More on 5d descriptions of 6d SCFTs

We propose new five-dimensional gauge theory descriptions of six-dimensional N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = (1,0) superconformal field theories arising from type IIA brane configurations includ-ing an ON0-plane. The new five-dimensional gauge theories may have SO, Sp, and SU gauge groups and further broaden the landscape of ultraviolet complete five-dimensional N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 supersymmetric gauge theories. When we include an O8−-plane in addition to an ON0-plane, T-duality yields two O7−-planes at the intersections of an ON0-plane and two O50-planes. We propose a novel resolution of the O7−-plane with four D7-branes in such a configuration, which enables us to obtain three different types of five-dimensional gauge theories, depending on whether we resolve either none or one or two O7−-planes. Such different possibilities yield a new five-dimensional duality between a D-type SU quiver and an SO − Sp quiver theories. We also claim that a twisted circle compactification of a six-dimensional superconformal field theory may lead to a five-dimensional gauge theory different from those obtained through a simple circle compactification.

In this work, we continue to expand our analysis by including a so-called ON 0 -plane in the type IIA brane setup. An ON 0 -plane in type IIB string theory is an object which is S-dual to a combination of an O5 − -plane and a D5-brane [12,13]. It also has a perturbative description which enables us to study its property [14][15][16][17]. T-duality implies that the ON 0plane also exists in type IIA string theory. It has been known that the inclusion of the ON 0plane realizes a further large class of 6d SCFTs [17]. The main aim of this paper is to take Tduality to the type IIA brane system with an ON 0 -plane as well as other orientifolds and obtain further new 5d gauge theories on new 5-brane webs. Indeed, we will find that, in some cases with an ON 0 -plane, T-duality yields new 5d gauge theories. In the previous cases without an ON 0 -plane, T-duality gave 5d quiver theories with mainly SU gauge groups. In the current cases, the T-duality yields quiver gauge theories with Sp and SO gauge groups, and widens the landscape of 5d gauge theories with different types of gauge groups.
It is possible to introduce different types of orientifolds such as an ON 0 -plane, an O8 − -plane and O6-planes simultaneously in the type IIA brane setup [17]. An ON 0 -plane is located at the intersection between the O8 − -plane and the O6-plane. T-duality may give two O7 − -planes located at the intersections of an ON 0 -plane and two O5 0 -planes each of which we define as a combination of an O5 − -plane and a D5-brane. Originally, it has been known that an isolated O7 − -plane consists of two 7-branes and can be resolved into a [1, −1] 7-brane and a [1,1] 7-brane by the quantum effects with respect to the string coupling [18]. When an NS5-brane is attached on an O7 − -plane, another resolution into a [0, −1] 7-brane and a [2, 1] 7-brane was proposed [4,5]. In this paper, we will propose further novel resolution of an O7 − -plane with four D7-branes at the intersection between an ON 0 -plane and an O5 0 -plane and also between an ON − -plane, which we define as an object S-dual to an O5 − -plane, and an O5 − -plane. The resolution of the O7 − -plane in the presence of the different orientifolds give rise to different five-dimensional gauge theories. Namely, we will have three dual five-dimensional gauge theories depending on either we decompose no, one or two O7 − -plane. The difference will yield a new five-dimensional duality between a D-type SU quiver theory and an SO-Sp quiver theory.
We also consider yet another generalization by changing the geometry of the compactification. In the analysis so far, we have focused on a simple circle compactification, namely a 6d SCFT on S 1 × R 4,1 . However, it is also possible to consider a circle compactification incorporated by a symmetry of a type IIA brane setup. In particular, we will consider a twisted circle compactification where the identification is done with a reflection. In other words, we put a 6d SCFT on M × R 3,1 where M stands for a Möbius strip with an infinite JHEP10(2016)126 width. The geometry has a S 1 boundary with a cross-cap, and T-duality along the S 1 yields a pair of an O7 − -plane and an O7 + -plane [19]. The resolution of the O7 − -plane yields new 5d gauge theories again.
The organization of the paper is as follows. In section 2, we first describe properties of an ON 0 -plane in a 5-brane web in type IIB string theory. We propose a microscopic picture of an ON 0 -plane incorporated in a 5-brane web and also conjecture a 5d duality associated to it. In section 3, we start with 6d SCFTs on M5-branes probing a D-type singularity. The system can be described by O6-planes and D6-branes with multiplet half NS5-branes on them. Although its 5d gauge theory description has been known, we take a different route to obtain the same 5d theory on 5-branes webs involving two ON 0 -planes, which make clear the role of ON 0 -planes. In section 4, we explore the simplest type IIA brane setup with an ON 0 -plane. Namely we introduce an ON 0 -plane at the end of D6-branes split by NS5-branes. In section 5, we combine the system analyzed in section 3 and section 4. Namely, we consider the type IIA brane setup with O6-planes and an ON 0 -plane. The presence of the two types of the orientifolds in fact implies the existence of an O8 − -plane. The circle compactification yields three types of 5d gauge theories, depending on either we resolve no O7 − -planes, we resolve two O7 − -planes or we resolve only one of the two O7 − -planes. Finally in section 6, we consider a twisted circle compactification of the type IIA brane setup with D6-branes and NS5-branes.

5-brane webs with an ON 0 -plane
In this section, we first describe various properties of an ON 0 -plane incorporated in 5-brane web diagrams in type IIB string theory.

