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

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.


Introduction
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][2][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 element in this is the existence of a topologically conserved U (1) current, j T = * Tr(F ∧ F ), 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 N f hypermultiplets in the doublet of SU (2). This theory is known to flow to a 5d fixed point provided N f ≤ 7 [1]. Furthermore, the global symmetry at the fixed point is enhanced from U (1) × SO(2N ) to E N f +1 [1]. From the gauge theory viewpoint the enhancement is brought about by instantonic particles.
The N f = 8 case is also interesting. In that case there is no 5d fixed point, but rather a 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 particles. A similar story is also believed to occur for the maximally supersymmetric theory, where the 6d theory is now the (2, 0) theory [5].
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 [6][7][8]. A more direct way is to find this from the superconformal index [9]. 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 S 4 × S 1 . Then the representations of the superconformal group are labeled by the highest weight of its SO L (5) × SU R (2) subgroup. We will call the two weights of SO L (5) as j 1 , j 2 and those of SU R (2) as R. Then following [10] the index is: 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 topological symmetries. More to the point, conserved currents are part of a BPS supermultiplet which contains a scalar operator with charges j 1 = j 2 = 0, R = 1, and in the Adjoint of the corresponding global symmetry. Thus, it contributes to the index a term of x 2 χ[Ad]. Therefore one can calculate the index, and from the x 2 terms infer the expected enhancement.
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 [28]. 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) + N f F with N f ≤ 8. This of course does not prove that these theories have a 6d or 5d fixed point. Since every N = 2 conformal 4d gauge theory obeys this condition, and as the Higgs branch is the same as in 5d, one can largely borrow the classification of [29], 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 U Sp(2N ) and SO(N ) in sections 3 and 4 respectively. Section 5 covers the exeptional groups. We end in section 6 with our conclusions.
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 discrete θ parameter.
When denoting the charges of states under the SU R (2) R-symmetry, we use R for the maximal weight of an SU R (2) multiplet, and r for the U R (1) ⊂ SU R (2) charge of a specific 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 .
In the N = 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 [28] 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 matter symmetry.
In the N = 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 [28] these form a single supermultiplet whose structure is identical to that expected from the Kaluza-Klein (KK) modes of a 6d N = 2 energy-momentum supermultiplet. This 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 discuss the results.

The case of U Sp groups
In this section we discuss the case of a U Sp(2N ) gauge theory. The 1-instanton is in an SU (2) subgroup of the U Sp(2N ) gauge group, breaking it to SU (2) × U Sp(2N − 2). Under this breaking the Adjoint of U Sp(2N ) decomposes into the Adjoint of SU (2), the Adjoint of U Sp(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 supermultiplet. The bifundamental provides 4(N − 2) fermionic zero modes that are in the 2N − 2 of the unbroken U Sp(2N − 2) gauge group, and in the 2 of SU R (2).
The 4(N − 2) zero modes can be split to 2N − 2 raising operators, B j , (with charge (2)). The full spectrum of states is generated by acting with raising operators on the ground state. However, we must also enforce U Sp(2N − 2) gauge invariance. The basic gauge invariant operator is given by J ij B i B j (where J ij is the invariant antisymmetric tensor of U Sp(2N − 2)). Starting from the ground state, with r = − N −1 2 , we can act on it with J ij B i B j operators leading to the N states: with r-charges − N −1 2 , − N −1 2 + 1, ....., N −1 2 . Taking the product with the broken current supermultiplet gives a multiplet whose lowest state is a scalar with R-charge R = N +1 2 . This is a BPS operator though not a conserved current. From its charges, we expect this operator to contribute x N +1 to the index. This agrees with results found from index calculations, where the 1 instanton contribution for a pure U Sp(2N ) gauge theory indeed behaves as: [10,14].
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 Z 2 center unbroken. We must also demand gauge invariance under this symmetry. It turns out that the ground state is even for a vanishing θ parameter, and odd for θ = π. This can be understood either directly from the path integral [28], or alternatively from the exact instanton counting [13].
From the results for SU (2) it is now straightforward to generalize to the case of U Sp(2N ). As the instanton is in an SU (2) subgroup, we again conclude that the ground state is even under Z 2 only if θ = 0. We must also consider the behavior of the fermionic operators B i under this symmetry. As these arise from SU (2) doublets, each one transforms under this Z 2 gauge symmetry. Thus, a state is even under Z 2 if it is made from an application of an even number of B i 's when θ = 0, or an odd number when θ = π. All the states in (2) contain an even number of B i operators and thus are kept when θ = 0, but are projected out when θ = π.
It is straightforward to generalize this result by adding N f hypers in the fundamental of U Sp(2N ). These provide 2N f zero modes, which are gauge singlets, whose application on the previous states furnish it with the Dirac spinor representation of SO(2N f ). Enforcing the Z 2 gauge invariance eliminates half of them so the end result is one state with R = N −1 2 and in a Weyl spinor representation of SO(2N f ), where the chirality of the spinor is determined by the θ angle. Thus, we find no broken current supermultiplets for U Sp(2N )+ N f F . This is consistent with the expectations from instanton analysis and brane webs in the presence of orientifolds [30].

