Dualities and 5-brane webs for 5d rank 2 SCFTs

We consider Type IIB 5-brane configurations for 5d rank 2 superconformal theories which are classified recently by geometry in arXiv:1801.04036. We propose all the 5-brane web diagrams for these rank 2 theories and show dualities between some of different gauge theories with explicit duality map of mass parameters and Coulomb branch moduli. In particular, we explicitly construct 5-brane configurations for $G_2$ gauge theory with six flavors and its dual $Sp(2)$ and $SU(3)$ gauge theories. We also present 5-brane webs for $SU(3)$ theories of Chern-Simons level greater than 5.


Introduction
Higher-dimensional gauge theories are in general not renormalizable as the gauge coupling becomes infinitely strong at high energies. However, those theories may make sense in the ultraviolet (UV) region when they have a non-trivial fixed point in UV. Such a phenomenon has arisen in the context of five-dimensional (5d) gauge theories with eight supercharges. From the field theoretic point of view, a necessary condition for the existence of UV complete 5d theories is that the metric of the Coulomb branch moduli space, which may be computed from the effective prepotential, must be non-negative [2,3]. The condition revealed a possibility of the existence of some UV complete 5d gauge theories and their existence has been also confirmed by explicitly constructing the 5d gauge theories from M-theory compactifications on non-compact Calabi-Yau threefolds [2][3][4][5] and also from 5-brane webs in type IIB string theory [6,7]. However, it had turned out that 5-brane web diagrams can realize 5d gauge theories that lie beyond the bound given in [3]. For example, one can add the hypermultiplets in the fundamental representation (flavors) up to N f = 2N + 4 for an SU (N ) gauge theory [8,9] 1 although the original bound was N f = 2N . Recently, the field theoretic condition for the existence of UV complete 5d gauge theories has been revisited and it was claimed in [12] that the original condition discussed in [2,3] should be relaxed. Namely, the metric of the Coulomb branch moduli space should be non-negative only on a "physical" Coulomb branch moduli space where the tension of monopole strings is non-negative. The SU (N ) gauge theory with N f = 2N + 4 flavors, which has the 5-brane web realization, indeed satisfies the new criteria. Not only that, the new condition in fact led to a large class of new UV complete 5d theories in [12]. Although the criteria is a necessary condition for the existence of UV complete 5d gauge theories, most of the new rank 2 gauge theories found in [12] have been also constructed geometrically using M-theory compactifications on non-compact Calabi-Yau threefolds in [1], which confirms their existence. Furthermore, the geometric construction implies intriguing dualities including 5d G 2 gauge theories. For example, the identical physics is described by the G 2 gauge theory with six flavors, the SU (3) gauge with six flavors and the Chern-Simons (CS) level 4 and the Sp(2) gauge theory with 4 flavors and two hypermultiplets in the antisymmetric representation.
It is then natural to ask if the 5d rank 2 gauge theories constructed by geometries in [1] also admit a realization by 5-brane web diagrams in type IIB string theory. In this paper, we propose 5-brane web diagrams for all the 5d rank 2 gauge theories whose existence is geometrically confirmed in [1]. In particular, we start with a 5-brane web for the pure G 2 gauge theory [13], and add more flavors to explicitly construct new 5-brane web diagrams for the G 2 gauge theory with six flavors, the SU (3) gauge theory with six flavors and the CS level 4 and the Sp(2) gauge theory with four flavors and two antisymmetric hypermultiplets. Their 5-brane diagrams have a periodic direction implying a 6d UV fixed point. Furthermore, the dualities among the G 2 , SU (3) and Sp(2) gauge theories may be understood from S-duality by rotating the 5-brane web diagrams accompanied with Hanany-Witten transitions by moving 7-branes. The explicit realization of the dualities gives us the duality maps for the Coulomb branch moduli and parameters among these three theories.
It is worth noting that in constructing 5-brane webs for an SU (3) gauge theory with higher CS level, in particular, pure SU (3) gauge theory with the CS level 7 which is dual to pure G 2 gauge theory, one may naively attempt to construct the theories with higher CS level by increasing the charge difference of the external 7-branes. In this way, the external 7-brane inevitably collide each other as moving each 7-brane to infinity. Up to the CS level 6, a suitable handling the monodromy cut of 7-branes would give a 5-brane web which the resulting 7-branes are no longer collide and thus can be taken to infinity. But for an SU (3) theory with CS level 7 or higher, such procedure involving a 7-brane going across another 7-brane often leads to ill-defined 5-brane web such that a 7-brane after going across monodromy cut of other 7-brane bends toward the center of the 5-brane web due to 7-brane charge changes from the monodromy. Hence, though not conventional, a 5-brane construction for pure SU (3) theory with CS level 7 seems best understood from the S-duality of pure G 2 gauge theory.
Starting from these 5-brane webs for the G 2 theories of various flavors or their dual SU (3) or Sp(2) theories, one can add different choices of hypermultiplets in an appropriate representation and to build up the tree of all theories connected by flavor decoupling or adding. In addition, a finer understanding of the 5-brane web for G 2 theory without flavor allows us to find the web for SU (3) theory without flavor and CS level 9. These leads to the construction of the 5-brane webs for the whole family rank 2 gauge theories constructed in [1]. A 5-brane web diagram for 5d SU (3) theory of the CS level 3 2 and with one hypermultiplet in the symmetric representation, can be constructed with O7 +and O7 − -planes, which reveals a new 5-brane structure for this marginal theory. From the constructed 5-brane web diagrams we can also obtain the duality map between the SU (3) gauge theory with nine flavors and the CS level 3 2 and the Sp(2) gauge theory with eight flavors and single antisymmetric hypermultiplet.
The organization of this paper is as follows. In section 2, we construct 5-brane web diagrams for the G 2 gauge theories with 0, 2 and 6 flavors and see the dualities to an SU (3) or an Sp(2) gauge theory with or without flavors. The duality map is obtained in each case. We also present a web diagram of the Sp(2) gauge theories from the viewpoint of SO (5).
In section 3, we consider deformations from the diagram in section 2 and obtain the other SU (3) gauge theories and their dual Sp(2) gauge theories with the duality map. Section 4 is devoted for another deformation to the Sp(2) gauge theory with three hypermultiplets in the antisymmetric representation from the viewpoint of SO (5). In section 5 we propose a 5-brane web diagram for the pure SU (3) gauge theory with the CS level 9. We will then summarize our results in section 6. Appendix A explains some subtle identification of the inverse of the squared gauge coupling for SO(5) gauge theories with spinors. In appendix B, we summarize all the 5-brane webs for rank 2 theories which were constructed using geometries in [1].

G -SU (3)-Sp(2) sequence
In this section, we first consider dualities involving G 2 gauge theories with flavors. A dual description of a G 2 gauge theory is given by an SU (3) gauge theory and/or an Sp(2) gauge theory depending on flavors [1]. We have constructed 5-brane web diagrams for G 2 gauge  Figure 1. (a): A 5-brane web diagram for the pure G 2 gauge theory which is realized with an O5plane. The orange line represents the O5-plane. The convention of the O5-plane is different from the one used in [13]. Here, the monodomry branch cut from a fractional D7-brane is extends from left to right whereas it extends from right to left in [13]. (b): The gauge theory parameterization of the pure G 2 diagram. theories in [13] and here generalize the construction to the case for the G 2 gauge theory with six flavors which may have a 6d UV completion. We will see the dualities from the viewpoint of 5-brane web diagrams.

Without matter
Before considering the duality involving G 2 gauge theory with flavors, we start from the case without matter. The pure G 2 gauge theory is dual to the pure SU (3) gauge theory with the CS level 7 [1]. We can also see the duality from the viewpoint of 5-brane webs.
Let us first review a 5-brane web diagram for the pure G 2 gauge theory. Two types of the 5-brane web diagram for the pure G 2 gauge theory have been proposed in [13]. In order to see the duality to the pure SU (3) gauge theory with the CS level 7, it is useful to consider the pure G 2 diagram with an O5-plane.
The strategy to realize the 5-brane web diagram for the pure G 2 gauge theory was as follows. We first start from a 5-brane web diagram for the SO(7) gauge theory with a hypermultiplet in the spinor representation. Then the Higgsing associated to the spinor matter yields the pure G 2 gauge theory at low energies. Hence if we apply the Higgsing procedure to the 5-brane web diagram for the 5-brane web of the SO(7) gauge theory with one spinor, the resulting diagram should be a 5-brane web for the pure G 2 gauge theory. The diagram obtained in this way is depicted in Figure 1(a). Figure 1(b) shows the parameterization for the Coulomb branch moduli a 1 , a 2 and the inverse of the squared gauge coupling m 0 .
From the 5-brane web for the pure G 2 gauge theory in Figure 1(a), we can see the duality to the pure SU (3) gauge theory with the CS level 7. To see the duality, we first take the S-duality which corresponds to the π 2 rotation for the diagram in Figure 1(a). In terms of geometry it corresponds to the fiber-base duality [1,7,14,15]. Application of the π 2 rotation to the diagram in Figure 1   postulated the S-dual object of an O5 ± -plane as an ON ± -plane [16][17][18][19][20][21]. We claim that this 5-brane web in Figure 2 represents pure SU (3) gauge theory with Chern-Simons level 7. We justify this claim by comparing the area of the compact faces of the web diagram with the effective prepotential or the tension of monopole string. This claim can also be justified from the decoupling of two flavors from 5-brane description for the SU (3) gauge theory with CS level 6 and two flavors, which has a clear 5-brane interpretation as a Higgsing of a quiver description SU (2) − SU (3) 3 − SU (2). We will discuss more detail in section 2.2.
We now compute the area from the 5-brane web in Figure 2 and compare it with the effective prepotential or the tension of the monopole string for the pure SU (3) gauge theory from the diagram. For that we first assign gauge theory parameters to the diagram as in Figure 3(a). a 1 , a 2 , a 3 (a 1 + a 2 + a 3 = 0) are the Coulomb branch moduli and m 0 is the inverse of the squared gauge coupling. Then the area of the faces in Figure 3 We then compare the area (2.1)-(2.4) with the effective prepotential of the pure SU (3) gauge theory with the CS level 7. In general, the effective prepotential of a 5d gauge theory with a gauge group G and matter is given by a cubic function of the Coulomb branch moduli φ i [3,4,22] where m 0 is the inverse of the squared gauge coupling, κ is the classical Chern-Simons level and m f is the mass parameter for hypermultiplets in the representation R f . We also defined h ij = Tr(T i T j ) and d ijk = 1 2 Tr (T i {T j , T k }) where T i are the Cartan generators of the Lie algebra associated to G. Then the effective prepotential for the pure SU (3) gauge theory with the CS level 7 becomes where we changed the basis of the Coulomb branch moduli into the Dynkin basis by using the relation in (2.6). The tension of the monopole string of the pure SU (3) gauge theory with the CS level 7 is given by taking a derivative of (2.6) with respect to the Coulomb branch moduli φ 1 , φ 2 . Hence the tension is given by We can compare the tension (2.8), (2.9) with the area (2.1)-(2.4) to see the pure SU (3) gauge theory realized by the web in Figure 2 have the CS level 7. It turns out that D3brane does not cover each face of the diagram in Figure 3(b). The comparison between the effective prepotential of the pure G 2 gauge theory and the area of the faces in [13] revealed that one face which D3-brane is wrapped on is 1 + 2 + 2 3 and the other face is 4 . The coefficient 2 in front of 3 may be interpreted by the effect of including the mirror image.
Since we have a single 5-brane web diagram for the pure G 2 gauge theory and the pure SU (3) gauge theory with the CS level 7, it is also possible to obtain an explicit duality map between the parameters of the two theories. The length of a line in the diagram in Figure 2 can be written by the two parameterizations. Since it is a single line the length written by the two parameterizations should be the same. Then we obtain the following duality map where we used the Dynkin basis also for the Coulomb branch moduli of the pure G 2 gauge theory by using (2.24).

