (1) → USp(12) which also requires two flavor singlet currents with instanton charges ±2. For Nf = 4, we again find conserved currents only if Ns ≥ 2. Again, for a 5d fixed is a flavor singlet. This suggests an enhancement of UT (1) → SU(2). The case of Ns = 3 is also interesting as it may have a 6d fixed point. In this case, we find two conserved currents For Nf = 2 we find conserved currents only if Ns ≥ 4. Since for Ns = 4 there is a Higgs branch leading to two copies of SU(2) + 8F , this theory is the only relevant case. USpF (4). This seems to combine US(1) × USpF (4) to the affine B3(1). In that case, we find two conserved currents of charges ±4 under US(1) and in the 5 of For Nf = 1, we again find conserved currents only if Ns ≥ 4. With that number of case is Ns = 4 where we find two conserved currents of charges ±4 under US(1) and in the 2 of SUF (2). The smallest global symmetry consistent with that is SU(4), which also requires two flavor singlet currents with instanton charges ±2. Finally, we consider the Nf = 0 case. Again we find conserved currents only if Ns ≥ 4. under US(1). This suggests an enhancement of UT (1) × US(1) → SU(2)2. so the only relevant case is Ns = 4 where we find two conserved currents of charges ±4 We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 4. There is a single spinor representation of SO(11). This representation is pseudoreal and we can add half-hypers. The fermionic raising operators, provided by a spinor hyper, are in the (1, 8) of the unbroken SU(2) × SO(7) gauge group. A half-hyper in the spinor representation gives 8 fermionic zero modes, in the (1, 8), that can be combined to form 4 fermionic raising operators. Their application on the ground state generate 16 states that decomposes to the 1 ⊕ 7 ⊕ 8 under the SO(7) unbroken gauge group. Next, we state our results for conserved currents. A Higgs branch analysis suggest that we cannot have more than 52 spinors. The case of 52 spinors cannot have a 5d fixed point though it may have a 6d one. Nevertheless, we find no conserved currents in this case. For Ns = 2 We find a conserved current only if Nf ≥ 1. When Nf = 1 this conserved current is a flavor singlet, and is expected to lead to an enhancement of UT (1) → SU(2). of USpF (4). This suggests an enhancement of USpF (4) × UT (1) → USp(6), which also is a Higgs branch leading to SU(2) + 8F so this theory is still interesting as it may lift to 6d. We find several conserved currents with charges (1, 1, 14), (1, 5, 1), and (5, 1, 1) under where both SUS(2)’s get lifted to A(22), and USpF (6) get lifted to A(52). SUS1 (2) × SUS2 (2) × USpF (6). This suggests an affinization of all the flavor symmetries When Ns = 32 , we find conserved currents only when Nf ≥ 5. Since for Nf = 5 there is a Higgs branch leading to SU(2) + 8F , this is the only interesting case. In that case, we find two conserved currents, one a flavor singlet and one in the 5 of SUS(2). The current in the 5 is consistent with SUS(2) getting lifted to the affine A(22), but the singlet does not appear to fit in this group. For Ns = 1, we find conserved currents only when Nf ≥ 5. When Nf = 5, we find a conserved current which is a flavor singlet. This suggests an enhancement of UT (1) → 12 dimensional representation of USpF (12). This suggests an enhancement of USpF (12) × UT (1) → USp(14), which also requires two flavor singlet currents with instanton charges ±2. Finally, for Nf = 7, there is a Higgs branch leading to two copies of SU(2) + 8F so this theory may have a 6d fixed point though not a 5d one. In that case, we find several one suggests an enhancement of USpF (14) to the affine A(123). We expected the remaining current to also lift US(1) to 6d though again, fully determining this, requires additional input. The 10 current, in particular, doesn’t appear to fit in either group. The last case to consider is Ns = 12 . We find that all the conserved currents come from the ground state which is a gauge SO(7) singlet. Thus, the conserved current spectrum is in the 16 of USpF (16). This can be accommodated in a finite group, USp(18), if in SUF (2) × SOS(4) × SU(2) SOS(4) × USp(6)∗ USpF (10) × SOS(2) × SU(2) SOS(2) × USp(14)∗ USpF (14) × SU(2) global symmetry is USp(2Nf ) × SOS(2Ns) × UT (1). Written are the found conserved currents with their representation under the USp(2Nf ) × SOS(2Ns) global symmetry, and minimal enhanced symmetry consistent with these currents. Only cases where conserved currents were found, and currents that are flavor singlets with instanton number ±2. addition there are two