With antisymmetric matter
We can add matter in the antisymmetric representation of U Sp(2N ). Under the breaking of U Sp(2N ) → SU (2) × U Sp(2N − 2) the antisymmetric of U Sp(2N ) decomposes to a singlet, the antisymmetric of U Sp(2N − 2), and a state in the (2, 2N − 2). Therefore, similarly to the Adjoint, the antisymmetric provides 4N − 4 fermionic zero modes that can be combined to form 2N − 2 fermionic operators, A i , in the 2N − 2 of the unbroken gauge U Sp(2N − 2), that raise by 1 2 the charge under U AS (1) ⊂ SU AS (2). We can form a U Sp(2N − 2) invariant by J ij A i A j whose repeated application on (2) charges it in the N dimensional representation of SU AS (2). Alternatively, we can form a U Sp(2N −2) invariant by J ij A i B j , and applying this on the ground state gives a new state, as it changes the charges of both U AS (1) ⊂ SU AS (2) and U R (1) ⊂ SU R (2) by 1 2 . Additional applications of J ij A i A j and J ij B i B j charges this state in the (N − 1, N − 1) dimensional representation of (SU R (2), SU AS (2)). Additional applications of J ij A i B j generate new states until we apply N − 1 such operators 2 . Tensoring this with the broken current supermultiplet, we find a multitude of states in representations (k + 2, k) for k = 1, 2.., N , under (SU R (2), SU AS (2)). In particular this includes a state in the (3, 1), given by: which corresponds to a conserved current. We also need to take into account the effect of the θ angle. As all these states are made from an application of an even number of fermionic raising operators, they are kept only if θ = 0 while for θ = π they are projected out. Therefore there are no conserved currents for θ = π.
We thus find that U Sp 0 (2N ) + AS has a conserved current which is an SU AS (2) singlet suggesting an enhancement of U T (1) → SU (2) while U Sp π (2N ) + AS has no enhancement, at least up to the limitations of this method. This is in accordance with the identification of U Sp 0 (2N ) + AS and U Sp π (2N ) + AS as the rank N E 1 andẼ 1 theory, respectively [3].
It is now straightforward to generalize by adding N f fundamental flavors. As previously stated this will just charge the instanton under the spinor representation of SO(2N f ). This suggests an enhancement of Finally, we consider the case of N f = 8. The conserved current is now in the spinor of SO (16). However, there is no finite Lie group which contains U (1) × SO (16), and whose Adjoint contains states with charges 128 1 , 128 −1 . On the other hand, there is an affine Lie group, E 8 , that has this spectrum 3 . This suggests that this theory lifts to a 6d theory with E 8 global symmetry in agreement with the results of [4,32].
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 U Sp(4) with 2 antisymmetrics and N f flavors. This theory has a Higgs branch leading to SU (2) + 2N f F , so as long as N f < 4 a 5d fixed point is not ruled out.
First consider the case of N f = 0. We still have the previous conserved current, given by: where we use the index f = 1, 2 for SU (2) ⊂ U Sp AS (4). This state is in the 2 −1 representation of U (1) × SU (2) ⊂ U Sp AS (4). We can also apply J ij A f 1 i A f 2 j generating an additional conserved current, now in the 2 1 representation. These two form the 4 of U Sp AS (4). This conserved current is kept provided θ = 0. In that case, we expect an enhancement of U T (1) × U Sp AS (4) → U Sp (6). Note that this enhancement also requires a U Sp AS (4) singlet conserved current with instanton charge ±2.
Adding flavor will give it also charges under the spinor of SO(2N f ). When N f = 1 the global symmetry associated with the flavor is U F (1) and the conserved current acquires the charge ± 1 2 depending on the value of the θ angle. Thus, we still expect an enhancement of U (1) × U Sp AS (4) → U Sp(6), just with the U (1) being a combination of U T (1) and U F (1).
When N f = 2 the global symmetry associated with the flavors is SU (2) × SU (2), and the conserved currents are in the (2, 1) or (1, 2), depending on the θ angle. The smallest global symmetry consistent with this is U Sp (8). This also requires conserved currents with instanton number ±2 and in the (1, 3) of U Sp AS (4) × SU (2). Assuming these states are indeed present, the global symmetry of this theory is SU (2) × U Sp (8).
When N f = 3 the global symmetry associated with the flavors is SU (4), and the conserved currents are in the 4 or4, depending on the θ angle. The smallest global symmetry consistent with this is U Sp (12). This also requires a conserved current, and it's conjugate, with instanton number 2 and in the (1, 10) of U Sp AS (4) × SU (4). Assuming these states are indeed present, the global symmetry of this theory is U Sp (12).
Finally, we consider the case of N f = 4. The global symmetry associated with the flavors is now SO (8), and the conserved currents are in the 8 S or 8 C , depending on the θ angle. There is no finite Lie group, containing U (1) × U Sp(4) × SO(8), whose Adjoint contains (4, 8) ±1 . However, this is contained in the affine Lie group A (2) 11 So this theory may lift to 6d.
There are two other theories in this family that may have a 6d fixed point. One is U Sp(4) + 3AS = SO(5) + 3V which we deal with in the next section when we discus SO(N ) groups with vector matter. The second is U Sp(6) + 2AS where we find a conserved current in the 10 of U Sp AS (4) which is kept if θ = 0. This cannot span any finite Lie group rather leading to the affine C

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 U Sp(2N ) → SU (2) × U Sp(2N − 2) the symmetric of U Sp(2N ) decomposes to symmetrics of SU (2) and U Sp(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 (SU R (2), SU S (2)), for k = 1, 2.., N .
The groups SU R (2) × SU S (2) form a larger group SO R (5), which is the R-symmetry of the maximally supersymmetric theory, and these states form a single representation of SO R (5) which is the N − 1 symmetric traceless representation. The final result is given by tensoring these two states. Finally, we need to consider the effect of the θ angle. If θ = 0 then these states are kept, but are projected out when θ = π.
For θ = 0 the resulting spectrum is consistent with the expected 6d lift. We remind the reader that 5d maximally supersymmetric U Sp 0 (2N ) theory is expected to lift to the 6d (2, 0) theory of type D N +1 where the compactification is done with a twist in the outer automorphism of SO(2N +2) [31]. The (2, 0) theory of type D N +1 has a short multiplet for every SO(2N +2) invariant polynomial. This short multiplet contains a symmetric traceless spacetime tensor that is in the D − 2 symmetric traceless representation of SO R (5) where 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 and N + 1. 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 which we expect to appear as instanton states. 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 antiperiodic 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 U Sp(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 given by tensoring the previously mentioned KK states with a state in the N −1 symmetric traceless representation of SO R (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.
The 5d U Sp π (2N ) maximally supersymmetric theory is expected to lift to the 6d (2, 0) theory of type A 2N 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 Z 2 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 the degrees 3, 5, ..., 2N − 1, 2N + 1 operators which are sensitive to the twist. Nevertheless, we find no states coming from the 1 instanton in this case.

