Rank-3 antisymmetric matter on 5-brane webs

We discuss Type IIB 5-brane configurations for 5d $\mathcal{N}=1$ gauge theories with hypermultiplets in the rank-3 antisymmetric representation and with various other hypermultiplets, which flow to a UV fixed point at the infinite coupling. We propose 5-brane web diagrams for the theories of $SU(6)$ and $Sp(3)$ gauge groups with rank-3 antisymmetric matter and check our proposed 5-brane webs against several consistency conditions implied from the one-loop corrected prepotential. Using the obtained 5-brane webs for rank-3 antisymmetric matter, we apply the topological vertex method to compute the partition function for one of these $SU(6)$ gauge theories.


Introduction
String theory is a useful tool to study various aspects of supersymmetric field theories. For example, 5-brane web diagrams in type IIB string theory proposed in [1,2] can realize ultraviolet (UV) complete five-dimensional (5d) theories with eight supercharges. By using 5-brane web diagrams, it is possible to explicitly see non-perturbative features of 5d theories such as dualities. We can also compute 5d Nekrasov partitions by applying the topological vertex [3,4] to 5-brane webs, which makes use of a chain of string dualities between 5-brane webs in type IIB string theory and non-compact Calabi-Yau threefolds in M-theory [5].
Since 5-brane web diagram is a powerful tool to study 5d theories, it is important to to see how large class of 5d theories 5-brane web diagram can realize. Original 5-brane web diagrams basically yield 5d SU (N ) gauge theories with hypermultiplets in the fundamental or bi-fundamental representation. The class of gauge theories realized on 5-brane webs can be further expanded by introducing an orientifold. An orientifold can change the gauge group into SO(N ) or Sp(N ) [6,7] or it can also introduce different representations such as the symmetric or antisymmetric representation of SU (N ) [7][8][9]. Recently it has been noticed that 5-brane web diagrams can provide more exotic theories which are typically not realized by brane configurations. An O5-plane may introduce the spinor representation of SO(N ) (7 ≤ N ≤ 12) gauge theories [10] or it can even yield G 2 gauge theories with hypermultiplets in the fundamental representation [11]. In particular, as for 5d rank 2 theories, the authors showed in [12] various 5-brane realizations of all the rank 2 theories which are geometrically constructed in [13].
It is then natural to ask if it is possible to still expand the class of 5d gauge theories which 5-brane web diagrams can construct. In this paper we argue that 5-brane web diagrams may yield further new type of gauge theories which are SU (6) or Sp(3) gauge theories with half-hypermultiplets in the rank-3 antisymmetric representation. The strategy to obtain the rank-3 antisymmetric representation of SU (6) is to make use of a 5-brane web diagram for the SO(12) gauge theory with a half-hypermultiplet in the conjugate spinor representation. Since the decomposition of the conjugate spinor representation under SU (6) × U (1) includes the rank-3 antisymmetric representation of the SU (6) which is not charged under the U (1), decoupling the degrees of freedom associated to the U (1) should yield a 5-brane diagram of the SU (6) gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation. A similar method was used to obtain the fourdimensional (4d) Seiberg-Witten curve for the SU (6) gauge theory with a hypermultiplet in the rank-3 antisymmetric representation in [14]. The extension of the construction can introduce more half-hypermultiplets in the rank-3 antisymmetric representation until four half-hypermultiplets or two hypermultiplets in the rank-3 antisymmetric representation.
Since 5d gauge theories with rank-3 antisymmetric matter are realized using 5-brane webs, it is also possible to compute the 5d Nekrasov partition functions. As an illustration, we explicitly compute the Nekrasov partition function for an SU (6) gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation.
We can then introduce matter in the fundamental representation and the rank-2 antisymmetric representation to 5-brane webs in addition to rank-3 antisymmetric matter. We can realize many of the SU (6) gauge theories with rank-3 antisymmetric representation matter that have a six-dimensional (6d) UV completion in the list in [15] which were obtained from the analysis of effective prepotentials. Moreover a Higgsing associated to a hypermultliplet in the rank-2 antisymmetric representation of marginal SU (6) gauge theories with rank-2 and rank-3 antisymmetric matter yields 5-brane diagrams for Sp(3) gauge theories with matter in the rank-3 antisymmetric representation which also have a 6d UV completion. From the construction of the 5-brane webs we also find dualities and propose explicit 6d theories for some of the marginal theories.
The organization of the paper is as follows. In section 2, we propose 5-brane web diagrams of SU (6) gauge theories with half-hypermultiplets in the rank-3 antisymmetric representation. From the obtained diagram we compute the Nekrasov partition function for an SU (6) gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation. We extend the construction of 5-brane webs in section 3 by adding hypermultiplets in other representations. In particular we propose 5-brane webs for 5d SU (6) gauge the-ories with rank-3 antisymmetric matter that have a 6d UV completion. In section 4, we Higgs the diagrams obtained in section 3 to construct 5-brane webs for marginal Sp(3) gauge theories with rank-3 antisymmetric matter. Finally we find explicit 6d UV complete theories for some of the marginal SU (6) gauge theories from the 5-brane webs and discuss dualities involving marginal SU (6) gauge theories with a half-hypermultiplets in the rank-3 antisymmetric representation in section 5.
2 SU (6) gauge theories with rank-3 antisymmetric matter In this section, we propose 5-brane webs for SU (6) gauge theories with half-hypermultiplets in the rank three antisymmetric representation. UV complete 5d SU (6) gauge theories can have at most two hypermultiplets in the rank-3 antisymmetric representation [15]. We will obtain brane webs with all possible number of massless rank-3 antisymmetric halfhypermultiplets in this section.

Decoupling from SO(12) gauge theory with conjugate spinor matter
One way to obtain the rank-3 antisymmetric representation of SU (6) is using the decomposition of the spinor or the conjugate spinor representation of SO (12) under SU (6) × U (1). We here consider the decomposition from the conjugate spinor for later use. In this case, the decomposition of the conjugate spinor representation under the SU (6) × U (1) is given by where the subscript stands for the U (1) charge 1 . The twenty dimensional representation 20 0 is the rank-3 antisymmetric representation of SU (6). The 5d N = 1 SO(12) gauge theory with a conjugate spinor also contains a vector multiplet in the adjoint representation of SO (12). The decomposition of the adjoint representation of SO (12) under the SU (6) × U (1) is given by Since 6 −2 ,6 2 , 15 2 and 15 −2 are charged under the U (1) of the SU (6) × U (1), the fields in those representations acquire large mass when we give a large vev to the Coulomb branch modulus for the U (1). Therefore, when the vev for the Coulomb branch modulus of the U (1) in the SO(12) gauge theory becomes infinitely large, the low energy effective field theory should be described by the SU (6) gauge theory with a hypermultiplet in the rank-3 antisymmetric representation. This method was made use of to obtain the Seiberg-Witten curve for the 4d SU (6) gauge theory with rank-3 antisymmetric matter in [14].  We can apply this procedure to a 5-brane web for the SO(12) gauge theory with a conjugate spinor for obtaining a brane web for the SU (6) gauge theory with rank-3 antisymmetric matter. In order to simplify the discussion, we start from the 5d SO(12) gauge theory with a half-hypermultiplet in the conjugate spinor representation. Then the decoupling procedure will lead to an SU (6) gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation at low energies.
A 5-brane web is constructed by a combination of (p, q) 5-branes in type IIB string theory [1,2] and it realizes a 5d theory on the brane web. As for the brane configuration, our convention is that a D5-brane extends in the (x 0 , x 1 , x 2 , x 3 , x 4 , x 5 )-directions and an NS5-brane extends in the (x 0 , x 1 , x 2 , x 3 , x 4 , x 6 )-directions in the ten-dimensional spacetime in type IIB string theory. A (p, q) 5-brane extends in the (x 0 , x 1 , x 2 , x 3 , x 4 )-directions and also in a one-dimensional space in the (x 5 , x 6 )-plane. The one-dimensional space is given by a line with slope q p in the (x 5 , x 6 )-plane. 7-branes in the (x 0 , x 1 , x 2 , x 3 , x 4 , x 7 , x 8 , x 9 )directions may be also introduced in the configuration by ending a (p, q) 7-brane on top of an external (p, q) 5-brane. 7-branes are useful to see a global symmetry of the theory realized on a web and also to consider a Higgsing. Since a non-trivial structure of the brane appear in the (x 5 , x 6 )-plane, we only write the configuration in the two-dimensional plane where we choose the horizontal direction as the x 5 -direction and the vertical direction as the x 6 -direction.
