JHEP05(2019)203 - − 7 O Springer April 4, 2019 May 20, 2019 May 29, 2019 : : : d,e Received Accepted Published theories with additional , N T -plane. We T-dualize the brane 0) superconformal field theories and Futoshi Yagi , − ) duality of Type IIB string theory, e 8 Z , O = (1 Published for SISSA by https://doi.org/10.1007/JHEP05(2019)203 -planes. The two situations give us two − N 7 O Kimyeong Lee c,e [email protected] , [email protected] . 3 Sung-Soo Kim, futoshi , a,b 1509.03300 The Authors. Brane Dynamics in Gauge Theories, Field Theories in Higher Dimensions, c

We propose 5d descriptions of 6d , [email protected] Korea Institute for Advanced Study, 85 Hoegi-ro Dongdaemun-gu, Seoul, 02455, Korea E-mail: [email protected] Universidad Aut´onomade Madrid, Cantoblanco, 28049 Madrid, Spain School of Physics, UniversityNo. of 4, Electronic Science section and 2, Technology NorthSchool of of Jianshe China, Mathematics, Road, Chengdu, Southwest SichuanWest Jiaotong 610054, zone, University, China high-tech district, Chengdu, Sichuan 611756, China Department of Physics, School4-1-1 of Kitakaname, Science, Tokai Hiratsuka-shi, University, Kanagawa 259-1292,Departamento and de Japan Te´orica Instituto F´ısica UAM/CSIC, de Te´orica F´ısica b c e d a Open Access Article funded by SCOAP tion of 5d theories whichflavors, is which obtained turns by out a not generalization to of beKeywords: 5d in the class ofSupersymmetry Type and IIA Duality brane construction generically. ArXiv ePrint: leading to completely differentties group between structure. 5d gauge Thesetheory. theories lead We whose also to UV explore a completion Higgsingthe very is of same rich the the flavors class 6d same from theories of 6d thecompletion and dual superconformal duali- is their 5d field the 5d theories same counterparts. gives rise 5d Decoupling to superconformal another field dual theory. 5d Finally theories we whose propose UV the 6d descrip- from either resolving bothdifferent or ways one to of read the offsecond two a type 5d originates from gaugegroup different theory ranks distributions from of of essentially the D5 the 5dof or quiver same the theory. D7-branes, 5-brane web web The shifting diagram. diagram, last which the one The is gauge a comes part from of the the 90 SL(2 or 45 degree rotations arising from Type IIAdiagram brane along configurations a with compactification circle an planes. and The obtain gauge a theory 5-brane descriptionmal web of field the diagram theory resulting with is 5d two not5d theory unique, for gauge and a theories. we given There argue 6d are that superconfor- three the sources non-uniqueness which leads lead to to various the dual 5d dualities. One type comes Abstract: Hirotaka Hayashi, 6d SCFTs, 5d dualities and Tao web diagrams JHEP05(2019)203 38 9 5 ... 15 − T ) 45 m 3 − N 36 ] quiver 8’s configuration 14 N 11 D SU(2 [2 = 1 − − = 1 33 a + 8 T ) a ) N − N 8 N m 6 2 9 39 O − SU(2 N + 7 and + 8 and one tensor multiplet N ] quiver + 8 and SU(2 12 + 8 flavors N = − · · · − N [ + 8 flavors 20 53 − N 22 ) 46 f N T N – 1 – ) N N = = 2 15 47 = 2 m f f = 2 f − 22 49 28 N − · · · − f N SU(2 N ) N N − 2 N ) Tao theory N SU(2 SU( 24 34 37 29 − N 17 − 37 29 T T SU quivers ) ) SU(2 Tao theories N Q theory with N 6 − 51 T 0 ) − Sp( SU( N SU quiver theories 10 − − + 2) + 1) gauge theory with and + 1) theory with ] SU gauge theory with flavors Q Sp( 5 53 8] m N N × N , − 46 − ,T ) gauge theory with A ) gauge theory with 4 T N N quiver 4.2.4 Higgsed cases 4.1.4 Higgsed cases 4.2.1 5d4.2.2 SU quivers 5d4.2.3 SU quivers with an 5d antisymmetric dualities hypermultiplet 4.1.1 5d4.1.2 SU quviers 5d4.1.3 Sp 5d dualities 6.3 Generalization 6.1 Global symmetry6.2 from 7-branes 5d 5.1 6d [1 5.2 6d [8] 5.3 6d [8+ 4.2 6d SU quivers with an antisymmetric hypermultiplet 4.1 6d Sp 3.2 5d SU( 3.3 Equivalence of3.4 the two 5d S-dualities descriptions and3.5 and mass their parameters flavor Distribution decoupling duality 3.6 limits Chain of duality and exotic example 2.2 5d Sp( 2.3 Equivalence between2.4 Sp an SU S-dualities theories and their flavor decoupling limits 3.1 5d SU 2.1 5d SU( 7 Discussions A A brief introduction of Tao diagram 6 6d description of 5d 5 Special cases: the 6d quiver gauge theories of 4 5d dualities from 6d: generalization 3 6d SU( Contents 1 Introduction and summary 2 6d Sp( JHEP05(2019)203 . – 1 7 S = 1 + 2) N N ] as well 23 = 1 super- + 8 flavors is – N N 21 = 2 f + 2) gauge theory N N + 8 hypermultiplets in the ] by the analysis of the instanton N 31 ], which has been conjectured to = 2 0) SCFT is to study a 5d , 30 ]. f N 26 – + 2) gauge theory with = (1 N 24 N ]. The system consists of NS5-branes, D6- 29 , – 2 – 28 1 ] and also their torus compactifications [ 20 – 0) SCFTs have been analyzed recently, for example, the ], the elliptic genus computation of the self-dual strings [ ) gauge theory with , 6 14 ) minimal conformal matter theory and the 5d SU( N – + 8 flavors. 1 +4 = (1 N N . After the compactification, we obtain a certain 5d 1 ,D N S = 2 +4 f ) minimal conformal matter theory. One then compactifies the 6d the- N N ) minimal conformal matter was also obtained in [ +4 D N +4 ]. The quantization under a one-instanton background can yield instanton states. One + 8 hypermultiplets in the fundamental representation. The corresponding N ]. It starts from a brane configuration in Type IIA string theory developed 34 0) superconformal field theories (SCFTs) possess many interesting as well ,D – N , ,D 27 31 +4 +4 ], realizing a 6d Sp( N = 2 N f D = (1 D 29 , N The main aim of this paper is to generalize the connection between the 6d SCFT A new way to see a direct connection between a 6d SCFT and a 5d theory was dis- Another important way of analyzing a 6d N The same claim that the UV completion of the 5d SU( 28 1 ], the anomaly polynomials [ the 6d ( operator in the 5d5d theory. theories The [ instanton operatormay analysis construct is a Dynkin also diagram adiagram of useful is an way affine, enhanced to then flavor explore the symmetry 6d 5d from SCFTs theory the from is instanton states. conjectured to If have the a Dynkin 6d UV completion. between the 6d ( gauge theory with and the 5d theoryType to IIA a string broader theory constructed class in of [ 6d SCFTs realized by a brane configuration in Moreover, it possible to seewith the 5-brane web diagramweb yields diagram a has 5d a SU( in particular a feature class of of angive so-called infinitely a Tao expanding 5d web spiral theory diagram whose shape. introduced UV completion in Hence, is [ it a is 6d SCFT. This way gives us a direct connection covered in [ in [ fundamental representation coupled to ais tensor multiplet. the ( The fixed pointory of on the a 6d theory circleconfiguration with in Wilson Type lines IIA and string perform theory T-duality becomes along a the 5-brane circle. web in Then Type the IIB brane string theory. come instantons in the 5d theory.in In the other 5d words, theory, and the they KKSCFT become modes at massless are the at dynamically UV the generated fixed UV pointof fixed of 6d point, the SCFTs, namely 5d it theory. we is recover Although the typically this 6d difficult picture to is identify useful what for is the a analysis 5d description of a 6d SCFT on supersymmetric quantum field theorySCFT, whose we UV first move completion to a isexpectation tensor the branch value where 6d (VEV), scalars SCFT. and in then Given tensorWilson multiplets lines perform a acquire along a 6d a the vacuum circle compactificationsymmetric quantum with field gauge theory. or From this flavor point of view, the Kaluza-Klein (KK) modes be- Various aspects of 6d classification from F-theory [ 13 as compactifications on other Riemann surfaces [ 6d as mysterious features. Fordescription. instance, They they may have may tensionless self-dualStill, not strings they have as are fundamental an expected degrees ultraviolet of tounderstanding freedom. (UV) be of Lagrangian well-defined the local 6d quantum SCFTs field will theories. deepen Therefore, our a understandings better of quantum field theories. 