A microscopic description of an ON 0 -plane in a 5-brane web
An ON 0 -plane was originally formulated as an object which is S-dual to a D5-brane on top of an O5 − -plane and it arises as an orbifold fixed point in string compactifications [12,13]. The ON 0 -plane was further made use of as a building block for brane configurations in type II string theory [14][15][16][17]. We here particularly focus on the behavior of the ON 0 -plane in a 5-brane web diagram in type IIB string theory and propose a microscopic description of the ON 0 -plane as a combination of an NS5-brane and an ON − -plane which we define as an object S-dual to an O5 − -plane. We will utilize this microscopic description in the later sections.
In order to understand the microscopic description of an ON 0 -plane in a 5-brane web, let us first start from the behavior of fundamental strings between two D5-branes on top of an O5 − -plane. The two D5-branes on top an O5 − -plane give an SO(4) gauge theory and the quantization of fundamental strings between the two D5-branes should give gauge fields in the adjoint representation of the SO(4) gauge group. Let us denote the two D5-brane D5 1 by D5 1 and D5 2 . Fundamental strings which connect D5 1 directly to D5 2 yield gauge fields in roots ±(e 1 − e 2 ) where e 1 and e 2 are the orthonormal basis in R 2 . There are also another type of fundamental strings which originate from D5 1 , pass through D5 2 and also the O5 −plane, and then end on the mirror D5 2 or vice versa. The fundamental strings yield gauge JHEP10(2016)126 fields in roots ±(e 1 +e 2 ). On the other hand, fundamental strings which connect D5 1 to the mirror D5 2 or fundamental strings which connect D5 2 to the mirror D5 2 are projected out by the orientifold action. Those fundamental strings supply all the roots of the SO(4) Lie algebra. The configuration of the fundamental strings is drawn in figure 1 (a). The S-dual picture gives rise to D1-strings which connect two NS5-branes and an ON − -plane but the configuration of the connections should be essentially the same as how the fundamental strings connect between the two D5-branes and the O5 − -plane. See figure 1 Performing T-duality in directions transverse to D1-strings but along the NS5-branes, one finally obtains a 5-brane web with an ON − -plane. We propose that the brane configuration in figure 2 (a) is a microscopic description of an ON 0 -plane in the system of 5-brane webs. The two coincident D5-branes separate the ON − -plane into two pieces. Indeed, we have a pair of an ON − -plane and an NS5-brane in the left part in figure 2 (a) which may form an ON 0 -plane after a suitable tuning. A shorthand way to write the brane configuration is depicted in figure 2 (b). In the later sections, we will often make use of the picture in figure 2 (b). Whenever we write figure 2 (b), we always mean that the precise configuration is the one in figure 2 (a).
A simple generalization by adding multiple NS5-branes to figure 2 is straightforward and given in figure 3 (a). We also put two semi-infinite D5-branes at the right end of the brane configuration of figure 3 (a) since we will often encounter this case. Now, we discuss the worldvolume theory realized on the D5-branes in this brane configuration. As discussed in [17], N D5-branes suspended between an NS5-brane and an   Figure 3. A microscopic description of an ON 0 -plane with multiple NS5-branes. We also add two semi-infinite D5-branes at the right end for later use. We will call this configuration split D5-branes on an ON 0 -plane. plane give SU(N 1 )×SU(N 2 ) gauge group factor with N 1 +N 2 = N . Here, N 1 is the number of D5-branes which connect the NS5-brane adjacent to the ON 0 -plane to the NS5-brane in the ON 0 -plane from the right, while N 2 is the number of D5-branes which end on the NS5brane next to the ON 0 -plane, pass through the NS5-brane in the ON 0 -plane, is reflected by the ON − -plane inside the ON 0 -plane and end on the NS5-brane in the ON 0 -plane from the left. Both of the gauge group have bi-fundamental matter, which couples to the other adjacent gauge node instead of each other. In our case, N 1 = N 2 = 1 and thus, we can formally write the 5d quiver gauge theory from this brane configuration with N − 1 NS5-branes is (2.1) where we have N − 2 SU(2) gauge nodes and we denoted the n hypermultiplets in the fundamental representation as [n]. In [1,20,21], it is discussed that the "SU(1)" gauge node together with a bi-fundamental hypermultiplet between the "SU(1)"' and the SU (2) give two flavors for the SU (2). Using this claim also to our case, we find that we can interpret that the above quiver gauge theory is interpreted as 2.2 A transition between O5 − − O5 + -planes and an ON 0 -plane Let us then consider the S-dual of the brane configuration in figure 3. The S-duality amounts to the 90 • rotation and one obtains a 5-brane web in figure 4 (a). We conjecture here that the brane conifguration in figure 4 (a) is connected to the brane configuration in figure 4 (b) by a deformation of parameters of the worldvolume theory realized on the 5-branes. That is, we can deform the brane configuration from (a) to (b) in figure 4 without changing the UV fixed point of the theory. Hence we claim that the 5d theory realized on the 5-brane web in figure 4 (a) is dual to the 5d theory realized on the 5-brane web in figure 4 (b). The transition may look more natural when one moves all the D5-branes to the place of the O5-plane. Then, (a) and (b) in figure 4 reduce to (a) and (b) in figure 5, respectively. In figure 5 (a), we see that an "unsplit" NS5-brane is attached to the O5 − -plane. In fact, an"unsplit" NS5-brane can be split into two fractional NS5-branes on the O5 − -plane without any cost of energy in certain cases. After the splitting, an O5 + -plane should be created since the NS5-brane changes the sign of the RR-charge of the O5 − -plane. In addition, the splitting can create certain number of D5-branes between the two fractional NS5-branes. It is argued in [22], the split configuration has the same energy as the unsplit confguration if the number of created D5-branes in between the two fractional NS5-branes is less than or equal to N −2. If we consider the case where the bound of the number of created D5-branes is saturated, we obtain figure 5 (b). Essentially, our conjecture is to claim that we obtain figure 4 (a) if we move all the D5-branes away from O5 − plane in figure 5 (a). Another support for the conjecture comes from checking the duality of the realized 5d theories on the two configurations. The figure 4 (b) gives a 5d Sp(N − 2) gauge theory with N f = 2N flavors. On the other hand, the S-dual diagram to figure 4 (a), which is nothing but the brane web in figure 3, gives the quiver gauge theory in (2.2). The latter theory is S-dual to a 5d SU(N − 1) gauge theory with N f = 2N flavors and the Chern-Simons level ±1. In fact, the Sp(N − 2) gauge theory with N f = 2N flavors is dual to the SU(N − 1) ±1 gauge theory with N f = 2N flavors [3,5],

JHEP10(2016)126
Therefore, the two brane configurations in figure 4 indeed give two dual 5d theories, which implies that the two brane webs in figure 4 are equivalent to each other. The above arguments strongly support the equivalence of the two 5-brane webs in figure 4. By combinining the S-duality, the 5-brane web in figure 4 (b) is further dual to the 5-brane web in figure 3. Hence, if there is a brane configuration of figure 4 (b) or figure 3 in a local part of a 5-brane web, we can replace it with the one of figure 3 or the one of figure 4 (b) respectively since they are equivalent to each other. More schematically, the S-duality relates the following two configurations (2.4) Although we will often use the duality (2.4) in later sections to modify 5-brane webs in a schematic way, the precise replacement is given by the the transition between figure 3 and figure 4 (b).

Duality between a D-type SU quiver and an SO − Sp linear quiver
In order to demonstrate one of interesting implications of this conjecture, we consider a brane configuration depicted in figure 6 (a). 1 This is interpreted as the following 5d quiver JHEP10(2016)126 gauge theory, (2.5) The case of k = 1 reduces to (2.2). When we take S-dual of the brane configuration in figure 6 (a) and locally use the deformation from (a) to (b) in figure 4, we obtain figure 6 (b). This brane configuration corresponds to a 5d quiver gauge theory where we have k Sp(N − 2) gauge nodes and k − 1 SO(2N ) gauge nodes. Therefore, we can expect that these two theories are dual to each other, which means, they have identical UV fixed point. We can support the conjecture by comparing the dimension of the Coulomb branch moduli space and the number of parameters of both theories. The dimension of the Coulomb branch moduli space of the D-type quiver theory (2.5) is (2.7) The dimension of the Coulomb branch moduli space of the SO-Sp quiver theory (2.6) is which precisely agrees with (2.7). The number of the parameters of the D-type quiver theory (2.5) is  which again exactly agrees with (2.9).
In this case, one can see more refined support from the instanton operator analysis given in [2,23]. All the SU(N c ) gauge nodes 2 of the 5d D-type quiver theory (2.5) satisfy the condition N f = 2N c where N f is the number of flavors coupled to the SU(N c ) gauge node. Therefore, the one-instanton states create two Dynkin diagrams whose shape is exactly the quiver [2]. Namely, the one-instanton states give rise to two D N Dynkin diagrams, which implies that the enhanced flavor symmetry is at least SO(2N ) × SO(2N ). The D-type quiver theory (2.5) also has a flavor symmetry SU(2k), hence the expected enhanced flavor symmetry at the 5d UV fixed point is SO(2N ) × SO(2N ) × SU(2k). On the other hand, the SO(2N ) × SO(2N ) flavor symmetry is perturbatively seen in the 5d SO-Sp quiver theory (2.6). The remaining part of flavor symmetry, SU(2k), is not clear from the one-instanton analysis but we can use 7-branes in the 5-brane web in figure 6 (b). In general, one can think of attaching 7-branes to the ends of external 5-branes in a 5-brane web when the charge of the 7-branes is the same as that of the external 5-branes. In figure 6, we have 2k external NS5-branes and hence they can end on 2k [0, 1] 7-branes. The global symmetry of the 5d theory may be understood from a symmetry realized on 7-branes in the corresponding 5-brane web. Therefore, the presence of the 2k [0, 1] 7branes implies that we have an enhanced flavor symmetry SU(2k) from the instantons of the SO-Sp quiver theory (2.6). In summary, the 5d SO-Sp quiver theory (2.6) is expected to have an enhanced flavor symmetry SO(2N ) × SO(2N ) × SU(2k) at the 5d UV fixed point. The flavor symmetry precisely agrees with the enhanced flavor symmetry of the 5d D-type quiver theory (2.5), determined by the instanton operator analysis.