With matter
In section 2.1, we saw that the 5-brane web diagram of the pure G 2 gauge theory in Figure  1(a) is S-dual to the 5-brane web diagram of the pure SU (3) gauge theory with the CS level 7 in Figure 2. In this section, we consider a 5-brane web diagram of the G 2 gauge theory with two hypermultiplets in the fundamental representation (flavors). The G 2 gauge theory with two flavors (G 2 + 2F) is dual not only to the SU (3) gauge theory with two hypermultiplets in the fundamental representation and the CS level 6 (SU (3) 6 +2F) but also to the Sp(2) gauge theory with two hypermultiplets in the antisymmetric representation and the non-trivial discrete theta angle (Sp(2) π + 2AS) [1]. The two dualities are also indeed seen from the viewpoint of 5-brane webs. The 5-brane web in Figure 1(a) for the pure G 2 gauge theory is obtained by Higgsing a 5-brane web for the SO(7) gauge theory with one spinor. In order to introduce one flavor in the G 2 gauge theory we can consider a Higgsing of an SO(7) gauge theory with a spinor or a flavor in addition to one spinor which we use for the Higgsing. A hypermultiplet in the vector representation of SO(7) becomes one hypermultiplet in the fundamental representation of G 2 after the Higgsing, while a hypermultiplet in the spinor representation of SO(7) becomes one flavor and a singlet of G 2 . For later convenience, we introduce two flavors to the pure G 2 gauge theory by considering a Higgsing of the SO(7) gauge theory with three spinors. The 5-brane web diagram obtained by the Higgsing is given in Figure  (  The matter is originated from two spinors of a SO(7) gauge theory before the Higgsing. (b): Another 5-brane web diagram for the G 2 gauge theory with two flavors and two singlets, which is obtained by performing a flop transition to the diagram in Figure 4(a).  4(a). One can perform a "generalized flop transition" [13,23] for the Sp(0) part in Figure  4(a) and then the diagram becomes the one in Figure 4 We can also assign the gauge theory parameters to the length of the 5-branes as in Figure 5(a). a 1 and a 2 are the Coulomb branch moduli of the G 2 gauge theory. Since the G 2 gauge theory with two flavors originates from the SO(7) gauge theory with three spinors where two spinors are attached to the left side and one spinor, which is used for the Higgsing, is attached to the right side the diagram of the SO(7) diagram before the Higgsing. Hence, the inverse of the squared gauge coupling of the G 2 gauge theory with two flavors can be read off similarly to the case of the SO(7) gauge theory with two spinors. As explained in appendix A, the inverse of the squared gauge coupling, m 0 , in this case can be computed by extrapolating the leftmost (1, −1) 5-brane and the rightmost (1, 1) 5-brane. Hence m 0 of the G 2 gauge theory with two flavors can be chosen in Figure 5.
The diagram contains two more parameters n 1 , n 2 depicted in Figure 5(b), which are determined by the position of asymptotic 5-branes. The two parameters are related to the ① ② ③ ④ ⑤ Figure 6. A labeling for the faces in the diagram of the G 2 gauge theory with two flavors. mass parameters for the two flavors. From the viewpoint of the Sp(0), n 1 is the inverse of the squared gauge coupling of the Sp(0) and n 2 is the mass parameter for one flavor of the Sp(0). Hence, the two parameters are associated to the flavor symmetry U (1) × SO(3). The SO(3) arises from the flavor D5-brane whose height is n 2 on top of an O5 − -plane. On the other hand, the two flavors of the G 2 gauge theory yield an Sp(2) flavor symmetry. Therefore, we need to change the basis given by the embedding U (1) × SO(3) ⊂ Sp(2) to obtain the mass parameters m 1 , m 2 of G 2 + 2F from n 1 , n 2 , 2 One can check the validity of the parameterization by computing the prepotential of the G 2 gauge theory with two flavors. From the parameterization, we can compute the tension of the monopole string by the area of the faces in Figure 6. The label of the faces are given in Figure 6. The area of the faces is given by We then compare the area (2.16)-(2.20) with the tension of the monopole string computed from the effective prepotential (2.5). In order to compute the prepotential of the G 2 gauge theory with two flavors, we need to determine the phase corresponding to the diagram in Figure 4(b). The phase can be fixed from the requirement that the length of any 5-brane in the diagram is positive. Then the parameterization given in Figure 5(a) and (2.15) implies the phase for the flavor with the mass parameter m 1 and for the flavor with the mass parameter m 2 . The effective prepotential of the G 2 gauge theory with two flavors on this phase becomes where we used the Dynkin basis for the Coulomb branch moduli. Then the tension of the monopole string of the G 2 gauge theory with two flavors is given by taking a derivative of the effective prepotential (2.24) with respect to the Coulomb branch moduli φ 1 , φ 2 . Note that as in the case of the pure SU (3) gauge theory with the CS level 7, a D3-brane will not be wrapped on arbitrary five faces but needs to be wrapped on a particular linear combination. In particular we need to consider 2 + 3 + 2 × 4 + 5 for the tension given by a derivative with respect to φ 2 . We indeed find the agreement between the linear combination of the area (2.16)-(2.17) and the tension of the monopole string, The equalities (2.25) and (2.26) implies the correctness of the parameterization in Figure  5(a) and (2.15) and they also reconfirm that the diagram in Figure 4(b) gives rise to the G 2 gauge theory with two flavors.

Duality to SU (3) 6 + 2F
As we saw the duality between the pure G 2 gauge theory and the pure SU (3) gauge theory with the CS level 7 from the 5-brane webs in section 2.1, it is also possible to see the duality between the G 2 gauge theory with two flavors and the SU (3) gauge theory with two flavors and the CS level 6 from the 5-brane web in Figure 4(b). Applying the S-duality to the diagram in Figure 4(b) yields a diagram in Figure 7. Since the diagram contains three color D5-branes, the diagram may be interpreted as an SU (3) gauge theory. The diagram contains two more parameters except for the gauge coupling, which are determined by the position of asymptotic 5-branes. Hence the parameters are associated to mass parameters of some matter of the SU (3) gauge theory. We argue that the matter is two hypermultiplets in the antisymmetric representation, which is equivalent to the antifundamental representation. Let us see a 5-brane web diagram for the SU (3) gauge theory with one hypermultiplet in the antisymmetric representation and the CS level 1 2 [24], which is depicted in Figure 8(a). Since the antisymmetric representation of SU (3) is equivalent to the antifundamental representation, one can deform the diagram in Figure  8(a) into the one from which we can explicitly see the presence of one flavor. When we move the (0, 1) 7-brane in Figure 8(b) according to the indicated arrow, the HW transition gives a diagram in Figure 8(c). This is nothing but a diagram giving rise to the SU (3) gauge theory with one flavor and the CS level 1 2 . This diagram can be obtained by a Higgsing from the SU (3) − SU (2) quiver theory where the CS level of the SU (3) is −1 [25]. The theory has an SU (2) flavor symmetry associated to the two external NS5-branes extending in the upper direction in Figure 8. One can perform a Higgsing associated to the SU (2) flavor symmetry, which corresponds to the tuning of the length of the 5-branes indicated by the purple × in Figure 8(d). The diagram exactly reduces to the one in Figure 8(a). Namely, the Higgsing of the quiver SU (3) −1 − SU (2) associated to the flavor symmetry SU (2) can yield the SU (3) gauge theory with a hypermultiplet in the antisymmetric representation and the CS level 1 2 . Extending the idea of the Higgsing, an SU (3) gauge theory with two antisymmetric hypermultiplets can originate from a Higgsing of the SU (2)−SU (3) 3 −SU (2) quiver theory. Namely a Higgsing of the two SU (2) gauge theories may yield two hypermultiplets in the antisymmetric representation for the SU (3). The SU (2) − SU (3) 3 − SU (2) quiver theory can be realized by using an ON − -plane [21,26] as in Figure 9(a). The two color branes for one of the SU (2) gauge group are given by the orange lines in Figure 9(a) and the two color branes for the other SU (2) gauge group are represented by red segments in Figure 9(a). The theory also has an SO(4) SU (2) × SU (2) flavor symmetry associated to the two external NS5-branes extending in the upper direction with an ON − -plane in Figure 9(a). We then perform two Higgsings which break the SU (2) × SU (2) flavor symmetry. The first Higgsing is realized by tuning the length of the 5-branes indicated by the purple × in Figure 9(b). Then the second Higgsing is achieved by tuning the length of the 5-branes indicated by the purple × in Figure 9(c). Then the resulting diagram becomes the one in Figure 9(d). After the two Higgsings, only one of the two color branes for each SU (2) gauge group remains and the two SU (2) gauge groups are broken. In order to connect to the diagram in Figure 7, we perform one flop transition and obtain the diagram in Figure 9(e). Although the diagram in Figure 9(e) is written with an ON − -plane. One can change the ON − -plane into an ON − -plane by moving a fractional D7-brane which may be put at the end of an external NS5-brane on an ON − -plane [13,27]. Therefore, the Higgsings of the two SU (2) in the quiver theory SU (2) − SU (3) 3 − SU (2) yields the diagram in Figure 7, implying that the SU (3) theory contain two hypermultiplets in the antisymmetric representation. Furthermore, the Higgsing of one SU (2) in Figure 8 increased the CS level by 3 2 . Hence it is natural to expect that the Higgsing of the two SU (2) through the process 9(b)-9(d) increases the CS level by 3 2 × 2 = 3. Hence, the CS level after the Higgsing will be 6. In summary, the diagram in Figure 7 may yield the SU (3) gauge theory with two antisymmetric hypermultiplets, or equivalently two flavors, and the CS level 6.
Let us confirm that the diagram in Figure 7 gives rise to the SU (3) gauge theory with two flavors and the CS level 6 from the computation of the effective prepotential. For that we assign gauge theory parameters for the length of 5-branes in Figure 7. The inverse of the squared gauge coupling m 0 = l 1 +l 2 2 and the Coulomb branch moduli a 1 , a 2 , a 3 , (a 1 + a 2 + a 3 = 0) are given in Figure 10, which is the same parameterization as that for the pure SU (3) gauge theory with the CS level 7 in Figure 3(a).
In order to see the parameterization of the mass parameters m 1 , m 2 , let us first recall how the mass parameter for antisymmetric hypermultiplet appears in Figure 8(a). The length associated to the mass parameter m is depicted in Figure 11(a). It is give by twice as long as the distance between the location of an O7 − -plane and the origin of the height for the SU (3) color branes. The location of the O7 − -plane can be determined by the intersection point between the line of the (0, 1) 5-brane and the line of the (2, −1) 5-brane. In terms of the length in the diagram for the SU (3) −1 − SU (2) quiver theory, the mass parameter is twice as long as the distance between the origin of the height for the SU (3) color branes and the origin of the height for the SU (2) color branes as in Figure 11(b). The distance between the two origins of the SU (3) and the SU (2) is nothing but the mass parameter for the bifundamental matter. Therefore, the antisymmetric mass is originated from the bifundamental mass before the tuning.
We can now generalize the discussion for the case with two antisymmetric hypermultiplets. Namely, a mass parameter is twice as long as the distance between the origin of the location of the SU (3) color branes and the origin of the location for the one of the SU (2). The orange dotted line in Figure 11(c) is originated from the origin for the SU (2) color branes given by the orange lines in Figure 9(a). One the other hand, the red dotted line in Figure 11(c) is originated from the origin for the SU (2) color branes given by the red lines in Figure 9(a). Therefore, the two mass parameters are related to the distance between the origin of the SU (3) and the orange dotted line or the red dotted line as in Figure 11  level 6. We again use the labels for the faces in Figure 6. Then the area of the five faces is given by where we used the Dynkin basis (2.7). The area corresponds to the tension of the monopole string which can be computed by taking a derivative of the effective prepotential with respect to Coulomb branch moduli. The effective prepotential of the SU (3) gauge theory with two flavors and the CS level 6 may be calculated from the general formula (2.5). The condition that the length of the Figure 7 is positive implies the following phase −a 1 − m 1 < 0, −a 2 − m 1 > 0, −a 3 − m 1 > 0, (2.32) for one antisymmetric hypermultiplet with mass m 1 and