1 Introduction and summary JHEP05(2019)203 . - 1 − S 7 (1.1) (1.2) O , ] , ] n n 2 + 8 + 8 N -planes and an [ N is the fundamental − − [2 R − theories, one by one 1)) ], we find that a circle − 1)) always yields a Tao web SU 27 n 1 − . When -plane after the T-duality. S R n − 7 + 8( O + 8( N N SU( The two descriptions, in fact, express -planes, we may split the both SU(2 3 stands for the anti-symmetric represen- -plane. As in [ − 7 − A 8 O − · · · − O -plane, and hence the T-duality induces two − 8 − · · · − ] to various 6d SCFTs that can be constructed – 3 – O gauge theory for two resolved 8 27 + 16) -planes or only one N − + 16) SU ]. The renormalization group (RG) flow triggered by 7 2 N O SU( -plane in Type IIA string theory. The 6d theories on the − − 8 hypermultiplets in the representation -plane. The D6-branes are divided into pieces by NS5-branes, , leading to the 5d theory which has the 5d UV fixed point. -plane, and thus yield different 5-brane web diagrams. On SU(2 O − k − 8 − + 8) 8 ]. As we have two ∞ The first type of the 5d duality that we argue comes from the O O N 35 means + 8) -plane can be split into two 7-branes by the non-perturbative effects SU( R − ] N 7 k -planes unresolved will not give a 5d description due to the explicit existence of the − O − ) 7 N SU(2 O − Sp duality. ) SU( is a positive integer and the subscript N − − n A Sp( -planes. An In the process of going down to 5d from 6d, we find that there are several different [1] gauge theory for an In our notation, [ Keeping two d − 2 3 6 d 7 6 It is also worthy notingpoint, that we given can these decouple dual flavors 5dby of theories sending one which the of have the the masses same 5d to 6d gauge UV theories, fixed say representation, we will omit it for simplicity. planes or only one oftwo them different into 5d a gauge pairSp theories, of 7-branes. an the other hand, sinceconfiguration, the these two two 5d theories web are diagrams dual arise to each essentially other from with the the same same 6d UV 6d completion. brane (i) SU choice of whether weThe resolve starting two 6d braneO setup includes an in the string coupling [ choices which yields 5dclaim theories that those with 5d differentUV theories gauge completion are as groups dual a and 6d tothis SCFT. matter. each 5d We other find duality. We, in that there however, the are sense three that different types they of have the the origin same for various nodes, depending onSCFTs the realized Higgsing. by We Type willcompactification IIA consider of branes such the with system two followed an largediagram by families T-duality leading of along to the 6d athe certain Tao 5d web diagram theory enables which one have to the read 6d off UV a fixed gauge point. theory description Furthermore, of the 5d theory. where tation. With theone sufficient may number Higgs of the thethe theories flavors Higgs at as the VEV in end [ theories yields node may a of have different the different 6d quiver ranks, theory theories, and at fundamental low hypermultiplets energies. will be The attached to corresponding gauge and and D8-branes introduce someworldvolume theory hypermultiplets on the in D6-branes the yieldIt a fundamental is 6d theory representation. possible on to theby The apply tensor the branch the brane of new a system method 6d withtensor SCFT. [ an branch of the 6d SCFTs we will consider are in the following two classes: branes, D8-branes and an JHEP05(2019)203 - − 8 O ) plane, p, q ]. However, this 31 gauge theories with theories, and then the -plane is located in the Sp -plane. In this case, it is ) duality. In particular, − Sp 7 − Z 7 , O O -plane induces a point-symmetric − ] which is an example in the class of 7 O 34 -planes maximally apart, the position of -planes and the transverse space to the ) and their Higgsed theories, we find that − − gauge theories and 7 7 ] only includes instanton states whose total 1.2 O O -plane is a real one-dimensional space. After 31 SU − – 4 – 8 O ) and ( -planes by introducing Wilson lines. In other words, − 1.1 7 where two -duality. After the 45 degree rotation in the ( ) plane is a part of the SL(2 O 1 S duality proposed in [ p, q The second type of the 5d duality arises from a choice of TST Sp The third type of the 5d dualities arises from utilizing the − -plane becomes two − -planes, the different distributions give a different 5d quiver theory SU 8 − 7 O O )-duality. Z , -plane, the fundamental region is either the left side or the right side of the ) duality in Type IIB string theory. The rotation of a 5-brane web in a real two- − Z 8 , -planes is a real two-dimensional space. Each From the 6d quiver theories of ( O − 7 of 5d theories whichnote have that the we same will 6dthose find exotic UV a 5d fixed sort theories point. oftype do “exotic” not when Furthermore, examples it show one after an is applies ais enhancement interesting the chain not of to of a instanton the contradiction the operator flavor since symmetry dualities. analysis the to performed Namely, analysis an in in affine [ [ another dual picture.rotation We by also 45 degrees find or thatthe the 5-brane there web is may another stillgives admit dual another a class frame 5d of corresponding gauge dual to theory 5d description. a theories. In thatcombinations case, of the this above rotation three types of the 5d dualities can give rise to various dual pairs (iii) SL(2 SL(2 dimensional space called theS-duality ( corresponds to theany rotation 5-brane by webs 90 degrees. obtained after Therefore, the the T-duality 90 from degree rotation the of 6d brane configuration leads to the duality explicitly at thedistributions level can of be the more 5-brane naturally web understood diagrams.pictures. as certain For Again, instance, mass decoupling deformations these flavors in different in theirin S-dual the the same sense way that as the (i) 5d above theories yield dual another to 5d each dualities other have the same 5d UV fixed point. of a line passing throughalong the the two circumference of the each D5-brane along thissplitting circumference the can two be distributedwith by different ranks proper of Wilson the lines.UV gauge fixed groups. After point Again, and the thus different they 5d are theories dual have to the each same other. 