Type IIA brane setup with O6-planes
In type IIA string theory, 6d SCFTs are realized on D6-branes fractionated by NS5-branes. The D6-branes extend in the x 0 , x 1 , · · · , x 6 directions whereas the NS5-branes extend in the x 0 , x 1 , · · · , x 5 directions, which is summarized in table 1. Hence, the NS5-branes cut the D6-branes into pieces in the x 6 direction as in figure 7. When we consider N D6-branes and k NS5-branes, the resulting 6d SCFT on the tensor branch is where we have k − 1 SU(N ) gauge nodes and the number in the bracket [·] represents the number of hypermultiplets in the fundamental representation of the SU(N ). The same 6d Figure 7. The type IIA brane configuration realizing a 6d SU(N ) linear quiver theory. Each horizontal line represents an D6-branes and each orange circle represents an NS5-brane. We chose the horizontal direction as the x 6 direction.
SCFT can be also realized on k M5-branes probing an A N −1 singularity. M5-branes extend in the x 0 , x 1 , · · · , x 5 directions, and the A N −1 singularity is a point in the non-compact four-dimensional space spanned by x 7 , x 8 , x 9 , x 11 . This is the base setup for 6d SCFTs in the type IIA construction or M-theory.
It is possible to generalize this configuration by including an O8 − -plane [10,11]. The 5d descriptions of such 6d SCFTs were extensively discussed in [1,[4][5][6] by utilizing Tduality. The main aim of this paper is to consider yet another generalization by including different types of orientifolds and to determine corresponding 5d gauge theory descriptions.
In this section, we first include O6 ± -planes in the same direction as D6-branes in table 1 where we have two semi-infinite O6 − -planes on the both sides in the x 6 direction. The same 6d SCFT can be also realized on k M5-branes probing an D N singularity. After a circle compactification of the M-theory configuration, we obtain a system with an O6 − -plane on top of N D6-branes with k NS5-branes on top of each other. On a generic tensor branch of the theory, the k full NS5-branes split into 2k factional NS5-branes due to the presence of the orientifold six-plane. The fractional NS5-branes can change the sign of the RR charge of the orientifold. Namely, when an O6 − -plane crosses the fractional NS5-brane, it changes into an O6 + -plane. Similarly, when the O6 + -plane crosses the fractional NS5-brane, it changes into an O6 − -plane. Furthermore, the condition of the cosmological constant of massive type IIA background implies that the number of D6-branes on top of the O6 + -plane must be N − 4.
Let us understand this phenomenon in two steps. The original configuration is k full NS5-branes which divides one O6 − -plane on N D6-branes into two pieces in the x 6 direction. Since the k NS5-branes are put on the O6 − -plane, we in fact have 2k fractional NS5-branes. It is possible to distribute the k pairs of fractional NS5-branes along the x 6 direction. In this case, we only have O6 − -planes since the RR charge of the O6 − -plane does not change when it crosses two fractional NS5-branes. Finally, we split the pair of fractional NS5-branes into separated fractional NS5-branes on an O6 − -plane on top of D6-branes. In this process, some amount of D6-branes is also created between the fractional NS5-branes. The number of the created D6-branes is completely fixed by the condition on the cosmological constant or equivalently the anomaly cancellation condition of the corresponding 6d gauge theory with tensor multiplets. As a result, N − 4 D6-branes on top of the O6 + -plane are created between two adjacent factional NS5-branes. The brane configuration of the system is depicted in figure 8. Note that we have 2k factional NS5-branes in this case.
In the end, as a 6d SCFT on the tensor branch, we have a 6d alternating quiver theory Figure 8. The type IIA brane configuration realizing an 6d SO − Sp alternating quiver theory. The horizontal dotted lines denote an O6 − -plane or an O6 + -plane. Each NS5-brane is a fractional NS5-brane. The number of D6-branes includes the mirror images.
where we have k Sp(N − 4) gauge nodes and k − 1 SO(2N ) gauge nodes. The two [SO(2N )] at the two ends of the quiver represent the flavor symmetry. Instead of O6 ± -planes, it is also possible to construct 6d SCFTs by using O6 ± -planes. 3 We are interested in a 5d description of the 6d theory (3.2). In fact, the 5d description has been known already and it can be obtained by considering a different S 1 compactification from the original M-theory picture. Namely, when we compactify, for example, the x 5 direction on S 1 , then we can regard this S 1 as the M-theory circle. The circle compactification yields k D4-branes sitting at the D N singularity. The 5d gauge theory description on the D4-branes is an affine D-type quiver theory [25] 5d where we have four SU(k) gauge nodes and N − 3 SU(2k) gauge nodes.
In the remaining part of this section, we take a different route to arrive at the same 5d description (3.3). This way in fact makes use of an ON 0 orientifold in [14][15][16][17]. This example also nicely illustrates a gauge theory description from a brane configuration involving the ON 0 -plane, which we will use in the later sections.
The starting point is the brane configuration in figure 8 realizing the 6d theory of (3.2). We compactify the x 5 direction on S 1 and then perform T-duality along the S 1 . Then, we obtain a 5-brane diagram given in figure 9. As for 5-brane web diagrams, we always write down the (x 6 , x 5 ) two-dimensional plane where the vertical direction is the x 5 direction and the horizontal direction is the x 6 direction. Note that each O6 − -plane becomes two O5 − -planes, and each O6 + -plane becomes two O5 + -planes. In between the upper and lower O5 − -planes, we have N D5-branes. In between the upper and lower O5 + -planes, we have N − 4 D5-branes.
Here, we take S-duality taking into account the transition discussed at (2.4). We then arrive at a configuration where we have two ON 0 -planes at the both ends, k pairs of 3 The number of D6-branes put on either O6 − -planes or O6 + -planes is determined by the condition of the cosmological constant. Here it is important to note that the O6 − -plane exits only when the value of the cosmological constant is odd [24]. For example, when one considers an NS5-brane on O6 − which splits into two fractional NS5-branes, the resulting 6d SCFT on the tensor branch is an SO(2N + 1) gauge theory with 2N − 7 hypermultiplets in the vector representation. With multiple fractional NS5branes on D6/ O6 system, one exemplary 6d quiver theory with minimal odd cosmological constant is Figure 9. The 5-brane web diagram which is T-dual to the circle compactification of the type IIA brane setup in figure 8. The number of the fractional NS5-branes is 2k. The vertical direction is the x 5 direction and the horizontal direction is the fractional D5-branes in the horizontal direction and also N − 2 NS5-branes in the vertical direction as in figure 10. The 5d gauge theory description on the 5-brane configuration can be read off from the rules in [17], and it is exactly the same as (3.3). For example, it is easy to see that we have N − 3 SU(2k) gauge nodes from 2k D5-branes between N − 2 NS5-branes. The 2k fractional D5-branes between an ON 0 -plane and an NS5-brane at the two ends give rise to the two-pronged SU(k) gauge nodes at the two ends of the quiver. Hence, the (S-or/and) T-duality after the circle compactification of the original type IIA brane configuration given in figure 8 gives a different way to show the 5d description (3.3) from the 6d (D N , D N ) conformal matter theory on S 1 .

