Duality and enhancement of symmetry in 5d gauge theories

We study various cases of dualities between $\cal{N}$$=1$ 5d supersymmetric gauge theories. We motivate the dualities using brane webs, and provide evidence for them by comparing the superconformal index. In many cases we find that the classical global symmetry is enhanced by instantons to a larger group including one where the enhancement is to the exceptional group $G_2$.


Introduction
Gauge theories in 5d are non-renormalizable and so seem to require a UV completion. However, in the N = 1 supersymmetric case, and for specific gauge and matter content, it is possible that the theory flows to a UV fixed point removing the necessity for a UV completion [1][2][3]. This follows since for 5d N = 1 gauge theories the low-energy prepotential on the Coulomb branch is at most cubic, and receives only one-loop corrections. Thus, the effective coupling takes the following rough form: where g 2 0 is the bare Yang-Mills coupling and c is the full Chern-Simons coupling which includes both the classical value and one-loop corrections. If the matter content is such that the right hand side of (1) is positive everywhere on the Coulomb branch, then one can take the limit g 2 0 → ∞ and a fixed point may exist.
The simplest example is an SU (2) gauge theory with N f < 8 flavors which exhibits another feature of 5d gauge theories, enhancement of symmetry. Besides the flavor symmetry, every non-abelian gauge group has an associated conserved current given by: j ∼ T r F ∧F , which is topologically conserved. The particles charged under it are instantons which are particles in 5d. In the SU (2) gauge theories with N f < 8 flavors it is believed that there is an enhancement of the classical global symmetry U (1) × SO(2N f ) to E N f +1 [1]. This stems from a string theory description as well as from their index which forms characters of E N f +1 [4][5][6][7].
In the case of pure SU (2) there is another theory, dubbedẼ 1 , with no enhanced symmetry. This theory differs from the case with the E 1 symmetry by a discrete θ angle as π 4 (SU (2)) = Z 2 [2]. This discrete parameter also exist for general U Sp(2N ) as π 4 (U Sp(2N )) = Z 2 . If fundamental flavors are present then this angle can be changed by switching the mass sign for an odd number of flavors, and so is no longer physical.
In some cases a fixed point may exist even though the effective coupling blows up, and thus there is a singularity, away from the origin of the Coulomb branch. Quiver theories provide such an example as in these theories when going along the Coulomb branch of one group, the other one will eventually become strongly coupled, and a singularity is encountered. However, it is argued in [8,9] that the theory may still have a fixed point, and the singularity is due to a state becoming massless. Then the theory is better described in terms of a dual theory, and thus one achieves a continuation past infinite coupling.
A concrete realization of this is given by using brane webs [8,10]. These can be used to describe such quiver theories as for example the web of figure 1. Going on the Coulomb branch, by expanding one of the faces of the web, one sees that the other face shrinks, and eventually a strong coupling singularity is encountered. Nevertheless, one can now do an S-duality resulting in the web of figure 1 (c). Note, that at that point a D-string becomes massless implying that an instanton of the quiver theory becomes massless. Hence, this suggests that quiver theories can exist as microscopic 5d theories, and that their strong coupling singulaities can be resolved by switching to a dual weakly coupled description. A simple example of this is the SU (2) × SU (2) theory with a hypermultiplet in the bifundamental representation, whose dual is SU (3) with two fundamental hypers shown in 1. For a complete characterization of the duality we also need to state the SU (3) Chern-Simons level and the θ angle for each SU (2) 1 . As worked out in [11], the angles for the SU (2)×SU (2) theory are (π, π) and the Chern-Simons level for the SU (3) is 0. We will denote these as SU 0 (3)+2F and SU π (2)×SU π (2) where a bifundamental is understood to exist whenever a × is written.
A natural question then is can we find evidence for this duality. One can test these duality conjectures by comparing the superconformal indices [12] of the two theories which must match if the theories are dual. Indeed, This was done in [11] for this case, as well as several generalizations, finding complete agreement. In this paper we continue to explore this subject motivating several additional dualities, and interesting cases of enhancement of symmetry. The main tool is the superconformal index which we calculate to reveal the full global symmetry, and compare it between proposed dual theories.
This article is organized as follows. Section 2 reviews the definitions and the methods for calculating the 5d superconformal index. In section 3 we discuss the generalization of the duality for SU (2) × SU (2) by adding two flavors, that is 1F +SU (2)×SU (2)+1F and SU π (2)×SU (2)+2F . Section 4 concentrates on symmetry enhancement in SU (2) × U Sp (6). Section 5 deals with generalizations by adding an SU (3) group, that is to theories of the form SU (2) × SU (3) × SU (2). Section 6 comprises our conclusions. Finally, in the Appendix we discusse the identification of the gauge theory from the web, particularly the determination of the CS levels and θ angles.