5-branes in
for the other antisymmetric hypermultiplet with mass m 2 . Here we used a 1 + a 2 + a 3 = 0 and expressed the weight of the antisymmetric representation by the weight of the antifundamental representation. Then the effective prepotential becomes Then we find the expected relation between the area (2.27)-(2.31) and the tension of the monopole string, The equalities (2.35) and (2.36) confirms that the SU (3) gauge theory has the CS level 6 and two flavors. By comparing the parameterization of the G 2 gauge theory and that of the SU (3) gauge theory, we can also obtain the duality map between the parameters. The duality map is given by where we put the subindex standing for the representation for the matter. The labeling of the number for the masses is the same as before. We will use this convention for writing duality maps hereafter. We note that if one decouples two hypermultiplets from the 5-brane web in Figure 9(e) (or equivalently Figure 7), then the resulting diagram is same as the 5-brane web in Figure  2, which we claimed a 5-brane web for the pure SU (3) theory with CS level 7. This hence provides a support of our construction of 5-brane web for the SU (3) gauge theory with the CS level 7, discussed in section 2.1, as the decoupling of two flavors would increase the CS level by 1.

Duality to Sp(2) π + 2AS
We have seen that the 5-brane diagram of the G 2 gauge theory with two flavors is S-dual to the diagram of the SU (3) gauge theory with two flavors and the CS level 6. In fact, the G 2 gauge theory with two flavors admits another dual description given by the Sp(2) gauge theory with two hypermultiplets in the antisymmetric representation and the non-trivial discrete theta angle. We will argue that this duality can be also seen from the 5-brane web diagram. We first start from the 5-brane web for the G 2 gauge theory with two flavors in Figure  4(b). In order to see the duality, we first perform two flop transitions and obtain a diagram in Figure 12(a). We then move a (1, −1) 7-brane and a (1, 1) 7-brane according the arrows in Figure 12(b). After moving the two 7-branes the diagram becomes the one in Figure  13(a). At this stage, we apply the S-duality to the diagram in Figure 13(a). Then the resulting configuration contains a pair of a (1, −1) 7-brane and a (1, 1) 7-brane in the same 5-brane chamber as in Figure 13(b). The (1, −1) 7-brane and the (1, 1) 7-brane may form an O7 − -plane [28] and we obtain the configuration in Figure 14(a). Since we have four color D5-branes with an O7 − -plane, the theory realized by the diagram in Figure 14  may be an Sp(2) gauge theory. A next question is whether the diagram in Figure 14(a) contains two hypermultiplets in the antisymmetric representation of the Sp(2). As we saw in section 2.2.1, the presence of the antisymmetric hypermultiplets can be understood from a Higgsing of an SU (2) 0 − Sp(2)−SU (2) 0 quiver theory also for an Sp(2) gauge theory. For that let us first see how the antisymmetric hypermultiplet of an Sp(2) gauge theory can appear from a 5-brane web. A 5-brane diagram for the Sp(2) gauge theory with one antisymmetric hypermultiplet and the zero discrete theta angle is given in Figure 15(a) [24]. The two external (1, 1) 5-branes in Figure 15(a) realizes an SU (2) flavor symmetry from the one antisymmetric hypermultiplet. It is also possible to see that the diagram for the Sp(2) 0 gauge theory with one antisymmetric hypermultiplet in Figure 15(a) can be obtained from a Higgsing of the Sp(2) 0 − SU (2) 0 quiver theory. A 5-brane web diagram of the Sp(2) − SU (2) quiver theory with the zero discrete theta angle for both gauge groups is depicted in Figure 15(b). The diagram shows an SU (2) × SU (2) flavor symmetry generated non-perturbatively from the viewpoint of the quiver theory. We can then perform a Higgsing associated to one of the SU (2) flavor symmetry by tuning the length of 5-branes indicated by the purple × in Figure 15(b). Then the Higgsing precisely yields the diagram in Figure 15(a). Therefore, the Sp(2) gauge theory with a hypermultiplet in the antisymmetric representation and the zero discrete theta angle can be obtained from the Higgsing of the Sp(2) 0 − SU (2) 0 quiver theory. We now apply the Higgsing argument to a diagram with an ON − -plane. Namely we consider a Higgsing of an SU (2)−Sp(2)−SU (2) quiver theory to obtain a 5-brane diagram of an Sp(2) gauge theory with two hypermultiplets in the antisymmetric representation. An SU (2)−Sp(2)−SU (2) quiver theory is realized by a diagram in Figure 16(a). In Figure  16(a), the two SU (2) gauge theories are realized from an ON − -plane. The D5-branes in the red color represents one SU (2) and the D5-branes in the orange color gives another SU (2). From the parallel external (0, 1) 5-branes, the diagram exhibits an SO(4) × SO(4) flavor symmetry. It turns out that we need to consider the Sp(2) gauge theory with the discrete theta angle π in order to connect to the diagram in Figure 14(a). The discrete theta angle of the Sp(2) gauge group can be more easily seen from the diagram in Figure  16(b). When we take the length of the middle (4, −1) 5-brane to be infinitely large, then the diagram is decomposed into two parts. The left part gives an Sp(2) gauge theory and the right part yields the two disconnected SU (2) gauge theories. Then the Sp(2) gauge theory realized in the left part of the diagram in Figure 16(b) has the discrete theta angle π [24,25]. On the other hand, the diagram for the two disconnected SU (2) gauge theories exhibits an SO(4) SU (2) × SU (2) flavor symmetry. Therefore the two SU (2) gauge theories both have the zero discrete theta angle.
From the diagram in Figure 16(a), we consider two Higgsings which break the two SU (2) gauge groups. The Higgsing will also reduce the flavor group from SO(4)×SO(4) → SO(4). We in particular consider a Higgsing which breaks the SO(4) flavor symmetry associated to the parallel (0, 1) 5-branes going in the upper direction. The Higgsing can be understood by two steps where each step is associated to each SU (2) in the SO(4). The first Higgsings is carried out by tuning the length of the 5-branes with the purple × in Figure 17(a) and the second Higgsing is done by setting the length of the 5-branes with the purple × in Figure 17(b) to be zero. The two Higgsings leave only one color brane for each SU (2) gauge groups and the two SU (2) gauge groups are broken. The resulting diagram after the Higgsing is depicted in Figure 17(c). The diagram in Figure 17(c) is in fact equivalent to the diagram in Figure 14(a) by moving an (0, 1) 7-brane which may be put at the end the external (0, 1) 5-brane extending in the upper direction [13,27]. Since the original Sp(2) gauge theory has the non-trivial discrete theta angle, the theory after the Higgsing will also has the discrete theta angle π. Therefore, we can conclude that the diagram in Figure 14(a) may realize the Sp(2) gauge theory with two hypermultiplets in the antisymmetric representation and the discrete theta angle π. Namely, the deformation of the diagram also shows that the G 2 gauge theory with two flavors is dual to the Sp(2) gauge theory with two antisymmetric hypermultiplets and the discrete theta angle θ = π. Let us also check if the diagram in Figure 14(a) or equivalently in Figure 14(b) yields the Sp(2) π gauge theory with two antisymmetric hypermultiplets from the calculation of the prepotential. The gauge theory parameterization for the Sp(2) π gauge theory with two antisymmetric hypermultiplets is given in Figure 18. m 0 is the inverse of the squared gauge coupling, a 1 , a 2 are the Coulomb branch moduli of the Sp(2) π gauge theory. n 1 , n 2 are related to the mass parameters for the two antisymmetric hypermultiplets. Note that  Figure 19. A label for the faces in the diagram for the Sp(2) π gauge theory with two hypermultiplets in the antisymmetric representation. n 1 , n 2 are related to the chemical potentials for the SO(5) flavor symmetry. On the other hand the mass parameters for the two antisymmetric hypermultiplets correspond to the chemical potentials for the Sp(2) flavor symmetry. Hence the two mass parameters are given by With the parameterization in Figure 18 and (2.42), we can express the area of the faces whose label is depicted in Figure 19 in terms of the parameters of the Sp(2) gauge theory with two hypermultiplets in the antisymmetric representation and the non-trivial discrete theta angle. The area of the five faces is given by A linear combination of the area (2.43)-(2.47) corresponds to the tension of the monopole string and it can be computed also from a derivative of the effective prepotential with respect to a Coulomb branch modulus. The effective prepotential is computed from the general formula in (2.5) and the diagram in Figure 14(a) corresponds to the phase for one antisymmetric hypermultiplet with mass m 1 and for the other antisymmetric hypermultiplet with mass m 2 . Then the effective prepotential for the Sp(2) π gauge theory with the two antisymmetric hypermultiplets on the phase becomes where we used the Dynkin basis for Sp(2) Then taking the derivative of (2.50) with respect to φ 1 , φ 2 should correspond to the linear combination 2 + 3 + 2 × 4 + 5 and 1 respectively. Indeed the explict comparison between (2.43)-(2.47) and (2.50) gives By comparing the parameterization in Figure 18 and (2.42) for the Sp(2) gauge theory with two antisymmetric hypermultiplets and the discrete theta angle π with the parametrization of the G 2 gauge theory with two flavors in section 2.2, one can obtain the duality map between the Sp(2) gauge theory and the G 2 gauge theory. Note that the S-dual of the diagram in Figure 14(b) yields a diagram of the G 2 gauge theory two flavors. The parameterization in section 2.2 can be translated to the parameterization for the Sdual diagram as in Figure 20. Again n 1 , n 2 in Figure 20 are related to the mass parameters by (2.15). Then, the comparison between the two parameterizations gives the duality map between the Sp(2) gauge theory and the G 2 gauge theory Combining the map (2.37)-(2.41) with the map (2.54)-(2.58) yields the map between the SU (3) gauge theory with two flavors and the CS level 6 and the Sp(2) gauge theory with two antisymmetric hypermultiplets and the non-trivial discrete theta angle