Rank 3 antisymmetric tensor
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 U Sp(6) and one with gauge group U Sp (8). The rank 3 antisymmetric representation of U Sp(2N ) is pseudoreal so one can add a half-hyper. Specifically for U Sp(6), the representation is 14 dimensional and because of the anomaly of [33] adding a half-hyper is inconsistent unless one also adds a half-hyper in the fundamental. The possible candidates for a 5d fixed point are 1 2 , 1 and 3 2 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 U Sp(6) → SU (2) × U Sp(4), the 14 decomposes into states in the (1, 4) and (2, 5) of SU (2) × U Sp (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 application on the ground state charge it in the 4⊕4 of the U Sp(4) gauge symmetry. Out of these two, only one is kept while the other is projected out, depending on the θ angle. Now we can discuss the possible cases. First if we have just a half-hyper in the 14 and the 6 then we essentially just need to repeat the original analysis, now with a ground state that is in the fundamental of U Sp(4). It is not difficult to see that there are just two U Sp(4) invariant states given by: These form a doublet of SU R (2) so there are no conserved currents. The generalization by adding N f fundamental hypers is immediate. Like in previous cases, the additional zero modes charge this state with the Dirac spinor representation of SO(2N f ) ⊂ SO(2N f + 1). Enforcing Z 2 gauge invariance reduce it to a Weyl spinor of SO(2N f ). However, we now recall that there is an additional ground state that is not invariant under Z 2 . When flavors are added, this ground state also contributes a Weyl spinor of SO(2N f ), but with opposite chirality. These combine to form an SO(2N f + 1) spinor. So the end result is we find an SU R (2) doublet that is in the spinor representation of SO(2N f + 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 U Sp(4). It is not difficult to see that there are just two conserved currents, given by: which form a doublet of SU AS (2). The minimal enhancement consistent with this is U T (1)× SU AS (2) → SU (3). We can further generalize by adding flavors in the fundamental of U Sp(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(2N f + 1).
. Such a spectrum cannot fit in a finite Lie group, but can fit in the affine Lie group E (2) 6 . Next we turn to the case of one full hyper in the 14 of U Sp (6). We find two conserved currents with charges ± 3 2 under U T AS (1). Depending on the θ angle, one of them is 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(2N f ). Half of these are projected out leaving two SO(2N f ) spinors of opposite chirality and with charges ± 3 2 under U T AS (1).
A Higgs branch analysis suggests that we can have at most 4 flavors and still have a 5d fixed point. For N f = 1 we get two conserved currents with opposite charges under SO F (2) and U T AS (1). The minimal enhancement consistent with this is U (1) 2 → SU (2) 2 . For N f = 2 we get two conserved currents of opposite charges under U T AS (1), one in the 2 of one SU F (2) and the other in the 2 of the other. The minimal enhancement consistent with We get two conserved currents in the 4 and 4 of SU F (4). The minimal enhancement consistent with this is U (1) 2 × SU F (4) → SU (6) which also requires two flavor singlet conserved currents with instanton number ±2.
For N f = 4 we get two conserved currents, one in the 8 S and the other in the 8 C of SO F (8). This suggests an enhancement of U (1) 2 × SO(8) → E 6 assuming we also get two conserved currents with instanton number ±2 and in the 8 0 (1) . Finally, we consider the case of N f = 5. In this case we get two conserved currents in the 16 and16 of SO F (10). This cannot fit in a finite Lie group, but can fit in the affine Lie group E (1) 6 . Next, we consider the case of 3 2 hypers in the rank 3 antisymmetric. We do not find any conserved current in this case.
The last case we consider is U Sp(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 [33] (see for example [29]). The effect of the half-hyper is to furnish the ground state in the 64 of the unbroken U Sp(6) gauge symmetry. We find no conserved current. These naturally form 2(N − 4) raising operators B aj (where a = 1, 2 ∈ SU (2), i = 1, ..., N − 4 ∈ SO(N − 4)) from which we can form a gauge invariant by: ab δ ij B ai B bj . Starting from the ground state, with r = − N −4 2 , and acting with these operators, leads to the N − 3 states:

The case of SO groups
whose r-charges are: Tensoring with the broken current supermultiplet we do not get a conserved current, rather a BPS operator which we expect to contribute x N −2 to the index. Indeed, this matches the contribution one finds from the explicit index calculation.
Next, we generalize by adding N f hypers in the vector representation. These add 4N f fermionic zero modes in the (2, 1) of the unbroken SU (2)×SO(N −4) gauge group, and the 2N f of the U Sp(2N f ) flavor symmetry. From these we can form 2N f raising operators C a in the (2, 1) of the unbroken gauge symmetry and N f From these we can form the gauge invariant ab C a C b . Applying this on (7) yields a new state which is in the rank 2 symmetric representation of representation should, on the one hand, be given by a symmetric product of the rank 2 symmetric representation of SU (N f ), 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 ( ab C a C b ) l should be the conjugate of the one given by applying ( ab C a C b ) N f −l . We end up with the N f + 1 states: where we have collectively denoted the states in (7) as |B . Their charges under given by 1, , , ,....., where the last Young diagram has N f rows. We next need to combine them into U Sp(2N f ) representations. We claim that these states build the rank and that all U Sp(2N f ) representations can be build from products of the fundamental representation. In order to get a state with lowest U (1) charge −N f we must multiply N f fundamentals. Furthermore, for that state to be an SU (N f ) 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 N f irreducible antisymmetric representation of U Sp(2N f ). In addition we may be able to form an invariant from both C a and B aj . Since only the B's are charged under SO(N − 4) they must independently combine to form an invariant under that group. This can be done in two ways. First, we can contract two B's using δ ij , 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 discussed operator. The second option is to contract N − 4 indices with the epsilon tensor. 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 N − 3 dimensional representation of SU (2), so that it can form an SU (2) gauge invariant, 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 N f < N − 4 there are not enough C's to form this invariant and the 1-instanton comprises one state with R-charge N −2 2 and in the rank N f irreducible antisymmetric representation of U Sp(2N f ).
When N f = 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 This is a singlet under U Sp(2N f ) 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 U T (1) → SU (2).
. This requires also two U Sp(2N f ) singlets with instanton charge ±2. The enhancement spectrum seen here is consistent with the results from the explicit index analysis and brane webs [30].
Finally when N f = N − 2 this state acquires charges in the rank 2 irreducible antisymmetric representation of U Sp(2N f ). There is no finite Lie group with this content, but there is an affine Lie group, A 2N f −1 , with this spectrum. This is in accordance with the 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 hyper are 8 zero modes coming from an SU (2) Adjoint, and 4(N − 4) zero modes coming from doublets of SU (2) that are also in the (2, N − 4) of the unbroken SU (2) × SO(N − 4) 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, A ai , that are in the (2, N − 4) of the unbroken SU (2)×SO(N −4) gauge group. These can be combined with the previous B ai operators to form the operator B α ai (where α = 1, 2 and . This is more than just a notational convenience as SU R (2)×SU AS (2) is enhanced to SO R (5) of the maximally supersymmetric theory. In addition to its gauge charges, The operator B α ai is charged in the We next need to apply these zero modes on the ground states, enforcing SU (2)×SO(N − 4) invariance, and compose the results into SO R (5) representations. As previously stated the basic gauge invariant one can make is δ ij ab B α ai B β bj . Due to the symmetry properties of the involved operators, this operator is in the 3 2 of U (1)×SU (2) ⊂ SO R (5). By repeatedly applying it on the ground states we generate the states: These states are in the sym 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 SO R (5). This can be seen by starting with the decomposition 5 = 1 2 + 1 −2 + 3 0 , and doing the symmetric multiplication. After removing the trace, one can see that we get the spectrum of (10).
Alternatively, we can contract the SU (2) ⊂ SO R (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 δ ij ab B α ai B β bj on it generates a single state in the rank N − 6 symmetric traceless representation of SO R (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 SO R (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)  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 DN 2 , 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 SO R (5).
For the N odd case, the theory is expected to lift to the 6d (2, 0) of type A N −1 with a Z 2 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 at the massless level matching the degree 2, 4, .., N − 1 invariant polynomials of SO(N ) for odd N . 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 instanton contribution.