The superconformal index
The superconformal index is a characteristic of superconformal field theories [12]. It 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. Thus it achieves being a characteristic of a superconformal theory as besides this merging the numbers of BPS operators cannot change under continuous deformations. Besides directly counting the operators the index can also be evaluated by a functional integral where the theory is considered on S d−1 × S 1 .
Specifically for 5d field theories 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 [4] the index is: Here 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. The index can be evaluated from the previously mentioned path integral using the method of localization. In the case at hand the localization procedure was done in [4]. The result is that the index can be divided into two parts. The first is the perturbative part coming from the one loop determinant one gets when evaluating the saddle point. It depends on the field content of the theory. We will only be interested in hypermultiplets and vector supermultiplets which contribute: where m i are the fugacities associated with the i'th flavor and α are gauge fugacities. The sum in (3) is over the roots of the Lie groups and the first sum in (4) is over the weights of the appropriate flavor representations. This builds what is called the one particle index. In order to evaluate the full perturbative contribution one needs to put this in a plethystic exponent which is defined as: where the · represents all the variables in f (which in our case are just the various fugacities). The second part comes from instantons. At the north or south pole of S 4 the localization conditions are somewhat more lax than elsewhere on the sphere, and point-like instantons (anti-instantons) localized at the north (south) pole are consistent with the localization conditions. Therefore they must also be included in the index. This is done by integrating over the full instanton partition function.
Finally in order to calculate the full index we take the perturbative result given by (5) with the one particle index as f . This needs to be multiplied by the instanton contributions and integrated over the gauge group.
The contributions of the instantons are expressed as a power series in the instanton number k: where we have called the U (1) Inst. fugacity a. These express the contributions of insantons localized at the north pole. Likewise there will be contributions of the south pole, which is just the complex conjugate of that for the north pole. So the full instanton contribution is given by |Z inst | 2 . Thus, calculating the instanton contributions reduces to calculating Z k which is generically the hardest part of the computation. We will expand the index in a power series in x, and calculate to a finite order. This has the advantage as Z k ≈ x c(k) where c is an increasing function of k. Hence, to a finite order in x only finitely many instantons are needed. The partition functions Z k are the 5d version of the Nekrasov partition function for the k instantons [13] which is expressed as an integral over what is called the dual gauge group 2 . The contributions to the integrand come from the gauge degrees of freedom and from flavors charged under the group. The exact form of these, for the group and matter contents that we will need, can be found in [4,7,11]. As these are quite lengthy we will not reproduce them here.
The integral can then be evaluated using the residue theorem once supplemented with the appropriate pole prescription which determines which poles should be taken. The poles can be classified depending on whether they originate from the contributions of the gauge group, matter content, or are poles at zero or infinity. The prescription for gauge group associated poles can be found in [4,7,11]. Matter representations other than the fundamental also add poles to the integral, and the correct prescription for dealing with them can be found in [7]. Finally, there can be poles at zero or infinity whose prescription will be mentioned shortly.
There are several problems encountered when calculating instanton contributions. The most pertinent to our case are two issues that appear for U (N ) groups and are thought to occur because of the failure of the U (1) part to decouple. First, there is a sign discrepancy between the U (N ) and SU (N ) results of (−1) κ+ N f 2 where κ is the bare Chern-Simons level. Second, there are sometimes contributions from decoupled states that must be removed. A thorough discussion of these problems can be found in [5][6][7]11].
The first problem was dealt with by changing the signs by a factor of (−1) κ+ N f 2 . Dealing with the second one requires identifying the decoupled states and removing their contributions. This can be easily achieved if there is a brane web description where it is manifested by the existence of parallel external legs. There is a decoupled D-string state associated with these legs which is the state we need to mod out.
From a field theory perspective this is seen as a lack of invariance under the superconformal group x → 1 x , and under flavor symmetries if these are not realized explicitly in the integrand. For example, in the case of SU (2) × SU (2) + 2F , which we discuss later, the integrand shows a global U (1) bifundamental symmetry, which is the correct global symmetry for U (2) × U (2) + 2F , but the bifundamental global symmetry is actually SU (2). The invariance under both the superconformal group and the full classical global symmetry is only achieved once these states are modded out. The removal of these states is generally achieved by: where the sum runs over the decoupled states, and m i , q i are their flavor and topological charges respectively. The previously mentioned poles at zero or infinity are related to these decoupled states and only appear where these states are present. As a result these poles can be either included or not, and the change is then absorbed in the removal factor. The expression (7) is valid when all these poles, that are within the contour, are included.