Type IIA brane setup with an ON 0 -plane
In the previous section, we included O6-planes in the D6-NS5-brane system and we found the 5d description of the 6d D-type conformal matter theory on S 1 . The brane configuration in type IIB string theory with ON 0 -orientifold, which yielded the 5d affine D-type quiver gauge theory, includes ON 0 orientifolds. The ON 0 -plane appeared as an object which is S-dual to a combination O5 − -plane + a full D5-brane in type IIB string theory. Figure 11. Type IIA configuration with an ON 0 -brane which is denoted by an empty circle. The presence of the ON 0 -brane gives rise to a 6d D-type quiver gauge theory. 2N is the number of D6-branes and k is the number of NS5-branes.
The existence of the ON 0 -plane is not restricted in type IIB string theory. In fact, Tduality implies that an ON 0 -plane also exists in type IIA string theory. Here we consider yet another case where we include an ON 0 -plane in the brane configuration in type IIA string theory. Namely, we consider a type IIA brane configuration with an ON 0 orientifold in addition to D6-branes and NS5-branes [17]. The ON 0 -plane extends in the same direction as those of NS5-branes in table 1 but is located at a different position in the x 6 direction.
Let us focus on the brane configurations which do not involve other kinds of orientifolds. The IIA brane configuration with an ON 0 -brane is given in figure 11. We have k NS5branes, and also we have 2N D6-branes in between two NS5-branes, or in between one ON 0 -plane and one NS5-brane. It is known that the presence of an ON 0 -brane gives rise to a D-type quiver theory [17] on a tensor branch of the corresponding 6d SCFT. The anomaly free 6d quiver theory is (4.1) We have 2 SU(N ) gauge nodes and k − 1 SU(2N ) gauge nodes. The global symmetry of the 6d theory is SU (2N ). We now derive a 5d gauge theory description of the 6d D-type quiver theory (4.1) after a circle compactification along one of the worldvolume directions of NS5-branes. After the S 1 compactification and performing T-duality along it, we obtain a 5-brane web with an ON 0 -plane on one end as well as 2N D5-branes and k NS5-branes on S 1 . The resulting 5-brane web diagram is depicted in figure 12.
Here, we take S-duality taking into account the transition explained in (2.4). Then, we have k + 1 D5-branes on top of each O5 − -plane and k − 1 D5-branes on top of each O5 + -plane. The total number of fractional NS5-branes is 2N . The resulting 5d gauge theory realized by the 5-brane web in figure 13 is 5d where there are N Sp(k − 1) gauge groups and N SO(2k + 2) gauge groups. Hence, our claim is that when we circle compactify the 6d theory given by (4.1), then the resulting 5d gauge theory description is the elliptic alternating quiver theory given by (4.2). This can be thought of as a natural generalization of the 5d elliptic quiver with SU nodes which arises as a circle compactification of the simple D6-NS5-brane system.
The T-dual description to the circle compactification of the type IIA brane setup in figure 11. The ON 0 -plane and the NS5-branes are on S 1 in the x 5 direction and the identification is denoted by ||. We can support this claim by comparing the number of parameters in the 6d theory with those of the 5d theory. Let us count the parameters of the 6d theory given by (4.1). We first count the number which become the number of the Coulomb branch moduli in the 5d theory after the circle compactification. In 6d, we count the sum of the number of tensor multiplets and the vector multiplets in the Cartan subalgebra. The number of the tensor multiplets corresponds to the number of the gauge nodes in (4.1), which is k + 1. From the brane picture in figure 11, they come from one ON 0 -plane and k NS5-branes. The two SU(N ) gauge nodes and k − 1 SU(2N ) gauge nodes further supply the 6d vector multiplets in the Cartan subalgebra. Hence the sum finally becomes k + 1 tensor multiplets After the circle compactification, both the 6d tensor multiplets and the 6d vector multiplets in the Cartan subalgebra becomes 5d vector multiplets in the Cartan subalgebra. The number of the 5d vector multiplets in the Cartan subalgebra or the dimension of the JHEP10(2016)126 0 1 2 3 4 5 6 7 8 9 Coulomb branch moduli space from the 5d elliptic alternating quiver theory given in (4.2) is This precisely agrees with the expectation from the parameter counting of the 6d theory (4.3). Let us then look at the matching associated with the global symmetry. In 6d, the flavor symmetry is SU(2N ). The circle compactification will give the affine A 2N −1 structure whose finite part is SU(2N )×U(1) I where U(1) I is a flavor symmetry associated with the Kaluza-Klein (KK) modes from the S 1 compactification. Therefore, the number of the parameters in the 5d theory should be 2N − 1 + 1 I = 2N where 1 I denotes the parameter associated to the U(1) I . In a 5d theory, parameters of the theory comes from the gauge couplings and also mass parameters of matter. The elliptic alternating theory given by (4.2) has 2N gauge couplings from the N Sp(k − 1) gauge groups and N SO(2k + 2) gauge groups. It is also important to note that we cannot introduce mass parameters for the hypermultiplets which connect Sp(k − 1) gauge group and SO(2k + 2) gauge group. Therefore, the total number of the parameters is 2N , which exactly agrees with the expectation.
One can in fact see the global symmetry SU(2N ) directly from the web diagram in figure 13. The 5-brane web has 2N external NS5-branes and a [0, 1] 7-brane can be put on the ends of the NS5-branes. These 2N [0, 1] 7-branes imply that the 5d theory have an SU(2N ) global symmetry.
In section 3 and 4, we introduced O6-planes and an ON 0 -plane respectively in the type IIA brane setup. As a generalization of the preceding sections, we introduce O6-planes and an ON 0 -plane simultaneously into the type IIA brane configuration. The worldvolume configuration that we consider is shown in table 2.
It was discussed in [17] that if one introduces orientifold planes of different kinds among ON 0 -, O6 ± -, and O8 ± -planes, one needs to consider all of the three orientifolds at the same time to describe the full system. It is also consistent from the point of view that the orientifold projections along the transverse directions with respect to each orientifold. We here consider brane configurations with an O8-plane, O6-planes and D6-branes divided by NS5-branes where an ON 0 -plane is stuck at the intersection between the O8-plane and the O6-plane or D6-branes. It is worth noting that the O8-plane and D8-branes are fractionalized when they end on an O6-plane.

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There are four possible configurations in which an ON 0 -plane can be included: (i) (O8 − , O6 − ), (ii) (O8 − , O6 + ), (iii) (O8 + , O6 − ), and (iv) (O8 + , O6 + ). Each type IIA brane configuration is anomaly free [17] and thus leads to a 6d SCFT on a tensor branch. Each configuration gives different 6d quiver theories associated with a Dynkin diagram. Different types of quiver diagram depend on the relative sign of the RR-charge between an O8-plane and an O6-plane. For instance, if the relative sign between an O8-plane and an O6-plane is opposite, then the brane configuration leads to a D-type quiver. If the relative sign is same, on the other hand, the resulting configuration is an A-type quiver [17].
In this section, we study possible 5d gauge theory descriptions of these 6d theories via a circle compactification. As explained in [1,[4][5][6] as well as the previous sections, the process that we propose to obtain a 5d description involves not only T-duality but also S-duality as well as the resolution of O7 orientifold planes into a pair of 7-branes. As resolving an O7plane is possible only for an O7 − -plane [18], obtaining a 5d description for the configuration with O7 + -planes is limited. It is also not clear whether an S-dual picture of an O7 + -plane leads to a sensible gauge theory description. Therefore, we will not discuss the brane configurations with an O8 + -plane among the possible IIA brane configurations above, as our proposal for obtaining 5d gauge theory description may not be fruitful for those configuration other than a natural IIB configuration having two O7 + -planes coming from T-duality. We will instead focus on the cases involving an O8 − -plane. In the following subsections, we study possible 5d gauge theory descriptions for (i) (O8 − , O6 − ) and (ii) (O8 − , O6 + ) cases.

5d gauge theory descriptions for
Let us first consider case (i), the configurations where an O8 − -plane, an ON 0 -plane and an O6 − -plane are intersecting with each other as shown in figure 14. For simplicity, we introduce eight D8-branes so that the net RR-charge is zero. We also consider the case of even number of D6-branes in the brane setup. These 2N D6-branes are split by 2k fractional NS5-branes. As discussed in earlier sections, due to the NS5-branes, the O6planes split in such a way that O6 − -and O6 + -planes are alternating, which leads to SO and Sp gauge groups. A representative brane configuration is given in figure 14. This ON 0 − O6 − − O8 − -plane configuration yields the following 6d SCFT on a tensor branch: where the number of Sp gauge group is k, and that of SO gauge group is k − 1.
The number of the parameters associated in the theory is given as follows. The number of the 6d vector multiplets in the Cartan subalgebra and the 6d tensor multiplets is The global symmetry of the system is SO(4N ) × SU(8) × U(1) and hence the rank of the global symmetry is 2N + 8. In the following subsections, we discuss three different ways of obtaining 5d gauge theory descriptions. ON 0