6d It is, in fact, possible for one to show O 5-brane web configuration with respecttwo-dimensional to plane. the Then, point the whereinfinite fundamental the mirror region lines is with a each regionpossible line bounded to passing allocate through by a different two different numbers parallel of D5-branes between the left side and the right side (ii) Distribution duality. how we allocate D5-branesan after the T-duality. Inplane the since Type the IIA transverse branethe space configuration T-duality, with to the the resulting two 5d theories shouldHence, be also we dual obtain to a eachthe 5d and same duality have the between flavors same 5d in 5dIn the UV particular, sense fixed the point. that 5d thedual dual pairs 5d that theories we have consider the in same this 5d paper. UV fixed point. In the same way, we decouple the flavors in the dual 5d theories, JHEP05(2019)203 - − 7 + 8 + 2) = 1. . In O gauge n 3 N N -plane. − Sp 7 , we focus O 5 ) with , we propose a 1.1 6 = 1 in section ) theory with 2 n Tao theory by matching can give a 5d SU( N 1 N S T , we first start the discussion ) with -plane become two sets of four Tao” theory. Although the Tao 2 − 1.2 8 N ] has appeared recently. T O 36 + 8 and one tensor multiplet N = 2 f – 5 – ) with their various Higgsings, we find that it is N 1.2 we give a brief introduction of Tao web diagrams. A Tao theory may not have a direct connection to a Type ) and ( N T -plane on top of each other after the Higgsing. An important 1.1 , we extend our analysis to 5d dualities for general 6d quiver Tao theory and we find that it is not generically written by a − 8 4 orientifold. The T-duality along the N ) and also the ones after various Higgsings. In section O T gauge node with an anti-symmetric hypermultiplet or an − -plane after the T-duality. The combination of four D7-branes with 8 1.2 − O 7 ]. We will call the 5d theory as 5d “ SU O 30 ] in terms of an O8 orientifold, D8-branes, D6-branes, and NS5-branes. ) gauge theory with ) and ( N 29 1.1 , theory [ 28 N -plane is special in the sense that the S-dual of this combination gives the same T − 7 -plane on top of each other. In this case, we find that the dual 5d theories can become While we are writing this paper, a related work [ The organization of this paper is as follows. In section As described above, we can relate various brane configurations in Type IIA string Among the 6d theories ( O − 8 Various brane constructions indone Type IIA in string [ theoryIn for this 6d section, anomaly wehypermultiplets free analyze theories in a were the circleconfiguration fundamental compactification has representation an of and a a 6d tensor Sp( multiplet whose brane 6d description of the 5d Type IIA brane setup. In appendix 2 6d Sp( exotic example. In section theories of ( on a specialO case where theanother Type class IIA that has brane not configuration been has obtained in eight the D8-branes other and sections. an In section the global symmetry as well as the number ofof certain the multiplets 5d between dualities the that twoWe arise theories. then from the move simplest onto 6dthis the setup case, next corresponding we to simplest ( will example see that the is appearance ( of the three types of the 5d dualities as well as an in Type IIA string theory.obtained The by 5d adding theories some from thatthe flavors 5d class at of the the end Taoweb web nodes diagrams diagram of can for the be a 5d generalIIA linear 5d brane quiver system, theory we realizing will propose a 6d description of the 5d node at either or bothNamely, ends these of special the 6d quiver, depending theories on yield the yet splitting another type class oftheory of an to dual 5d Tao theories. webanother class diagrams of in Tao web Type diagrams that IIB do not string arise from theory. the T-duality We from a also brane system note that there exists point in this case isD7-branes with that an the eight D8-branesan with the 7-brane configuration. Namely,plane we even can after the still S-duality. interpretquiver Due may it to have this an as feature, four we find D7-branes that and the resulting an S-dualized 5d the enhancement of the flavor symmetrywith to the an higher affine instanton type number. when we include instanton states particularly interesting to focus oneight a D8-branes case with where an the Type IIA brane configuration includes instanton number is 1. Therefore, we argue that the exotic 5d theories should also show JHEP05(2019)203 ) = f p, q N ) theory: n ) minimal +4 N ) gauge theory flavors and the , as depicted in N +16) ,D − f theory with N + 1) gauge theory 8 +4 N (4 + 8 flavors and one 0 O N N D6 branes (its mirror SO D N 0) Sp( , + 2) N = 2 N = (1 f N N ) minimal conformal matter in the ) N +4 ( N ] for Sp ,D + 1) theory with the same number of 29 , +4 + 8 flavors N 28 N N ], the 5d global symmetry has an additional D + 2) gauge theory with 27 , and the other is 5d Sp( N = 2 – 6 – + 8) D8 branes on top of f 2 f N or 5d Sp( N ) gauge theory with N 4 − . N κ ) 5-brane web diagram, the resulting 5d theories also NS5 ], However, we claim that the same brane configuration ] that the superconformal indices for seemingly different + 4 p, q 27 34 D6 N 0) Sp( N , 2 = +8)D8 N κ = (1 theory with together with (2 0 orientifold planes in Type IIB string theory. As the decoupling limit N O8+(2 − − 8 gauge theory with the same rank and the same number of flavors. There 7 O 6 + 2) O x + 8 flavors and a tensor multiplet consists of 2 ) theory of the CS level 9 Sp , NS5 n 8 , N 7 N x = 2 flavors. Thus, it implies that the two theories are dual to each other. We here . Left: type IIA brane realization of the ( + 8 flavors as shown in [ + 16). We note that, as discussed in [ f f is an SU( factor associated to the KK modes. We discuss the two 5d theories using ( N N N κ N I ) Consider the 6d + 8 flavors and the zero CS level, From here on, we use the following short hand notation for denoting the CS level of the SU( n 4 N with images included) suspended betweenorientifold a plane NS5 brane and its mirror NS5 braneSU( through an 5-brane web diagrams in details. 2.1 5d SU( In Type IIA description, 6d brane configuration [ Depending on the choice2 of Wilson lines, weflavors. either Although get the 5d introduction SU( symmetric of phase, the both Wilson have the lineSO(4 same breaks global the symmetry global asU(1) symmetry, that in of the its 6d “mother” theory, both pictures. Depicted inhave an the S-dual ( picture. tensor multiplet. The UV fixedconformal point for matter this theory theory. is known as We 6d compactify ( this theory on a circle to obtain 5d theories. with attempt to give an accountthe for resolution this of equivalence from theof perspective flavors of yields different pattern anotherfor of 5d resulting theories 5d of theories 5d with UV less fixed flavors point, if such one a decouples dual exactly picture the still same holds flavors in the with 2 also yields a 5d was an interesting observation [ 6d theories are equivalent:Chern-Simons (CS) one level is 5d SU( Figure 1 tensor branch. Right: the quiver diagram of the 6d theory. JHEP05(2019)203 − 7 can + 2) O for a − N C with a 7 O 1 S planes, the -planes into − − ) gauge group 7 7 = 1 SU( , on planes which are N O 5 O N x − 7 O 1] 7-brane, and − and the vertical direction , ]. For instance, 6 + 8) D7-branes and two x 35 N ) gauge theory with one tensor for a [1 N B 7 8 9 × × × × × × 6 × 0) Sp( C B , = (1 C B -planes may break the 6d Sp( A N − for a D7 brane, = 1. Left: the brane configuration with two – 7 – 7 A 1] 7-branes. As one resolves the O N − 0 1 2 3 4 5 × × × × × × × × × × × × × × × × × × × × × × × × , . 