We used a brane web description to determine the number of such decoupled states where there is one for every pair of parallel external branes. We then used the web as well as the constraints coming from x → 1 x invariance for the 1-instanton to fully determine m i . Then the full partition function is determined via (7). As a consistency check we verified that all the partition functions we used are invariant under x → 1 x , and form characters of the classical global symmetry.
Finally, in the case of SU (2), there are two different ways one can calculate the index depending on whether one uses the expressions for U groups or for U Sp groups which stems from the fact that SU (2) = U Sp (2). Since the moduli space is realized differently in both cases the dual gauge groups and integrands are different even though the final results must agree. We denote these two different approaches as the U and U Sp formalisms. With the exception of section 5, we have employed the U formalism to calculate SU (2) instantons. In the U formalism the group is regarded as U (2) and reduction to SU (2) is done by setting the overall U (1) fugacity to 1. As previously explained, one also has to remove additional remnants of this U (1), such as decoupled states, to get the correct result.
In the U formalism one can naturally add a CS level. This again follows because the theory considered is U (2) where such a term is possible in contrary to SU (2). When reducing to SU (2) one finds that this CS level determines the θ angle of the SU (2), where in general changing the CS level by one changes the θ angle by π. By explicitly comparing the resulting par-tition function with the one evaluated with the U Sp formalism, where a θ angle can be naturally accommodated, one finds that CS level 0 corresponds to θ = 0 while CS level 1 corresponds to θ = π. The addition of flavor shifts this identification by 1 2 . So, for example, for N f = 2 CS level 0 corresponds to θ = π and for N f = 3 CS level 1 2 corresponds to θ = π and CS level − 1 2 corresponds to θ = 0. When flavors are present the difference between the angles can be undone by redefining the flavor fugacities. Nevertheless, it can be important if the flavors are provided by bifundamentals.

Adding more flavor
In this section we consider the extension of the duality between SU π (2) × SU π (2) and SU 0 (3) + 2F by adding additional flavors. The generalization to one extra flavor, that is to SU π (2) × SU (2) + 1F , was already considered in [11], where the dual was proposed to be SU ± 1 2 (3) + 3F . We extend this to the case of two extra flavors 3 . We now have a choice on the SU (2) × SU (2) side of whether to have the two flavors under the same group or one under each. The starting point for the two cases are the brane webs shown in figure 2 and 3. Examining their S-duals we conjecture that: In both cases, the classical global symmetries do not agree, but there is an instanton driven enhancement leading to the same quantum symmetries. The classical global symmetry of SU π (2)×SU (2)+2F consists of the topological symmetries, U I 1 (1) for the flavored group and U I 2 (1) for the unflavored group, and the flavor symmetries which are SU M (2) for the bifundamental and SU F 1 (2)×SU F 2 (2) = SO(4) for the two flavors. The classical global symmetry of 1F + SU (2) × SU (2) + 1F consists of two topological U (1)'s, two flavor U (1)'s, and the SU M (2) of the bifundamental. Both SU (3) theories have a topological U T (1), a baryonic U B (1) and an SU (4) flavor symmetry.
In the case of (8), the 1-instanton of the flavored SU (2) gauge group leads to an enhancement of . This can be understood as this gauge group sees effectively 4 flavors and so, ignoring the gauging of the first SU (2) for a moment, leads to an E 5 = SO(10)  global symmetry. However, an SU (2) inside this SO(10) is actually a gauge symmetry leading to the breaking SO(10) This doesn't match the global symmetry of the SU (3) theory, but on that side there is an enhancement of a combination of U I (1) and U B (1) to SU (2). The appropriate combination is the diagonal if κ = 1 and the anti-diagonal if κ = −1. This enhancement is related by flow, when the flavors are given a mass, to the enhancement in SU ±3 (3) found in [11]. Thus, this theory also has U (1) × SU (2) × SU (4) global symmetry. Note that in this example both theories have undergone symmetry enhancement, where the enhanced symmetry on one side is realized perturbativly on the other side.
In the case of (9) there is an enhancement of the bifundamental SU (2) and two U (1)'s, which are combinations of the topological and flavor ones for both groups, to SU (4). As we will show from the index calculation, this is brought by the (1,0) + (0,1) + (1,1)-instantons, and is similar to the enhancement to SU (4) of the SU 0 (2) × SU 0 (2) theory found in [11]. There is no enhancement on the SU (3) side and so the symmetries match, both theories having a U (1) 2 × SU (4) global symmetry.