S-dual picture in type IIB setup
We now consider a type IIB brane configuration for this brane setup via a circle compactification and T-duality. As a result of T-dual action on the O8 − -plane, one has two O7 − -planes in the type IIB brane picture. As the O8 − -plane and eight D8-branes are fractional, the resulting two O7 − -planes and eight D7-branes are also fractional. With suitable Wilson lines, we allocate the O7 − -plane and four D7-branes on top of each other to make an O7/D7 combination whose net RR charge is zero, and we call it an O7 0 -plane. (See figure 15 (a).) The distance between the two O7 − -planes is inversely proportional to the radius of the compactifying circle. This is just a IIB description of the 6d theory on a circle. Note also that when one has the fractional O7 0 -plane and the ON 0 -plane at the same time, one must have another orientifold 5-plane [17]. Namely, the fractional O7 0 -plane has JHEP10(2016)126 to be located at the intersection between the ON 0 -plane and the O5-plane. The charge conservation further implies that the O5-plane has no net RR-charge, which should be a combination of an O5 − -plane and a D5-brane. We call it an O5 0 -plane in this article. Therefore, the fractional O7 0 -plane is located at the intersection between the ON 0 -plane and the O5 0 -plane in the 5-brane web. From the T-duality, the O5 0 -plane originates from the combination of the O6 − -plane and one D6-brane in the type IIA brane configuration. The RR-charge conservation implies that one O6 − -plane yields two O5 − -planes after the T-duality. Hence, we have one O5 0 -plane at the top of the diagram and another O5 0 -plane at the bottom of the diagram. (See figure 15 (a).) In order to obtain a genuine 5d gauge theory description for the 6d theory, we utilize the S-duality as discussed in section 2. Due to the transition (2.4), the 5-brane web after S-duality is given in figure 15 (b), which corresponds to a 5d quiver gauge theory interpretation as where the number of Sp gauge groups is N − 1, while that of SO gauge group is N − 2.
Let us then count the number of parameters of the 5d gauge theory and see its comparison with the 6d computation. The dimension of the Coulomb branch moduli space is given by which is the expected number including the symmetry arising from the KK modes, 1 I . Note that we can introduce a mass parameter only for the bi-fundamental hypermultiplet between the Sp(2k−1) gauge group and the SU(2k+1) gauge group. From the brane configuration and the parameter counting, (5.3) is a possible 5d description for the 6d theory (5.1).

The resolution of two O7 − orientifolds
We now consider the resolution of an O7 − -plane [18,26]   D7-brane and an O7 − -plane mutually commute with each other. It is straightforward to see from 7-brane monodromies that

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In the current brane setup, the O7 0 -plane is stuck at the intersection between an ON 0plane and an O5 0 -plane. As it is stuck at the intersection of the orientifold planes, the O7 0 -plane becomes fractional. From the 7-brane configuration in (5.6), a half-O7 0 -plane may be described effectively by the full ANA 7-branes. Since the half O7 0 -plane consists of three full 7-branes, it can be detached away from the orientifolds. Therefore, we propose that a half O7 0 -plane at the intersection between an ON 0 -plane and an O5 0 -plane can be resolved into the full ANA 7-branes. This resolution is also natural from the point of view of the (p, q) charge conservation at the interaction between an ON 0 -plane and an O5 0 -plane, involving the cuts due to ANA 7-branes. (See figure 16.) The monodromy created by the branch cuts of the ANA 7-branes is in fact exactly the S-duality transformation, Therefore, as the cuts of the ANA 7-branes are going through the intersection of the orientifolds, an O5 0 -plane is converted to an ON 0 -plane, or vice versa. After the resolution of the O7 0 -plane, a microscopic picture of the ON 0 -plane and the O5 0 -plane is that the branch cuts of the ANA(= AAB = CAA) 7-branes change an ON − -plane (and an NS5-brane) into an O5 − -plane (and a D5-brane) respectively.
We then turn around the cuts of 7-branes in a way that the alternating O5 + -and O5 − -planes go across these cuts. Since going across the cuts locally generate S-duality transformation as discussed above, these O5-planes are converted into ON 0 -planes taking into account the transition (2.4). The resulting web diagram is depicted in figure 17.
As done in [5], we can pull out the B, C 7-branes in such a way to yield a 5d gauge theory interpretation, which is to move the B, C 7-branes through the 5-branes of the same (p, q) charge so that no 5-branes are created in so doing. For example, take the B 7-brane and move it through the (1, −1) 5-brane as shown in figure 18. We move the B, C JHEP10(2016)126 7-branes until they are put as close as possible in Coulomb branch moduli chambers. This leads to the following two possible configurations for which one can give a 5d gauge theory interpretation.
• The first case is N ≥ 2k. In this case, as depicted in figure 19 for k = 1, the B and  Figure 19. A procedure leading to a 5d gauge theory from the web diagram after allocating 7branes in suitable positions. By turning around the cuts of 7-branes, one can see a 5d gauge theory description more clearly.
chamber. This yields the following D-type quiver theory Here we put all four A 7-branes back to the ON 0 -plane so that they are again fractionalized, giving each four flavors.
It is straightforward to see the parameters of theory precisely match with those of the 6d theory (5.2). The dimension of the Coulomb branch moduli space is given by where it includes the U(1) part coming from the KK mode, 1 I , as expected.
In this case, we can further see the consistency with the enhancement of the global symmetry by the one-instanton analysis given in [2]. As for the D-type Dynkin quiver (5.7), almost all the gauge nodes except for the rightmost one satisfy N f = 2N c for a SU(N c ) gauge node where N f represents the effective number of flavors attached to the SU(N c ) gauge node. On the other hand, the rightmost gauge node satisfies N f = 2N c + 2. The one-instanton analysis then implies that, in the case where N f = 2N c , the instanton states give rise to two Dynkin diagrams whose shapes are equal to the quiver diagram. In the case where N f = 2N c + 2, the two nodes given by the one-instanton states not only participate in the two Dynkin diagrams but also are the fundamental representation and the antifundamental representation of the flavor symmetry attached to the gauge node. Therefore, the one-instanton analysis of the D-type quiver theory (5.7) at least yields the flavor symmetry whose structure is described by the affine D 2N Dynkin diagram. The affine D 2N Dynkin diagram implies that the UV completion is a 6d theory with a flavor symmetry SO(4N ), which is consistent with the flavor symmetry of the 6d theory (5.1).
• The other case is when N < 2k. In this case, the B, C 7-branes are put together in a single Coulomb branch moduli chamber. By doing the trick converting B, C 7-branes back to an O7 − -plane, as depicted in figure 20, one finds that the corresponding 5d gauge theory of a D-type quiver is given by which includes the U(1) part coming from the KK mode, as expected.

The resolution of only one O7 − orientifold
Regarding the resolution of the O7 − orientifold planes, one can consider the case where only one out of the two O7 − -planes is resolved, which leads to an interesting UV duality in 5d. For instance, it was discussed in [5] that there are two seemingly different 5d descriptions whose UV completion is the same 6d conformal matter theory. Depending on whether one resolves one or two O7 − -planes which come from the T-dual action on an O8 − -plane in the type IIA brane setup, one has an SU gauge theory description for resolving two O7 − -planes, while one has an Sp gauge theory description for resolving only one O7 − -plane. In this subsection, as a generalization, it is interesting to study what dual description we get when we resolve only one O7 − -plane leaving the other O7 − as it is, like figure 21. As in the previous subsection, possible brane configurations depend on the number of D5-branes and NS5-branes. We first consider the case when 2N ≥ 2k + 1. After resolving only one O7 − -plane, say the bottom O7 − -plane in figure 15 (a), we repeat the procedure of moving B 7-brane through the chambers of 5-brane webs as explained in figure 18. For definiteness, an example of brane configuration of one O7 − resolution for N = 4 and k = 2 is shown in figure 22. Alternating O5 + -O5 − -plane gives alternating Sp − SO gauge groups. The resulting 5d gauge theory is given by (5.13) where there are 2k gauge nodes.