1 1] and [1 , O8-plane D6-brane D8-brane NS5-brane D5-branes, two NS5-branes, the (2 plane. Middle: the brane configuration with two N − 7 O ] 7-branes subject to the monodromy condition that the monodromy of + 8 flavors in the fundamental representation, which yields plane D5-branes away from the plane p, q - N - - - N O7 O7 ). The non-perturbative effect in string theory can resolve the A . Type IIB brane descriptions for 6d N -plane with four D7-branes is the minus of the identity [ = 10 flavors and zero CS level. . The brane configuration in the ten-dimensional spacetime in Type IIA string . The brane configuration that yields a 6d theory on a tensor branch of a 6d SCFT in Type − f 7 1 N -planes which are maximally separated apart along the T-dualized circle. The Wilson To have the 5d picture, we compactify one of the worldvolume direction of the NS5- O − 1] 7-brane. Right: pulling out 7-branes gives rise to a web configuration for SU(3) gauge theory 7 , line putting to 5d U( a pair of twothe [ be resolved into a pair of [1 brane that are parallelWilson line, to and the then worldvolume takeconfiguration of T-dual involves along D6-branes, the for compactifiedO direction. example The resulting 5-brane [1 with figure theory is summarized in table multiplet and 2 gauge theory with theis number the of flavors. T-dualized direction. Theplanes. horizontal A For direction dashed simplicity, is linereflection denotes cut the of branch an cutnon-perturbatively of resolved. 7-brane, Here while we denote a dash-dot line denotes the mirror Figure 2 Table 1 IIA string theory. JHEP05(2019)203 ]. 30 , 27 or [ ] and is named + 8 D5-branes 27 A N .) 2 = 10 flavors which is a circle f + 8 flavors. N N ]. (See figure = 2 37 f N + 2 color branes, implying that the rank of N – 8 – can be reorganized to be the leftmost web configuration 2 gauge theory with = 1 SU(3) gauge theory with 0 + 8 D7-branes give rise to semi-infinite 2 N 0) Sp(1) gauge theory with one tensor multiplet and 10 flavors. The N , + 2) N = (1 ) 5-brane configuration with 7-branes and 5-brane loops, one has a N p, q . This period can be expressed in terms of the parameter of the 5d theory, , confirming the zero CS level of the 5d theory. Therefore, the resultant 3 = 1 SU( 2 N . Tao diagram for 5d The web diagram for this case was already studied by the authors in [ Given such a ( spectrum of the compactification. For a short introduction, see appendix 5-brane loops has the necessarythe structures 6d to theory. be It is interpretedas of as given the a in shape figure circle of compactification anthe of infinitely instanton factor, repeated which spiral is web an expected withThe property a spectrum for constant coming a period, from circle compactification infinitely of expanding 6d 5-branes theory. are naturally identified as the KK and the angle between theshown two in lower external figure 5-branes are5d the theory same is as the one between as as a Tao web diagram. Such Tao diagram obtained by pulling all the 7-branes outside the resulting 5d gauge theory inthe this resulting process 5d has gauge theorytensor is multiplet increased contributes by the one. 5dgroup. vector For this, multiplet It one giving is can one worthcan also more noting be argue rank that that to measured the the the as obtained 6d of 5d the 5d NS5-brane gauge angle in theory difference the has web of zero configuration, two CS the vertically level. angle inclined between As 5-branes the the playing two CS upper a level external role 5-branes the gauge coupling finite asas pulling a out result all of theexternal the 7-branes. 5-brane. Hanany-Witten 5-branes effect Hence, are suchyielding naturally 2 that the generated a flavor 7-brane branesthe is of web attached the diagram, to theory. the 5-brane Due end loops to of induce the an two charge more conservation color at D5-branes. a junction This of means that the fundamental domain of the 5-branelines” web is diagram that expanded is towith confined the by the “the two-dimensional mirror mirror plane reflection reflection lines and forms the “5-brane boundary loops” made [ 5-dimensional by gauge NS5 theory description branes by pulling all the 7-branes to infinity, while keeping compactification of 6d rightmost brane configuration ongiven figure here, using 7-braneother monodromies. 7-branes gives a Pulling Tao out wed 7-branes diagram on passing the through rightmost. the branch cut of Figure 3 JHEP05(2019)203 . - ) 0) − n 4 , 7 = 1 + 1) O N N = (1 N 1] 7-branes plane − - , = 1 Sp( - O7 planes is resolved. N plane if the Sp( + 2) gauge theory 1] 7-branes. -plane. Both have , − − 7 N − ) gauge theory. This 7 7 O 1] and [1 n O O , +8. The increase of the ) theory. In this process planes which arise from N N − gauge group. As discussed -planes, and the other leads 1] and [0 7 , − ) gauge theory with one tensor = 2 O 7 Sp N f O N = 1. Left: the brane configuration with 0) Sp( , N B + 8 flavors = (1 plane - - ) gauge theory has flavors, the effect of the N = 10 flavors. plane with a pair of [1 N n planes leads to the 5d SU( O7 f − C – 9 – A N 7 − = 2 7 O plane, to retain an f O − N + 8 flavor and one tensor multiplet in two ways with -planes as well as the Wilson lines: one of which leads 7 − O N 7 O plane into a pair of two 7-branes induces an extra color D5 = 2 − ]. Therefore, one can choose whatever splitting of the + 1) gauge theory in the Coulomb branch is given in figure f 7 38 O N N , there is a discrete theta parameter for Sp( 2 plane in Type IIA brane picture of 6d Sp( Z − 8 + 8 flavors in the fundamental representation, which yields + 1) theory involving the resolution of only one O + 1) theory with +2) theory involving the resolution of two )) = plane N n plane N - N N - - - O7 O7 (Sp( A + 8 flavors, one may resolve only one of the two planes. Middle: the brane configuration with only one of two . Type IIB brane descriptions for 6d 4 π − N 7 +1) gauge theory with the same number of flavors, ) gauge theory with As O N N respect to the resolution ofto the the 5d SU( to the 5d Sp( the same number of flavorswhich as come the from 6d a theory, T-dual as of the D8-branes flavors in are the associated brane with picture D7-branes of the 6d theory. Note that as chosen earlier, and one can also resolve it2.3 with a pair of Equivalence [2 between Sp anIn SU the theories preceding and subsections, theirSp( we flavor have decoupling discussed limits a circle compactification of 6d parameter can be interpreted asgauge two inequivalent theory resolutions has of no the discrete flavors. theta When parameter the candifference Sp( be is absorbed not physical byplane. [ a For example, redefinition one of can resolve mass the parameters and the brane when 7-branes areSp( pulled out to infinity.rank of Hence, the gauge the group resultingbrane can 5d setup be for theory again the due is to 5d the Sp( contribution of one 6d tensor multiplet. A with 2 T-duality of the of reducing the 6d theoryare to located 5d, above the the Wilsonearlier, unresolved line a is resolution of introduced the such that all the D5-branes two Right: the resulting Sp(2) gauge theory with 2.2 5d Sp( While the quantum resolution of both Figure 4 multiplet and 2 gauge theory with the number of flavors. For simplicity, JHEP05(2019)203 5 ], 34 +1) N + 2) gauge N , we can take . In such Tao ) minimal con- 4 5 +4 = 10 flavors, and + 16) is the global N . f N ,D N −∞ SU(2) quiver theory with +4 × N or D ∞ +2) theory and the Sp( + 2) gauge theory. where SO(4 N N I U(1) SU(2). For instance, the brane config- [4] quiver theory which has four flavors × + 2) gauge theory. Therefore, we should × − N = 10 flavor is SU(2) + 16) f – 10 – SU(2) N N × for the Sp( has two “SU(1)”s linked to each SU(2) instead of four flavors. comes from the circle compactification. SU(2) 5 ∞ +7 which have the 5d UV fixed point, was computed [ I describes a 5d SU(3) theory with − N 2 2 ≤ S-dual + 1) gauge theory. For example, from the figure f N N + 7 can be easily explained if one considers the flavor decoupling of the ) 5-brane web by 90 degrees, namely, D5-branes become NS5-branes and N 2 p, q ≤ ) 5-brane web diagram, the S-duality structure of the theory is manifest as one f . The S-dual of SU(3) theory with p, q N + 2) theory should be either maximum or minimum. This comes from the fact that We note that in order to hold the 5d dualities for the less flavor cases, the CS level of Interestingly, the superconformal index for the 5d SU( As both have the same UV completion as the 6d SCFT, ( N The middle configuration of figure = 4 flavors to each SU(2) theory of the quiver. 