The discrete symmetries of the two theories also match. In particular, in (9) there is a symmetry of exchanging the two groups which has no analog in the SU (3) theory. However, that theory has charge conjugation symmetry with no analog on the quiver side. The duality identifies the two discrete symmetries, similarly to the case without the flavors [11].

Index calculation
In the rest of this section we calculate the indices for these 4 theories and compare them, giving further support to the above discussion.
We start with the case of SU π (2) × SU (2) + 2F . We use q, t for the instanton fugacities (t for the flavored group), z for the bifundamental SU M (2), and c, l for the SU (2) × SU (2) flavor symmetry. As can be seen from figure 2, There is a problem with parallel branes so we removed the two decoupled states by: These match the two decoupled D-strings seen in figure 2 (a). The flavor charges arise due to fermionic zero modes. Using (10) we calculate the index of this theory. We worked to order x 5 which requires the contributions from the (1,0)+(0,1)+(2,0)+(1,1)+(0,2)+(1,2) instantons. Other instantons do not contribute as they enter at higher order in x, or else they carry gauge charges and form gauge invariants only at higher orders. We find: where we have presented the results only to order x 3 to avoid over cluttering, although we calculated to order x 5 . One can read the resulting global symmetry from the x 2 terms. There are the perturbative currents spanning the classical U (1) × U (1) × SU (2) × SU (2) × SU (2) symmetry, and then there are also the 8 states coming from the (0,1)-instanton. These provide the necessary currents to enhance U (1) × SU (2) × SU (2) to SU (4) suggesting that the global symmetry is made of a U (1) (spanned by q), an SU (2) (spanned by l) and an SU (4) (spanned by z, c and t). Indeed, as we will show, the index can be written in characters of SU (4), at least to the order we are working in.
Next we turn to the SU 0 (3) + 4F theory. There are no problems with either parallel branes or signs. We use a for the instanton fugacity, and span the U F (4) by: We separate the index into a perturbative contribution, which is identical also in the SU ±1 (3) + 4F case, and an instanton contribution. The perturbative contribution is: Next are the instanton contributions. For SU 0 (3) + 4F only the 1instanton contribute at this order for which we find: where we labeled the SU (3) instanton fugacity by a.
It is now apparent that there is no enhancement in the SU 0 (3) + 4F (no x 2 term in (14)) so this theory cannot be dual to SU (2) × SU (2) + 2F . This is in accordance with the web picture which suggests the dual to be SU ±1 (3)+4F . The two theories differ by the contributions of their instantons. In the SU ±1 (3) + 4F there are parallel external branes and one must mod out the decoupled U (1) state by: where we have chosen a positive Chern-Simons level (the expression for the negative case can be generated by charge conjugating the result for the positive case). Using these we can calculate the instanton contribution for this theory where to this order we get contributions from the 1-instanton, entering at x 2 , and the 2-instanton, entering at x 4 . We find: Note the x 2 instanton contribution which enhances the diagonal instantonbaryonic symmetry to SU (2). Now comparing the x 2 terms one can see that the indices indeed match to that order if we take t = b p and l = √ abp.
Furthermore, the matching of the x 3 terms demands q = a b 3 p . With this mapping we find that the two indices match to order x 5 .
As suggested by the duality, the index can be written in characters of the quantum symmetry U (1) × SU (2) × SU (4): where we have used the notations χ y [d] for the character of the d dimensional representation of SU y (2), and χ q [d 1 , d 2 ] for the character of a state in a d 1 dimensional representation of SU F (2), a d 2 dimensional representation of SU (4) and with charge q under the remaining U (1). In terms of the classical U (1)'s, the remaining U (1) is spanned by √ a bp which we have normalized to be charge one (this is in terms of the SU (3) variables where in the SU (2)×SU (2) case it is spanned by q √ t). Finally, we note that for the 20 and 84 of SU (4) the dimension is not enough to fix the representation so we should add that these are the ones corresponding to the Cartan weights (0, 2, 0) and (2, 0, 2) respectively.
Next we turn to the 1F + SU (2) × SU (2) + 1F theory which the previous argument suggests should be dual to SU 0 (3) + 4F . As we are used to by now, there are decoupled D-strings that must be removed, the exact form depending on the chosen U (2) CS terms which is reflected in the web. We use the web shown in figure 3 so the required correction is: (18) where we have used z again for the bifundamental fugacity, t and q for the instanton fugacities and l and j for their respective flavors. In the field theory this corresponds to taking κ = ( 1 2 , 1 2 ). There is also another web, not related by an SL(2, Z) transformation to the one in figure 3, with a different spectrum of decoupled states which corresponds to the case of κ = ( 1 2 , − 1 2 ) in the field theory (this is similar to the flavorless case with θ 1 = θ 2 = 0 [11]). We have checked that both methods give the same results at least to the order we are working in.