flavor singlet currents with instanton charges ±2. However, this theory has a Higgs branch leading to SU(2) + 8F so we do not expect a 5d fixed point. We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 5. For SO(12) there are two different, self conjugate, spinor representations. Both are pseudoreal so half-hypers in these representations are possible. The fermionic raising operators, provided by a spinor hyper, are in the (1, 8S/C ) of the unbroken SU(2) × SO(8) gauge group, depending on the chirality of the spinor. A half-hyper in the spinor representation gives 8 fermionic zero modes, in the (1, 8S/C ), that can be combined to form 4 fermionic raising operators. Their application on the ground state generates 16 states that decomposes to the 8V ⊕ 8C/S under the SO(8) unbroken gauge group. Next, we state our results Like for SO(11), the maximal number of spinor half-hypers, in any combination of chirality, is 52 . When all of these are of the same chirality, the theory has a Higgs branch leading to the maximally supersymmetric SU(2) gauge theory so this theory may have a 6d fixed point, but probably not a 5d one. Nevertheless, we find no conserved currents in The other possibilities, Ns = 2, Nc = 12 and Ns = 32 , Nc = 1, with Nf vectors, have a Higgs branch leading to USp(6)+ 12 T AS+1AS+(2Nf + 12 )F and USp(6)+1T AS+(2Nf +3)F have a 5d fixed point while the Nf = 1 case may have a 6d one. This suggests an enhancement of UC (1) × UT (1) → SU(2)2. For the Nf = 0, Ns = 2, Nc = 12 case, we find a conserved current which is in the (2, 2) of SUS(2)2. This suggests an enhancement of SUS(2)2 × UT (1) → SU(4). For the Finally, for Nf = 1, Ns = 2, Nc = 12 , we find a conserved current which is in the (2, 2, 2) of SUS(2)2 × SUF (2). This can fit in a finite group leading to an enhancement of SUS(2)2 × SUF (2) × UT (1) → SO(8), assuming there are additional flavor singlet currents the 2 of SUF (2) and with charge ±3 under UC (1). This can fit in a finite group leading to an enhancement of SUF (2) × UC (1) × UT (1) → SU(4), assuming there are additional flavor singlet currents with instanton charges ±2. Next we discuss the case of 4 spinor half-hypers. First consider the case where all of is a flavor singlet. This suggests an enhancement of UT (1) → SU(2). Adding vectors will now furnish this current in the rank Nf antisymmetric irreducible SUF (2). This suggests an enhancement of UT (1) × SUF (2) → SU(3). For Nf = 2, the current is now in the 5 of USpF (4), and we expect an enhancement of UT (1) × USpF (4) → an enhancement of UT (1) × USpF (6) → F4 which also requires two flavor singlet currents with instanton charges ±2. For Nf = 4, there is a Higgs branch leading to SU(2) + 8F , and indeed the conserved current is now in the 42 of USpF (8). This cannot be accommodated in a finite Lie group, but rather in the affine E6(2), as could be expected from a theory with a 6d fixed point. Moreover, we find two additional currents with charges (1, 5, 1), and (5, 1, 1) under SUS1 (2) × SUS2 (2) × USpF (8). This suggests that both SUS(2)’s also get lifted to A(22). If instead we consider Ns = 32 , Nc = 12 then we find no conserved currents unless an affinization of SUS(2) to A(2) while USpF (8) appears not to affinize at this level. 2 interesting case. In this case, we find a conserved current in the 5 of SUS(2). This suggests The last case to consider is Ns = Nc = 1. In this case we find no conserved currents unless Nf ≥ 2. For Nf = 2 this current is a flavor singlet, and we expect an enhancement of UT (1) → SU(2). For Nf = 3, the additional vector fermionic zero modes charge the current in the 6 of USpF (6). This suggests an enhancement of USpF (6) × UT (1) → USp(8) which also requires two flavor singlet currents with instanton charges ±2. For Nf = 4, there is a Higgs branch leading to SU(2) + 8F so this case is again interesting from a 6d perspective. We find several conserved currents with charges 1(0,0), suggests an affinization of USpF (8) to A(72). Again, in that light we expect the remaining current to affinize US(1) and UC (1) though it is unclear how the flavor singlet current fits We next consider the case of three half-hypers. From all the relevant cases, we find find two conserved currents with charges ±1 under US(1). These can fit in a finite group, particularly, UT (1) × US(1) → SU(2)2. However, this theory has a Higgs branch leading to SU(2) + 8F so we do not expect a 5d fixed point. Next we discuss the case where there are two half-hypers. There are just two possibilities, either the two half-hypers are