A 5-brane web for the SO(12) gauge theory with a half-hypermultiplet in the spinor or the conjugate spinor representation has been proposed in [10] and we depict the diagram in Figure 1. The "Sp(0)" part in-between the (2, −1) 5-brane and the (2, 1) 5-brane yields "Sp(0)" instantons and they can be interpreted as a half-hypermultiplet in the spinor or the conjugate spinor representation depending on the discrete theta angle of the Sp(0).
In order to explicitly see if the configuration contains the spinor or the conjugate spinor representation, we may consider a diagram after a generalized flop transition for the Sp(0) part, which can distinguish the discrete theta angle of the Sp(0) [16]. It turns out that the generalized flop transition in the case of the conjugate spinor representation of SO(12) yields the diagram depicted in Figure 2. To see that, we identify a weight of a representation for the matter in the theory from the length of an internal 5-brane in the diagram. We first label the height of the six color D5-branes as a 1 , a 2 , a 3 , a 4 , a 5 , a 6 as in Figure 2 and identify them with the six Coulomb branch moduli of the SO (12). This parameterization corresponds to being in a Weyl chamber specified by positive roots With the parameterization, the length of the (1, 1) 5-brane depicted as a red line in Figure  2 is 1 2 (a 1 − a 2 − a 3 − a 4 + a 5 + a 6 ). Also the length of the (0, 1) 5-brane that is reflected in the O5-plane depicted as a blue line in Figure 2 is 1 2 (−a 1 + a 2 + a 3 + a 4 + a 5 + a 6 ). Hence, a string with the length 1 2 (a 1 − a 2 − a 3 − a 4 + a 5 + a 6 ) connecting two D5-branes in the diagram yields a hypermultiplet for a weight 1 2 (e 1 − e 2 − e 3 − e 4 + e 5 + e 6 ), (2.4) while a string with the length 1 2 (−a 1 + a 2 + a 3 + a 4 + a 5 + a 6 ) connecting two D5-branes through the O5-plane in the diagram yields a hypermultiplet for a weight 1 2 (−e 1 + e 2 + e 3 + e 4 + e 5 + e 6 ), (2.5) where e i , (i = 1, · · · , 6) are the orthonormal basis of R 6 . The weight of (2.4) and (2.5) are indeed weights in the conjugate spinor representation of SO (12). Combining the 5-brane lines corresponding to the weight (2.4) and (2.5) with 5-brane lines for the positive roots of (2.3) gives a half of the weights of the conjugate spinor representation which are given by 1 2 (−e 1 + e 2 + e 3 + e 4 + e 5 + e 6 ), 1 2 (e 1 − e 2 + e 3 + e 4 + e 5 + e 6 ), 1 2 (e 1 + e 2 − e 3 + e 4 + e 5 + e 6 ), 1 2 (e 1 + e 2 + e 3 − e 4 + e 5 + e 6 ), 1 2 (e 1 + e 2 + e 3 + e 4 − e 5 + e 6 ), 1 2 (e 1 + e 2 + e 3 + e 4 + e 5 − e 6 ), Hence, the diagram in Figure 2 yields hypermultiplets corresponding to the weights (2.6) and (2.7) or equivalently a half-hypermultiplet in the conjugate spinor representation of SO (12). In order to obtain a diagram for the SU (6) gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation, we need to take a limit where the Coulomb branch modulus for the U (1) in the decomposition SU (6) × U (1) ⊂ SO(12) becomes infinitely large. It is in fact straightforward to identify this U (1) degree of freedom from the diagram in Figure 2. Due to the presence of the O5-plane, individual height of the six color D5-branes can be independent parameters. Therefore, the U (1) part can be decoupled when we separate the brane configuration in the upper half-plane infinitely far away from the O5-plane. In this limit, the O5-plane is infinitely far from the other brane configuration and strings between them are decoupled. The resulting brane diagram without the O5plane is depicted in Figure 3 and the diagram should realize an SU (6) gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation. From the diagram in Figure 3, the right part of the digram is identical to the one for the pure SU (6) gauge theory. Hence the matter contribution comes only from the left part of the diagram.
It is straightforward to construct a 5-brane diagram for an SU (6) gauge theory with a hypermultiplet in the rank-3 antisymmetric representation. When we obtained a halfhypermultiplet in the rank-3 antisymmetric representation, we started from the diagram which realizes the SO(12) gauge theory with a half-hypermultiplet in the conjugate spinor representation. Hence, we can start from the SO(12) gauge theory with a hypermultiplet in the conjugate spinor representation in order to obtain a diagram for a rank-3 antisymmetric hypermultiplet. The proposed diagram in [10] for the SO(12) gauge theory with a conjugate spinor is depicted in Figure 4(a). The discrete theta angle for the two Sp(0) parts should be chosen so that the diagram contains matter in the conjugate spinor representation of SO (12). In this case, we can only realize massless hypermultiplet in the conjugate spinor representation. We then perform generalized flop transitions for the two Sp(0) parts in the diagram in Figure 4(a) and decouple the U (1) degree of freedom. The procedure yields a diagram in Figure 4(b) which should realize an SU (6) gauge theory with a hypermultiplet in the rank-3 antisymmetric representation. Since we started from massless matter, the rank-3 antisymmetric hypermultiplet after the decoupling is also massless. It is indeed natural that the diagram in Figure 4(b) gives a massless hypermultiplet in the rank-3 antisymmetric representation given that a half-hypermultiplet in the rank-3 antisymmetric representation comes from the left part of the diagram in Figure 3. The diagram consists of two copies of the left part of the diagram in Figure 3 and hence it should give two  half-hypermultiplets in the rank-3 antisymmetric representation, which correspond to a massless hypermultiplet in the rank-3 antisymmetric representation.

Monopole string tension
In the previous subsection, we obtained 5-brane diagrams for SU (6) gauge theories with rank-3 antisymmetric matter. We give further support for the claim by comparing the monopole string tension computed from the diagram in Figure 3 with that calculated from the prepotential in the gauge theory. We have not yet determined the Chern-Simons (CS) level for the theories and the CS level can be also fixed from the monopole string tension computation. We first compute the monopole string tension from the diagram in Figure 3. A monopole string in a 5d theory can be realized by a D3-brane stretched on a face bounded by 5-brane segments in the corresponding 5-brane web. Hence the tension of the monopole string is given by the area of the face on which the D3-brane is stretched. In order to compute the area, we label the height of the six color D5-branes as a 1 , a 2 , a 3 , a 4 , a 5 , a 6 as in Figure 5(a). Contrary to the diagram for the SO(12) gauge theory in Figure 2, the overall height is irrelevant and the parameters satisfy 6 i=1 a i = 0, which can be solved by setting On the other hand, the inverse of the squared classical gauge coupling m 0 is the length of D5-branes in the limit where all the Coulomb branch moduli are turned off. It turns out that the length of the top D5-brane is parameterized by m 0 + 7a 1 as in Figure 5(a). We can then compute the area of faces in the diagram in Figure 5(b). Note that the region 1 is connected to 2 and it forms a single face on which a D3-brane is stretched. Similarly the region 7 is connected to 6 . Therefore we have in total five faces in the diagram in Figure 5 SU (6). The area of the five faces parameterized by m 0 , a i , (i = 1, · · · , 6) is We can compare the area (2.9)-(2.13) with the monopole string tension computed from the effective prepotential. In general the effective prepotential on a Coulomb branch of a 5d gauge theory with a gauge group G and matter f in a representation R f is given by [17][18][19] (2.14) Here, m 0 is the inverse of the squared gauge coupling, κ is the classical Chern-Simons level and m f is a mass parameter for the matter f . r is a root of the Lie algebra g associated to G and w is a weight of the representation R f of g. Furthermore, we defined where T i are the Cartan generators of the Lie algebra g.