The decoupled states in (18) correspond to the three possible different D-strings connecting the three parallel NS5-branes in 3 (a). The charges of these states under the instanton and bifundamental symmetries can be inferred by examining their behavior under changing of the positions of the external NS5-branes (where moving the first and last branes corresponds to changing the coupling constants of the two groups and moving the middle one is related to changing the bifundamental mass). The additional flavor charges arise from fermionic zero modes.
The index can be written in characters of the global U (1) × U (1) × SU (4) symmetry where it reads: where the notation χ SU (4) [d] stands for the d dimensional representation of SU (4). For the remaining U (1)'s we have used the notation of the SU (3) theory though they can be easily mapped to the corresponding quiver ones. Like in the previous case some of the SU (4) representations are ambiguous, and are the same as stated above.
4 Enhancement of symmetry in SU (2)×U Sp (6) In this section we explore enhancement of symmetry in theories of the form SU (2) × U Sp(2 + 2M ). The cases M = 0, 1 where covered in [11], and for M > 2 one doesn't expect a UV fixed point to exist [1], so we concentrate on the case M = 2, that is SU (2) × U Sp (6). As we are mainly interested in symmetry enhancement, we take the SU (2)'s θ angle to be 0, and leave the U Sp(6)'s angle unspecified for the moment.
There are several problems with calculating the instanton contributions for this theory. First, for U Sp(6) we must use the U Sp formalism and one then encounters problems when evaluating digroup instantons [11]. As a result, we will ignore their contributions seeing what we can learn just from states neutral under the U Sp(6) topological symmetry. Thus, we consider only instantons of the SU (2) theory. These are essentially identical to instantons of SU (2) + 6F with part of the SO(12) global symmetry identified with the U Sp(6) gauge symmetry. We use the U Sp formalism to take the SU (2) instantons into account, but the Sp formalism suffers from a problem here 4 . Specifically, the result one finds for the 2-instanton partition function of U Sp(2)+6F is not x → 1 x invariant similarly to what happens in the problem with parallel legs in the U formalism. This can be fixed by correcting the partition function by: where we have denoted the instanton fugacity by q. Using this one can recover the index for U Sp(2) + 6F as predicted in [4] and evaluated by [5][6][7].
We evaluate the index to order x 5 , requiring the contributions of the (1,0)+(2,0)+(3,0)+(4,0)-instantons. The lowest order terms in the index are: One can see the conserved currents of the classical global symmetry as well as instanton contributions are exactly the ones necessary to enhance U I (1) × SU M (2) → G 2 , where the spanning is such that: 7 = z 2 + 1 + 1 z 2 + (z + 1 z )(q + 1 q ). Using this it is possible to show that the index can be written in G 2 characters as: where we employed the notation χ[d] for the d-dimensional representations of G 2 . As there are two 77 dimensional representations of G 2 , both appearing in the index, we have added their Cartan weights. This strongly suggests that the theory has an enhancement of symmetry to G 2 .
So far we have not considered states charged under the instanton U (1) of the other group, and thus the results are independent of the U Sp(6)'s θ angle. Including these states requires dealing with the problems of digroup instantons in the U Sp formalism. We postpone this for future study.

Inserting an SU (3) group
In this section we concentrate on generalizations where we add an SU (3) group between the two SU (2)'s so that the gauge group is SU (2) × SU (3) × SU (2). Next, we need to choose the level of the SU (3) CS term. There are two possible choices for which there is a brane web without self intersecting branes. These are the level 0 case, shown in figure 4 (a), and the ±1 case, shown in figure 5 (a). Figures 4+5 (b), show the web after a large mass has been given to the bifundamentals. From this the gauge content becomes evident, and it is possible to read the CS level, as explained in the Appendix. A natural question is then whether there are more discrete parameters, particularly the θ angles. Each SU (2) group has such a discrete parameter, but there are massless flavors in the theory, the two bifundamentals. The bifundamentals imply the θ angles can be absorbed into their mass sign by doing charge conjugation. However, this changes the sign of the SU (3) CS level, and also changes both angles simultaneously. Thus, when the CS level is zero there is a single discrete parameter given by the relative angle, θ 1 − θ 2 . When the CS level is non-zero both angles are physical, but a theory with CS level and angles (θ 1 , κ, θ 2 ) is related by charge conjugation to one with (θ 1 + π, −κ, θ 2 + π) and so is physically equivalent. This is also reflected in the brane webs, which one can deform so as to change both angles. However, it is not possible to change one of them, while keeping the SU (3) CS term fixed. The angles can now be determined from the webs in figures 4, 5 (b) as explained in the Appendix. One can also draw webs corresponding to other choices of the angles, but these don't appear to have gauge theory duals, and will not be considered here.