5 f then its S-dual theory is(denoted the [4] in the square bracket “[ ]”) onSuch SU(1) each has the gaugethe instanton group factor other which hand, as can using in bea 7-brane bi-fundamental used monodromy figure hypermultiplet to analysis, connecting express it between the can the period be SU(1) of shown and the that SU(2) Tao such also web SU(1) represents diagram. two factor On flavors. together with In the ( rotates the ( NS5-branes become D5-branes.theory A which simple yields example theuration quiver would on theory be the of the right SU(2) S-duality of of figure the SU(3) theory and the 5dthe Sp( mass of the flavorsalso only take to the + same flavor decoupling limit for2.4 the 5d SU( S-dualities 5d theories from thedeforming critical the flavors. mass of Namely, a one flavor can to take an the extremeSU( value mass of decoupling either limitwe by need to decouple the flavors exactly in the same way between the 5d SU( or elliptic genera, of both theories do agree ortheory not. with less flavors showing the equivalence of theflavors two indices. In the web diagram, this equivalence with the symmetry of 6d theory and U(1) formal matter theory, and thus theare same dual global symmetry, to we claim eachwould that other be these at two equivalent. theories the For UV. instance, Hence, it some would physical be observables interesting like to index see functions the partition functions, Figure 5 N global symmetry for both theories is SO(4 JHEP05(2019)203 − 7 ; (b) O (2.1) 1 2 + 8 fla- N = ) gauge theory f (c) ]. This S-duality N N 27 + 8 flavors, + 8 flavors, N N = 1 + 2) gauge theory with planes or only one [4] quiver theory. a = 2 N = 2 − − f N 7 f N + 1 SU(2) gauge nodes and 4 O N . N = 9 flavors and the CS level SU(2) [4] f − − N SU(2) quiver theories are all dual to } + 8 and × ) gauge theory with SU(2) N SU(2) N − plane = - (b) - {z ], one can decouple flavors of a given Tao di- f theory with + 1 nodes O7 – 11 – 27 − · · · − 1 2 N U(1) anomaly free global symmetry. N × SU(2) quiver theory. The resulting flavor decoupled + 2) gauge theory with = 0; (c) [3] + 1) gauge theory with × k SU(2) | N + 8) N [4], a quiver theory with − + 1), and the SU(2) N − [4] N SU(2) As discussed in [ + 8 ], SU( + 2) gauge theories, we can decouple the same number of flavors. + 8 ], Sp( N N N = 9 flavors and + 1), Sp( , as discussed, the constant period of spiral shape is expressed in terms [ 2 f [ 2 2 N N − · · · − − − ) gauge theory with + 8 and one tensor multiplet, on a circle: (a) N = 1 hypermultiplet in antisymmetric representation together with one tensor N + 2) + 1) a SU(2) + 8 flavors is given by the SU(2) quiver theory N N + 1) or SU( . Mass decoupling limit. (a) SU(3) N = 2 − N f N flavors are attached to each boundary node. N We summarize the 5d theories obtained by compactifying the 6d Sp( [4] Sp( SU( = 2 (i) f (ii) (iii) 3 6d SU( In this section, wevors and consider six-dimensional SU( multiplet. This theory has SU( agram, which yields 5dIn supersymmetric Sp( gauge theoryThe which same has idea applies the totheories 5d the from UV SU(2) SU( fixed point. each other, and are expected to have the same 5d UV fixed point. The (i) and (ii)plane are when reducing obtained to by 5d, resolving and (iii) eitherDecoupling is both S-dual flavors. two of (i). Therefore, (iii) is also (S-)dual to (ii). with The period for these Tao diagramof is each expressed gauge in terms node. of a product of the instanton factors of the instanton factorperiod of is the expressed gauge in theory.generalizes terms to of It higher instantons holds rank factors for cases ofN the so the that S-dual quiver the theory picture, S-dual [ the of constant the 5d SU( Figure 6 Sp(2) theory with diagrams of figure JHEP05(2019)203 ) ], N 39 and = 1. , we 0 a 2 -plane n N − 7 O . Depending planes based 2 − + 8 and 7 , where we chose U(1) actually en- O N 8 × = (N+8) f N SU , there is an additional D5-branes are placed on = 3, the anti-symmetric n 1 N . Accordingly, the charge of direction. Compared to 1 1 S (N) S SU = 4, anti-symmetric representation N planes, where A − ) gauge theory with depends on the Wilson line of the SU( 1 7 N = 1 and with one tensor multiplet is realized n O a N – 12 – . Compared to figure 7 + 8, N -plane(s), we obtain two different 5d descriptions. We planes depends on this distribution. Likewise, the D7- NS5 − = 7 − 7 f O plane. Moreover, odd number of D6-branes are also allowed ) D5-branes are place on the right hand side of this 5-brane. O planes here are attached to the 5-branes. As discussed in [ N D6’s n − − 8 N 7 − NS5 O O (N+8) D8’s(N+8) N - ) 5-brane connecting two SU(2). The discussion below is similar to that in section O8 + × p, q SU gauge theory with flavors is small, there can be the enhancement. When × , because the N . 0 U(1) actually enhances to SU(12). When 2 ) gauge theory with . Left: type IIA brane set up for 6d SU( n NS5 × N − ]. The decomposition in this case is, however, more non-trivial than that on discussed Here, we would like to consider the quantum resolution for one of the We compactify one of the directions which NS5-brane extends and take T-duality along When 35 + 8 on [ in section one needs to properlyone choose of the the charge 7-branes of should the be resolved attached two to 7-branes the in 5-brane such that a is way attached that to the the left hand side andWe ( note that thisgauge “distribution” group labeled which by wethe introduce 5-branes when attached we to compactifybranes the which on are T-dual ofN the D8-branes are also distributed in analogous way into in this case. this direction. Then, wethe obtain fundamental region the to IIB behave brane the additional setup half ( of depicted the in compactified figure 3.1 5d SU 6d SU( by the brane setNS5-brane on up top depicted of in the figure is the real representation.on two Therefore, half we hypermultiplets. can Inhances consider this to the case, SU(12) the SU(2) global globalon symmetry symmetry resolving SU(12) acting both oralso only discuss one various 5d dual descriptions corresponding to this 6d theory. representation is equivalent to theory (anti-)fundamental is representation. interpreted as Therefore, SU(3) thisSU(11) gauge the- theory with 12 flavors. In this case, the global symmetry Figure 7 Right: corresponding quiver diagram. JHEP05(2019)203 .) Then Once the 1] 7-brane 10 6 − , 1) 5-brane as − , = [1 1] 7-brane is denoted . We note that a 5d B − , B 10 1) 5-brane. -planes. − 1] 7-branes as the total mon- − , . − 7 , 7 O C 1) 5-brane which is connected -planes generates new 5-branes 7-brane along the direction of − − D5 , 7 ) B n 1] and [1 O D7 − ) 1) 5-brane. A [1 0 , N n ( − D5 plane is attached to (1 , NS5 − +8 7 N ( O B - C . By moving - – 13 – -plane into [3 7-brane becomes fee so that it is not attached to O7 9 − plane can be decomposed into O7 7-branes. 