of the same chirality or opposite ones. In either case, we find a conserved current only when Nf ≥ 6. Specifically, for the Nf = 6 case, we find a conserved current which is a flavor singlet in both cases. This suggests an enhancement of UT (1) → SU(2). When Nf = 7, this current is charged in the 14 of USpF (14). So in both cases, we expect an enhancement of USpF (14) × UT (1) → USp(16) which also requires two flavor singlet currents with instanton charges ±2. A(125), but the first one does not appear to fit in this group. theory probably doesn’t have a 5d fixed point, but may have a 6d one. In this case, we find an enhancement of USpF (16) to the affine A(125). We expect the first two to affinize US(1) though it will probably require going to higher instanton order to verify this. For Ns = 12 , Nc = 12 , we find two conserved current one a flavor singlet while the other is in the 119 of USpF (16). The last one suggests an enhancement of USpF (16) to the affine Finally, there is the case of a single half-hyper. Going over all the relevant cases, we find no conserved current. We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 6. For SO(13) there is a single spinor representation. This representation is pseudoreal so half-hypers in this representation are possible. The fermionic raising operators, provided by a spinor hyper, are in the (1, 16) of the unbroken SU(2) × SO(9) gauge group. A halfhyper in the spinor representation gives 16 fermionic zero modes, in the (1, 16), that can be combined to form 8 fermionic raising operators. Their application on the ground state group. Next, we state our results for conserved currents. A Higgs branch analysis suggests that we cannot have more than two spinor halfhypers. For Ns = 1, we find conserved currents only for Nf ≥ 3. For Nf = 3, this current is a flavor singlet, and we expect an enhancement of UT (1) → SU(2). For Nf = 4, this current acquires the representation 8 under USpF (8). This suggests an enhancement of USpF (8) × UT (1) → USp(10), which also requires two flavor singlet currents with instanton 10, and 440 under USpF (10) × US(1). The last one suggests an enhancement of USpF (10) to the affine A(92). We expect the first two to also affinize US(1), though the current 10 does not appear to fit in either group. In the case of Ns = 12 , we find conserved currents only when Nf ≥ 7. For Nf = 7, this conserved current is a flavor singlet. This suggests an enhancement of UT (1) → SU(2). USpF (16)×UT (1) → USp(18) which also requires two flavor singlet currents with instanton SOS(4) × F4∗ USpF (4) × SOS(2) × SOC (2) × SU(2) SOS(2) × SOC (2) × USp(8)∗ USpF (12) × SOS(2) × SU(2) USpF (12) × SU(2) SOS(2) × USp(16)∗ The classical global symmetry is USp(2Nf ) × SOS(2Ns) × SOC (2Nc) × UT (1). Written are the found conserved currents with their representation under the USp(2Nf ) × SOS(2Ns) × SOC (2Nc) global symmetry, and minimal enhanced symmetry consistent with these currents. where conserved currents were found, and where a 5d fixed point is not ruled out, are shown. Minimal enhanced symmetry USpF (6) × SOS(2) × SU(2) SOS(2) × USp(10)∗ USpF (14) × SU(2) global symmetry is USp(2Nf ) × SOS(2Ns) × UT (1). Written are the found conserved currents with their representation under the USp(2Nf ) × SOS(2Ns) global symmetry, and minimal enhanced symmetry consistent with these currents. Only cases where conserved currents were found, and currents that are flavor singlets with instanton number ±2. does not fit in a finite Lie group, but rather in the affine A(127). This is consistent with the Higgs branch analysis where this theory reduces to two copies of SU(2) + 8F . We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 7. USpF (8) × SOS(2) × SU(2) SOS(2) × USp(12)∗ global symmetry is USpF (2Nf ) × SUS(Ns) × US(1) × UT (1). Written are the found conserved currents with their representation under the USpF (2Nf ) × SUS(Ns) × US(1) global symmetry, and minimal enhanced symmetry consistent with these currents. Only cases where conserved currents were found, and where a 5d fixed point is not ruled out, are shown. ∗ This enhancement also requires two conserved currents that are flavor singlets with instanton number ±2. For SO(14) there are two complex spinors where each one is the complex conjugate of the other. The fermionic raising operators, provided by a spinor hyper, are in the (1, 16) of the unbroken SU(2) × SO(10) gauge group. Next, we state our results for conserved currents. A Higgs branch analysis suggests we must have Ns ≤ 1 so the only relevant case is Ns = 1. The