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. Generically, under the breaking SO(N ) → SO(4) × SO (N − 4), the spinor of SO(N ) decomposes into 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: where the sum goes over all types of fermionic zero modes, those coming from the Adjoint, vector matter and spinor matter. Here, x i stands for the fugacity of the global U (1) raised by the corresponding fermionic zero mode. Finally, we collectively denote by χ i the 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 x i '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 SU R (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 the cases of N = 3, 5 were already covered in the previous cases. The case of N = 4 is not a simple group and so won't be discussed here. Finally, the case of SO(6) = SU (4) with only spinor matter was covered in [28]. 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.

SO(7)
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 N f = 4, the maximal number of spinors is N s = 1 (if N s = 2 the theory as a Higgs branch where it reduces to SU (2) + 8F ). We recall that for N s = 0 there is a conserved current in the 8 of U Sp F (8). The addition of spinors also gives it charges under the global symmetry 4 The plethystic exponent is defined as P E[f (·)] = exp ∞ n=1 1 n f (· n ) where the dot represents all the variables in f . associated with the spinors. Particularly, for N s = 1, it acquires the representation 2 of SU S (2). We find no additional conserved currents, from the 1-instanton, besides this. The minimal global symmetry consistent with this is U Sp (12). This also requires conserved currents with instanton number ±2 and in the (1, 3) of U Sp F (8) × SU S (2). Assuming these states are indeed present, the global symmetry of this theory is U Sp (12).
When N s = 2, we get a conserved current in the (5,8) of U Sp S (4) × U Sp F (8). This cannot fit in a finite Lie group, but can form an affine one, A 12 . For N f = 3, the maximal number of spinors is N s = 3 (if N s = 4 the theory has a Higgs branch where it reduces to two copies of SU (2) + 8F ). We recall that for N s = 0 there is a conserved current which is a U Sp F (6) singlet. The addition of spinors gives it charges under the spinor global symmetry. For N s = 1 this state acquires charges under the 2 of SU S (2). The minimal global symmetry consistent with this is SU (3). For N s = 2, the conserved current is now in the 5 of U Sp S (4). The minimal global symmetry consistent with this is SO (7). When N s = 3, the conserved current is now in the 14 of U Sp S (6). The minimal global symmetry consistent with this is F 4 , which also requires two extra conserved currents with instanton number ±2.
Finally, for N s = 4 the conserved current is now in the 42 of U Sp S (8). There is no finite Lie group that can accommodate this structure, but there is an affine group, E 6 , that can. Also it turns out that there is an additional conserved current in the (1, 14) of U Sp S (8)×U Sp F (6). Again this appears to suggest an enhancement to an affine group A 5 . Thus, in this case, both the global symmetries appear to be affinized which is consistent with this theory lifting to 6d.
For N f = 2 we find no conserved currents unless N s ≥ 4. When N s = 4, this current is in the (1, 4) of U Sp S (8) × U Sp F (4). This appears to suggest an enhancement of U T (1) × U Sp F (4) to U Sp (6). This also requires two states with instanton number ±2. This is the maximal number of spinors allowed in this case. For N s = 5, there is a Higgs branch 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 U Sp S (10) × U Sp F (4) which is consistent with an affine group, in this case C 7 . For N f = 1 we find no conserved currents unless N s ≥ 4. When N s = 4, this current is a singlet under the flavor symmetry. This should lead to an enhancement of U T (1) → SU (2). For N s = 5 it acquires charges in the fundamental of U Sp S (10) suggesting an enhancement to U Sp(12) (again this also requires states with instanton number ±2). Finally, in the case of N s = 6, the conserved current is in the rank 2 irreducible antisymmetric of U Sp S (12). This cannot be accommodated in a finite Lie group, but rather in the affine A 11 . This is consistent with a Higgs branch analysis, as for this case the theory can be reduced to SU (2) + 8F . Strangely, SU F (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 form an R-charge singlet we must act on the ground state, whose r-charge is − 3 2 , with 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.
Written are the found conserved currents with their representation under the U Sp(2N f ) × U Sp(2N s ) 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. * * This enhancement also requires two conserved currents that are in the (1, 3) with instanton number ±2.