7 . − B O C 7 C 7-branes are separated by the NS5-brane as depicted D7 O 0 7-brane is attached to the (1 D5 n . A T-dual description of figure and C -plane. In this case, however, one can rearrange the 7-branes so as n -planes are resolved into two 7-branes and one of which is B plane attached to (1 − B − 7 D5 , the 7 − O and 7 , the quantum resolution of 9 O NS5 O 2 B Figure 8 - . This kind of transforming procedure is useful in obtaining various (1,-1) O7 9 , where two 1] 7-brane is denoted by 1] 7-brane so that , 10 , . Resolution of the 1) 5-brane using Hanany-Witten transition, one can transform the configuration = [1 − , 7-brane as in the middle of figure and a [1 C -plane is resolved, 7-branes induce new configurations. Imagine, for example, a 5-brane As discussed in section It is also possible for one to resolve the B − B 6 7 it leads to a 5-branefield web diagram theory depicted description on is the possible right side only of when figure the newly generatedodromy 5-branes is are the D5-brane, sameto as yield that the of same an configuration with side of figure attached to a 5-brane is7-brane moved as along described. the direction To of obtainone the a can charge 5d move of gauge around the theory one 5-brane descriptiontoward outside of to for the this be the 5-brane brane a D7-branes loop. configuration, free passing (see through the D5-branes arrows as attached to well D7-branes as in 7-branes figure the 5-brane. As aon the result, right of figure five-dimensional (5d) field theory descriptions after resolving to make a closed 5-brane loop. The resulting brane configuration is depicted on the left and O junction composed of an NS5-brane,to a D5-brane and athe (1 (1 into a brane configuration where the Figure 9 by before the resolution. Fordepicted example, on when the left an of figure JHEP05(2019)203 10 + 7 + 7 and (3.1) D5 n ) N 0 N . This n -planes − = ]. = 7 11 +7 f O 39 N f ( N N , 1) ] , 2 n 1) . With a suitable ( D5 , 3 1) N +1 ] charge of the D7- − n B n 2 = 1 − p, q a N D5 N ( + 3 ( .” We will discuss this N +4 N n N 1) D5 [2 n, 2 D5 = 3 C − , then we get the configuration 1) 0 N 1] 7-brane which is essential to 0 ( gauge theory with 8 n -plane out of the two n , ) ( gauge theory with should have the identical 6d UV − 1) n 0 , + 7 and 7 = 1 and with one tensor multiplet 0 (2 n − O should be chosen in such a way that a 1) N , + 1) 1 N n + 1) + 1 = D5 n N . 2 1) f is resolved. Right: web-diagram obtained by N N distribution duality -plane from 0 N + 8, n 8 ( N ( − 3.5 7 SU( = 1 and with one tensor multiplet compactified N – 14 – O a − = N 0 f 7-branes, which convert the [ , are constrained by 0 N D5 n + 1) C + 8, 1) D7 1) n , N D7 n ) +1 0 planes in figure and n n = B 2 SU( N − ( f B 7 +7 − N ) flavor group, the brane configuration on the right of figure N ( ] O N f ( , we can decompose only one N N . As discussed in the previous subsection, we can take a D7-brane 1) 1]. This leads to the configuration on the right of figure , − 1 2.2 , is also described by 5d SU( C 11 n 1 2 + 3 + 1) gauge theory with S D7 = [0 -plane. Notice that the number of fundamental hypermultiplet is reduced N n ( = 1. = 1 antisymmetric hypermultiplet coming from the 5-brane connected to 1) ) gauge theory with − N [3 a 7 a N 0 N D5 n ( D7 N O N 1) . Left: the two . Suppose we resolve the upper n ( 8 . In other words, the theories with different ) gauge theory with 1 The resulting 5d description is a 5d SU( N S , we claim that any resulting 5d quiver gauge theory is a possible 5d description for 6d by one, since wehave a used brane up configuration a withthat D7-brane an 6d antisymmetric to hypermultiplet. SU( convert Therefore, tocompactified we a on conclude [0 flavors and brane to be configuration corresponds to adding the flavor branes to theflavor one discussed and in [ unresolved 3.2 5d SU( Analogous to section in figure on the left ofthrough figure to locate between on fixed point. Wedistribution denote duality more this in equivalence detail as in section “ where as the distributionall parameter, the flavors the and integer ranksn of the gauge groupsSU( of this theory should be positive. With such the number of the flavor D7-branes Wilson line for thegives SU( rise to the 5d quiver gauge theory Figure 10 moving all the 7-branes outside. provided that the two distribution parameters, the number of the color D5-branes JHEP05(2019)203 ) R n N (3.2) (3.3) D5 ) 0 n and . ] L R +7 n n ]. By giving N ( . flavor branes n , respectively. − 3 1) R R , n = 1 flavors from the 2 = 1 n − n 3 . This also shifts a ( D5 a R − N − n +3 11 N + 1) n ]; and B n 3 n N L 3 + 3 − N D5 n ( N − N ) and + 7 and + 3 R [2 D5 n + 3 N = 1. N − - ) to be shifted by D5 a C + flavors and = n R N n O7 n N L f [2 ) − N n N n 1) ( and 2 , − − (2 L − 0 − + 1 ) n n + 4 N − + 1 + 7 + 7 and − D5 n flavors from the brane configuration, which 0 N flavors out of the left flavors of the quiver n N N N SU( R − L = 3 + 1 n = = 0 × n SU( f and hence reduces the theory to be n – 15 – f N + 3 − N N and R n L n + 1) gauge theory with L SU( n n 0 n − D5 − = 1 and one tensor multiplet, which are summarized as with 1) L D7 0 a + 1) n + 1) D7 n ) N 0 R n flavors out of the right flavors of (i), [2 n n N , let us take B + 1) N − R ( n L flavors, corresponding to moving these n +7 SU( + 8, 10 n N R ( − N n SU( ] ) obtained from the flavor decoupling should have the identical 5d L + 1) = . 5d SU( + 1) to be − flavors from the left - ], and n gauge theory with ] f C and N 3.3 L 0 N O7 N , we can decouple − N N n L − n N flavors of the brane configuration on the right of figure − 10 + 1) 0 +3 − n D7 0 D5 n N ) and ( + 3 Figure 11 n 5d SU( − n n + 3 3.2 n + 7 N In a similar fashion, we can decouple some flavors of (ii). Say, we give the positive For instance, in figure 5d SU( 5d [3 5d [3 (i) (ii) UV fixed point, giving rise to another3.4 non-trivial 5d duality. S-dualities andBy mass using parameters S-duality, we canpoint. obtain In more this 5d subsection, descriptions we which consider have S-dual the description identical for UV the fixed quiver theory discussed in the CS level of SU( As the way wetheories decouple ( the flavors from (i) and (ii) are same, we expect that two 5d infinite masses to right theory (i), [3 heavy masses to upward in figure reduces the left and rightThis flavors to also be alter 3 therespectively. CS After levels the of flavor decoupling, SU( we get These two descriptions areFurthermore, dual in by the considering senseparameters that the to they be flavor have the infinity, decouplingthe same we identical limit 6d will 5d UV again fixed by fixed point. obtain taking point. the some dual theories, of which the then have mass again In the previous subsections,gauge we theory discussed with two different 5d descriptions for the 6d SU( 3.3 Equivalence of the two 5d descriptions and their flavor decoupling limits JHEP05(2019)203 ) 3.1 (3.5) (3.6) (3.4) by a [6]. 12 − 0 1] 7-branes , . In this mid- . SU(4) 12 12 . − 0 12 , SU(4) [3] , − − [5] − . 