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We now check the parameters of this 5d quiver theory and compare them with the 6d theory that we started with. There are one SU gauge group, k Sp gauge groups, and k − 1 SO gauge groups which lead to the total dimension of the Coulomb branch moduli space to be JHEP10(2016)126 where we have 2k instanton fugacities from the 2k gauge nodes, and one bi-fundamental field connecting the first SU gauge group and the adjacent Sp gauge group. Again it agrees with the 6d parameters as expected. When 2N < 2k + 1, one may repeat the same procedure, which leads to the case where the B 7-brane will eventually hit the upper O5 0 -plane. In this case, however, it is not clear whether there is a well-defined gauge theory description. To obtain a proper gauge theory description, we might need to convert the 7-branes to a full or fractional O7 − -plane. In order for one to have a fractional O7 − -plane of a consistent configuration with the (p, q) charge conservation, one needs a C 7-brane to hit the O5 0 -plane. Here, on the other hand, we have the B 7-brane hitting the O5 0 -plane, which does not lead to a gauge theory description.

5d gauge theory descriptions for
We now consider the case (ii) (O8 − , O6 + ) which is a 6d SCFT which can be described by a type IIA brane setup involving an ON 0 − O6 + − O8 − system. On top of a fractional O8 −plane with 8 D8-branes, an ON 0 -plane is located, and D6-branes/O6 + -planes are extended along the x 6 direction as shown in figure 23. Like the previous cases with an O6 − -plane, D6-branes and an O6-plane are suspended between fractional NS5-branes, leading to a configuration with alternating O6 − -and O6 + -planes.
In figure 23, there are two types of configurations for D6-branes between the ON 0plane and the first NS5-brane in this case [17]. The first configuration is that (2N − 8) D6-branes are suspended between the first NS5-brane and the NS5-brane inside the ON 0 . Figure 23. A type IIA brane setup involving the intersection of ON 0 , O6 + , and O8 − -planes. There are 2k − 1 fractional NS5-branes and the number of D6-branes includes their mirror pairs. The other one is that 2N D6-branes are extended from the first NS5-brane and jump over the NS5-brane, passing through the ON − -plane and then attach back to the NS5-brane or the mirror NS5-brane inside the ON 0 . These two types of D6-branes contribute to different Sp gauge groups. The numbers of the D6-branes in figure 23 are fixed by the cosmological constant condition, or equivalently the anomaly free condition.

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The resulting brane configuration then yields a D-type quiver with two-pronged Sp gauge nodes of different ranks. It follows then that the 6d SCFT on a tensor branch is given by The number of the 6d vector multiplets in the Cartan subalgebra plus the number of the tensor multiplets are given by N − 3 + N + 1 two split Sp s+tensors The global symmetry is SO(4N ) × SO (16) and thus the rank of the global symmetry is 2N + 8. In general, the number of tensor multiplets is equal to the number of gauge nodes. Since we have two-pronged Sp gauge nodes in the left part of the quiver (5.16), the distance between the NS5-brane and the ON − -plane, which form the original ON 0 plane, is also related to the tensor branch moduli of one of these gauge groups.

S-dual picture in type IIB setup
We now discuss a 5d description for this 6d SCFT on a tensor branch, by compactifying it on a circle and taking a T-duality. A straightforward type IIB setup is given in figure 24. As mentioned above, one NS5-brane is away from the ON − -plane and one fractional O7 0 -plane is located at the intersection between the ON − -plane and an O5 − -plane. This picture gives rise to a 5-brane configuration that D5-branes jump over this NS5-brane and reconnect to JHEP10(2016)126 2N-4 Figure 24. A IIB version of figure 23 after a circle compactification and T-duality. The vertical direction is the compactification direction, where two fractional O7 0 (= O7 − + 4D7 s)s are located at the two corners. Figure 25. A procedure leading to a 5d gauge theory description for the IIB brane configuration given in figure 24. (a) A ON 0 is refined to be an ON − -plane and an NS5-brane. (b) Taking (2.4) into account, we obtain S-dual description.
the same NS5-brane after passing through the ON − -plane. See figure 25 (a). The more precise configuration is the one explained in section 2.
After the S-duality, taking into account the transition (2.4), we obtain a 5-brane configuration with two ON − -planes on the left and on the right, which comes from the S-dual of the O5-plane. See figure 25 (b). This may lead to a 5d affine D-type quiver gauge theory, where there are two Sp(k − 2) as well as two Sp(k) gauge groups in each end of the affine D-type quiver, and (N − 1) SO(4k) gauge groups and (N − 2) Sp(2k − 2) gauge groups in the middle of the affine D-type quiver. The dimension of the Coulomb branch moduli space of the proposed 5d quiver gauge theory (5.18) is given by  20) which also agrees with the number of the rank of the flavor symmetry of the 6d theory as expected.

The resolution of two O7 − orientifolds
As done in subsections 5.1.2 and 5.1.3, we also consider the resolutions of O7 − -planes into a pair of 7-branes for either both or one O7 − -plane(s). The procedure of pulling out the B and C 7-branes together with flavor 7-branes from the orientifolds and then allocating these 7-branes throughout 5-branes is essentially same as that explained in subsections 5.1.2 and 5.1.3. Here we hence discuss subtle issues or differences in obtaining 5d descriptions, compared with subsections 5.1.2 and 5.1.3.
Let us consider resolutions of two O7 − -planes of the IIB brane configuration in figure 24 or more precisely in figure 25 (a). We focus on the configuration near one of the two O7 −planes. Consider a lower half corner of figure 25 (a), where where we resolve O7 − -plane of the O7 0 -plane to be B and C 7-branes. Together with four D7-branes, they are initially stuck at the intersection between ON − and O5 − -planes and thus factional branes. As explained in section 5.1.2, these branes can leave the intersection as a combination of full 7-branes, for instance, (C, A, A) 7-branes. Recalling 5-brane configurations associated with ON − -planes discussed in section 2, one easily sees the proper 5-brane configuration of intersecting ON − and O5 − -planes is given by a configuration shown in figure 26 (a). D5/NS5-branes jump over NS5/D5-branes to pass through the orientifolds and reconnect to the NS5/D5-branes. Using 7-brane monodoromies, we can rearrange the 7-branes to be (A, A, B) so as to convert the O5 − -plane into the ON − -plane as shown in figure 26 (b). Then as done before, the B 7-brane can be allocated by crossing (1, −1) 5-branes, with proper changes of 5-branes charges when the cut of B passes through. We also turn around the cuts of the two A 7-branes, which makes the nearest 5-brane from the ON −plane a generic NS5-brane, as shown in figure 26 (c). We then consider this NS5-brane as the NS5-brane to form an ON 0 -plane so that the resulting 5-brane configuration is of a 5-brane configuration with the ON 0 -plane. See figure 26 (d).
In the same fashion, one can repeat the procedure for the remaining O7 − -plane in the other corner of the intersection between an ON − -, an O5 − -, and an O7 − -plane. Putting together two resolved 7-branes at both corners gives rise to a 5d gauge theory description. We note that as in section 5.1.2, two 5d gauge theory descriptions are possible depending on the relation between N and k. • When N ≥ 2k − 1, the corresponding 5d gauge theory description is given by (5.21) where there 2k gauge nodes in the quiver. To compare to the 6d theory that we started with, we count the parameters of the resulting 5d theory. The dimension of the Coulomb branch moduli space is given by  Figure 27. Left: a Type IIB brane configuration with ON 0 , O5 + , and two O7 0 . Right:A resulting 5d gauge theory description after resolving two O7 − -planes. Here the brane configuration assumes that the number of D5-branes (N ) and that of NS5-branes (k − 1) satisfy N ≥ 2k − 1.
which again agrees with the number of the rank of the flavor symmetry of the 6d theory plus an extra parameter accounting for the U(1) I KK instanton. See figure 27. It is also possible to perform the instanton operator analysis to see a subgroup of the enhanced flavor symmetry at the UV fixed point. In the 5d quiver (5.21), most of the gauge nodes satisfy N f = 2N c whereas only one gauge node at the right end satisfies N f = 2N c +2. The rule in [2] implies that the one-instanton states create a Dynkin diagrams where two of the quiver diagrams of (5.21) are connected by the Dynkin diagram of SU(2N − 4k + 2). The resulting Dynkin diagram is an affine D 2N Dynkin diagram. This agrees with the fact that the original 6d theory (5.16) has the SO(4N ) flavor symmetry.
• When N < 2k−1, we find that (B, C) 7-branes are located in a chamber of the Coulomb branch moduli space, which we can recombine to form an O7 − -plane, as shown in figure 28. Hence, the corresponding theory is given by an D-type quiver with two-pronged SU gauge nodes of different ranks in one end, and an Sp gauge node in the other end of the quiver (5.24) where there are N SU-type gauge nodes in the quiver. The dimension of the Coulomb branch moduli space reads which includes the U(1) part coming from the KK mode, as expected.