SO(8)
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 N f = 5 we can have at most a single spinor of any chirality as for N s = 2 or N s = N c = 1 there is a Higgs branch leading to SU (2)+8F . Recall that when N s = N c = 0 there is a single conserved current in the 10 of U Sp F (10). The addition of spinor matter charge it also under the spinor flavor symmetry. Thus, for N s = 1 it acquires the 2 of SU S (2) so we find a single conserved current in the (10, 2) of U Sp F (10) × SU S (2). This should lead to an enhancement of U T (1) × SU S (2) × U Sp F (10) → U Sp (14). This also requires conserved currents with instanton number ±2 and charges (1, 3).
The cases of N s = 2 and N s = N c = 1 are also interesting as they may possess a 6d fixed point. For N s = 2, we find a conserved current in the (10, 5) of U Sp F (10) × U Sp S (4). This does not fit in a finite Lie group, but rather in the affine A 14 . For N s = N c = 1, we find a conserved current in the (10, 2, 2) of U Sp F (10) × SU S (2) × SU C (2). This again does not fit in a finite Lie group, but rather in the affine A Alternatively, for N s = N c = 1, we find a conserved current in the (1, 2, 2) of U Sp F (8)× SU S (2) × SU C (2). This should lead to an enhancement of U T (1) × SU S (2) × SU C (2) → SU (4). For N s = 3, N c = 0, we find a conserved current in the (1, 14 ) of U Sp F (8) × U Sp S (6). This should lead to an enhancement of U T (1) × U Sp S (6) → F 4 , which also requires conserved currents with instanton number ±2 that are flavor symmetry singlets. The last remaining case is N s = 2, N c = 1 for which the spinor fermionic zero modes charge the current in the (5, 2) of U Sp S (4) × SU C (2). So we find a conserved current in the (1, 5, 2) of U Sp F (8) × U Sp S (4) × SU C (2). This should lead to an enhancement of U T (1)×U Sp S (4)×SU C (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. There are three relevant cases. For N s = 4, N c = 0, we find two conserved currents, one in the (42, 1), and the other in the (1, 42) of U Sp F (8) × U Sp S (8). These appear to affinize both U Sp(8)'s to E There are just two other cases, not related by triality to the previous cases, where we find a conserved current. The first is N f = N s = N c = 2 where we find a single conserved current which is a flavor singlet. This should lead to an enhancement of U T (1) → SU (2). The second being N f = 3, N s = N c = 2 where we find a conserved current in the (6, 1, 1) of U Sp F (6)×U Sp S (4)×U Sp C (4). This should lead to an enhancement of U T (1)×U Sp F (6) → U Sp (8), which also requires conserved currents with instanton number ±2 that are flavor symmetry singlets.
The case of N f = N s = 3, N c = 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 U Sp F (6) × U Sp S (6) × U Sp C (4). This should affinize U Sp F (6) × U Sp S (6) to C (1) 6 . Incidentally, U Sp C (4) is not affinized, at least not 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 2.
. Written are the found conserved currents with their representation under the U Sp(2N f ) × U Sp(2N s ) × U Sp(2N c ) 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. * * This enhancement also requires two conserved currents that are in the (1, 3) with instanton number ±2.

SO(9)
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. We find only a handful of cases where there are conserved currents. First for N f = 6, N s = 1, there is a Higgs branch leading to SU (2) + 8F so this theory has a potential 6d origin. Recall that for N f = 6, N s = 0, there is a conserved current in the 12 of U Sp F (12). The additional fermionic zero modes charge it in the 3 of SU S (2) so we find a conserved current in the (12, 3) of U Sp F (12) × SU S (2). This cannot be contained in a finite Lie group, but rather in an affine one, A 14 . For N f = 5, we can have at most one spinor, as for N s = 2 there is a Higgs branch leading to two copies of SU (2) + 8F . Again, recall that for N s = 0 there is a single conserved current which is a U Sp F (10) singlet. The additional spinor zero modes charge it under SU S (2), and we find a conserved current in the (1, 3) of U Sp F (10) × SU S (2). This suggests an enhancement of U T (1) × SU S (2) → U Sp (4). For N s = 2, it is now charged in the (1,14) of U Sp F (10) × U Sp S (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 to affine Lie groups, particularly, A 9 and A (2) 4 . For N f = 4 we find a conserved current only if N s ≥ 2. Specifically for N s = 2 we find a conserved current in the (8, 1) of U Sp F (8) × U Sp S (4). This should lead to an enhancement of U T (1) × U Sp F (8) → U Sp(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 N s = 3 there is a Higgs branch leading to SU (2) + 10F .
When N f = 3 we again find a conserved current only when N s ≥ 2. This current is a flavor singlet, and should lead to an enhancement of U T (1) → SU (2). This is the maximal allowed number of spinors as for N s = 3 there is a Higgs branch leading to SU (2) + 8F . Indeed, in that case, we find a conserved current in the (1, 21) of U Sp F (6) × U Sp S (6). Thus, U Sp S (6) appear to be affinized to C The only other relevant case where we find a conserved current is N f = 1, N s = 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 SU F (2) × U Sp S (8) suggesting an enhanced affine Lie group E (2) 6 . 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.