The sign for the one-loop correction terms in (2.14) is fixed from the parameterization of the Coulomb branch moduli in the diagram in Figure 5(a). Namely, the positive roots are given by e i − e j , (1 ≤ i < j ≤ 6), and the positive weights of the rank-3 antisymmetric representation are reduced from the positive weights of the conjugate spinor representation in (2.6) and (2.7), e 1 + e i + e j , (2 ≤ i < j ≤ 6). On this phase, the effective prepotential for the SU (6) gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation becomes where κ is the CS level. We then rewrite the effective prepotential (2.15) in terms of the Coulomb branch moduli φ i , (i = 1, · · · , 5) in (2.8), and the monopole string tension is given by taking the derivative of the effective prepotential with respect to the φ i . Then the comparison with (2.9) which corresponds to taking the derivative with respect to φ 1 , yields κ = 5 2 . Hence the diagram in Figure 3 realizes the SU (6) gauge theory with a halfhypermultiplet in the rank-3 antisymmetric representation and the CS level κ = 5 2 . Fixing the CS level to 5 2 , the other comparison between the area and the monopole string tension may be interpreted as support for our claim that the diagram in Figure  It is also possible to make a comparison between the area and the monopole string tension for the diagram in Figure 4(b). We checked the agreement and the CS level of the SU (6) gauge theory realized by the diagram in Figure 4(b) is κ = 3.

Nekrasov partition function
As we have seen in the previous subsection. the computation of the monopole string tension confirmed that the diagrams in Figure 3 and in Figure 4(b) realize the SU (6) gauge theory with N TAS = 1 2 and κ = 5 2 and the SU (6) gauge theory with N TAS = 1 and κ = 3 respectively. We can now use power of 5-brane web diagrams to compute various physical quantities from the 5-brane web diagrams. One important applications is to compute the Nekrasov partition function or the topological string partition function from the 5-brane webs using the topological vertex [3,4]. Although the topological vertex was originally formulated to compute the all genus topological string partition function for toric Calabi-Yau threefolds, we can also apply the topological vertex to non-toric diagrams obtained from a Higgsing of toric diagrams [20][21][22][23] and also to diagrams with an O5-plane [24].
By using the techniques, it is straightforward to apply the topological vertex for the 5-brane diagrams in Figure 3 and in Figure 4(b). We here illustrate the computation by using the diagram in Figure 3 and we calculate the Nekrasov partition function for the SU (6) gauge theory with N TAS = 1 2 and κ = 5 2 . In order to compute the topological string partition function for the diagram in Figure 4(b), we first assign Young diagrams Y 0 , Y 1 , · · · , Y 6 to the horizontal lines as in Figure 6.
Then the application of the topological vertex to the diagram in Figure 3 yields  Figure 6. A labeling of Young diagrams assigned to the horizontal lines in Figure 3.
are contributions of the left part and the right part of the web in Figure 6 respectively when we cut the digram at the horizontal lines with the Young diagrams Y i , (i = 1, · · · , 6) and they are given by Moreover, A i , (i = 1, · · · , 6), q and g are defined by where is the unrefined Ω-deformation parameter. We argue that the topological string partition function (2.18) is the Nekrasov partition function for the the 5d SU (6) gauge theory with N TAS = 1 2 and κ = 5 2 . It is instructive to look into the partition function by the order of the instanton fugacity q. In terms of the expansion by q, the Nekrasov partition function is given by Z pert represents the perturbative part of the partition function given by the order q 0 in (2.18). On the other hand the Z k stands for the k-instanton partition function. Let us first look at the perturbative part. This is obtained from the contribution of Y 1 = Y 2 = · · · = Y 6 = ∅ in (2.18) and is given as where we used the identity with PE representing the Plethystic exponential and Note that in order to obtain the exact expression for the perturbative part we still need to sum over the Young diagram Y 0 . We can still evaluate the summation in terms of an expansion by A 1 . Namely when we sum over the Young diagram until |Y 0 | ≤ k, the expression is exact until the order A k 1 . The summation of the Young diagram Y 0 until |Y 0 | = 7 yields the expression We observed that the the series expansion by A 1 gives an expression which stops at the order A 1 inside the Plethystic exponential as far as we checked. Hence, we claim that O(A 1 8 ) term is actually exactly zero. Indeed, the partition function (2.28) is exactly equal to the perturbative part of the partition function of the SU (6) gauge theory with a halfhypermultiplet in the rank-3 antisymmetric representation. We can also see that the charge of the BPS states counted by the perturbative partition function agrees with the charge of the positive weights used in the prepotential computation for (2.15). Next, we compute the 1-instanton part. The 1-instanton part can be read off from the coefficient of the q 1 order part in (2.18) divided by the perturbative part given in (2.28). Hence the order q 1 contribution is given by combinations where |Y i | = 1 for one of the Y i , (i = 1, · · · , 6) and the others are trivial. Furthermore, we still need to sum over Y 0 and evaluate the summation in terms of a series expansion by A 1 2 . A 1 is a good expansion parameter since the explicit summation of Young diagrams in (2.18) involves only positive powers of A 1 . For example, the contribution from |Y 1 | = 1, |Y j | = 0 (j = 2, 3, 4, 5, 6) to the 1-instanton part is given by We again observed that the stop of the series expansion by A 1 in the numerator of (2.29) and we claim that the O(A 1 8 ) term is actually exactly zero. Similarly we can also compute the other combinations of the Young diagrams which contribute to the 1-instanton part. Summing up all the contributions from the other combinations of the Young diagrams which contribute to the 1-instanton part, we obtain This is the explicit expression for the 1-instanton part of the partition function for the SU (6) gauge theory with N TAS = 1 2 and κ = 5 2 . The two instanton contribution can be written in the following form: Here, the lower indices {1, 1} indicates the contribution from the Young diagrams The contributions from the vector multiplets are given as The expression (2.18) contains factors with A1 in the denominator. We perform a series expansion by A1 only for the numerator of (2.18). and while the contributions from the hypermultiplet in rank-3 antisymmetric tensor representation are Here, we have introduced the following U (5) characters and the following U (4) characters Analogous to the case of the perturbative and the 1-instanton contribution, we claim that the O(A 11 1 ) term in (2.31) is exactly zero. (6) gauge theories with N TAS = 3 2 and 2 We have constructed 5-brane webs for an SU (6) gauge theory with one or two halfhypermultiplets in the rank-3 antisymmetric representation. It is natural to ask if we can add more half-hypermultiplets in the rank-3 antisymmetric representation. For that it is useful to take a different view for the diagrams in Figure 3 and Figure 4(b). In fact, the diagram for the SU (6) 5 2 3 gauge theory with N TAS = 1 2 in Figure 3 may be also obtained 3 The subscript of the SU gauge group represents the Chern-Simons level.   Figure 7(a). In the diagram we introduced 7-branes ending on the external 5-branes to see the Higgsing explicitly. The Higgsing procedure can be done as follows. The SU (6) 5 2 − SU (3) 0 quiver theory has an SU (2) × SU (2) flavor symmetry, which can be seen from parallel external 5-branes in the diagram in Figure 7(a), and we will give vevs to hypermultiplets associated to the flavor symmetry. For that we first set the length of 5-branes depicted as red lines in Figure 7(a) to be zero. The resulting diagram is given by the one in Figure 7(b). Then giving the vevs corresponds to decoupling pieces of 5-branes in between 7-branes which are drawn as blue lines in Figure 7(b). After this procedure, coincident 5-branes end on the same 7-brane and then the diagram is equivalent to the one in Figure 3. The Higgsing from the SU (3) 0 − SU (6) 3 − SU (3) 0 quiver theory to SU (6) 3 gauge theory with N TAS = 1 may be obtained in similar way. We first shrink the length of the red lines in a diagram for the SU (3) 0 − SU (6) 3 − SU (3) 0 quiver theory in Figure 8(a), which gives rise to the diagram in Figure 8(b). Then decoupling the blue lines in Figure 8(b) reduces to the diagram for the SU (6) 3 gauge theory with N TAS = 1 in Figure 4

SU
To summarize, the Higgsing of the SU (6) Hence we can add rank-3 antisymmetric matter by coupling more SU (3) 0 gauge nodes and then Higgsing them. For the original theory to be UV complete, one can consider two more types of quiver theories which are given by SU (6) The former one is an D 4 quiver theory and the latter one is an affine D 4 quiver theory. The Higgsing of the D 4 quiver theory will yields the SU (6) 1 2 gauge theory with N TAS = 3 2 and the Higgsing of the affine D 4 quiver theory will give the SU (6) 0 gauge theory with N TAS = 2 which is supposed to have a 6d UV completion [15]. The latter Higgsing realizes a renormalization group flow from an affine D 4 Dynkin quiver theory which has a 6d UV completion to another 5d theory which also has a 6d UV completion.