Duality among marginal theories
In section 2.2, we started from the diagram of the G 2 gauge theory with two flavors and discussed that the diagram can be deformed into the one for the SU (3) gauge theory with two flavors and the CS level 6 and also into the one for the Sp(2) gauge theory with two hypermultiplet in the antisymmetric representation and the discrete theta angle θ = π.
In order for the G 2 gauge theory to have a UV completion, we can add four more flavors to the G 2 gauge theory with two flavors. The UV completion of the G 2 gauge theory with six flavors is not a 5d SCFT but is supposed to be a 6d SCFT [12,29]. It is in fact straightforward to extend the discussion of the dualities in section 2.2 by adding four more flavors to the diagram for the G 2 gauge theory with four flavors. The G 2 gauge theory with six flavors is dual to the SU (3) gauge theory with six flavors and the CS level 6 and is also dual to the Sp(2) gauge theory with four flavors and two hypermultiplets in the antisymmetric representation [1] and we will see the dualities from the 5-brane web diagram.
Let us first add four flavors to the diagram for the G 2 gauge theory with two flavors in Figure 4(b). When we added two flavors to the diagram for the pure G 2 gauge theory, we introduced the flavors which originate from two spinors in the SO(7) gauge theory before the Higgsing, This time, we introduce four flavors which originate from four hypermultiplets in the vector representation of SO(7) before the Higgsing. The introduction of the flavors can be done by adding four D7-branes to the diagram for the G 2 gauge theory with two flavors as in Figure 21(a). When we turn over the branch cuts of D7-branes in the horizontal directions, external 5-branes will cross each other. In order to obtain a consistent 5-brane web for the G 2 gauge theory with six flavors, we perform the flop transitions and move the (1, −1) 7-brane and the (1, 1) 7-brane as we did when we obtained the diagram for the Sp(2) gauge theory. The diagram after the flop transitions and moving the 7-branes is given in Figure 21(c). From this diagram, we let the (1, −1) 7-brane and the (1, 1) 7-brane cross the branch cuts of the D7-branes as in Figure 21(c). The charge of the 7-branes changes and the diagram becomes the one in Figure 21(d). At this stage, we have a pair of a (1, 1) 7-brane and a (1, −1) 7-brane, which can form an O7 − -plane. Therefore, the final diagram in Figure 21(e) contains a pair of an O7 − -plane and an O5-plan in the vertical direction. The periodicity in the vertical direction implies that the theory has a 6d UV completion, which is consistent with the result in [12,29].
Duality to SU (3). From the diagram of the G 2 gauge theory with six flavors, it is also possible to get the duality to the SU (3) gauge theory with four flavors and the CS level 4, and to the Sp(2) gauge theory with two antisymmetric hypermultiplets and four flavors. We first consider the duality to the SU (3) gauge theory with six flavors and the CS level 4. We first start from the diagram of the G 2 gauge theory with six flavors in Figure 21(a). Turning over the branch cuts of the D7-branes in the horizontal direction yields a diagram in Figure 22.
One the other hand, the S-dual to the diagram in Figure 21(a) gives rise to a diagram in Figure 23(a). As in the case for the G 2 gauge theory with two flavors in section 2.2.1, the existence of the three color D5-branes in Figure 23(a) implies that the diagram yields   between the two theories. The gauge theory parameterization for the G 2 gauge theory with six flavors is given in Figure 24. m 0 is the inverse of the squared gauge coupling, a 1 , a 2 are the Coulomb branch moduli and m 3 , m 4 , m 5 , m 6 are the mass parameters. n 1 , n 2 are also related to the two mass parameters by (2.15). It is also possible to obtain the gauge theory parameterization for the SU (3) gauge theory with six flavors and the CS level 4 by extending the parametrization in section 2.2.1. The parameterization for the SU (3) gauge theory is given in Figure 25. The inverse of the squared gauge couping m 0 is given by m 0 = l 1 +l 2 2 . a 1 , a 2 , a 3 , (a 1 + a 2 + a 3 = 0) are the Coulomb branch moduli and m 3 , m 4 , m 5 , m 6 are the mass parameters for the additional four flavors. The two other The comparison between the two parameterizations in Figure 24 and in Figure 25 gives rise to the duality map Duality to Sp (2). We can also see the duality to the Sp(2) gauge theory with two antisymmetric hypermultiplets and four flavors. For that we start from the diagram in Figure 21(c) which gives the G 2 gauge theory with six flavors. When we send the 7-branes to infinitely far, the diagram becomes the one in Figure 26. Then we consider applying the S-duality to the diagram in Figure 21(c). Then the resulting configuration becomes the one in Figure 27(a). Then we can move the four (0, 1) 7-branes according the arrow in   (2) gauge theory with two hypermultiplets in the antisymmetric representation and the four hypermultiplets in the fundamental representation. An equivalent diagram when we send the 7-branes in Figure 27(b) to infinitely far is also depicted in Figure 27(d).
We can also see the parameterization of the both two theories by generalizing the parameterization in section 2.2.2 and also obtain the duality map between the two theories. The gauge theory parameterization for the G 2 gauge theory realized in Figure 26 is given in Figure 28. m 0 is the inverse of the squared gauge coupling, a 1 , a 2 are the Coulomb branch moduli and m 3 , m 4 , m 5 , m 6 are the mass parameters. n 1 , n 2 are related to the two other mass parameters by (2.15). On the other hand, the gauge theory parametrization for the Sp(2) gauge theory realized in Figure 27(d) is depicted in Figure 29. The inverse of the squared gauge coupling is given by m 0 = l 1 +l 2 2 , a 1 , a 2 are the Coulomb branch moduli and  m 3 , m 4 , m 5 , m 6 are the mass parameters for the four flavors. n 1 , n 2 are related to the mass parameters for two antisymmetric hypermultiplets and the relation is given by (2.42).
The comparison between the two parameterizations in Figure 28 and in Figure 29 yields the duality map between the G 2 gauge theory and the Sp(2) gauge theory, where we defined Combining the map (2.67)-(2.72) between the SU (3) gauge theory and the G 2 gauge theory with the map (2.74)-(2.79) between the Sp(2) gauge theory and the G 2 gauge theory, we can also obtain the map between the Sp(2) gauge theory and the SU (3) gauge theory,