SO(10)
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 N f = 7, N s = 1 there is a Higgs branch leading to SU (2) + 8F so this theory may have a 6d origin. Indeed, we find two conserved currents with charges ±2 under U S (1) and in the 14 of U Sp F (14). This suggests an enhancement to the affine Lie group A (2) 15 . For N f = 6, the maximal allowed number of spinors is N s = 1. In that case, we find two conserved currents of charges ±2 under U S (1). This suggests an enhancement of U T (1) × U S (1) → SU (2) 2 . The case of N s = 2 has a potential 6d origin, and we find several conserved currents with charges (1, 1) 4 , (1, 1) −4 , (1, 5) 0 , and (65, 1) 0 under U Sp F (12) × SU S (2) × U S (1). The last two suggest an enhancement of U Sp F (12) to the affine A (2) 11 and of SU S (2) to A (2) 2 . In that light, we expect the first two currents to affinize U S (1), though understanding exactly what it leads to requires additional input.
For N f = 5 we find a conserved current only if N s ≥ 2. A Higgs branch analysis reveals that the only relevant case is N s = 2 in which we find a conserved current which is in the 10 of U Sp F (10). This suggests an enhancement of U Sp F (10) × U T (1) → U Sp (12) which also requires two flavor singlet currents with instanton charges ±2.
For N f = 4, we again find conserved currents only if N s ≥ 2. Again, for a 5d fixed point, the only relevant case seems to be N s = 2 where we find a conserved current which is a flavor singlet. This suggests an enhancement of U T (1) → SU (2). The case of N s = 3 is also interesting as it may have a 6d fixed point. In this case, we find two conserved currents of charges 6 2 and6 −2 under SU S (3) × U S (1). This suggests an affinization of U S (3) to C For N f = 2 we find conserved currents only if N s ≥ 4. Since for N s = 4 there is a Higgs branch leading to two copies of SU (2) + 8F , this theory is the only relevant case. In that case, we find two conserved currents of charges ±4 under U S (1) and in the 5 of U Sp F (4). This seems to combine U S (1) × U Sp F (4) to the affine B 3 . For N f = 1, we again find conserved currents only if N s ≥ 4. With that number of spinors, there is always a Higgs branch leading to SU (2) + (4N s − 9)F so the only relevant case is N s = 4 where we find two conserved currents of charges ±4 under U S (1) and in the 2 of SU F (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 N f = 0 case. Again we find conserved currents only if N s ≥ 4. With that number of spinors, there is always a Higgs branch leading to SU (2)+(4N s −11)F so the only relevant case is N s = 4 where we find two conserved currents of charges ±4 under U S (1). This suggests an enhancement of We summarize our results for the conserved currents, and the expected symmetry enhancement, for theories with a potential 5d fixed point, in table 4.

SO(11)
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 5 2 spinors. The case of 5 2 spinors cannot have a 5d fixed point though it may have a 6d one. Nevertheless, we find no conserved currents in this case.
For N s = 2 We find a conserved current only if N f ≥ 1. When N f = 1 this conserved current is a flavor singlet, and is expected to lead to an enhancement of U T (1) → SU (2). For N f = 2, the additional fermionic vector zero modes charge this current in the 4 of U Sp F (4). This suggests an enhancement of U Sp F (4) × U T (1) → U Sp (6), which also requires two flavor singlet currents with instanton charges ±2. Finally, for N f = 3, there 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 SU S 1 (2) × SU S 2 (2) × U Sp F (6). This suggests an affinization of all the flavor symmetries where both SU S (2)'s get lifted to A (2) 2 , and U Sp F (6) get lifted to A (2) 5 . When N s = 3 2 , we find conserved currents only when N f ≥ 5. Since for N f = 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 SU S (2). The current in the 5 is consistent with SU S (2) getting lifted to the affine A 2 , but the singlet does not appear to fit in this group.
For N s = 1, we find conserved currents only when N f ≥ 5. When N f = 5, we find a conserved current which is a flavor singlet. This suggests an enhancement of U T (1) → SU (2). For N f = 6 the additional vector fermionic zero modes furnish this current with the 12 dimensional representation of U Sp F (12). This suggests an enhancement of U Sp F (12) × U T (1) → U Sp (14), which also requires two flavor singlet currents with instanton charges ±2. Finally, for N f = 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 conserved currents with charges 1 4 , 1 −4 , 1 0 , and 90 0 under U Sp F (14) × U S (1). The last one suggests an enhancement of U Sp F (14) to the affine A (2) 13 . We expected the remaining current to also lift U S (1) to 6d though again, fully determining this, requires additional input. The 1 0 current, in particular, doesn't appear to fit in either group.
The last case to consider is N s = 1 2 . 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 identical to the case of N s = 0. More specifically, there are no conserved currents if N f < 7. For N f = 7 there is a conserved current which is a flavor singlet. This suggests an enhancement of U T (1) → SU (2). Finally, for N f = 8 there is a conserved current which is in the 16 of U Sp F (16). This can be accommodated in a finite group, U Sp (18), if in 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.  Like for SO (11), the maximal number of spinor half-hypers, in any combination of chirality, is 5 2 . 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 this case. The other possibilities, N s = 2, N c = 1 2 and N s = 3 2 , N c = 1, with N f vectors, have a Higgs branch leading to U Sp(6) + 1 2 T AS + 1AS + (2N f + 1 2 )F and U Sp(6) + 1T AS + (2N f + 3)F respectively. From the analysis of section 3.3, we conclude that the case of N f = 0 may have a 5d fixed point while the N f = 1 case may have a 6d one.
For the N f = 0, N s = 2, N c = 1 2 case, we find a conserved current which is in the (2, 2) of SU S (2) 2 . This suggests an enhancement of SU S (2) 2 × U T (1) → SU (4). For the N f = 0, N s = 3 2 , N c = 1 case, we find two conserved currents with charges ±3 under U C (1). This suggests an enhancement of Finally, for N f = 1, N s = 2, N c = 1 2 , we find a conserved current which is in the (2, 2, 2) of SU S (2) 2 × SU F (2). This can fit in a finite group leading to an enhancement of (8), assuming there are additional flavor singlet currents with instanton charges ±2. For N f = 1, N s = 3 2 , N c = 1, we find two conserved currents in the 2 of SU F (2) and with charge ±3 under U C (1). This can fit in a finite group leading to an enhancement of SU F (2) × U C (1) × U T (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 them are of the same chirality, and N f = 0. In that case, we find a conserved current which is a flavor singlet. This suggests an enhancement of U T (1) → SU (2).
Adding vectors will now furnish this current in the rank N f antisymmetric irreducible representation of U Sp F (2N f ). So for N f = 1, we find a conserved current in the 2 of SU F (2). This suggests an enhancement of U T (1) × SU F (2) → SU (3). For N f = 2, the current is now in the 5 of U Sp F (4), and we expect an enhancement of U T (1) × U Sp F (4) → SO(7). Finally, for N f = 3 the conserved current is in the 14 of U Sp F (6). This suggests an enhancement of U T (1) × U Sp F (6) → F 4 which also requires two flavor singlet currents with instanton charges ±2.
For N f = 4, there is a Higgs branch leading to SU (2) + 8F , and indeed the conserved current is now in the 42 of U Sp F (8). This cannot be accommodated in a finite Lie group, but rather in the affine E (2) 6 , 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 SU S 1 (2) × SU S 2 (2) × U Sp F (8). This suggests that both SU S (2)'s also get lifted to A 2 . If instead we consider N s = 3 2 , N c = 1 2 then we find no conserved currents unless N f ≥ 4. As for N f = 4 there is a Higgs branch leading to SU (2) + 8F , this is the only interesting case. In this case, we find a conserved current in the 5 of SU S (2). This suggests an affinization of SU S (2) to A (2) 2 while U Sp F (8) appears not to affinize at this level. The last case to consider is N s = N c = 1. In this case we find no conserved currents unless N f ≥ 2. For N f = 2 this current is a flavor singlet, and we expect an enhancement of U T (1) → SU (2). For N f = 3, the additional vector fermionic zero modes charge the current in the 6 of U Sp F (6). This suggests an enhancement of U Sp F (6)×U T (1) → U Sp (8) which also requires two flavor singlet currents with instanton charges ±2.
We next consider the case of three half-hypers. From all the relevant cases, we find only one where there is a conserved current. This case is N f = 6, N s = 1, N c = 1 2 , and we find two conserved currents with charges ±1 under U S (1). These can fit in a finite group, particularly, U T (1) × U S (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 N f ≥ 6. Specifically, for the N f = 6 case, we find a conserved current which is a flavor singlet in both cases. This suggests an enhancement of U T (1) → SU (2). When N f = 7, this current is charged in the 14 of U Sp F (14). So in both cases, we expect an enhancement of U Sp F (14) × U T (1) → U Sp (16) which also requires two flavor singlet currents with instanton charges ±2.
Finally, for N f = 8, there is a Higgs branch leading to two copies of SU (2) + 8F so this theory probably doesn't have a 5d fixed point, but may have a 6d one. In this case, we find different currents in both cases. In the case of N s = 1, N c = 0, we find several conserved currents with charges 1 4 , 1 −4 , and 119 0 under U Sp F (16) × U S (1). The last one suggests an enhancement of U Sp F (16) to the affine A (2) 15 . We expect the first two to affinize U S (1) though it will probably require going to higher instanton order to verify this.
For N s = 1 2 , N c = 1 2 , we find two conserved current one a flavor singlet while the other is in the 119 of U Sp F (16). The last one suggests an enhancement of U Sp F (16) to the affine A (2) 15 , but the first one does not appear to fit in this group. 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.