The resolution of only one O7 − orientifold
We now consider the case where we resolve only one O7 − -plane out of the two O7 − -planes. The way that one can obtain a gauge theory description is similar to section 5.2.2, except that the unresolved O7 − -plane is located at the intersection of ON − and O5 − -planes. See figure 29. As discussed in section 5.1.3, 5d gauge theory description is possible when 2N ≥ 2k−1, and is given by an D-type quiver with two-pronged Sp gauge nodes of different ranks, Figure 29. Left: a type IIB brane configuration with ON 0 , O5 + , and two O7 0 . Right: a resulting 5d gauge theory description after resolving only one O7 − -plane. Here the brane configuration assumes that the number of D5-branes (N ) and that of NS5-branes (k − 1) satisfy 2N ≥ 2k − 1.
where there are 2k gauge nodes (i = 1, 2, · · · , k −1). The dimension of the Coulomb branch moduli space for this quiver is given by where the 2k instanton factors are from the 2k gauge nodes. Again it agrees with the rank of the flavor symmetry of the 6d theory as expected.

Beyond circle compactification
So far, we have considered a simple circle compactification of 6d SCFTs and determined their 5d gauge theory description. In this section, we consider another one-dimensional compactification of a 6d SCFT. In this case also, it turns out that the different onedimensional compactification still yields a 5d gauge theory. The starting setup is a 6d SU(N ) gauge theory with N f = 2N flavors. This 6d theory can be realized either on N D6-branes with two NS5-branes which fractionate the N D6branes in the x 6 direction in type IIA string theory or on an M5-brane probing an A N -type JHEP10(2016)126  singularity in M-theory. It is well-known that the standard circle compactification yields a 5d elliptic quiver theory where we have N SU(2) gauge nodes connected by bi-fundamental hypermultiplets.
Let us consider a different one-dimensional compactification. We start from the configuration of the N D6-branes fractionated by the two NS5-branes in type IIA string theory, which is a special case with two NS5-branes in figure 7. Then, we compactify the brane setup along the x 5 direction. In the case of the circle compactification, we identify the configuration at x 5 = 0 with that at x 5 = 2πR where R is the radius of the S 1 . This time we consider an identification with a reflection with respect to an axis along x 6 = 0. Here we choose x 6 = 0 as the middle point between the two NS5-branes, and hence it is a symmetry of the brane configuration. We call this compactification a twisted circle compactification. The schematic picture of the geometry in the x 6 -x 5 -plane is depicted in figure 30. In other words, the compactification geometry in the x 5 -x 6 -plane is a Möbius strip with an infinite width. Due to the identification, the original two NS5-branes are connected to each other and become a single NS5-brane.
The geometry of the compactification in the x 6 -x 5 -plane can be seen in another way.  fundamental region. It is also possible to take a different fundamental region. Let us choose the right half of the double cover. Then, it can be regarded as a semi-infinite cylinder with a cross-cap at one boundary as in figure 32. Now we perform T-duality along the x 5 direction to go to a 5d description. In fact, it is known that the T-duality along the S 1 with a cross-cap produces a pair of an O − -plane and an O + -plane [19]. Therefore, the brane configuration after the T-duality is that we have a pair of an O7 − -plane and an O7 + -plane and N fractional D5-branes and also a single NS5-brane. The schematic picture is depicted in figure 33.
In order to read off the gauge theory content from the diagram, we choose a different fundamental region as in figure 34. When N is even, then we have N 2 D5-branes between two NS5-branes. When N is odd, then one of the color D5-brane is stuck on the O7 +plane and we have N −1 2 D5-branes away from the O7 + -plane between the two NS5-branes. However, we have always N flavor D5-branes.
By the strong coupling effect of the string coupling, an O7 − -plane splits into two 7branes. After the resolution of the O7 − -plane, we obtain a 5-brane web configuration with one O7 + -plane. The splitting process is depicted in figure 35.
We have N semi-infinite D5-branes and N 2 +1 parallel finite D5-branes. This is nothing but a 5-brane web that yields a 5d SO(N + 2) gauge theory with N v = N hypermultiplets in the vector representation. Therefore, we propose that the UV completion of the 5d JHEP10(2016)126 O7 + O7 − Figure 34. The same 5-brane web diagram as the one in figure 33 but we take a different fundamental region. When N is odd, we have one fractional color D5-brane stuck on the O7 + -plane. The total number of the flavor D5-branes is always N .
O7 + It was originally discussed that the 5d SO(M ) gauge theory theory has 5d fixed point when the number of the hypermultiplets in the vector representation satisfies N v ≤ M − 4, based on the the requirement that the effective gauge coupling is non-negative in all the region of the Coulomb branch moduli space [7]. However, it was recently noticed that 5-brane web configurations predict the existence of more 5d theories with a 5d fixed point. Indeed it was pointed out that for the 5d SO(M ) theories, there exists a brane configuration involving an O7 + -plane with N v = M − 3 [8], which suggests a new 5d fixed point beyond the bound from [7]. Our analysis shows that when one adds one more hypermultiplet in the vector representation of the SO(M ), the UV completion of the 5d theory is a 6d SCFT, namely the 6d (A M −3 , A M −3 ) minimal conformal matter.
One can see another support for the claim from the flavor symmetry of the system. Before the twisted circle compactification, the flavor symmetry is SU(2N ) = A 2N −1 . The JHEP10(2016)126 Figure 36. Left: a Tao brane configuration for SO (2) theory with N v = 0. As a rank one theory, there is only one D5 brane above the O7 + -plane. One takes a branch cut of a [1, 1] 7-brane, and let us pull out a [1, −1] 7-brane along the direction of its (p, q) charge. This 7-brane then must pass through the cut of the [1, 1] 7-brane, which then turns to a [3, 1] 7-brane. When passing the bottom of the fundamental region, it appears as again a [1, −1] 7-brane on the right hand side of the O7 + due to the action of O7 + which will pass through the cut off the [1, 1] 7-brane then becomes again a [3, 1] 7-brane. A repeated application of pulling 7-branes leads to a Tao configuration with a constant period. Right: a Tao diagram of the brane configuration for SO(3) theory with N v = 1. In this case, a factional D5-brane is stuck at the orientifold and one can introduce the cut of a [1, 0] 7brane in addition to the cut of a [1, 1] 7-brane. In the same way, by pulling out a [1, 1] 7-brane which experiences the 7-brane cuts, one see that it also makes a Tao configuration. the [1, 0] 7-branes. standard circle compactification yields the affine A 2N −1 algebra. Therefore, a natural expectation after the twisted circle compactification is that we obtain the twisted affine algebra A For a 5d theory on a 5-brane web, the flavor symmetry is realized on 7-branes. We have now N D7-branes attached to the N semi-infinite D5-branes and also a pair of an O7 − -plane and an O7 + -plane. When N D7-branes are on top of the O7 − -plane, we obtain an SO(2N ) symmetry. On the other hand, when N D7-branes are on top of the O7 + -plane, we obtain an Sp(N ) symmetry. Hence, the full structure of the flavor symmetry should include both the SO(2N ) and the Sp(N ) symmetry. There is no finite group which includes both and the symmetry structure is in fact the twisted affine algebra A 2N −1 . Indeed, it has been known that the N Dp-branes with a pair of an Op − -plane and an Op + -plane yields the twisted affine algebra A 2N −1 [27]. This also agrees with the one-instanton analysis in [28]. We note that, following [1,9], one sees that the web diagram for SO (2) with N v = 0 and SO (3) with N v = 1 shows a rotating spiral shape (Tao sturcture), as shown in figure 36. A straightforward generalization for generic SO(M ) does not seem to work, but it does not mean that there is no Tao structure, as Tao diagram may appear depending on the number of 7-brane cuts which other 7-branes are passing through.
It is straightforward to generalize the analysis to a 6d SU(N ) quiver theory given in (3.1). We consider a 6d theory realized on the brane setup given by figure 7. As natural generalization, we first consider 2k NS5-branes as well as N D6-branes. Then we compactify the configuration on an S 1 in the x 5 direction with identification by a reflection and then perform T-duality along the S 1 . The analysis is essentially the same in the case JHEP10(2016)126 O7 + Figure 37. The 5-brane web configuration after after compactifying the brane setup in figure 7 on an S 1 with the twist and performing T-duality along the x 5 direction. Figure 38. A schematic picture of the 5-brane web configuration after splitting the O7 − -plane in figure 37 and pulling the two 7-branes outside of the 5-brane loops. For simplicity, we draw the case of k = 2.
with k = 1, and we obtain a 5-brane web given in figure 37. We have N 2 D5-branes, 2k NS5-branes and a pair of an O7 − -plane and an O7 + -plane.
Again at the quantum level, the O7 − -plane splits into two 7-branes. This process creates 5-brane loops. Then, we try to pull out the two 7-branes outside of the 5-brane loops so that the final 5-brane web configuration yields a 5d gauge theory description. The process is essentially the same as the one done in section 4.1.2 in [5]. We will not repeat the analysis here and simply present the final 5-brane web configuration in figure 38. The resulting 5d theory is 5d SO(N +2k)−SU(N +2k −4)−SU(N +2k −8)−· · ·−SU(N −2k +4)−[N −2k +2] (6.1) The number of the SU gauge nodes is k − 1. Hence, we claim that the 5d SO-SU quiver theory (6.1) has the 6d UV completion given by the 6d SU(N ) quiver theory with the 2k − 1 gauge nodes.
The case with 2k + 1 NS5-branes is also straightforward. This time, one of the NS5branes connect O7 + plane and O7 − plane, as in the left of figure 39. Analogous procedure ends up with the right of figure 39, whose 5d theory is The number of the SU gauge nodes is k. In this case, the left gauge node is the SU(N + 2k − 1) gauge group with matter in the symmetric representation which we denote by [S] JHEP10(2016)126 Figure 39. The web diagram with odd number of NS5-branes.
in (6.2). Hence, we claim that the 5d SU quiver theory (6.2) with symmetric matter has the 6d UV completion given by the 6d SU(N ) quiver theory with the 2k gauge nodes.
In particular, when k = 1, one obtains a 5d SU(N + 1) gauge theory with N − 1 fundamental hypermultiplets and also a hypermultiplet in the symmetric representation. A 5-brane web analysis implies that one can add N −2 fundamental matter to a 5d SU(N +1) gauge theory with one symmetric matter when the UV completion is a 5d SCFT [8]. The analysis here means that when one adds one more flavor, the UV completion is the 6d SU(N ) quiver theory with 2 gauge nodes.