Realization as SO(5) + 2V + 4S
In section 2.2.2 and section 2.3, we have seen the realization of the Sp(2) gauge group from four D5-branes with an O7 − -plane. In fact, the diagram may be deformed to a diagram which can be interpreted as an SO(5) gauge theory. This is consistent with the fact that there is an isomorphism so(5) sp(2) at the level of the Lie algebra.
To see the deformation, we start from the diagram in Figure 21(a) for the G 2 gauge theory with six flavors. In section 2.3 we have seen this diagram can be deformed into the one in Figure 27(c), yielding the Sp(2) gauge theory with two antisymmetric hypermultiplets and four flavors. Here we consider a different deformation. First, the diagram in Figure 21(a) can be written as the one in Figure 30(a). Applying flop transitions yields the diagram in Figure 30(b). From the diagram in Figure 30(b), we move the (1, 1) 7-brane according to the arrow in Figure 30(c), giving rise to the diagram in Figure 30(d). Then, performing further flop transitions changes the diagram finally into the one in Figure 30(e). The diagram in Figure 30(e) is exactly the diagram for the SO(5) gauge theory with two hypermultiplets in the vector representation and four hypermultiplets in the spinor representation, which is equivalent to the Sp(2) gauge theory with two hypermultiplets in the antisymmetric representation and four hypermultiplets in the fundamental representation when we do not see the global structure. Therefore, one can also consider a sequence of decoupling hypermultiplets in terms of the SO(5) viewpoint.
From the deformation from Figure 30(a) to Figure 30(e), one can determine the duality map between the G 2 gauge theory with six flavors and the SO(5) gauge theory with two hypermultiplets in the vector representation and four hypermultiplets in the spinor representation. To determine the duality map we compare the diagram in Figure 30 with the diagram in Figure 30(c). The parameterization for the G 2 gauge theory with six flavors is given in Figure 31. a 1 , a 2 are the Coulomb branch moduli and m 0 is the inverse of the squared gauge coupling. n 1 , n 2 are related to two mass parameters by (2.15), namely The other mass parameters m 3 , m 4 , m 5 , m 6 appear directly in the diagram in Figure 31.
On the other hand, the parameterization for the SO (5)   four spinors is given in Figure 32. a 1 , a 2 are the Coulomb branch moduli and m 1 , m 2 are the mass parameters for the two hypermultiplets in the vector representation. n 1 , n 2 , n 3 , n 4 are the mass parameters for the four hypermultiplets in the spinor representation. Since the mass parameters for the two flavors in section 2.2 originate from the mass parameters of the two spinors of the SO(7) gauge theory before the Higgsing, we choose the mass parameters for the four spinors similarly to (2.15), namely On the other hand, the length corresponding to m 0 , the inverse of the squared gauge coupling, is different from that for the G 2 gauge theory with two flavors or the SO(7) gauge theory with two spinors. As explained in appendix A, m 0 for the SO(5) gauge theory with four spinors, attaching two spinors on the both sides of the diagram, is given by By comparing the parameterization in Figure 31 with the parameterization in Figure 32 with the deformation depicted in Figure 30(b), we can determine the duality map between the G 2 gauge theory with six flavors and the SO(5) gauge theory with two vectors and the four spinors. The duality map is given by where and we used the Coulomb branch moduli in the Dynkin basis of SO(5) We can also see the map between the Sp(2) gauge theory with two antisymmetric hypermultiplets and four flavors discussed in section 2.3 with the SO(5) gauge theory from the comparison of the duality map (2.74)-(2.79) with (2.90)-(2.94). In order to obtain a simple map, we first rename the mass parameters for the G 2 gauge theory by The map (2.98)-(2.102) is reasonable since the SO(5) gauge theory is equivalent to the Sp(2) gauge theory when we ignore the global structure. Note that there is a minus sign for the map (2.101). This is because the diagram for the Sp(2) gauge theory with two antisymmetric hypermultiplets and four flavors in section 2.3 has the discrete theta angle π as it was obtained by adding four flavors to the Sp(2) π gauge theory with two antisymmetric hypermultiplets. On the other hand, the SO(5) gauge theory in the diagram in Figure 30(c) has zero discrete theta angle. Hence the minus sign in (2.101) is necessary to change the discrete theta angle. In this section, we consider deformations that lead to theories of gauge groups SU (3) and Sp (2) which are dual to each other without involving G 2 . We start with the marginal theory Sp(2) + 2AS + 4F in the G 2 − SU (3) − Sp(2) sequence and decouple hypermultiplets in the antisymmetric representation. After decoupling one antisymmetric hypermultiplet, we obtain Sp(2) + 1AS + 4F, and then adding more flavors yields another marginal theory Sp(2) + 1AS + 8F, which is dual to SU (3) 3 2 + 9F, and it will be discussed in section 3.1. We also discuss yet another deformation by decoupling the remaining antisymmetric hypermultiplet and then obtain Sp(2) + 10F, which is dual to SU (3) 0 + 10F, and it will be discussed in 3.4. As explained in the previous section 2.3, a 5-brane configuration for G 2 + 6F can be deformed to display a 5-brane configuration for Sp(2) + 2AS + 4F. For example, Figure  21 shows a deformation from the diagram of G 2 + 6F and the last diagram is S-dual to the diagram for Sp(2) + 2AS + 4F given in Figure 27(c). In this section, we first discuss decoupling of a hypermultiplet in the antisymmetric representation (AS) by starting from a 5-brane web, for instance, Figure 21(e) or equivalently Figure 33(a). There are four flavor D7-branes stacked on top of an O7 − -plane, and there are two external NS5-branes in Figure 33(a). The height of the four D7-branes gives the mass of flavors while the position of the two NS5-branes along the horizontal axis is related to the mass of two antisymmetric hypermultiplets. The precise relation between the mass of antisymmetric hypermultiplets and the length in the diagram in Figure 33(a) was obtained in (2.42). In particular, the distance between the two NS5-branes parameterizes two times of the mass of one antisymmetric hypermultiplet, 2m 2 . The distance from the ON-plane to the center of mass position of the two NS5-branes parametrizes the mass of the other antisymmetric hypermultiplet, m 1 . Let us consider a case where we decouple one AS by taking m 1 → ∞.
To this end, we first need to perform a flop transition in such a way as depicted from Figures 33(a) to 33(b). When we move from the diagram in Figure 33(a) to the one in Figure 33(b), we also transform the ON-plane into an ON-plane by moving a fractional D7-brane, where the precise process can be found in [27]. Then we can take the limit m 1 → ∞ with m 2 fixed, which can be also realized by sending the horizontal position of the ON-plane to infinitely right. The process is depicted from Figure 33(b) to Figure  33(d). When m 1 becomes larger compared to the diagram in Figure 33(c), we conjecture that the upper right configuration may involve two (2, 1) 5-branes off from the ON-plane 4 which preserve the charge conservation, and then eventually the diagram may be effectively described without the ON-plane as in Figure 33(d) in the limit m 1 → ∞. This leads to a brane configuration for Sp(2) + 1AS + 4F in Figure 33(d). We note that as shown in [23], some of the transitions in the deformation shown in Figure 33 corresponds to different phases of the Seiberg-Witten curve which can be obtained from the diagrams in Figure 33.
Duality between Sp(2) + 1AS + 8F and SU (3) 3 2 + 9F We first see a duality between Sp(2) + 1AS + 4F and SU (3) 7 2 + 5F starting from the configuration in Figure 33(d). From the perspective of the 5-brane web, this duality can be seen as re-arrangement of 7-branes as depicted in Figure 34. By resolving the O7 − into two 7-branes of the charge [1, −1] (blue dot) and [1,1] (pink dot), one finds that the resulting diagram is given by Figure  34(a), where 4 D7-branes (four red dots) are allocated in the upper part for convenience. After applying a flop transition, the brane configure becomes Figure 34(b), where we also take the [2, 1] 7-brane (black dot) inside the 5-brane loops. From this configuration it is possible to move 7-branes around to obtain a brane configuration where the presence of an SU (3) gauge group is manifest. Firstly, we take the [1, 1] 7-brane (pink dot) outside along the arrow in Figure 34(b), resulting in the web diagram in Figure 34(c). We then take out the remaining two 7-branes of the charge [1, −1] and [2,1] across the lower D5-brane in 4 In [23], a similar decoupling limit is discussed, that is the decoupling process from the rank 1Ẽ1 theory to the E0 theory from the perspective of a brane configuration in the presence of an O5-plane. Due to the generalized flop transitions, the 5-brane configuration for the pure Sp(1)π gauge theory was deformed to have two long NS5-branes near the O5-plane, and taking a limit where the length of the NS5-branes become infinitely long may effectively yields a brane configuration with the long NS5-branes which end on an [0, 1] 7-brane.  5 It was discussed in [1,12] that there is some subtlety in the CFT limit of some SU (2) × SU (2) quiver descriptions. This can be also explicitly seen from 5-brane webs. As a representative example, we consider SU (3) 5 2 + 7F (or Sp(2) + 1AS + 6F). A 5-brane web diagram for a mass deformed configuration of SU (3) 5 2 + 7F is given in Figure 35(a). Its S-dual transformed web is given in Figure 35 It is straightforward to add more flavors to duality relation between SU (3) 5 2 + 7F and SU (2) π × [SU (2) + 5F]. With more flavors, the following theories are S-dual to each other: The corresponding web diagrams for the marginal case are given in Figure 36. We note that there is another decoupling from the quiver theory which yields different dualities. For example, for the [SU (2) This theory turns out to be dual to SU (3) 1 + 8F and also to Sp(2) + 8F, which we will discuss in more detail in section 3.4. The decoupling of a flavor from the quiver theory [SU (2) + 1F] × [SU (2) + 5F] is, in particular, interesting as it allows three different ways of decoupling of a flavor. Recall that an SU (2) theory with a flavor can lead to the pure SU (2) theory with different discrete theta angles, SU (2) 0 and SU (2) π , depending on taking the mass of the flavor to be ±∞. By decoupling a flavor in the first SU (2), one hence finds two quiver gauge theories, SU (2) 0 × [SU (2) + 5F] and SU (2) π × [SU (2) + 5F]. Here the latter theory SU (2) π × [SU (2) + 5F] is dual to SU (3) 3 2 + 7F and also to Sp(2) + 1AS + 6F as we discussed before. In the previous subsection, we saw the duality between the SU (3) gauge theory with nine flavors and the CS level 3 2 and the Sp(2) gauge theory with one antisymmetric hypermultiplet and eight fundamental hypermultiplets. Since we have the diagrams for the two theories, it is possible to obtain the duality map between the parameters of the two theories. Before obtaining the duality map between the marginal theories, let us start from an easier example by decouple eight flavors from the both theories. We decouple the eight flavors by sending the mass of the flavors to +∞ and redefine the gauge coupling. Then the decoupling yields a duality between the SU (3) gauge theory with one flavor and the CS level 11 2 and the Sp(2) gauge theory with one antisymmetric hypermultiplet and the non-trivial discrete theta angle. We first obtain the duality map between them.

Duality map between
The 5-brane web diagram and the gauge theory parameterization for the Sp(2) π gauge theory with one antisymmetric hypermultiplet is given in Figure 37. a 1 , a 2 are the Coulomb branch moduli of the Sp(2) gauge theory and m is the mass parameter for the antisymmetric hypermutliplet. It turns out that the inverse of the squared gauge coupling is given by Let us confirm the choice of the parameters by comparing the area of the faces in the diagram 37 and the tension of a monopole string computed from the effective prepotential of the Sp(2) gauge theory. The area for the faces labeled in Figure 38 becomes On the other hand, the effective prepotential of the Sp(2) gauge theory with one antisymmetric hypermultiplet can be calculated by using (2.5). The phase associated to the antisymmetric hypermultiplet from the diagram in Figure 37 is Hence the effective prepotential of the Sp(2) gauge theory becomes where we used the Coulomb branch moduli in the Dynkin basis of Sp(2) given by (2.51). Then the derivative of the prepotential (3.10) with repsect to the Coulomb branch moduli yield the tension of a monopole string. A D3-brane can be wrapped on a face which confirms the gauge theory parameterization in the diagram in Figure 37 and (3.3). The diagram in Figure 37 can be deformed into the one for the SU (3) gauge theory with one flavor and the CS level 11 2 , The deformation is essentially given in Figure 37 and the resulting web and also the gauge theory parameterization for the SU (3) gauge theory are depicted in Figure 39. a 1 , a 2 , a 3 with a 1 + a 2 + a 3 = 0 are the Coulomb branch moduli and m is the mass parameter for the one flavor. The inverse of the squared gauge coupling Let us also compare the area of the faces in Figure 39 with the tension of a monopole string for completeness. The faces labeled in Figure 40 yield the area On the other hand, the phase for the SU (3) 11 2 gauge theory with one flavor is given by (3.16) and the effective prepotential becomes where we used the Coulomb branch moduli in the Dynkin basis (2.7). The area of the faces (3.14) and (3.15) agrees with the derivative of the prepotential (3.17) with respect to the Coulomb branch moduli φ 1 , φ 2 by Since we know the deformation between the diagrams for the Sp(2) π gauge theory with one antisymmetric hypermultiplet and the SU (3) 11 2 gauge theory with one flavor as well as their parameterization, comparing the two diagrams may give the duality map between the two parameterization. The duality map then is given by AS .
Or if we express the Sp(2) gauge theory parameters in terms of the SU (3) gauge theory parameters, the map becomes (3.36)

+ 9F
For a marginal theory, which can be viewed as a 6d theory on a circle, it may be natural to assume that the bare coupling of the marginal theory would be the radius of the compactification circle. It is then expected that bare couplings of each dual theory are equal to each other as they would correspond to the same radius. For instance, the bare couplings of SU (3) 0 + 10F and Sp(2) + 10F are equal to each other, which can be also explicitly seen from their 5-brane webs as done in [32]. However, from the duality maps we obtained, some dual theories which are marginal have had different m 0 . For instance, the bare coupling of SU (3) 4 + 6F is different from that of Sp(2) + 2AS + 4F as in (2.81). Another example is the bare couplings of SU (3) 3 2 + 9F and Sp(2) + 1AS + 8F as shown in (3.25). It is then natural to ask which m 0 is related to the period of a circle associated to the circle compactification of a 6d theory. In this subsection, we consider 5-brane configurations of SU (3) 3 2 + 9F and Sp(2) + 1AS + 8F and compare their bare couplings with the period of the diagrams.
Marginal theories whose 5-brane configuration can be constructed without an orientifold are described by a particular 5-brane configuration of special properties: a shape of an infinite rotating spiral with constant period, call it a Tao web diagram [9,33]. Since a Tao diagram is periodic, the period associated to the diagram can be read off from the configuration of a Tao diagram.
Consider a 5-brane web for Sp(2) + 1AS + 8F. For instance, Figure 41(a) is an example of a 5-brane web configuration for Sp(2) + 1AS + 8F. Applying the Hanay-Witten transitions explained in [33], one can readily get a Tao web diagram for Sp(2) + 1AS + 8F [30] depicted in Figure 41 average of the asymptotic distances on the center of the Coulomb branch moduli from two pairs of the external 5-branes: The length l i in Figure 41(b) can be expressed by the gauge theory parameters as .