same analysis also suggests that we must have Nf ≤ 5. We find conserved currents only when Nf ≥ 4. When Nf = 4 this current is a flavor singlet, and we expect an enhancement of UT (1) → SU(2). When Nf = 5, this current is in the 10 of USpF (10). This suggests an enhancement of USpF (10) × UT (1) → USp(12), which also requires two flavor singlet currents with instanton charges ±2. Finally, for Nf = 6, there is a Higgs branch leading to SU(2) + 8F so this case is still interesting as there may be a 6d fixed point. In this case, we find several conserved an enhancement of USpF (12) to the affine A(121), and we expect the first two to also affinize US(1), although fully determining this requires additional input. We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 8. The case of exceptional groups In this section we discuss the case of exceptional groups. Like the case of SO groups with spinors, this is especially interesting as conventional instanton counting methods are unavailable in this case. In the pure YM case this can be circumvented owing to the identification of the Nekrasov partition function [35] with the properly symmetrized Hilbert series of the corresponding instanton moduli space [36]. Fortunately, methods for the calculation of the Hilbert series for the instanton moduli spaces of exceptional groups are known [37]. Using their results we find that the 1-instanton Nekrasov partition function has the following form: ZG1 = (1 − xy)(1 − xy ) k=0 irreducible k-symmetric product of the Adjoint (in Cartan weights these are given by [0, k] for G2, [k, 0, 0, 0] for F4, [0, k, 0, 0, 0, 0] for E6, [k, 0, 0, 0, 0, 0, 0] for E7 and [0, 0, 0, 0, 0, 0, 0, k] for E8). This result has the following simple interpretation. The infinite series and the denominator come from applying the 4hG bosonic zero modes on a single instanton ground state. This state in turn contributes xhG to the index. Next, we discus each group in turn, stating our results. Instantons of G2 are contained in an SU(2) subgroup breaking G2 → SU(2)×SU(2). Under this breaking the Adjoint 14 of G2 decomposes into the Adjoints of both SU(2)’s as well as a state in the (2, 4) representation. Thus, the full state space is given by acting on the We can form an SU(2) gauge invariant by contracting two Bi operators. By applying these on the ground state we get a triplet corresponding to one state with R-charge 1. Tensoring with the broken current supermultiplet, we find a single state which, while not a conserved current, is a BPS state. We expect this state to contribute the term x4 to the Next, we generalize by adding hypermultiplets in the 7 of G2. Each one of these contributes a raising operator in the 2 of the unbroken SU(2) gauge group. These can form an invariant by contraction with ij , whose application on the previous state furnishes it with the rank Nf irreducible antisymmetric representation of the USp(2Nf ) flavor symmetry. Next we ask whether one can form a conserved current state by contracting both types of zero modes. In order to get a conserved current we need an R-charge singlet, necessitating two Bi’s. As these are fermionic zero modes, the SU(2) charges must form an antisymmetric product so the product can be either in the singlet or the 5 dimensional representation. The singlet is just the previous state so if we are to find a conserved current it must come by contracting the Bi’s in the 5 dimensional representation of SU(2) and canceling it against flavor zero modes. To get the 5 dimensional representation we need a symmetric product of 4 states in the 2, and since these are fermionic zero modes, they must be antisymmetric in their flavor charges. Therefore, we conclude that for Nf < 4 there are no conserved currents. When to the enhancement UT (1) → SU(2). When Nf = 5 there is a conserved current which is in the 10 of the USpF (10) flavor symmetry. This should lead to the enhancement UT (1) × USpF (10) → USp(12). This also requires conserved currents with instanton number ±2 that are flavor symmetry singlets. seems to lead to the affine A(121). This suggests that this theory lifts to 6d. tensor of the USpF (12) flavor symmetry. This cannot form a finite Lie group, but rather We can also consider the maximally supersymmetric case where we add a hypermultiplet in the 14 of G2. The fermionic zero modes supplied by the Adjoint hyper form 4 raising operators which together with Bi form a doublet of SU(2) ⊂ SOR(5). The basic gauge invariant one can form is given by contracting two such operators. This results in a state, with