We here make use of the Higgsing procedure to construct 5-branes webs for an SU (6) gauge theory with N TAS = 3 2 , 2. We first start from the construction of a web for the SU (6) gauge theory with N TSA = 3 2 and κ = 1 2 which will be obtained from a Higgsing of the SU (6) [25][26][27][28] and it is depicted in Figure 9(a). One of the SU (3) 0 gauge nodes in Figure 9(a) is given by the left part of the diagram in Figure  9(a) and two of the SU (3) 0 gauge nodes are realized by the right part of the diagram in Figure 9(a) using an ON-plane. We can also see an SU (2) × SU (2) flavor symmetry from the left part and also an SO(4) × SO(4) ∼ = SU (2) 4 flavor symmetry from the right part. Then we apply the same Higgsing procedure in Figure 7 to the diagram in Figure 9(a). The Higgsing associated to the flavor symmetry yields the diagram in Figure 9(b). We claim that the diagram in Figure 9(b) gives rise to the SU (6) gauge theory with three half-hypermultiplets in the rank-3 antisymmetric representation and κ = 1 2 . In order to obtain a diagram for the SU (6) 0 gauge theory with N TAS = 2, we start from a diagram for the SU (6) 0 −[SU (3) 0 ] 4 affine D 4 quiver theory in Figure 10(a). The two ONplanes in Figure 10(a) realizes the four SU (3) 0 gauge nodes coupled to the middle SU (6) gauge theory, Applying the Higgsing done in Figure 7 to the both sides of the diagram in Figure 10(a) gives rise to the diagram in Figure 10(b). We argue that the diagram in   (6) gauge theory with two massless hypermultiplets in the rank-3 antisymmetric representation with zero Chern-Simons level. After performing Sduality to the diagram in Figure 10(b), which is equivalent to rotating the diagram by 90 degrees, the diagram contains two O5-planes on the upper side and the lower side. The two O5-planes implies a periodic direction in the vertical direction, suggesting a 6d UV completion.
We can confirm the claim by comparing the area with the monopole string tension as done in section 2.2. In order to compute the area of the faces in the diagram in Figure  9(b), we label the height of the six color D5-branes as a 1 , a 2 , a 3 , a 4 , a 5 , a 6 with 6 i=1 a i = 0, which are the Coulomb brach moduli of the SU (6) 1 2 gauge theory with N TAS = 3 2 . The inverse of the squared gauge coupling m 0 is determined by the non-trivial length of the D5-brane after turning off the Coulomb branch moduli. Then the length of the top color D5-brane in Figure 9(b) is m 0 + 2a 1 . The parameterization is summarized in Figure 11(a). With this parameterization we can compute the area of the faces of the diagram in Figure  9(b). A labeling of the faces is given in Figure 11(b). As in the case of the diagram in Figure 5(b), some of them are connected to each other. In fact, 1 , 2 and 3 are a single face and the area of the region 3 should be doubled due to the presence of the ON − -plane [11,12]. Simliarly, the region 7 , 8 and 9 are connected to each other and the area of the region 9 needs to be doubled. Then we have in total five faces and the corresponding area is given by We then compare the area (2.40)-(2.44) with the monopole string tension calculated, using the effective prepotential (2.14). The parameterization in Figure 11(a) fixes the phase of the SU (3) 1 2 gauge theory with N TAS = 3 2 . Positive roots are e i − e j , (1 ≤ i < j ≤ 6) and positive weights are e 1 +e i +e j , (2 ≤ i < j ≤ 5) and e 1 +e i +e 6 , (i = 2, 3), −e 1 −e i −e 6 , (i = 4, 5). Then the effective prepotential (2.14) is given by After rewriting (2.45) in terms of the Coulomb branch moduli φ i , (i = 1, · · · , 5) in (2.8), taking the derivative of (2.45) with respect to the φ i gives the monopole string tension. Indeed we found that which supports the claim that the diagram in Figure 9(b) yields the SU (6) gauge theory with three half-hypermultiplets in the rank-3 antisymmetric representation and the Chern-Simons level κ = 1 2 .
Extending the comparison to the case of the SU (6) 0 gauge theory with N TAS = 2 realized in the diagram in Figure 10(b) is straightforward. We checked that the area of the faces in the diagram in Figure 10(b) reproduces the monopole string tension calculated from the effective prepotential of the SU (6) gauge theory with two massless hypermultiplets in the rank-3 antisymmetric representation and zero Chern-Simons level.
3 Marginal SU (6) gauge theory with rank-3 antisymmetric matter In this section, we provide more 5-brane diagrams for SU (6) gauge theories with rank-3 antisymmetric matter by including hypermultiplets in other representations. In particular we present 5-brane web diagrams for SU (6) marginal theories with half-hypermultiplets in the rank-3 antisymmetric representation and other matter which have the UV completion as a 6d theory. Possible SU (6) marginal theories with rank-3 antisymmetric hypermultiplets and other hypermultiplets are classified in [15] based on the Coulomb branch analysis. Though we do not find all the marginal SU (6) marginal theories with rank-3 antisymmetric matter listed in [15], 5-brane webs for the marginal theories that we found precisely agree with the matter content and also show the periodic structure which supports that the theories can be understood as 6d theory on a circle with or without a twist.
Moreover, one can put the 7-branes appearing in our 5-brane webs into the 5-brane loops, and then from which one can read off a global symmetry of the theory. Such characterization of a global symmetry is only possible for symmetry group of ADE type [29][30][31]. For instance, given a 7-brane configuration where one allocates 7-branes into a 5-brane loop, the corresponding (non-abelian part of) global symmetry is read off from the Kodaira classification where the following shorthand notation is used to denote the 7-brane charges For other types of global symmetry, one may infer it from possible maximal subgroups of ADE type via various Hanany-Witten transitions on a given 5-brane web. We remark that in the way, we perform 7-brane monodromy analysis for those 5-brane webs which do not have orientifolds to find global symmetries for the marginal theories, and we see that the obtained global symmetries are consistent with those given in [15].
In Table 1, we summarize marginal 5-brane web diagrams that we obtained. We note that as decoupling of hypermultiplets from the marginal theories, one can also perform decoupling of hypermultiplets on 5-brane webs, as discussed in [12], which would give rise to various 5-brane webs for other genuine 5d SCFTs.  antisymmetric hypermultiplets was already discussed in section 2.4, and the corresponding web diagram is given in Figure 10.

5-brane webs for SU (6) + 1 TAS with various hypermultiplets
Following section 2, it is straightforward to get a 5-brane web diagram for an SU (6) gauge theory with one rank-3 antisymmetric hypermultiplet. For instance, in Figure 5, we presented a 5-brane web for the SU (6) gauge theory with one rank-3 antisymmetric hypermultiplet, which has the Chern-Simons level κ = 3. It is then possible to express a 5-brane web for the SU (6) theory with one rank-3 antisymmetric hypermultiplet which is of the Chern-Simons level κ = 0 by suitably choosing the asymptotic (p, q) charges for the external 5-branes, as depicted in Figure 12(a). Notice that asymptotic 5-brane charges for this SU (6) 0 gauge theory with one rank-3 antisymmetric hypermultiplet is the same as those for the pure SU   SU (6) 0 + 1TAS + 10F. A 5-brane web for SU (3) 0 +10F theory is constructed in [33], and then it follows that a 5-brane web for the SU (6) 0 + 1TAS + 10F theory can be constructed by adding 10 flavors in the same way as done for 5-brane web for the SU (3) 0 + 10F theory. The resulting 5-brane web is depicted in Figure 13. We note that as shown in Figure 13(b), the 5-brane web of the SU (6) 0 + 1TAS + 10F theory has an infinite repeated periodic structure, named Tao web diagrams [22], which is expected as the 5-brane web for the SU (3) 0 + 10F theory also has the periodic structure [33]. The period of a Tao web diagram is expressed in terms of the coupling and mass parameters, which corresponds to the inverse of the compactification radius. Hence such Tao web diagrams imply that these 5d theories are realized as a 5-brane for a Kaluza-Klein (KK) theory where a 6d theory is compactified on a circle associated with the period on the Tao web diagram. We will discuss the 6d origin of the SU (6) 0 + 1TAS + 10F theory later in section 5.1.