Next, we can do S-duality to both theories leading to the webs depicted in figures 6,7. From these we conjecture the following dualities 5 : In the case of (24), the classical global symmetries match, where in both cases it is U (1) 5 consisting of topological, baryonic and bifundamental U (1)'s. Nevertheless, in both cases we will show that there is an enhancement of U (1) × U (1) → SU (2) × SU (2). In the theory on the right this follows since each SU ±1 (3) sees 4 flavors leading to the same enhancement as in section 3. For the theory on the left the enhancement is brought about by the instantons of each SU (2) group. Concentrating on one of these for a moment, this SU (2) gauge group sees 3 flavors. If we ignore the gauging of SU (3), we would get an enhanced SU (5) symmetry. However, as an SU (3) inside it is actually a gauge symmetry only the commutant U (1) × SU (2) is realized as a global symmetry. The same thing also occurs in the other SU (2) gauge group leading to the said enhancement. Thus, the quantum global symmetry of these theories is U (1) 3 × SU (2) 2 .
In the case of (25), the classical global symmetries do not match, but the quantum symmetries match. In the theory on the left, The classical global symmetry is again U (1) 5 . Like the previous case, the SU (2) instantons lead to an enhancement of U (1) × U (1) → SU (2) × SU (2), but now there is one more enhanced SU (2) coming from the middle SU ±1 (3) (which sees effectively 4 flavors).
The theory on the right has classical global symmetry of U (1) 4 × SU (2). In addition there is an enhancement of U (1) × U (1) → SU (2) × SU (2) coming from the instantons of the SU (2) group. This follows as the SU (2) sees 4 flavors and, ignoring the gauging of SU (4), gives an enhancement to E 5 = SO(10). However, part of this symmetry is actually the gauge SU (4) = SO(6) and not a global symmetry. There are two possible embeddings of SO(6) inside the SO(10) depending on whether the latter is broken to SO(4) × SO(6) or SO(2) × SO(6) × SO(2) which are in one to one correspondence with the SU (2) θ angle. The choice SO(4)×SO(6) corresponds to the θ = 0 case, and indeed gives the said enhancement. Overall, the quantum symmetry in both theories is SU (2) 3 × U (1) 2 .
Finally, the discrete symmetries also match. In (25), the SU (2)×SU (3)× SU (2) theory is invariant under exchanging the two end groups which has no analogue in the SU (2) × SU (4) theory. However, this theory is charge conjugation invariant while the SU (2) × SU (3) × SU (2) theory is not. The duality should identify these symmetries. In (24), both theories are invari- ant under a combination of charge conjugation and exchanging the two end groups.

Index calculation
Now we want to test these conjectures by comparing the superconformal indices of the theories. As explained in section 2, The calculation is done from the U perspective with a correction for the sign and parallel legs problem. The θ angles are taken into account by the U (2) CS term. We start with the SU (2) × SU (3) × SU (2) theory of (24). We use the fugacity spanning shown in figure 8. As there are many instantons involved we worked only to order x 4 . We also break the index into several parts depending on the contributing sector so as to make the results more presentable. We find: for the perturbative part. Next we add the instantonic contributions starting with instantons of combined order 1: the (1,0,0)+(0,1,0)+(0,0,1)-instantons. Their contribution is: Figure 8: The fugacity allocations for SU π (2) × SU 0 (3) × SU 0 (2). The two circles are the SU (2)'s, the square is the SU (3) and the lines are the bifundamentals. The letter above the lines are the ones for the appropriate bifundamental fugacity, and the ones inside the circles are for the topological fugacities.
One can see that these provide the states necessary to enhance U (1) × U (1) → SO (4). To the order we are working, we also need the contributions of the (1,1,0)+(1,0,1)+(0,1,1)+(2,0,0)+(0,0,2)+(1,1,1) instantons. These provide: This completes the index to this order. Next we shall compare it with the one for 1F + SU 1 (3) × SU −1 (3) + 1F starting with the perturbative part: where the fugacities are allocated as in figure 9. Next are the instanton contributions. To the orders we are working in we only need the (1,0)+(0,1)+(1,1) instantons which contribute: One can see that the instantons provide exactly the needed states to bring about the enhancement of U (1) × U (1) → SO(4) as required for the two theories to be dual. The matching now requires b At order x 3 one can see that setting l = √ p √ p √ f render the two equal. With this the indices also match to order x 4 completing the matching.