r-charge −2, with these operators generates 14 states that form the rank 2 symmetric traceless representation of SOR(5). The general thought is that this theory lift to the 6d (2, 0) theory of type D4 where a Z3 twist in the outer automorphism of D4 is imposed when going around the circle [32]. Recall that SO(8) has invariant polynomials of degree 2, 6 and two distinct ones of degree 4. Under the Z3 twist the polynomials of degree 2, 6 are even, and so contribute at the massless level. This correctly reproduces the expected operators of G2 whose invariant polynomials are of degrees 2 and 6. On the other hand, the two polynomials of degree 4 are rotated by 1200 [38]. Thus, these states are all massive with mass ∼ gY21M MK3K , 2MKK where MKK is the mass of the first KK state given by the states corresponding 3 to the invariant polynomials even under the twist. The first instanton contribution should correspond to the state with mass MKK and so we expect it to give a KK supermultiplet where the lightest states having a mass of in the 14 of SOR(5). This indeed matches our results. Instantons of F4 are contained in an SU(2) subgroup breaking F4 → SU(2)×USp(6). Under this breaking the Adjoint 52 of F4 decomposes into the Adjoints of SU(2) and USp(6) as well as a state in the (2, 140) representation. Thus, the full state space is given by acting unbroken USp(6) gauge group. We can form a USp(6) gauge invariant by contracting two Bi operators. By applying these on the ground state we get a tower corresponding to one state with R-charge 72 . Tensoring this with the basic broken current supermultiplet, we get a BPS state, which we expect to contribute to the index as x9. Recalling that hF4 = 9, this matches the expectations from the partition function. We can generalize by adding hypermultiplets in the 26 of F4. Each one of these contributes a raising operator in the 6 of the unbroken USp(6) gauge group. We can form gauge invariants by contracting two such operators. Their application on the ground state will charge it under the flavor symmetry. We can ask whether we can form an invariant which will be an R-charge singlet so that we get a conserved current. Using numerical a flavor singlet suggesting an enhancement of UT (1) → SU(2). can flow on the Higgs branch to theories that are thought to possess neither a 5d nor a 120 of USpF (8). This cannot fit in either a finite or an affine Lie group. We can also consider the maximally supersymmetric case. Evaluating numerically, we find two distinct states in the rank 7 and 3 symmetric traceless representation of SOR(5). We expect this theory to lift to the 6d (2, 0) theory of type E6. The reduction to 5d is done with a Z2 outer automorphism twist under which the E6 invariant polynomials of degrees 2, 6, 8 and 12 are even, while those of degrees 5 and 9 are odd. Therefore the first instanton contribution should be the lowest mode of the states associated to the invariant polynomials of degrees 5 and 9, and so should be in the states given by tensoring the KK state with the rank 3 and 7 symmetric traceless representation of SOR(5), respectively. This indeed matches our results. Instantons of E6 are contained in an SU(2) subgroup breaking E6 → SU(2) × SU(6). Under this breaking the Adjoint 78 of E6 decomposes into the Adjoints of SU(2) and SU(6) as well as a state in the (2, 20) representation. Thus, the full state space is given by acting unbroken SU(6) gauge group. We can form an SU(6) gauge invariant by contracting two Bi operators. By applying these on the ground state we get a tower corresponding to one state with R-charge 5. This is a BPS state which we expect to contribute to the index as x12. Recalling that hE6 = 12, this matches the expectations from the partition function. We can generalize by adding hypermultiplets in the 27 of E6. Each one of these contribute a raising operator in the 6 of the unbroken SU(6) gauge group. We can ask whether we can form an invariant which will be an R-charge singlet so that we get a which we find a single conserved current which is a flavor singlet. This leads us to expect an enhancement of UT (1) → SU(2) in this case. We do not expect a fixed point, 5d or 6d, to exist when Nf > 4. We can also consider the maximally supersymmetric case. Unfortunately, the numerics in this case, as well as for the other E groups, proves to be quite time consuming, so we reserve this for future study. Instantons of E7 are contained in an SU(2) subgroup breaking E7 → SU(2) × SO(12). Under this breaking the Adjoint 133 of E7 decomposes into the Adjoints