The enhanced global symmetry of the SU (6) 0 + 1TAS + 10F theory can be read off from 7-brane monodromy analysis as shown in Figure 14. Starting from the 5-brane web for the SU (6) 0 + 1TAS + 10F theory given in Figure 13, one puts flavor D7-branes A's together as in Figure 14(a). As D7-branes can cross D5-branes, one can put all the D7branes inside 5-brane loops, which gives 7-brane configuration given in Figure 14(b). Using

7-brane monodromy analysis (counterclockwise) like
one can relocate the 7-branes to obtain the configuration in Figure 14(c), which leads to the 7-brane configuration yielding an SO(20) symmetry as D 10 = A 10 BC as shown in Figure  14(d). This agrees with the propsed global symmetry in [15]. It is in fact the same global symmetry structure as that for the SU (3) 0 + 10F theory [33], which is expected as their asymptotic 7-brane configurations are identical. the same as that of the SU (3) 3 2 + 9F theory, whose 5-brane web is constructed in [12,34]. Hence, in the same way, one can construct a 5-brane configuration for the SU (6) 3 2 +1TAS+ 9F theory by introducing 9 D7-branes such that the Chern-Simons level is κ = 3/2. For instance see Figure 15. As expected, it is also also a Tao diagram.
One can perform a similar 7-brane monodromy analysis to read off the global symmetry. As in Figure 16, one can rearrange the 7-branes with (3.3) and also AC = CX (2,1) , to find the global symmetry of the SU (6) 3 2 + 1TAS + 9F theory. Given a 7-brane configuration for the SU (6) 3 2 + 1TAS + 9F theory, for instance, Figure 16(a), one can use (3.3) to obtain from which one rearranges the 7-brane in the upper 5-brane loop chamber in Figure 16(b). The rearrangement of the 7-brane in the lower chamber in Figure 16(b) is obtained from where the first and third equalities are due to (3.3) and the second and fourth equalities come from (3.4). One then finally relocates three A's in the upper chamber to the lower chamber as well as brings two A's in front as depicted in Figure 16(b). The resulting configuration is given in Figure 16(c), from which we find that the non-Abelian part of the global symmetry 5 is E 8 × SU (2) . We note that as the 7-brane analysis is insensitive for an abelian symmetry, here U (1) is added by hand to match with the number of the mass parameters in the 5d theory, assuming that the rank-3 antisymmetric hypermultiplet is massive in general. We also note that this global symmetry is slightly different from the expected global symmetry reported in [15], which is E 2 . As our 5-brane configuration is the same as that of the SU (3) 3 2 + 9F theory, and also all the 7-branes can be put in two different 5-brane loops, it is expected to show the same global symmetry as that of the SU (3) 3 2 + 9F theory. Our 5-brane construction for the SU (6) 3 2 + 1TAS + 9F theory is, in fact, the theory of massless rank-3 antisymmetric hypermultiplet. It may be that the 5-brane configuration for the massless rank-3 antisymmetric matter does not capture further enhancement from SU (2) × U (1) to SU (3), since there are not enough 7-branes 6 . SU (6) 0 + 1TAS + 1AS + 4F. One can also introduce a hypermultiplet in the rank-2 antisymmetric representation in addition to a hypermultiplet in the rank-3 antisymmetric representation as depicted in Figure 17(a), which gives the SU (6) 0 + 1TAS + 1AS + 4F. As shown in Figure 17(b), the presence of the rank-2 antisymmetric matter can be understood since the diagram contains a configuration with an O7 − -plane attached to an NS5-brane [7]. By putting an O7 − -plane and four D7 branes together, one constructs a 5-brane web for the SU (6) 0 theory with a hypermultiplet in the antisymmetric representation and four flavors Figure 17(c), which has two orientifolds. Since the combination of an O7 − -plane with four D7-branes is S-dual invariant, its S-dual diagram shows clearly that it is of a periodic structure in the vertical direction, supporting the consistency that the SU (6) 0 + 1TAS + 1AS + 4F is marginal. 6 There is a similar case for the 5-brane configuration for the 6d E-string on a circle, which yields a 5-brane web for the 5d Sp(2) gauge theory with 8 flavors and one antisymmetric hypermultiplet. The expected global symmetry from 5d perspective is E8 × SU (2). Here, one has both 5-brane webs for massless [22] and massive [7,12,36] antisymmetric hypermultiplet. For the massless case, 7-brane analysis does not capture the SU (2) part, while the massive case see the full enhanced global symmetry, E8 × SU (2).

5-brane webs for SU (6) + 1/2 TAS with various hypermultiplets
In section 2, 5-brane webs for SU (6) theories with a half-hypermultiplet in the rank-3 antisymmetric representation is discussed. For instance, Figure 3 is a 5-brane web for the SU (6) + 1/2 TAS theory with the Chern-Simons level κ = 5/2. One can readily change the Chern-Simons level by adjusting the charges of the external 5-branes just as done for the SU (6) + 1 TAS theory. Below we construct 5-brane webs for marginal SU (6) + 1/2 TAS theories with various hypermultiplets.
SU (6) 0 + 1 2 TAS + 13F. Given a 5-brane web for the SU (6) 5 2 + 1/2TAS theory in Figure  3, one can modify the charge of the external 5-branes and then add 13 D7 branes in a way that it leads to the Chern-Simons level κ = 0. An example for 5-brane web for the SU (6) 0 + 1/2TAS + 13F theory is given in Figure 18(a). As it is a marginal theory, we expect it is of a certain periodic structure. In a similar way done in the marginal SU (6) + 1TAS theories with only flavors in section 3.2, we can move 7-branes and allocate the cuts of 7-branes to show a periodic structure as shown in Figure 18(b). By pulling out 7-branes across the cuts arranged in 18(b). One sees that it is a Tao diagram showing periodic web configuration as depicted in Figure 18(c).
The global symmetry in this case can be readily read off from the 7-brane configuration associated with the 5-brane web in Figure 18. In Figure 19, one can allocate all the D7branes in the innermost 5-brane loop but other 7-branes are confined to all other 5-brane loops. This leads to an SU (13) = A 13 symmetry, which is the same as the non-abelian part of the perturbative global symmetry of the theory. As it is the non-abelian part of the flavor symmetry, the expected global symmetry would be then SU (13) × U (1) × U (1) since the total number of the mass parameters of the theory is 14, agreeing with the global symmetry obtained in [15].
(a) C X (2,-1) Figure 19. SU (6) 3 + 1 2 TAS + 9F One can also construct the SU (6) theory with a half-hypermultiplet in the rank-3 antisymmetric representation and 9 flavors which has the Chern-Simons level κ = −3, by introducing 9 D7-branes in such a way that it has the Chern-Simons level κ = −3. See Figure 20. It can be shown that it is also a Tao diagram as depicted in Figure  20(b).