Conclusion and discussions
As a generalization of [5], we studied 5d gauge theory descriptions of various 6d N = (1, 0) SCFTs realized by type IIA brane setup with an ON 0 -plane with or without the presence of other orientifolds such as O6-and O8-planes.
An ON 0 -plane is an object which is S-dual to a D5-brane on top of an O5 − -plane. Brane configurations involving an ON 0 -plane lead to various 6d quiver gauge theories of a D-type or an A-type. 5d gauge theory description whose brane setup involving such ON 0 -plane, on the other hand, is not well studied. In this paper, we explored possible brane setups of ON 0 with/without other usual orientifolds. We gave a brane description for an ON 0 -plane with a detailed brane web diagram in section 2, which allows one to see various dual descriptions through S-duality.
Among many 6d SCFTs, especially 6d SCFTs whose brane setup is realized with different types of the orientifolds, ON 0 , O6, O8, leads to fruitful 5d gauge theory descriptions, We studied such 6d theories by first compactifying them on a circle and then taking a T-duality which yield a type IIB brane configuration with ON 0 , O5, O7-planes. Based on the type IIB brane setup, we proposed various kinds of 5d N = 1 quiver gauge theories whose UV fixed point is an identical 6d SCFT. The main techniques that we devised to obtain a natural 5d gauge theory description is S-duality in IIB brane setup and quantum resolutions of O7 − -planes.
An S-duality transformation acting on the 5-brane web diagram is often used to obtain a proper gauge theory description and is useful to relate a 5d theory to different 5d gauge theories with identical UV fixed point. In the web diagram, this S-duality transformation JHEP10(2016)126 is simply realized by rotating the (p, q) 5-brane web diagram by 90 degrees. When we include O5-planes as we discussed in section 4 and 5, the analysis is slightly more involved because we do not know the field theory description when we have an S-dual object of an O5 + -plane. Instead of directly taking an S-duality, we first recombine (as a revise procedure of splitting or as making two fractional NS5-brane unsplit) the fractional NS5branes attached to the O5-planes in such a way to avoid appearing O5 + -planes. After obtaining an ON 0 -plane as S-dual of an O5 0 -plane, we again split the D5-branes, which were originally the fractional NS5-branes mentioned above. This procedure enables us to obtain a proper 5d gauge theory description via S-duality. As for clearer understanding of splitting D5-branes on an ON 0 -plane, especially in the context of the relation to its S-dual picture, it is important to further investigate we need further investigation in the future.
Another technique is the resolution of an O7 − -plane. This resolution was originally proposed by [18] and applied to various brane setups to obtain non-trivial relation among 5d theories. This resolution was also generalized to the case where an O7 − -plane is attached to an NS5-brane [4,5]. In this paper, we proposed novel resolution of an O7 − plane located at two intersections of an ON 0 -plane and each O5 0 -plane. This is proposed by observing that the resolution of an O7 − -plane into a pair of [1,1] and [1, −1] 7-branes is consistent with the orientifold projection created by an O5-plane and an ON 0 -plane and by interpreting that one of them becomes the mirror image of the other. Due to this resolution, we obtained various highly non-trivial dualities among 5d gauge theories, one of which is the novel duality between a D-type SU quiver gauge theory and an SO-Sp quiver gauge theory. It would be interesting to give further evidence for these conjectured dualities as well as for the novel resolution of the O7 − -plane.
Still another technique is a twisted circle compactification which yields two different types of O7-planes after T-duality. They are an O7 − -plane and an O7 + -plane maximally separated apart along the direction of the compactification radius. Combined with the resolution of the O7 − -plane mentioned above, we obtain a 5d gauge theory with an SO gauge group.
Although we did not discuss in the main text, there are, of course, non-Lagrangian theories naturally arising along the procedure. For example, web diagrams leading to non-Lagrangian theory appear when one implements S-duality and the resolution of O7 −plane at the same. In this paper, however, we focused on 5d web diagram giving a gauge theory description and have given proposals for 5d gauge theory description for various 6d SCFTs. We checked that all the proposed 5d descriptions have the expected number of mass parameters and Coulomb branch moduli parameters, and they all agree with those of the 6d theories. It would be also interesting to give more evidences to support our conjectures. One of the future work in this context will be to compute the elliptic genus of the 6d SCFT corresponding to the original type IIA brane setup and compare it with the 5d partition function or 5d superconformal index computed from topological string partition function. However, the topological vertex formalism corresponding to a web diagram with an O5-plane and/or ON 0 -plane is not known. The study in [29] may give a clue to this issue.