Note that there is one U (1) combination left undetermined as there is no state charged under it to this order. The index can be written in terms of SO(4) characters as: where we used χ[d 1 , d 2 ] for the SO(4) representation of dimension d 1 under one SU (2) and d 2 under the other. For the U (1)'s we have used the SU π (2) × SU 0 (3) × SU 0 (2) notation though they can be easily transformed to the SU (3) 2 ones using the above relations. The three last U (1)'s seem to be l, b z and qt. Next we turn to the theory in (25), SU π (2) × SU −1 (3) × SU π (2). We use the fugacities and CS choices shown in figure 10. Again we divide the index into a perturbative part, one instanton part and higher instantons. The perturbative part is just given by (26). The one instanton part, including the (1,0,0)+(0,1,0)+(0,0,1) instantons, is: Figure 10: The fugacity allocations for SU π (2) × SU −1 (3) × SU π (2). The two circles are the SU (2)'s, the square is the SU (3) and the lines are the bifundamentals. The letter above the lines are the ones for the appropriate bifundamental fugacity, and the ones inside the circles are for the topological fugacities.
Index SU (2)×SU (4) where again the notation χ[d 1 , d 2 , d 3 ] represents the representation dimensions under each SU (2) where the first is spanned by a B 2 and the last by aB 2 . The index is written for SU (2) × SU (4) + 2F though it can be easily mapped to the SU π (2) × SU −1 (3) × SU π (2) theory by the above relations. The two remaining U (1)'s seem to be spanned by f and A √ a.

Two extra nodes
We now turn to generalizations of these dualities by the addition of an extra SU (3) group. Concentrating only on cases with gauge theory duals and without crossing external legs, we find 3 distinct cases. In one case, depicted in figure 11, the dual is an SU 1 (4) × SU −1 (4) gauge theory with 2 fundamentals for each group. In another case, shown in figure 12, the dual is an SU 1 (3) × SU 0 (3) × SU −1 (3) gauge theory with a fundamental hypermultiplet for each edge group. These two are the generalizations of (24). There is also a generalization of (25), illustrated in figure 13, where the dual is an SU (5)×SU (3) gauge theory with 3 fundamentals for the SU (5) and one for the SU (3). In all cases the dual is an SU (2)×SU (3)×SU (3)×SU (2) gauge theory differing by the choices of θ angles and CS terms. These can in turn be read from the web suggesting the following dualities: Interestingly, the difference between dualities (38) and (39) is in the orientation of the CS terms relative to the SU (2) θ angles. By changing the mass sign of all the bifundamentals, we can change both θ angles and the sign of the CS terms. Thus, we expect 8 physically different 2 . These are distinguished by the orientations of the CS levels and θ angles relative to one another and themselves. The remaining 5 appear not to posses gauge theory duals and won't be considered here.
In the rest of this section we begin exploring these dualities by matching the lowest order terms of the superconformal indices. Besides giving support for the dualities, the calculation also reveals the quantum global symmetry, and shows the profound effect of changing the sign of the CS level relative to the θ angles. Due to the large rank and the considerable number of instantons required, the calculation is quite complicated, and we only carried it to order x 3 .
We begin with case (38). Starting with the SU (4) 2 theory, using the fugacity spanning shown in figure 14, we find: where to this order there are perturbative contributions, and (0,1)+(1,0) instanton contributions. One can see that there appears to be an enhancement of the instantonic-baryonic-bifundamental symmetries of the two groups to SU (2) so that the theory has an SU (2) 4 × U (1) 3 global symmetry. Indeed the index can be concisely written as:    figure 15, with the CS level and θ angles chosen to be, from left to right, (π, − 1 2 , 1 2 , 0). We will separate the index into a perturbative part, that is identical in all three cases, and the instanton contributions. The perturbative part is: In this case, we get contributions of the (1,0,0,0)+(0,1,0,0)+(0,0,1,0)+(0,0,0,1)+(0,1,1,0) instantons. The full instanton contribution is: We see that the instantons provide sufficient conserved currents to enhance four U (1)'s to four SU (2)'s so that the global symmetry matches the one of the SU (4) 2 theory. Furthermore, setting z 2 = renders the two indices equal 8 .