Following the 7-brane analysis for the SU (6) 0 + 1 2 TAS + 13F theory in Figure 19, one easily sees that the 7-brane configuration for the SU (6) 3 + 1 2 TAS + 9F theory is readily manipulated to yields the non-abelian part of global symmetry is SU (9) = A 9 symmetry, which is the non-abelian part of the perturbative flavor symmetry of the theory. The expected global symmetry would be then SU (9) × U (1) × U (1). SU (6) 0 + 1 2 TAS + 1AS + 9F As done in [7] and also in section 3.2, one can introduce a rank-2 antisymmetric hypermultiplet by introducing a configuration of an NS5-brane ending on an O7 − -plane. A 5-brane web for the SU (6) 0 theory with a rank-3 antisymmetric half-hypermultiplet and one antisymmetric hypermultiplet and 9 flavors can be constructed as in Figure 21(a). It can be also shown that it is a Tao diagram as depicted in Figure  21(b), implying that the theory has a 6d UV completion. As drawn in Figure 22, the non-Abelian part of the global symmetry is SU (10). Figure 23. One can show that the corresponding 5-brane web diagram is a Tao diagram, though it requires delicate arrangements of 7-branes as shown in Figure  23(b). The non-abelian part of the global symmetry that we can see from the corresponding 7-brane configuration seems to be SU (8), which means that this 7-brane configuration may not show any enhancement other than the perturbative symmetry of the 8 flavors. Hence, the rank of the global symmetry from the 7-brane configuration is smaller than that of the global symmetry SO(16) × SU (2) × U (1) proposed in [15]. SU (6) 0 + 1 2 TAS + 1Sym + 1F In 5-brane web, a hypermultiplet in the symmetric representation is represented with an NS5-brane ending on an O7 + -plane [7]. Examples of  5-brane webs for marginal theories with symmetric matter, SU (3) 0 + 1Sym + 1F and SU (3) − 3 2 + 1Sym, are discussed in [12]. They are, in fact, instructive examples for constructing the SU (6) theories with a rank-3 antisymmetric half-hypermultiplet and a symmetric hypermultiplet as the asymptotic 7-brane configurations for both the SU (3) theories and the SU (6) theories are the same. We first consider a 5-brane configuration for the SU (6) 0 + 1 2 TAS + 1Sym + 1F theory. Using the fact that a decoupling of a symmetric hypermultiplet for an SU (N ) theory gives rise to the change of the Chern-Simons level κ by κ − 1 2 (N + 4), (for N = 6, κ → κ − 5), one has a 5-brane configuration for the SU (6) 0 + 1 2 TAS + 1Sym + 1F theory as follows: one prepares a 5-brane web diagram for  SU (6) −5 + 1 2 TAS + 1F and then attaches one external single 5-brane to an O7 + -plane, as shown in Figure 24. We know that its 7-brane charges are the same as those appear in a 5-brane web for the SU (3) 0 + 1Sym + 1F theory. (See Figure 49 in [12].) We note that unlike the 5-brane web for the SU (3) 0 + 1Sym + 1F theory, two 5-branes are attached to some 7-branes. It is however still possible to make this 5-brane configuration to have a periodic structure, by moving 7-branes inside the 5-brane loops and also by manipulating a pair of 7-branes to be converted into an O7 − -plane, as shown in Figure 25. It is therefore a 5-brane web with an O7 − -plane and an O7 − -plane.
The result is summarized in Table 2.
As we already constructed 5-brane diagrams for the SU (6) 0 gauge theory with N TAS = 1, N AS = 1, N F = 4 and the SU (6) 0 gauge theory with N TAS = 1 2 , N AS = 1, N F = 9, we consider the Higgsings (4.2) and (4.4). 7 There are two 14-dimensional representations of Sp(3). One is the rank-2 antisymmetric representation whose Dynkin label is [0, 1, 0] and the other is the rank-3 antisymmetric representation whose Dynkin label is [0, 0, 1]. We used 14 for the rank-3 antisymmetric representation. 8 The Chern-Simons level of an SU (6) gauge theory does not affect the IR Sp(3) gauge theory. This can bee seen for example from the effective prepotential computation. The Higgsing of SU (6) to Sp(3) using a vev for an antisymmetric hypermultiplet requires the tuning a6 = −a1, a5 = −a2 and a4 = −a3. Therefore the contribution to the effective prepotential from the Chern-Simons term becomes zero after the tuning. Sp(3) + 1 2 TAS + 19/2F We first consider the Higgsing from the SU (6) 0 gauge theory with N TAS = 1 2 , N AS = 1, N F = 9. A diagram for the SU (6) 0 gauge theory with N TAS = 1 2 , N AS = 1, N F = 9 is given by Figure 21. The Higgsing associated to the antisymmetric matter can be carried out diagrammatically as follows. The essential part involving the antisymmetric matter for an SU (6) gauge theory is depicted in Figure 28(a). From the diagram in Figure 28(a) we first perform flop transitions and move the (2, −1) 7-brane in the direction specified in Figure 28(b), which results in the diagram in Figure 28(c). In order to perform the Higgsing associated to the antisymmetric matter, we further do flop transitions until we obtain the diagram in Figure 28(d).
Note that the diagram in Figure 28(d) itself can be also understood from a Higgsing of a quiver theory involving an SU (6) gauge node. The Higgsing is depicted in Figure 29. The left diagram in Figure 29 has an SU (3) × SU (3) flavor symmetry associated to the external 5-branes. We can then partially Higgs the diagram by shrinking the lines in red of the left diagram in Figure 29. The tuning opens up a Higgs branch which is related to the space of deformations of pieces of 5-branes between 7-branes. After decoupling the pieces of 5-branes we end up with a diagram at low energies and it is given by the right diagram in Figure 29 which is the same diagram as the one in Figure 28(d). Then reading off the gauge theory content from the two diagrams in Figure 29 implies the following relation (4.5) In order to perform the Higgsing associated to the antisymmetric hypermultiplet, we need to further tune the length of lines in Figure 28(d). The lines which need to be shrunken are depicted in red in the left diagram in Figure 30. The tuning opens up a Higgs branch and decoupling pieces of 5-branes yields the right diagram in Figure 30. Hence, gauging the six horizontal D5-branes in the right diagram in Figure 30 gives rises to an Sp(3) gauge theory.
Therefore, the Higgsing associated to the antisymmetric matter can be achieved diagrammatically by replacing the right part of the diagram in Figure 21 with the diagram in Figure 30. Then we obtain a diagram for the Sp(3) gauge theory with N TAS = 1 2 , N F = 19 2 and it is given in Figure 31(a). By moving 7-branes of the diagram in Figure 31(a), it yields the diagram in Figure 31(b) and the diagram in Figure 31(b) shows that the diagram is a Tao diagram, implying that the the Sp(3) gauge theory with N TAS = 1 2 , N F = 19 2 has a 6d uplift.
Sp(3) + 1TAS + 5F It is also possible to perform the same Higgsing to the diagram for the SU (6) 0 gauge theory with N TAS = 1, N AS = 1, N F = 4 in Figure 17. By replacing the part involving the antisymmetric matter to the part giving the Sp(3) gauge group which is given by the diagram in Figure 30, we obtain a diagram depicted in Figure 32(a) for the Sp(3) gauge theory with N TAS = 1 and N F = 5 where one of the five hypermultiplets in the fundamental representation as well as a hypermultiplet in the rank-3 antisymmetric representation are massless. We can confirm that the Sp(3) gauge theory with N TAS = 1 and N F = 5 has a 6d UV completion from the diagram in Figure 32(a). From the diagram in Figure 32(a), we first move the (1, −1) 7-brane and the (1, 1) 7-brane inside the middle 5-brane loop together with the four flavor D7-branes. Then the (1, −1) 7-brane and the (1, 1) 7-brane form an O7 − -plane and the diagram becomes the one in Figure 32(b). The combination of an O7 − -plane and four D7-branes is S-dual invariant and hence after Sduality we obtain the diagram in Figure 32(c) which has a pair of an O7 − -plane and an O5 − -plane, showing periodicity in the vertical direction. The appearance of the periodicity in the vertical direction implies that the Sp(3) gauge theory with N TAS = 1 and N F = 5 has a 6d uplift.
5 Dualities and 6d uplift of marginal SU (6) gauge theories with rank-3 antisymmetric matter Since we have constructed 5-brane web diagrams which imply a 6d UV completion for the realized 5d theory on the web, it is natural to ask what is the 6d theory which completes 5d SU (6) or Sp(3) gauge theories with rank-3 antisymmetric matter at UV. In order to see the 6d uplift explicitly, we need to convert a 5-brane web diagram into some another configuration realizing a 6d theory. One way is to use T-duality and transform a 5-brane web into a brane configuration in type IIA string theory. In fact it is possible to convert the 5-brane web diagrams for the SU (6) gauge theory with N TAS = 1, N F = 10, κ = 0, the SU (6) gauge theory with N TAS = 1 2 , N F = 13, κ = 0 and the SU (6) gauge theory with N TAS = 1 2 , N Sym = 1, N F = 1, κ = 0 into type IIA brane system. Furthermore, in order to see the 6d uplift of the SU (6) 0 gauge theory with N TAS = 1 2 , N F = 13, it will be useful to first go to a dual frame which is given by a 5d quiver theory by moving 7-branes. The 6d uplift of the quiver theory has been known in [34,36,37] and we can make use of the result to see the UV completion of the the SU (6) 0 gauge theory with N TAS = 1 2 , N F = 13. Similar deformations by moving 7-branes will yield other dualities from SU (6) gauge theories with a half-hypermultiplet in the rank-3 antisymmetric representation and other matter.