of SU(2) and SO(12) as well as a state in the (2, 32) representation. Thus, the full state space is given by acting on the ground state, whose r-charge is −8, with the fermionic operator Bi in the 32 of the unbroken SO(12) gauge group. we can form an SO(12) gauge invariant by contracting two Bi operators. By applying these on the ground state we get a tower corresponding to one state with R-charge 8. Taking the direct product with the broken this matches our expectations from the partition function. We can generalize by adding hypermultiplets in the 56 of E7. This representation is pseudoreal so half-hypers are possible. From the Higgs branch analysis, we conclude that the maximal number of half-hypers possible for the existence of a 5d fixed point is 6. Each full hyper contributes a raising operator in the 12 of the unbroken SO(12) gauge group. A half-hyper contributes 12 fermionic zero modes which can be combined to form 6 raising 64 states transforming as the 32 ⊕ 320 of SO(12). We now ask, for a given number of half-hypers, can we form a gauge invariant state with 0 R-charge, which as we are now quite accustomed to gives a conserved current after tensoring with the broken current supermultiplet. From group theory it is clear that this can only happen when the number of half-hypers is even. First recall that SO(12) has a Z2 × Z2 center. This contains three distinct Z2 subgroups, each of which act on two of the basic representations: 12, 32, and 320, as −1 and on the other by 1. If a state is to be gauge invariant, it must also be invariant under the center. The previous analysis now implies that only a product of an even number of all three basic representations, or a product of an odd number of all of them, can contain a gauge invariant. In particular, when applied to the case of an odd number of half-hypers, we see that an SO(12) invariant state can only be made by an odd number of Bi’s and thus cannot be an This leaves only 3 cases that need to be worked out exactly. Unfortunately, the numerical analysis proves to be quite time consuming and we leave pursuing it to future work. Instantons of E8 are contained in an SU(2) subgroup breaking E8 → SU(2) × E7. Under this breaking the Adjoint 248 of E8 decomposes into the Adjoints of SU(2) and E7 as well as a state in the (2, 56) representation. Thus, the full state space is given by acting on the ground state, whose r-charge is −14, with the fermionic operator Bi in the 56 of the unbroken E7 gauge group. we can form an E7 gauge invariant by contracting two Bi operators. By applying these on the ground we get a tower corresponding to one state with R-charge 14. When tensored with the broken current supermultiplet, this results in a BPS state which we expect to contribute a term of x15 to the index. Recalling that hE8 = 30, this matches our expectations from the partition function. The fundamental 248 representation of E8 is identical to the Adjoint so upon adding a single hyper we get the maximally supersymmetric E8 Yang-Mills theory, which should In this paper we explored symmetry enhancement from 1 instanton operators for USp(2N ), SO(N ) and exceptional groups with various matter content. This line of thought can be generalized in several directions. First, one can attempt to explore more general 1 instanton operators, where one applies also bosonic zero modes or fermionic non-zero modes, or go to higher instanton number. We have seen that in some cases, in order to complete a symmetry group, conserved currents with higher instanton number are necessary. It will be interesting if these can also be verified by some modification of these methods. We have also concentrated on the case of a simple gauge group. An interesting direction is to generalize to quiver theories, extending the results of [29]. Also, we have adopted a rather broad criterion, and it is not clear if the theories checked actually have a 5d fixed point. In some cases it is known to exist as these theories can be engineered in string theory using brane webs [31]. It is interesting to know whether the other cases also flow to In some cases we have seen that the global symmetry is enhanced to an affine Lie group. The most straightforward interpretation of this is that these theories lift to a 6d theory with the appropriate global symmetry. In the case of USp(2N ) + AS + 8F the 6d UV theory is known. 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Gabi Zafrir. Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories, Journal of High Energy Physics, 2015, 87, DOI: 10.1007/JHEP07(2015)087