(3.38)
Then the period of the Tao diagram in Figure 41 is given by the sum of the length l i , (i = 1, 2, 3, 4) and it turns out to be equal to 2m    Figure 43. The deformation from a 5-brane web of SU (3) 3 2 + 9F to its Tao web diagram. The 5-brane configuration in Figure 42 can be deformed to the 5-brane web (a). Applying a particular successive Hanany-Witten transitions by moving 7-branes, we arrive at a Tao web diagram given in (d).
blue 7-branes to get Figure 43(a). And further performing Hanany-Witten transitions in a particular order described in Figure 43 yields the diagram in Figure 43(d). In Figure  43(d), we denoted the dotted lines for the monodromy cuts of some 7-branes. By letting all other 7-branes go through these monondromy cuts, 7-brane charges for those 7-branes are changed and in fact, these particular monodromy cuts are chosen so that all other 7-branes keep passing through the cuts and they form a spiral shape with a constant period. The periodic structure can be more explicitly seen in Figure 43. The length l i , (i = 1, · · · , 6) in Figure 43 are related to the mass parameters m i , (i = 1, · · · , 9) and m 0 for the pure SU (3) 0 gauge theory as where Then the diagram in Figure 43 implies that the period is the sum of l i , (i = 1, · · · , 9) and it yields Since two marginal theories are dual to each other, it is expected that they have the same period as the UV completion 6d theory on a circle whose radius directly related to the period of two different 5d descriptions. Namely, only the 2m 0 of Sp(2) + 1AS + 8F is directly equal to the period of the Tao diagram but the 3m 0 for SU (3) 3 2 + 9F needs a shift depending on the mass parameters as in (3.42) in order to form the period. of Sp(2)+1AS+8F by decoupling the hypermultiplet in the antisymmetric representation. The mass of the antisymmetric hypermultiplet is proportional to the distance between the two external NS5-brane in Figure 45(a). To decouple this antisymmetric hypermultiplet, we take this mass to infinity, or equivalently we take the distance be infinite. To this end, as depicted in Figure 45(a), we bring out the [2, 1] 7-brane outside the 5-brane loops and move it to infinitely far away from the diagram. The resulting web diagram is given in Figure 45(b), where we also moved the [1, 1] 7-brane to the right. Then we move the [1, −1] 7-brane next to the [1, 1] 7-brane by rotating the cut of the [1, −1] 7-brane so that it extends in the lower direction as depicted in Figure 45(c). It is then readily seen that one can recombine the two 7-branes of the charge [1, −1] and [1,1] to make an O7 − -plane and thus the resulting 5-brane configuration is a familiar configuration for Sp(2) + 8F as given in Figure 45 corresponding CS level for this SU (3) theory is 1. Hence the diagram gives Sp(2) + 8F and SU (3) 1 + 8F, and they are dual to each other [11,30].
From the perspective of S-duality, the decoupling corresponds to decoupling a flavor from the flavors associated with the SU (2) + 5F of the quiver [SU (2) + 2F] × [SU (2) + 5F]. For instance, SU (3) 1 + 8F can be obtained by taking one of the lower D7-branes in Figure  36 As discussed, we can consider adding more flavors to Sp(2)+8F and SU (3) 1 +8F in the same way. The marginal theory one can obtain in this way is Sp(2)+10F and SU (3) 0 +10F, which are dual to each other. SU (3) 0 + 10F is S-dual to [SU (2) + 4F] × [SU (2) + 4F]. The duality map between Sp(2) + 10F and SU (3) 0 + 10F has been already obtained in [32]. For book-keeping purpose, we summarize the map here. For convenience, we label the Sp(2) parameters with a prime ( ) and the SU (3) parameters without a prime. For each instanton factor (m 0 , m 0 ), the Coulomb moduli parameters (a i , a i ) and the mass parameters (m i , m i ), the duality map between Sp(2) + 10F and SU (3) + 10F is given as follows:

5-brane web for SU (3) 0 + 1F + 1Sym
There is another deformation from an SU (3) theory with a flavor. It is to add a hypermultiplet in the symmetric representation (Sym), which may yield SU (3) 0 + 1F + 1Sym. We note that this theory is a marginal theory as the prepotential contribution of a symmetric hypermultiplet can be effectively "equivalent" to that of eight hypermultiplets in the fundamental representation and a hypermultiplet in the antisymmetric representation (1Sym ∼ 8F + 1AS). It follows that SU (3) 0 + 1F + 1Sym would give an equivalent prepotential as that of SU (3) 0 + 9F + 1AS or SU (3) 0 + 10F, which has a 6d UV fixed point. However a 5-brane configuration for SU (3) 0 + 1F + 1Sym [26] is quite distinct from the brane configuration for SU (3) 0 + 10F. The SU (N ) theory with a symmetric hypermultiplet is described by the introduction of an O7 + -plane on which an NS5-bane ends [24]. For instance, see Figure 47, which shows a 5-brane web for SU (3) 0 + 1F + 1Sym in Figure  47(a) and a 5-brane web for SU (3) − 1 2 + 1Sym in Figure 47(b). Using this 5-brane web diagram for SU (3) 0 + 1F + 1Sym, we will show that the areas of the compact faces of the web diagram agree with the monopole tension from the effective prepotential.
In Figure 48 which is a 5-brane web diagram describing SU (3) 0 + 1F + 1Sym with the mass parameters m F and m Sym . We reflected 5-brane webs on the left against the O7 + -plane to the right below, and chose the right part as the fundamental region which looks similar to that of an SU (3) theory (it is the bold faced 5-brane web in the figure). In O7 + m -½ (1,1) (1,-1) (0,1) (3,-1) (3,1) Figure 49. A deformed web diagram for SU (3) 0 + 1F + 1Sym which enables one to decouple a hypermultiplet in the symmetric representation by taking its mass to −∞. and hence 1 , 2 can be expressed as A little bit of algebra then yields that the area of the compact faces 1 and 2 in Figure  48 are given by The effective prepotential is computed from (2.5). The phase of the parameters corresponding to the configuration of Figure 48 is The prepotential reads It is straightforward to see that the monopole string tension agrees with the area from the 5-brane web for SU (3) 0 + 1F + 1Sym: We close the subsection with a comment on decoupling of hypermultiplets. First we can take m F → ∞ to decouple a flavor, which leads to SU (3) − 1 2 + 1Sym as shown in Figure 47(b) 6 . We can see that the area after the flavor decoupling reproduces the monopole tensions of SU (3) − 1 2 + 1Sym from the corresponding prepotential. It is also possible to take the mass of a symmetric hypermultiplet to −∞ in order to decouple the hypermultiplet in the symmetric representation. For that, consider a deformed web diagram for SU (3) 0 + 1F + 1Sym depicted in Figure 49, where three color D5-branes are put in on the right. On the left, there is a (1, 1) 5-brane coming from the reflection of (−3, 1) 5-brane due to the O7 + -plane. The mass of the symmetric matter m Sym is given by the distance between O7 + -plane and the center of the Coulomb branch (denoted as a red line). Since the origin of the Coulomb branch moduli is above the location of the O7 + -plane, the distance between them is given by − 1 2 m Sym . By taking m Sym → −∞, one gets a web digram given in Figure 50

+ 1Sym
Here, we consider yet another marginal theory: SU (3) 3 2 + 1Sym. Similar to the 5-brane web configuration of Figure 49, one has a 5-brane web for SU (3) − 3 2 + 1Sym depicted in Figure 51. As discussed in the previous section, the mass of a symmetric hypermultiplet parameterizes the distance between O7 + -plane and the center of the Coulomb branch, as O7 + (0,1) O7 - (2,1) (2,-1) (4,1) (4,-1) (4,-1) Figure 52. A web diagram for SU (3) − 3 2 + 1Sym with two O7-planes by recombining a pair of 7-branes of charges [1,1] and [1, −1] into an O7 − -plane in Figure 51. As a result, an NS5-brane one the left goes through the branch cut of the O7 − -plane reappear as a (4, −1) 5-brane on the right, which is the solid blue line. This goes through again to the cut of O7 + -plane reappear as an NS5-branes on the left of the first NS5-brane. This makes a 5-brane configuration with infinitely many NS5-branes on the left and infinitely many (4, −1) 5-branes on the right. shown in Figure 51. It is then straightforward to see that taking it mass to −∞ which shifts the CS level by − 7 2 gives rise to SU (3) −5 . This 5-brane configuration has an intriguing aspect which is quite different from 5brane web for SU (3) 1 2 +1Sym+1F . As discussed in [26], one can recombine three 7-branes of the charges [1, −1], [0, 1] and [1,1] in Figure 49 to deform the 5-brane configuration to be a 5-brane configuration with an O7 + -plane and an O7 − -plane, connected by an NS5-brane. This hence makes the theory manifestly marginal. It is, in fact, a twisted compactification of a 6d theory [26]. One can attempt to recombine 7-branes in a 5-brane web diagram for SU (3) 3 2 + 1Sym. For example, see Figure 52. It is a 5-brane configuration with two different O7-planes but an NS5-brane is not connected to two O7-plane, rather the NS5brane is left away. As there are two O7-planes, this NS5-brane goes through the branch cut of an O7 − -plane reappears as a (4, −1) 5-brane on the other side of O7 − -plane, as shown in Figure 52. In fact, this configuration does not stop here. The (4, −1) 5-brane (the blue solid line in the figure) again goes through the branch cut of an O7 + -plane, and comes out as an NS5-brane on the left side of the first NS5-brane, which again reappear on the right side of the first (4, −1) 5-brane, and this pattern is repeated. This 5-brane configuration for SU (3) 3 2 + 1Sym with an O7 + -and O7 − -planes separated apart along the vertical direction of 5-brane plane, gives rise to a new kind of 5-brane configuration representing a twisted compactification of a 6d theory with an infinitely repeated 5-branes on the left and right sides of two O7-planes.
New 5-brane web diagram for 5d SU (N ) N/2 + 1Sym. It is straightforward to construct a 5-brane configuration for SU (N ) N 2 + 1Sym as depicted in Figure 53. We note that + 1Sym with two O7-planes. Due to two O7-planes, this 5-brane configuration has infinitely many (3 − N, 1) 5-branes on the left and infinitely many (N + 1, −1) 5-branes on the right. N ≥ 2. As before, 5d SU (N ) N 2 + 1Sym has a 5-brane configuration with two O7-planes with two sets of infinitely repeated 5-branes of the charges (3 − N, 1) and (N + 1, 1), on the left and right sides of the O7-planes, which is depicted in Figure 54. As the positions of these infinitely repeated 5-branes depend on the separation between two O7-planes, one can express the periodicity of the infinitely repeated 5-branes as a linear function of the vertical separation between two O7-planes.
We note that for SU (N ) N 2 + 1Sym, it is easy to see that decoupling of a symmetric hypermultiplet shifts the CS level κ by It follows that decoupling of a symmetric hypermultiplet from 5d SU (N ) N 2 + 1Sym gives either SU (N ) N +2 or SU (N ) 2 (modular the sign of the CS level).