6d uplift of SU (6) 0 + 1TAS + 10F
We start from the 5-brane web diagram in Figure 13 which realizes the SU (6) gauge theory with N TAS = 1, N F = 10, κ = 0. In section 3.2 we have seen that the SU (6) 0 gauge theory with N TAS = 1, N F = 10 has a 6d UV completion since it can be written as a Tao diagram which is given by the right diagram in Figure 13. Another way to see that the theory has a 6d UV completion is to form a pair of O7 − -planes placed in the vertical direction. For that after flop transitions we move the (1, 1) 7-brane and the (1, −1) 7-brane in the upper part and in the lower part inside 5-brane loops. Then each pair of the (1, 1) 7-brane and the (1, −1) 7-brane are put into the same 5-brane loops and two O7 − -planes are formed as in Figure 33(a). With the two O7 − -planes separated in the vertical direction, we can apply T-duality along the vertical direction which convert a pair of O7 − -planes into an O8 − -plane in type IIA string theory. Similarly a D5-brane becomes a D6-brane and an NS5-brane still remains to be an NS5-brane in type IIA string theory. Then the 5-brane web diagram in   Figure 33(b). It is straightforward to read off the gauge theory content from the brane system in Figure 33(b) and it is an Sp(2) − SU (2) quiver theory where ten flavors are coupled to the Sp(2) gauge group. Namely the brane configuration implies the following UV completion for the SU (6) 0 gauge theory with N TAS = 1, N F = 10. We can further support the 6d uplift (5.1) by counting the number of the parameters from both sides. We compactify the 6d theory on a circle and turn on holonomies for the SO(20) flavor symmetry. Hence we have ten parameters in addition to the radius of the circle, which gives eleven parameters in total. On the other hand the 5d theory has ten mass parameters for the ten flavors and also there is a gauge coupling for the SU (6) gauge theory. Therefore we have also eleven parameters in 5d, which matches with the number of the parameters obtained by a circle compactification of the 6d theory. Note that the rank-3 antisymmetric hypermultiplet of the SU (6) gauge theory is massless and there is no mass parameter for the rank-3 antisymmetric matter. Let us also see the matching of the number of Coulomb branch moduli. After a circle compactification a 6d tensor multiplet becomes a 5d vector multiplet. Hence two tensor multiplets in addition to the Cartan part for the Sp(2) and SU (2) vector multiplets yield 2 + 2 + 1 = 5 dimensional Coulomb branch moduli space in five dimensions. This agrees with the five Coulomb branch moduli of the SU (6) gauge theory.

5.2
Dualities and 6d uplift of SU (6) 0 + 1 2 TAS + 13F We then consider the SU (6) gauge theory with N TAS = 1 2 , N F = 13, κ = 0. The 5-brane diagram is given in Figure 18. From the diagram in Figure 18 we move D7-branes and perform flop transitions to arrive at the diagram in Figure 34 Figure 34(a), we move the (2, 1) 7-brane and the (1, −1) 7-brane along the arrows specified in Figure 34(a). Then the diagram becomes the one in Figure 34(b). We further move the (0, 1) 7-brand and the (1, 1) 7-brane in the diagram along the arrows depicted in Figure  34(b). The resulting diagram after the movement of the 7-branes is given in Figure 34(c). Then diagram in Figure 34(c) is nothing but a diagram for the [4] quiver theory. The 6d UV completion of the quiver theory has been discussed in [34,36,37] from manipulation of the 5-brane web and the 6d uplift is given by the SU (5) gauge theory with N F = 13, N AS = 1. We can form a pair of O7 − -planes from the diagram in Figure  34(c) and T-duality yields the type IIA brane system in Figure 35. Namely the brane configuration implies the following UV completion for the SU (6) 0 gauge theory with N TAS = 1 2 , N F = 13. We can also see that the number of the mass parameters and the Coulomb branch moduli from a circle compactification of the 6d theory agrees with the number of the mass parameters and the Coulomb branch moduli of the 5d theory.
Furthermore, since we can deform the diagram of the SU (6) gauge theory with N TAS = We can also see a relation to the 6d SU (6) gauge theory with N TAS = 1 2 , N F = 15. Note that applying a 5d limit to the 6d SU (6) gauge theory with N TAS = 1 2 , N F = 15 will yield a 5d SU (6) gauge theory with N TAS = 1 2 and some flavors. A 5d limit may be achieved by decoupling some Coulomb branch moduli of a 5d theory whose UV completion is given by the 6d SU (6)   In section 5.2, we have seen that a deformation of a 5-brane web implies that the SU (6) gauge theory with N TAS = 1 2 , N F = 13, κ = 0 is dual to quiver theories given by (5.3). A similar deformation of 5-brane webs of SU (6) gauge theories with N TAS = 1 2 can lead to other dualities. SU (6) −3 + 1 2 TAS + 9F We first consider the 5-brane diagram of the SU (6) −3 gauge theory with N TAS = 1 2 and N F = 9 which is given in Figure 20. From the diagram in Figure 20 moving some of the flavor D7-branes lead to the diagram in Figure 36(a). In the diagram in Figure 36(a), we move the (2, −1) 7-brane along the arrow and the diagram becomes the one in Figure 36(b). The moved (2, −1) 7-brane is now attached at the end of the external (2, −1) 5-brane in the upper right part of the diagram. From the diagram in Figure 36(b), we flop the line in blue and also move 7-branes in the blue circle to obtain the diagram in Figure 36(c). Finally, moving the (1, 1) 7-brane along the arrow in Figure 36(c) yields the diagram in Figure 36(d), which can be interpreted as SU (6) 0 + 1 2 TAS + 1AS + 9F In fact, the similar deformation can be applied to the SU (6) gauge theory with a hypermultiplet in the rank-2 antisymmetric representaiton in addition to N TAS = 1 2 and flavors. The first example is the SU (6) gauge theory wtih N TAS = 1 2 , N AS = 1, N F = 9 and κ = 0. The diagram of the theory has been given in Figure 21.
We then deform the diagram in Figure 21 to another diagram which can be interpreted as a quiver theory. For that it is enough to focus on a part of the diagram in Figure 21 which is given in Figure 37(a). Compared with the diagram in Figure 21, three D7-branes and a (0, 1) 7-brane are decoupled in the upper direction and the diagram yields the SU (6) gauge theory with N TAS = 1 2 , N F = 6, κ = 5 2 . From the diagram in Figure 37(a), we first move the (1, 1) 7-brane along the arrow to go to the diagram in Figure 37(b) and then move the (2, −1) 7-brane as in Figure 37(b). After the deformation, the resulting theory from the web in Figure 37(c) leads to the SU (2) − SU (5) 3 − [4F] quiver theory. Since we start the diagram with three D7-branes and the (0, 1) 7-brane decoupled, we need to reintroduce the 7-branes to the diagram in Figure 37(c) which yields the diagram in Figure 37(d). Then reintroducing the (0, 1) 7-brane adds a hypermultiplet in the antisymmetric representation of SU (5) and three D7-branes give three flavors to the SU (5). Hence, the deformations in Figure 37 imply that the SU (6) 0 gauge theory with N TAS = 1 2 , N AS = 1, N F = 9 is dual to the SU (2) − SU (5) 1 − [7F, 1AS] quiver theory. Another deformation by 7-branes may give further dual quiver theory such as The next example of dualities which involve SU (6) gauge theories with an antisymmetric hypermultiplet in addition to N TAS = 1 2 is the SU (6) gauge theory with N TAS = 1 2 , N AS = 1 and κ = 3 2 . The diagram has been obtained in Figure  23. In order to obtain a dual quiver description we can make use of the deformations from Figure 37(a) to Figure 37(c). From the diagram in Figure 3.3, decoupling one D7-brane and a (0, 1) 7-brane in the upper direction and also remove a D7-brane in the lower direction gives rise to the diagram in Figure 37(a). Hence we can reintroduce the 7-branes to the diagram in Figure 37(c) for a deformed diagram from the one in Figure 23. The final deformed diagram is depicted in Figure 38  5.4 6d uplift of SU (6) 0 + 1 2 TAS + 1Sym + 1F Here, we discuss the 5d SU (6) gauge theory with N TAS = 1 2 , N Sym = 1 2 , N F = 1, κ = 0. In [38], it is discussed that we obtain O7 + O7 - Figure 39. Type IIB web diagram corresponding to twisted circle compactification of 6d SU(3) quiver gauge theory. All the (p, 1) 5-branes are written vertically for simplicity.