SO(13)
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 half-hyper 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 generate 256 states that decomposes to the 44 ⊕ 84 ⊕ 128 under the SO(9) unbroken gauge group. Next, we state our results for conserved currents.
A Higgs branch analysis suggests that we cannot have more than two spinor halfhypers. For N s = 1, we find conserved currents only for N f ≥ 3. For N f = 3, this current is a flavor singlet, and we expect an enhancement of U T (1) → SU (2). For N f = 4, this U Sp(16) * Table 6: The enhancement of symmetry for the 5d theory SO (12) Written are the found conserved currents with their representation under the U Sp(2N f ) × SO S (2N s ) × SO C (2N c ) 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.
current acquires the representation 8 under U Sp F (8). This suggests an enhancement of U Sp F (8)×U T (1) → U Sp (10), which also requires two flavor singlet currents with instanton charges ±2.
Finally, for N f = 5, there is a Higgs branch leading to SU (2) + 8F so this theory may have a 6d fixed point. In that case, we find several conserved currents with charges 1 4 , 1 −4 , 1 0 , and 44 0 under U Sp F (10) × U S (1). The last one suggests an enhancement of U Sp F (10) to the affine A (2) 9 . We expect the first two to also affinize U S (1), though the current 1 0 does not appear to fit in either group.
In the case of N s = 1 2 , we find conserved currents only when N f ≥ 7. For N f = 7, this conserved current is a flavor singlet. This suggests an enhancement of U T (1) → SU (2). For N f = 8, this current is now in the 16 of U Sp F (16). This suggests an enhancement of U Sp F (16)×U T (1) → U Sp (18) which also requires two flavor singlet currents with instanton charges ±2. Finally, for N f = 9, we find a conserved current in the 152 of U Sp F (18). This does not fit in a finite Lie group, but rather in the affine A (2) 17 . 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.

SO(14)
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 N s ≤ 1 so the only relevant case is N s = 1. The same analysis also suggests that we must have N f ≤ 5. We find conserved currents only when N f ≥ 4. When N f = 4 this current is a flavor singlet, and we expect an enhancement of U T (1) → SU (2). When N f = 5, this current is in the 10 of U Sp F (10). This suggests an enhancement of U Sp F (10) × U T (1) → U Sp (12), which also requires two flavor singlet currents with instanton charges ±2.
Finally, for N f = 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 currents with charges 1 4 , 1 −4 , and 65 0 under U Sp F (12) × U S (1). The last one suggests an enhancement of U Sp F (12) to the affine A (2) 11 , and we expect the first two to also affinize U S (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 [34] with the properly symmetrized Hilbert series of the corresponding instanton moduli space [35]. Fortunately, methods for the calculation of the Hilbert series for the instanton moduli spaces of exceptional groups N f N s Currents Minimal enhanced symmetry 4 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.
are known [36]. Using their results we find that the 1-instanton Nekrasov partition function has the following form: where h G is the dual Coexter number of the group G and χ[k] stands for the character of the irreducible k-symmetric product of the Adjoint (in Cartan weights these are given by [0, k] for G 2 , [k, 0, 0, 0] for F 4 , [0, k, 0, 0, 0, 0] for E 6 , [k, 0, 0, 0, 0, 0, 0] for E 7 and [0, 0, 0, 0, 0, 0, 0, k] for E 8 ). This result has the following simple interpretation. The infinite series and the denominator come from applying the 4h G bosonic zero modes on a single instanton ground state. This state in turn contributes x h G to the index. Next, we discus each group in turn, stating our results.