0 SU(2) [6] ). In this way, we can obtain several − − 0 0 ), but for convenience we consider the 3.5 SU(3) | 3.4 − [1] 0 1] 7-branes along with the Hanany-Witten SU(3) SU(4) = 3, then the 5d quiver gauge theory ( − , − − m 0 0 – 16 – SU(3) | − [1] 0 1) 5-branes, NS5 branes, and D5 branes, respectively. SU(3) SU(4) , which is related to the left diagram of figure − − − = 6 and , 13 , we obtain the middle diagram of figure SU(3) [6] N − 12 SU(2) [5] , we first consider the S-duality transformation exchanging D5- 1] 7-branes and [1 − , 2.4 [3] 1) 5-branes into (1 . S-transformation of the 5d quiver gauge theory [6] , . We discuss the dual descriptions with a specific example, and generalization 3.1 Figure 12 Here, we can consider another type of S-duality, which transforms D5 branes, NS5 Parallel to section which is different from the usual5d S-daulized descriptions theory by ( different types of S-duality transformation. of the diagram. Weweb start diagram from on the the same left theorysimple of ( Hanany-Witten figure transition to moveNS5 the brane right inward. top After andupward taking left or downward, bottom the we D7-branes STS obtain to transformation the the and 5d then quiver gauge moving theory [0 where the corresponding web diagram depicted on the rightbranes, of and figure (1 We denote this as “STS transformation”, which roughly corresponds to a 45-degree rotation to the left diagramdle on diagram, figure we movetransition, [1 then we obtain the following 5d quiver gauge theory The corresponding web diagram is depicted on the leftbranes diagram and of NS5-branes to figure eachgram. other, which We corresponds denote this to type a of 90-degree S-duality rotation as of “S-transformation”. the Acting dia- the S-transformation section is then straightforward. Take becomes JHEP05(2019)203 or 12 [6]. − 0 SU(4) ). Analogously, − 3.5 0 SU(4) ) with different values of − , which describe the same 3.1 14 . It is worth emphasizing that 3.6 ) rather than ( 3.6 ) rather than – 17 – 3.5 . Same theory with different mass parameter Figure 14 , we discussed that 5d quiver gauge theories ( . STS-transformation of the 5d quiver gauge theory [6] 3.1 , we could have started from the diagram on figure 13 We remark here that a given gauge theory may have web diagrams withe different Figure 13 have an identical 6d UV fixed point, referred to the distribution duality. We devote 3.5 Distribution duality In section n ourselves in this subsectionviewpoint of to type discuss IIB the 5-brane distribution web diagram. duality in more details from the the S-transformation yields 5dthe quiver STS-transformation gauge gives the theory theory ( ( depending on the valuecan of be the mapped mass to parameters different of gauge theories the under flavors, the the identical same S-duality 5d transformation. gauge theory figure theory but with differentleads mass to parameters. an Thedescription interesting existence is yet of identical another up different dualresults to web under description. the configuration the value S-duality of Namely, transformation. the although For mass the instance, parameters, original it one is 5d may straightforward obtain to different see that ranges of mass parameters. For example, instead of starting from the left of figure JHEP05(2019)203 . ) ) n 16 3.1 − (3.7) ] +1 2 N n − ] flavors coupling + 3 2 2 n n . n ] 3 , it is also straight- − n − 3 + 2] n 3 and 3.3 n − N 3 1 − [2 ] n N − n N − [2 3 N 2 [2 by 1 in the quiver gauge the- − n − [2 ) − 0 n n ) N 2 − n n does not depend on the upper part − 1 2 [2 ) − n − − ) 15 0 n + 1 − N , which describes the 5d quiver gauge ) n − N N SU( 15 − − + 1 SU( SU( N 0 − − N – 18 – 1 1 SU( n n 1] 7-brane painted in red along the arrow in the + 2) SU( , − n 1 2 − . We can also consider another type of flavor decoupling + 1) + 2) 0 − n n SU( 15 . Derivation of the distribution duality. to downward. Then, we obtain the left diagram in figure − are dual to each other as long as all the numbers of flavor + 1) + 2) ] SU( SU( . In this process, the charges of 5-branes in the lower part of n 15 n n N − − ] ] 15 − 1 1 SU( SU( are related by the distribution duality. n n − − Figure 15 n − − + 6 ] ] n N N N N [3 − − − − + 3 + 5 + 3 + 6 ). Then, we move the [0 n n n n 3.1 ) with an arbitrary ). The repetition of this procedure yields that the quiver gauge theories ( 5d [3 5d [3 5d [3 5d [3 3.7 3.1 Up to here, we have discussed the flavor decoupling limit, which corresponds to moving As an example, suppose that we move one of the flavor branes coupled to SU( By considering the flavor decoupling limit discussed in section We start from the left diagram in figure • • • • 1] 7-brane. This changes the structure of the web diagram so that the resulting diagram , downward in the left of figure By considering the analogous procedure, we find the following two are related all the possible value the flavor branes upward in figure limit corresponding to moving flavor branes downward. analogous procedure We can further consider moving the flavor branes upward. Thus, this class of theories with forward to show that this distributionUV duality fixed holds points. also for Since the theof theories process the with depicted diagram, 5 in dimensional the figure discussionto is the parallel two even gauge after we groups. decouple We see that the following two theories are again related by the We find that thisory theory ( is the oneand obtained ( by shifting and the rank of the gauge groups are positive. leftmost diagram in figure the diagram are modified[0 because they went acrossgives the rise monodromy to cut the generated following by 5d the quiver gauge theory theory ( JHEP05(2019)203 ] 2 n 17 (3.8) − k ] − 2 may look : n − 17 + 3 17 n n 3 3 , or − − 16 , N N + 1] [2 [2 15 n any more. However, this 3 − − ] 2 n 2 k − n n is translated to moving the ) − 3 2 N n n − 17 ) [2 − . n N − N [1] − 1 [2 − − ) − SU( n + 1 0 ) − − n N 2 k SU(2) − − 1 + 1 SU( 1] 7-brane as in figure − n N , − N – 19 – 1 | SU( n 1] 7-brane in figure [1] + 2) , SU( SU(3) n − − 1 + 1) − 0 SU( n · · · − − . Derivation of the distribution duality 3. . Derivation of the distribution duality 2. ] + 2) + 1) SU( 1 n n n − ] − 1 SU( SU( k n − − − − Figure 17 Figure 16 ] ] N N N N − − − − + 4 + 6 + 3 + 3 2. , flavor branes upward. Therefore, the discussion here can be summarized as the n n n n 1 , 2 n 5d [3 5d [3 5d [3 5d [3 = 0 The deformation of moving the [0 Although the processes to move the [0 It would be also straightforward to see the following duality from figure It would be also possible to combine them with the flavor decoupling limit to move k • • • • and 1 D7-brane. This D7-brane corresponds to the flavor charged under the SU(3) gauge group for exotic, it looks oftenas quite an natural example when and wediagrams consider correspond see the to this the S-transformation in identical for the 5d both. S-dual theory frame. in Then, the Take we form figure observe that both n following duality: Contrary to the previous case,duality they keeps are the not total related numberof by of CS shifting levels. flavors, total In number this of sense, gauge this ranks, is and also total the analogue number of the distribution duality. JHEP05(2019)203 , 16 = 1 , (3.9) a 15 N = 14 and f N [6] [3] 00 1] 7-brane as in figure − − , 0 0 , we demonstrate that varous 5d “SU(1) SU(2) − SU(4) SU(3) 3.5 − ? − − 3 2 0 are actually connected by such process. 