Sp(2) gauge theory with 3AS
In the G 2 − SU (3) − Sp(2) sequence, G 2 + 3F is dual to Sp(2) + 2AS + 1F and it can be also understood as SO(5) + 2F + 1S. Its decoupling is in particular interesting because depending on how we decouple the fundamental hypermultiplet for Sp(2), it leads to two different discrete θ-angle for the Sp(2) theory. Moreover, it allows us to deform the theory by adding another hypermultiplet in the antisymmetric representation. Namely, we can properly decouple the flavor from Sp(2) + 2AS + 1F to obtain Sp(2) 0 + 2AS and then add one more hypermultiplet in the antisymmetric representation, which gives rise to another marginal theory Sp(2) 0 + 3AS or equivalently SO(5) 0 + 3F.
In this section, we consider the deformation leading to the Sp(2) 0 + 3AS marginal theory. Introducing three hypermultiplets in the antisymmetric representation in a 5-brane web is not yet clear, so it is better to change the brane configuration for Sp (2) to that for SO(5) 0 as adding an antisymmetric hypermultiplet to an Sp(2) theory is equivalent to adding a vector to an SO(5) theory. Now we start with a brane configuration for SO(5) + 2F + 1S given in Figure 55(a). There are three different possible decouplings as depicted in Figure 55. By decoupling a vector, we get SO(5) + 1F + 1S (Figure 55(b)). By decoupling a spinor taking its mass to negative infinity, we get SO(5) π + 2F (Figure 55(c)) Decoupling a spinor by taking the mass of the spinor matter infinite, we get SO(5) 0 + 2F (Figure 55(d)). For completeness, let us compare the area with the monopole tension from the effective prepotential of SO(5) gauge theories with antisymmetric hypermultiplets. We start from SO(5)+2F in Figure 55(d). By mass deformations, we can get a web diagram for SO(5) 0 + 2F given in Figure 56. One can then read off the monopole tension of the theory from the areas in the brane configuration: where the range of the parameters are given as We now compare the area of the two faces with the monopole string tension from the prepotential for SO(5)+2F in the chamber. The effective prepotential in the phase (4.3) is then given by where we used the Dynkin basis (2.96). One can see explicitly that the monopole tension computed from (4.4) is related to the area (4.1) and (4.2) by (4.5) SO(5) + 3F case. For the SO(5) theory with three hypermultiplets in the fundamental representation, a 5-brane web diagram with an O5-plane is depicted in Figure 57. Note that the two external 5-branes in Figure 57(a) are of the charges (1, 1) and (1, −1). Then a (1, 1) 7-brane and a (1, −1) 7-brane can end on the external 5-branes respectively and they can be combined to be an O7 − -plane as shown in Figure 57(b). It hence has a periodic direction in the 5-brane plane, and therefore it is a marginal theory, which can be understood as a twisted compactification [26]. The area of the compact faces in the web diagram in Figure 57(a) are then given by We now compare these area with the monopole string tension from the prepotential. The diagram in Figure 57(a) is in the phase a 1 ≥ a 2 ≥ m i , i = 1, 2, 3. Then the effective prepotential for SO(5) + 3F in this phase is given by As expected, one can readily see that the monopole string tension agree with the area of the faces of the web diagram in Figure 57,  Figure 58. The increase of the CS level by 2 is implemented by replacing one side of the diagram of the pure SU (3) gauge theory with the CS level 5 with an ON-plane. Hence, it is natural to guess that replacing another side of the diagram of the pure SU (3) gauge theory with the CS level 7 with an ON-plane may give rise to a diagram of the pure SU (3) gauge theory with the CS level 9. We then propose that the diagram in Figure 59 gives rise to a 5-brane web diagram for the pure SU (3) gauge theory with the CS level 9.
One can check the claim by computing the tension of the monopole string from the 5-brane web diagram in Figure 59. The tension is given by the area and we can compare the area with the result expected from the field theory. In order to write the area by the gauge theory parameters of the SU (3) gauge theory, we assign the Coulomb branch moduli a 1 , a 2 , a 3 , (a 1 + a 2 + a 3 = 0) and the inverse of the squared gauge coupling m 0 as in Figure  60. Then the area of the four faces in Figure 60 becomes Let us then compare the area with the tension of the monopole string of the pure SU (3) gauge theory with the CS level 9. Since we do not have matter, the theory have only one phase and the effective prepotential can be computed from (2.5) and it becomes where we changed the basis for the Coulomb branch moduli into the Dynkin basis in (5.5) by using (2.7). Then the tension of the monopole string is given by taking a derivative of the prepotential with respect to φ 1 and φ 2 . Hence the tension from (5.5) is Now we can compare the tension (5.6) and (5.7) with the area (5.1)-(5.4). As in the case of the comparison between the area and the monopole string tension for the pure G 2 gauge theory, we need to consider a linear combination among (5.1)-(5.4) to obtain the area of a face where D3-brane covers [13]. More specifically, the area corresponding to the tension (5.6) should be 2 1 + 2 + 2 3 while the area corresponding to the tension (5.7) is simply given by 4 . Indeed, it is straightforward to check the equalities from the explicit expressions of (5.1)-(5.4) and (5.6)-(5.7). This gives an evidence that the diagram in Figure 59 gives rise to the pure SU (3) gauge theory with the CS level 9.

Conclusion
In the paper, we proposed all the 5-brane webs of rank 2 superconformal theories classified via geometries in [1], and discussed their mutual dualities from the perspective of S-duality and the Hanany-Witten transitions arising by moving 7-branes. As many of 5-brane webs for such rank 2 theories are already known, our focus has been those theories newly proposed in [1], which did not have 5-brane descriptions. We explicitly constructed 5-brane webs for all the marginal theories. We compared the area of the web diagram for each theory with the monopole string tension calculated from the effective prepotential, which showed the perfect agreement. We also found the duality map among dual theories. For instance, explicit 5-brane webs are presented in section 2 for the G 2 gauge theories with six flavors (G 2 + 6F) and its dual theories, the Sp(2) gauge theory with four flavors and two hypermultiplets in the antisymmetric representation (Sp(2) + 2AS + 4F) and the SU (3) gauge theory with six flavors and Chern-Simons level 4 (SU (3) 4 + 6F). We also present 5-brane web from the viewpoint of the SO(5) theory with two hypermultiplets in the fundamental representation and four hypermultiplets in the spinor representation (SO(5) + 2V + 4S). The duality map among the theories in the G 2 -SU (3)-Sp(2) sequences are also discussed in sections 2.3 and 2.4.
From the G 2 − SU (3) − Sp(2) sequence, we also discussed various deformations: (i) One can deform the theory by decoupling one antisymmetric hypermultiplet from Sp(2) + 2AS + 4F and then by adding flavors which leads to the Sp(2) gauge theory with one antisymmetric and eight flavors (Sp(2)+1AS+8F) which is dual to the SU (3) gauge theory with nine flavors and the CS level 3 2 (SU (3) 3/2 + 9F). (ii) One can further decouple the antisymmetric hypermultiplet and add more flavors to get the Sp(2) gauge theory with ten flavors (Sp(2) + 10F) which is dual to the SU (3) gauge theory with ten flavors and the CS level 0 (SU (3) 0 +10F). (iii) One can also deform the theory to the SU (3) gauge theory with one symmetric hypermultiplet and one flavor with the zero CS level (SU (3) 0 + 1Sym + 1F) and also to the SU (3) gauge theory with only one symmetric hypermultiplet with the CS level 3/2 (SU (3) 3 2 + 1Sym). (iv) Another possible deformation is to deform the theory to the Sp(2) 0 theory with three hypermultiplets in the antisymmetric representation and the discrete theta angle zero (Sp(2) 0 + 3AS) or equivalently the SO(5) 0 theory with three vectors and the discrete theta angle zero (SO(5) 0 + 3F).
We note that the 5-brane web diagrams for the marginal SU (3) theories, SU (3) 0 + 1Sym + 1F and SU (3) 3 2 + 1Sym are obtained with an O7 + -plane. In particular, a 5brane web diagram for SU (3) 3 2 + 1Sym can be constructed with O7 + -and O7 − -planes with 5-branes appearing repeatedly on both sides of O7-planes with a periodic structure. This can be straightforwardly generalized to rank N , exhibiting a new 5-brane structure for 5d SU (N ) N 2 + 1Sym theory. We also note that, inspired by the brane web for the pure G 2 gauge theory with an O5-plane, we constructed a 5-brane web for the marginal SU (3) gauge theory with the CS level 9, which requires two ON -planes 7 . An ON -plane appears not only as the S-dual object of an O5-plane, but also naturally as Higgsing and decoupling of the D-type quiver theory with an ON 0 -plane. Thus, a 5-brane web with an ON -plane sometimes can allow a field theory description.
The duality between Sp(2) + 1AS + 8F and SU ( By decoupling hypermultiplets from the marginal theories, we can obtain 5-brane webs for 5d superconformal theories with various hypermultiplets. Various 5d theories with less number of hypermultiplet can be obtained from decoupling of hypermultiplets from another marginal theory. For example, we decoupled the symmetric matter to obtain a brane web for SU (3) − 7 2 + 1F. This decoupling generates various RG flows among rank 2 theories. Following a summary figure presented in [1], we also summarize the 5-brane webs for rank 2 theories, their duality relations, and RG flows in Figure 65.
It would be interesting to study the 6d origin of the marginal theories discussed in the paper. It seems generic that for a marginal SU gauge theory with non-zero CS level has the property that the compactification radius (or the period) is composed of a linear combination of the bare coupling and the mass parameters of the hypermultiplets, which may indicate some intriguing interplay between the compactification radius and the mass parameters. It would also be interesting to further study such relation from the perspective of its 6d origin. Another interesting future direction would be further confirm the duality relation from BPS operator counting from the gauge theories. For instance, one may compute superconformal indices for dual theories and confirm the duality map, which would be another consistency check for the duality maps that we obtained from 5-brane webs. Finally, pursuing 5-brane webs for superconformal theories of higher rank greater than 2 which may lead to new 5-brane perspective on higher Chern-Simons levels and hypermultiplet in other representations than what was discussed. For instance, the hypermultiplet in the rank 3 antisymmetric representation can be constructed [34].
A The gauge coupling for SO(2N + 1) gauge theory with spinors In section 2.4, the inverse of the squared gauge coupling m 0 for the SO(5) gauge theory with two vector hypermultiplets and four spinor hypermultiplets was defined by (2.89), using the parameters in Figure 32. The definition of m 0 in a web was in fact different from that of the G 2 gauge theory with two flavors given in Figure 5. The G 2 gauge theory with two flavors was obtained from the Higgsing of the SO(7) gauge theory with three spinors. Therefore, how to read off m 0 from the diagram was different between the SO(5) gauge theory and the SO(7) gauge theory. In this appendix, we give an explanation of the difference by using the effective prepotential of an SO(2N + 1) gauge theory with spinors.