G 2
Instantons of G 2 are contained in an SU (2) subgroup breaking G 2 → SU (2) × SU (2). Under this breaking the Adjoint 14 of G 2 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 ground state, whose r-charge is −1, with the fermionic operator B i in the 4 of the unbroken SU (2) gauge group. We can form an SU (2) gauge invariant by contracting two B i 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 x 4 to the index. Recalling that h G 2 = 4, this matches our expectations from the partition function.
Next, we generalize by adding hypermultiplets in the 7 of G 2 . 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 N f irreducible antisymmetric representation of the U Sp(2N f ) 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 B i '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 B i '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 N f < 4 there are no conserved currents. When N f = 4 there is one conserved current which is a flavor symmetry singlet. This should lead to the enhancement U T (1) → SU (2).
When N f = 5 there is a conserved current which is in the 10 of the U Sp F (10) flavor symmetry. This should lead to the enhancement U T (1) × U Sp F (10) → U Sp (12). This also requires conserved currents with instanton number ±2 that are flavor symmetry singlets. Finally, when N f = 6 the conserved current carries a rank 2 irreducible antisymmetric tensor of the U Sp F (12) flavor symmetry. This cannot form a finite Lie group, but rather seems to lead to the affine A (2) 11 . This suggests that this theory lifts to 6d. We can also consider the maximally supersymmetric case where we add a hypermultiplet in the 14 of G 2 . The fermionic zero modes supplied by the Adjoint hyper form 4 raising operators which together with B i form a doublet of SU (2) ⊂ SO R (5). The basic gauge invariant one can form is given by contracting two such operators. This results in a gauge invariant raising operator that is in the 3 of SU (2) ⊂ SO R (5). Acting on the ground state, with r-charge −2, with these operators generates 14 states that form the rank 2 symmetric traceless representation of SO R (5).
The general thought is that this theory lift to the 6d (2, 0) theory of type D 4 where a Z 3 twist in the outer automorphism of D 4 is imposed when going around the circle [31]. Recall that SO(8) has invariant polynomials of degree 2, 6 and two distinct ones of degree 4. Under the Z 3 twist the polynomials of degree 2, 6 are even, and so contribute at the massless level. This correctly reproduces the expected operators of G 2 whose invariant polynomials are of degrees 2 and 6.
On the other hand, the two polynomials of degree 4 are rotated by 120 0 [37]. Thus, these states are all massive with mass ∼ 1 where the lightest states having a mass of where M KK is the mass of the first KK state given by the states corresponding to the invariant polynomials even under the twist. The first instanton contribution should correspond to the state with mass M KK 3 and so we expect it to give a KK supermultiplet in the 14 of SO R (5). This indeed matches our results.

F 4
Instantons of F 4 are contained in an SU (2) subgroup breaking F 4 → SU (2) × U Sp (6). Under this breaking the Adjoint 52 of F 4 decomposes into the Adjoints of SU (2) and U Sp(6) as well as a state in the (2,14 ) representation. Thus, the full state space is given by acting on the ground state, whose r-charge is − 7 2 , with the fermionic operator B i in the 14 of the unbroken U Sp(6) gauge group. We can form a U Sp(6) gauge invariant by contracting two B i operators. By applying these on the ground state we get a tower corresponding to one state with R-charge 7 2 . Tensoring this with the basic broken current supermultiplet, we get a BPS state, which we expect to contribute to the index as x 9 . Recalling that h F 4 = 9, this matches the expectations from the partition function.
We can generalize by adding hypermultiplets in the 26 of F 4 . Each one of these contributes a raising operator in the 6 of the unbroken U Sp(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 analysis, we find that this is only possible if N f > 2. For N f = 3, the conserved current is a flavor singlet suggesting an enhancement of U T (1) → SU (2).
Incidentally, N f = 3 is the maximal possible number of flavors, since beyond this, one can flow on the Higgs branch to theories that are thought to possess neither a 5d nor a 6d fixed point. Indeed, solving numerically for N f = 4, we find a conserved current in the 120 of U Sp F (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 SO R (5). We expect this theory to lift to the 6d (2, 0) theory of type E 6 . The reduction to 5d is done with a Z 2 outer automorphism twist under which the E 6 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 SO R (5), respectively. This indeed matches our results.

E 6
Instantons of E 6 are contained in an SU (2) subgroup breaking E 6 → SU (2) × SU (6). Under this breaking the Adjoint 78 of E 6 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 on the ground state, whose r-charge is −5, with the fermionic operator B i in the 20 of the unbroken SU (6) gauge group. We can form an SU (6) gauge invariant by contracting two B i 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 x 12 . Recalling that h E 6 = 12, this matches the expectations from the partition function.
We can generalize by adding hypermultiplets in the 27 of E 6 . 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 conserved current. Using numerical analysis, we find that this is only possible if N f = 4 in which we find a single conserved current which is a flavor singlet. This leads us to expect an enhancement of U T (1) → SU (2) in this case. We do not expect a fixed point, 5d or 6d, to exist when N f > 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.

E 7
Instantons of E 7 are contained in an SU (2) subgroup breaking E 7 → SU (2) × SO (12). Under this breaking the Adjoint 133 of E 7 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 B i in the 32 of the unbroken SO(12) gauge group. we can form an SO(12) gauge invariant by contracting two B i 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 current supermultiplet, we find a single state with R-charge 9. Recalling that h E 7 = 18, this matches our expectations from the partition function.
We can generalize by adding hypermultiplets in the 56 of E 7 . 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 operators in the 6 1 of U (1) × SU (6) ⊂ SO (12). Applying them on the ground state yields 64 states transforming as the 32 ⊕ 32 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 Z 2 × Z 2 center. This contains three distinct Z 2 subgroups, each of which act on two of the basic representations: 12, 32, and 32 , 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 B i 's and thus cannot be an R-charge singlet. 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.

E 8
Instantons of E 8 are contained in an SU (2) subgroup breaking E 8 → SU (2) × E 7 . Under this breaking the Adjoint 248 of E 8 decomposes into the Adjoints of SU (2) and E 7 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 B i in the 56 of the unbroken E 7 gauge group. we can form an E 7 gauge invariant by contracting two B i 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 x 15 to the index. Recalling that h E 8 = 30, this matches our expectations from the partition function.
The fundamental 248 representation of E 8 is identical to the Adjoint so upon adding a single hyper we get the maximally supersymmetric E 8 Yang-Mills theory, which should lift to 6d.

Conclusions
In this paper we explored symmetry enhancement from 1 instanton operators for U Sp(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 [28]. 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 [30]. It is interesting to know whether the other cases also flow to 5d fixed points.
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 U Sp(2N ) + AS + 8F the 6d UV theory is known. It is interesting to see if we can understand the 6d lift also in the other cases.