The discrete symmetries also match as both theories are invariant under a combination of charge conjugation and a reflection of the groups. Next we move to the case of SU (3) 3 . To order x 3 , we get contributions of the (1,0,0)+(0,1,0)+(0,0,1)+(1,1,0)+(0,1,1)+(1,1,1) instantons. Using the fugacity spanning shown in figure 16, we find: One can see that the instantons provide additional conserved currents forming an enhanced SU  . The index can then be written as:  1 2 , − 1 2 , 0) in this case, only the instanton part is different. We find: (π, 1 2 ,− 1 One can see that the instantons provide the conserved currents to form an SU (3) × SU (3) global symmetry. Particularly, setting: , renders the two indices equal. The discrete symmetries also match as both theories are invariant under a combination of charge conjugation and group reflection. Next we move to the final case of SU (3) × SU (5). We use the fugacity spanning shown in figure 17. For the SU (3) × SU (5) theory we find: One can see that the SU (3) 1-instanton provides the conserved currents to form two enhanced SU (2)'s as expected from SU 0 (3) with 6 flavors. The full global symmetry is then, SU (3) × SU (2) 2 × U (1) 3 , and the index can be written as: where we used χ[d SU (3) , d SU 1 (2) , d SU 2 (2) ] for the characters of the appropriate representations.
Next we compare it with the index of the SU (2)×SU (3)×SU (3)×SU (2) theory. Again this differs from the previous cases only by the instanton part. We find: Table 1: Summary of the dualities studied in this article. Theory 1 and 2 stands for the two dual theory, and the last column specifies the quantum global symmetry.

Conclusions
In this article we have continued to explore duality and symmetry enhancement in 5d gauge theories. A summery of the dual pairs studied in this article with their global symmetry is shown in table 1.
We provided evidence for the duality between SU (2)×SU (2) theories with two additional fundamentals and SU (3) + 4F . In this duality the difference between the flavors under each group is mapped to the SU (3)'s Chern-Simons level. This leads us to conjecture that N f 1 F + SU (2) × SU (2) + N f 2 F is dual to SU ±(N f 1 −N f 2 ) (3) + (N f 1 + N f 2 )F which was argued to flow to a fixed point when N f 1 +N f 2 +2|N f 1 −N f 2 | ≤ 6 [3]. It is interesting if this has an analog on the quiver side, or that maybe it is possible that even theories violating the inequality exist where the duality allows a continuation past infinite coupling.
We have also explored symmetry enhancement in the SU 0 (2) × U Sp(6) theory suggesting that it has an enhanced G 2 symmetry. It is interesting to extend the calculation also to states charged under the U Sp(6) topological symmetry. Another interesting direction is to study the higher N generalizations U Sp(2N ) × U Sp(2(N + M )), particularly in the context of AdS/CFT. These theories have an AdS 6 dual [9], and it is interesting if we can understand some of their properties such as dualities and lack of a UV fixed point when M > 2 also from this perspective.
Finally, there are additional choices, without a gauge theory dual, that we have not studied. The web and index calculation suggests that these should have interesting enhanced symmetries. It will be interesting to also study these theories. original CS level from the resulting one. Figure 19 illustrates this in a simple example from which one also learns that integrating the flavor from bellow the web corresponds to giving a positive mass while integrating from above corresponds to a negative mass. Therefore, given a web for SU (N ) with N f flavors one can determine the CS level by integrating out the flavors. Then comparing the resulting web with the one in figure 18, doing an SL(2, Z) transformation if necessary, determines the CS level of the pure SU (N ) one has in the IR. By the preceding arguments this is related to the original one by: where N a (N b ) is the number of flavors integrated from above (below). This can be easily generalized to the case of quiver theories. Then there are deformations, corresponding to giving large masses to the bifundamentals, where the web decomposes into a series of individual gauge theories connected through one of their external legs. In this presentation it is easy to read the gauge and matter content, and determine the CS level through the previous method, remembering that now a bifundamental is integrated out. We will see several examples of this in section 5.
Finally, this method can also be used to determine the θ angle for SU (2) groups, using the connection between the θ angle and the U (2) CS level. In the pure case there are 3 different SU (2) webs not related by an SL(2, Z) Figure 19: Two webs for SU (3) with a single fundamental flavor. The middle webs show a low value of the flavor mass (compared to the mass of the Wbosons). These can be deformed by giving large masses to the flavor resulting in the upper and lower webs which differs by the sign of the mass. One can see that the resulting pure SU (3) in the upper web in (a) has a CS level of −1 while the one in the lower web has CS level 0. This shows that the CS level of the theory of the web in (a) is − 1 2 . Applying the same procedure on (b) shows it's CS level is 1 2 . From this we also determine that integrating a flavor from above, as in the upper webs, corresponds to a negative mass while integrating from below corresponds to a positive mass. transformation corresponding to different U (2) CS levels [11,15]. These are also given from the general web of figure 18. Using these we can determine the U (2) CS levels from the web and then translate this to the θ angles.