A.1 Decoupling of a spinor
We first discuss how decoupling one spinor of an SO(2N + 1) gauge theory affects the inverse of the squared gauge coupling by using the effective prepotential. The effective prepotential for the SO(2N + 1) gauge theory with N f vectors and N s spinors can be calculated from the general expression (2.5) and it is given by We then consider decoupling one spinor by sending m Ns → +∞. Then the terms involving the N s th spinor in the last line of (A.1) become − 1 12 Therefore, in the limit where m Ns → ∞, the effective prepotential (A.1) becomes A.2 Reading off gauge coupling from web Using the general formula for the shift of m 0 after decoupling a spinor, we identify the length corresponding to m 0 from a 5-brane web diagram for the SO(2N + 1) gauge theory with spinors.

A.2.1 One spinor case
For simplicity, we consider the SO(2N + 1) gauge theory with one spinor. We denote the inverse of the squared gauge coupling by m Ns=1 0 and the mass of the spinor by m 1 . The inverse of the squared gauge coupling after decoupling one spinor is denoted by m Ns=0 0 . A 5-brane web for the SO(2N + 1) gauge theory with one spinor is depicted in Figure 61. The mass parameter m 1 is related to a half of the length between the (N − 2, 1) 5-brane and the (1, 1) 5-brane on the O5-plane in Figure 61, which can be interpreted as a half of the inverse of the squared gauge coupling of the "Sp(0)" part. Hence a 5-brane web for the pure SO(2N + 1) gauge theory after decoupling the spinor can be realized by moving (1, 1) 5-brane at the right hand side to infinitely right. Then, it is straightforward to read of the inverse of the squared gauge coupling m Ns=0 0 of the pure SO(2N + 1) gauge theory in a symmetric phase (vanishing Coulomb branch parameter) since it is simply the distance between the (N − 1, −1) 5-brane on the left and the (N − 2, 1) 5-brane on the O5-plane in Figure 61.
Since we can identify m Ns=0 0 in the web in Figure Figure 62(a). Namely, we should use the "outside" point where the (1, 1) 5-brane ends as in Figure 62(a). For the SO(7) gauge theory, the relation becomes m Ns=1 0 = m Ns=0 0 + m 1 . The diagram for the SO(7) gauge theory is depicted in Figure 62(b). Unlike the case for the SO(9) gauge theory, we use the "middle" point between the NS5-brane and the (1, 1) 5-brane to define m Ns=1 0 as in Figure 62

A.2.2 Two spinor case
Next, we consider a case for the SO(2N + 1) gauge theory with two spinors. The two spinors are realized at one side as depicted in Figure 63. We denote the inverse of the squared gauge coupling for the SO(2N + 1) gauge theory with two spinors by m Ns=2 0 and we write the masses of the two spinors by m 1 and m 2 . These parameters can be read off from the web in Figure 63. Note that this parametrization is consistent with decoupling of the spinor matter with mass m 2 , which corresponds to sending the height of the flavor D5-brane extending in the left direction to infinitely far, keeping the length between the (N − 2, 1) 5-brane and the (1, 1) 5-brane on the O5-plane. It turns out that the inverse of the squared gauge coupling for the "Sp(0)" part is identified with m 1 + m 2 rather than 2m 1 + 2m 2 , which makes the way of reading off m Ns=2 Since m Ns=0 0 is given as depicted in Figure 63, we can determine the length corresponding to m0 Ns=2 by utilizing the relation (A.6). For example, a diagram for the SO(9) gauge theory with two spinors in the symmetric phase is depicted in Figure 64(a). In this case, the relation (A.6) is expressed as m Ns=2 0 = m Ns=0 0 + 2(m 1 + m 2 ). Therefore, the length corresponding to m Ns=2 0 is given by adding the distance between the (2, 1) 5-brane and the (1, 1) 5-brane on the O5-plane twice to m Ns=0 0 as in Figure 64 , we need to use the "outside" point which is given by extrapolating the NS5-brane as depicted in Figure 64 . Therefore, we should use the "middle" point between the NS5-brane on the left and the NS5-brane on the right in order to define m Ns=2 0 . The explicit length corresponding to m Ns=2 0 is drawn in Figure 64(c).

B The web diagrams for rank two SCFTs
In this appendix, we give the 5-brane web diagrams for all the 5d N = 1 SCFTs with rank 2 classified in [1]. In Figure 65, all the diagrams for such theories are listed. The theory in the box with gray color has 6d UV fixed point. In other words, certain 6d (1, 0) SCFTs compactified on S 1 give the 5d gauge theories inside the gray box. All the other 5d theories are obtained by RG flows triggered by relevant deformation, which is described by the arrows. The gauge theories inside the same box are the dual theories, which have identical SCFTs at their UV fixed point. 8 In some boxes, non-Lagrangian theories are given, which are specified by the Calabi-Yau geometry. We exclude 5-brane webs which can be obtained by an S-duality transformation or trivial Hanany-Witten transitions, unless they have manifest Lagrangian descriptions.
In each box in Figure 65, a Figure number from 66 to 137 is given, in which the corresponding web diagrams are depicted, where the external 5-branes are attached to 7-branes which should be understood as being taken to infinity. Note that, for all the diagrams we list in the appendix, we can move all the 7-branes to infinity by finite steps. In this way, a conventional attempt to construct 5-brane for pure SU (3) 7 theory is not very useful. One may consider a naive diagram for SU (3) 7 like the left diagram in Figure 87(b), but it does not lead to a "finite" diagram like the right diagram in (b). In other words, one cannot move the 7-branes to infinity in finite steps, and hence a 5-brane web for describing the pure SU (3) 7 theory may need to rely on unconventional 5-brane constructions. As presented in Figure 72 In some cases, more than one diagrams are depicted for one gauge theory. In order to emphasize the difference of such diagrams, we put brief explanations in the bracket after the name of the theory. For example, the second diagram in Figure 67 denoted as SU (3) 9/2 + 5F (= 1AS + 4F). This means that one out of 5 flavors are actually realized as the antisymmetric tensor representation using ON − -plane while the other 4 flavors are conventionally realized as fundamental representation using D7-branes. Although antisymmetric tensor and fundamental are identical for SU (3) gauge group, they nicely characterize the difference of the web diagrams.
Also, for the SO(5) gauge theories with hypermultiplets in spinor representation, we often obtain several diagrams due to the fact that the spinor representation can be realized both at the left hand side and the right hand side of the SO(5) gauge theory in the web diagram. For example, in Figure 68, the last diagram in the group (b) is denoted as "SO(5) + 2V + 2S(1S + 1S)". This means that one spinor is realized at the left hand side while the other one is at the right hand side. On the contrary, the diagrams in the group (a) and group (c) are explained as "2S & θ = 0" and "2S & θ = π", respectively. This means that the two spinors are realized at the left side for both cases. The remaining information θ = 0 or θ = π denote the discrete theta angle. Although discrete theta angle is not really defined for the SO(5) theory with spinors, it is used just to briefly specify the configuration at the right hand side of this diagram. The configuration for "θ = 0" is the same as the right half of the diagram in Figure 55(d) while "θ = π" is the one in Figure  55(c).
The explanations for the G 2 gauge theories are a little bit tricky. For example, in Figure 67, the first diagram is denoted as "G 2 + 5F (= 1S + 4V)". This actually means that this diagram for the G 2 gauge theory with 5 flavor is obtained by Higgsing one hypermultiplet in spinor representation of SO(7) gauge theory with two hypermultiplets in spinor representation and four hypermultiplets in vector representation. Since one out of two hypermultiplets in spinor representation disappear in the process of Higgsing, what remains in the G 2 gauge theory is 1 flavor coming from the spinor representation and the 4 flavors coming from vector representation. This information of the origins are briefly explained as "1S + 4V". Analogously, "G 2 + 5F (= 2S + 3V)" for the diagram below means that the 2 flavor originates from spinor representation while the 3 flavor originates from the vector representation of the parent SO(7) gauge theory.
All the external (p, q) 5-branes are terminated by (p, q) 7-branes in all the diagrams in the following Figures. In some cases, we can straightforwardly move (p, q) 7-branes to (p, q) direction and obtain the web diagrams consists only of (p, q) 5-branes. For other cases, (p, q) 7-branes go across the branch cut created by other (p, q) 7-branes. In this case, we need to properly move (p, q) 7-branes taking into account this monodromy as well as Hanany-Witten transition in order to move all the (p, q) 7-branes to infinity. Especially in the diagrams for the 6d theories, it is not possible to move all the (p, q) 7-branes to infinity at least in finite step unless they does not exist from the beginning as in Figure  137.                                    Figure 67. Various diagrams representing G 2 + 5F, SU (3) 9 2 + 5F, Sp(2) + 2AS + 3F, SO(5) + 2V + 3S. These diagrams are obtained from any of the diagrams in Figure 66 by a certain limit corresponding to a relevant deformation.