by the twisted circle compactification of where we have 2k SU (N ) gauge nodes. Although N was assumed to be even number for simplicity when this was diagrammatially derived in [38], we can generalized this relation to the case for odd N . The 5d SU (6) gauge theory with N TAS = 1 2 , N Sym = 1 2 , N F = 1, κ = 0 turns out to be related to the case N = 3, k = 2.
The corresponding web diagram of this theory is given in Figure 39. It is straightforward to see that Figure 25 can be obtained by the Higgsing of this web diagram. From the point of view of the 6d theory, this Higgsing has to be done at the left hand side and the right hand side so that it is compatible with the twist. Therefore, we conclude that 6 Summary and discussion In this paper, we explicitly constructed 5-brane webs for 5d SU (6) and Sp(3) gauge theories with hypermultiplets in the rank-3 antisymmetric representation. For an SU (6) gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation, we started from a 5-brane diagram for the SO(12) gauge theory with a half-hypermultiplet in the conjugate spinor representation, and utilized the embedding SO(12) ⊃ SU (6)×U (1) where the rank-3 antisymmetric representation of the SU (6) is not charged under the U (1). Then decoupling the degree of freedom associated with the U (1) yields a 5-brane configuration for the SU (6) gauge theory of the Chern-Simons level κ = 5 2 with a half-hypermultiplet in the rank-3 antisymmetric representation. We also confirmed the validity of the brane diagram by computing the monopole string tension. Using the topological vertex, we computed the Nekrasov partition function for the SU (6) 5 2 + 1 2 TAS theory up to two instanton orders and confirmed that the instanton part correctly captures the Chern-Simons level, which also supports our 5-brane construction of the SU (6) 5 2 + 1 2 TAS theory. This would be the first quantitative result for the partition function for the SU (6) 5 2 gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation.
By increasing the number of half-hypermultiplets in the rank-3 antisymmetric representation and also adding various hypermultiplets in other representations, we constructed 5-brane diagrams for 5d marginal SU (6) gauge theories with N TAS = 1 2 , 1, 2 hypermultiplets in the rank-3 antisymmetric representation, which are summarized in Table 1. The matter content of what we found for the 5d marginal SU (6) gauge theories with rank-3 antisymmetric hypermultiplets is in agreement with those classified in [15]. Moreover, global symmetries that one can read off from the 5-brane webs also support our 5-brane construction for the marginal SU (6) theories with half-hypermultiplets in the rank-3 antisymmetric representation. The 5-brane web diagrams also imply 6d uplifts or dualities for some of the marginal theories. Interestingly, as discussed in section 5, some SU (6) gauge theories with half-hypermultiplets in the rank-3 antisymmetric representation are dual to quiver theories.
For marginal Sp(3) gauge theories with half-hypermultiplets in the rank-3 antisymmetric representation, we used a Higgsing of marginal SU (6) gauge theories involving a hypermultiplet in the rank-2 antisymmetric representation. Possible Higgsings are discussed in (4.2)-(4.4). Since we know two of the UV SU (6) theories of the three Higgsings, we explicitly realized 5-brane configuration for the Sp(3) + 1 2 TAS + 19 2 F theory in Figure  31, and for the Sp(3) + 1TAS + 5F theory in Figure 32.
Although we have constructed 5-brane webs for many of the marginal SU (6) gauge theories with rank-3 antisymmetric matter in Table 1 which are classified in [15], there are some marginal theories that we did not find their 5-brane web configurations. We note that not having a 5-brane web for a marginal theory does not imply that 5-brane webs for its descendent theories are not constructed. For instance, consider SU (6) 0 + 3/2TAS + 5F or SU (6) 2 +3/2TAS+3F. As discussed in section 2.4, a 5-brane web for the SU (6) 1 2 +3/2TAS theory is given in Figure 9(b). In fact, one can introduce flavors to the 5-brane web diagram properly to make the CS level to be that of the marginal theories of interest. For SU (6) 0 + 3/2TAS + 5F, one may find a configuration with 3 D7-branes added above and 2 D7-branes below, so that the resulting configuration has the CS level 0. It is also not so difficult to find that a little manipulation of the 7-branes allows a pair of 7-branes which can be converted to an O7 − -plane, and hence together with an ON − -plane, it yields a configuration with two orientifolds horizontally separated. On the other hand, we were not able to make this 5-brane web to be a conventional 5-brane configuration with O7 − -plane and O5-plane vertically apart, after performing the S-duality. Hence, we did not include such 5-brane configuration in Table 1.
It is also worth noting that there may be some intrinsic issues on 5-brane realization of marginal theories. For example, our construction utilizes 5-brane web diagrams and there will be some restriction to theories which 5-brane web on a plane can realize. Second, the marginal theories in [15] were classified based on only necessary conditions and hence it may be still possible that some of them may not have a UV completion. We here make some comments on marginal SU (6) gauge theories which we have not constructed from 5brane webs in this paper. One class of such theories is SU (6) gauge theories with N TAS = 1 2 , N AS = 2 and flavors. Note that a half-hypermultiplet in the rank-3 antisymmetric representation can arise from a Higgsing from [SU (6) where the CS level for the SU (6) gauge node needs to be chosen so that the quiver theory has a UV completion. The quiver theory (6.1) has a shape of the E 6 Dynkin diagram and it is difficult to realize the quiver theory from 5-brane web diagrams on a plane. Therefore the Higgsed theories, which are SU (6) gauge theories with N TAS = 1 2 , N AS = 2, would be also difficult to be obtained by 5-brane webs on a plane. We will need trivalent gauging for web diagrams considered in [39].
There are also marginal SU (6) gauge theories which only have the rank-2 antisymmetric representation as matter. The SU (6) gauge theories have three hypermultiplets in the rank-2 antisymmetric representation and the CS level can be |κ| = 0, 1, 2, 3 [15]. Since a hypermultiplet in the rank-2 antisymmetric representation arises from a Higgsing from [SU (6) The quiver theory (6.2) has a shape of the affine E 6 Dynkin diagram which might have a 6d UV completion for some specific CS level for the SU (6) gauge node. In particular when the CS level for the SU (6) gauge node is zero, the Higgsed theory becomes the SU (6) gauge theory with N AS = 3, κ = 3 since each Higgsing which gives the rank-2 antisymmetric matter increases the CS level by one as in (4.5). The SU (6) theory is exactly one of the marginal SU (6) theories classified in [15]. Since this quiver is an affine E 6 Dynkin quiver it is difficult to realize it by a 5-brane web on a plane, implying that the Higgsed theories would be also difficult to be obtained by a brane web on a plane. In order to construct the affine E 6 quiver theory, we will need trivalent gauging for web diagrams again. 5-brane webs we found for marginal theories show the periodicity either as a Tao diagram [22] or as a 5-brane configuration with two orientifolds. It clearly suggests that they are a realization of 6d theory on a circle with/without some twists. For some marginal theories, we discussed their 6d uplifts. It would be interesting to find 6d uplifts for other marginal theories as well as possible dual quiver descriptions.
In this paper we have focused on half-hypermultiplets in the rank-3 antisymmetric representation for SU (6) and Sp(3) gauge theories. It would be also interesting to generalize the result to rank-3 antisymmetric matter for other gauge theories such as SU (7) or Sp(4) gauge theories which were also discussed in [15].