0 0 XI SU(6) SU(7) SU(4) SU(5) − − − − – 20 – and section , we write various web diagrams, which are related that depending on the value of the mass parameters 3.4 : [6] : [9] : [12] : [12] 19 are also related by the distribution dualities. X 3.4 and diagram III VI , XII , , , X II IX V VIII , , , . Distribution duality in the S-dual frame XI I factor and thus, does not have the standard field theory de- ↔ IV 00 VIII , VI VII and Figure 18 IV ↔ II , we can show that diagram In the original frame, we see that We again consider the 5d theories corresponding to 6d SU(6) with 17 The last one includes “SU(1) scription and moreover, does not lookHowever, by fit using with the the analogous distribution processor duality to discussed move one previously. of the [0 We note that to each other eitherinterpreted by also mass as deformation the or mass the deformation distribution in the duality, which S-dual is frame. sometimes Combining what we studied inquiver section gauge theories have identical 6d UV fixed point. and with a tensor multiplet. In figure the identical S-duality transformation. 3.6 Chain of duality and exotic example and moving it“distribution simply duality”, corresponds which change to theoriginal distribution changing frame, of is the the actually rank mass just of thethe of the mass same observation deformation gauge this seen in group in the flavor. in section S-dualof the frame. the Therefore, This flavors, is the the essentially same 5d gauge theory can be mapped to different gauge theories under JHEP05(2019)203 X IX III VI – 21 – . Chain of dualities. Figure 19 XII II V VIII XI I VII IV JHEP05(2019)203 2 and (4.1) N , 2 ] n < f + 8 = 1 falls in to (3.10) N N n ] to some of these [2 [3] [3] [3] 31 − SU(2) − − − − 1 1)) 1. The global symmetry − | [5] [3] [1] n SU(2) SU(2) n > − SU(3) SU(2) − − − 0 − − + 8( . The case with ]. It is straightforward to repeat N [5] 20 SU(2) SU(3) SU(2) SU(2) − 29 SU(2) SU(2) − , for the general 6d quiver theories. We 0 − − − SU(2 0 − − 3 1 2 gauge node, which is a generalization of 28 0 1 | [1] Sp SU(3) and SU(3) 2 SU(3) SU(3) SU(3) SU(2) SU(2) − − · · · − − – 22 – 0 − − − − − 0 | [1] . The simple generalization of the setup in section + 16) SU(3) 2 SU(3) SU(2) SU(2) SU(2) SU(2) SU(2) N − , the central gauge node SU(3) has flavor − − − − − − 0 III SU(2 -plane introduced in [ − SU(2) SU(2) SU(2) SU(2) SU(2) SU(2) SU(3) − 8 − − − − − − − O + 8) ] using instanton operators, some of these nicely shows affine SU(14) . Hence, we concentrate on the cases with : [3] : [3] : [3] : [3] : [3] : [3] : [5] N 31 SU quiver theories I 2 V III IV VII XII Q , VIII SU(2 II − − ) N Sp( Here, we comment on the global symmetry. These theories are all expected to have In the S-dual frame, the gauge theory description is possible only for the following: d 6 realizes a 6d linear quiver and its Type IIAthe brane case configuration in is section given in figure will see that the 5d reduction yields4.1 various 5d dualities. 6d Sp We first focus on 6dthe quiver 6d theories quiver with considered one in section 4 5d dualities fromIn 6d: this generalization section, webranes, consider D8-branes a an generalessentially 6d the same quiver analysis theory done in constructed section by NS5-branes, D6- S-dual description of diagram thus, does not show affinein symmetry. this case. We claim It thatinclude would the one-instanton be higher analysis interesting instanton is to contribution notsymmetry for generalize enough is these the obtained. case method to with check instanton the operators expected to affine global monodromy analysis for these diagramsapply all the show method this [ expected globalsymmetry and symmetry. thus, When consistent we with thewith claim SU(14) that global these symmetry. 5d theories However,5d have strange 6d quiver to UV gauge say, fixed application theories point of fails [ to reproduce the expected affine structure. For example, the We omitted the case wherethis the way, we diagram obtain does various 5d not quiver have gauge clear theories which field have theory identical 6d interpretation.SU(14) UV In global fixed point. symmetry at UV fixed point. Indeed, we can explicitly check that 7-brane JHEP05(2019)203 ). corresponds 4.1 and the total ]. When one 2 n NS5 20 ], the D6-brane 40 NS5 1] where the number , ··· , 1 , 2 , NS5 ) is associated to a symmetry on ··· D6’s 2N+8n D6’s , 2 4.1 , + 1. NS5 ··· n , 2N+8(n-1) 5 1 . The 6d gauge theory content can be read − n entries of 1’s. A general Higgsing corresponds − 2 NS5 n n – 23 – + 8 , n NS5 + 8 N ··· NS5 , N 1) + 4 . A process of the 6d Higgsing. 1 = 2 − l − k n 2N+8 D6’s NS5 · l (2 . Therefore, the brane configuration in figure U(1). The number of the tensor multiplets is n N , n, n 1] with 2 n =1 × l , X ) Figure 21 +8 n ··· 2N D6’s ··· - N , D6-branes at the end. It is also possible to let one semi-infinite D6- n, + 8 O8 n . Type IIA brane configuration for the 6d linear quiver theory ( N + 8 N + 8 D8-branes in total. In this picture, each D6-brane at the end connect an NS5 N with a condition Figure 20 . Then, there appears fractionated D6-branes between D8-branes. The motion of l k 21 NS5 In general, such a Higgsing is classified by a Young diagram as in [ We can further generalize the 6d quiver theory by Higgsing. The flavor symmetry of is l to a Young diagram [1 to a Young diagram [ of off by moving D8-branes to the left in the brane configuration until no D6-branes end on hence the flavor symmetry isover reduced. NS5-branes, Due the to ranks the of appearance some of gauge the groups jumping are ofrepresents also D6-branes a reduced. Young diagramrepresent the by number a of vector D6-branesthe ending with Young on diagram non-increasing is D8-brane. 2 numbers, The total each number component of the boxes in that only one D6-braneshould connect can to connect other D6-brane anfigure between NS5-brane NS5-branes with by jumping athe some D6-branes D8-brane NS5-branes between as [ D8-branes in describesAfter the sending vev the of fractionated the D6-branes corresponding into hypermultiplets. infinity, we have decoupled D8-branes and semi-infinite 2 brane become finite and endwe on have one 2 D8-brane.NS5-brane Since each with D6-brane a end D8-brane. onbrane A one on Higgs D8-brane, two branch or of more the D8-branes. 6d Since theory the arises condition by of attaching preserving one supersymmetry D6- implies is generically SU(2 rank of the gauge groups is the brane construction of the canonical 6d quiver ( JHEP05(2019)203 . 23 (4.2) 1], we . The , . N+4n-nm D7 ), and see 20 ). The T- ). ··· 3.1 . , ) 4.2 1 4.2 4.1 , +1 2 l , lk 1 ··· ) and ( − =1 , l X n 2 4.1 N+(n-1)(4-m) D5 -planes as in figure , − − D6's 7 NS5 ] 1) l+1 ··· 1 | O , lk k − l=1 n-1 Σ 1 [ n − , which corresponds to the first , n + 8( D8's N 1 k 2N+8(n-1)- N . ··· , N+4-m D5 1 22 NS5 − SU(2 - - O7 O7 N D5 , n, n D6’s – 24 – n − · · · − + 8 color D5 branes originated from the 2nd gauge NS5 ··· ) D8's n N n, n-1 k k 2N+8-k − ] 1 N+4+m D5 − | . The + 8 n NS5 k [ N 4.1 D8's n k 2N D6’s compactification of the canonical 6d quiver theory ( - SU(2 1 S O8 gives us a 5-brane configuration with two − 1 ) . Type IIA brane configuration for the 6d linear quiver theory ( ] S N n | . Type IIB brane configuration after performing a T-duality to figure k [ N+(n-1)(4+m) D5 Sp( d 6 Figure 22 Figure 23 We will consider a circle compactification of the 6d quivers of ( N+4n+nm D7 We first focus onduality an along the After the T-duality, wenumber of have D5-branes the in distributiongauge node the ambiguity of middle discussed the column 6d in quiver is section always Type IIA brane configuration is depicted in figure the 5d descriptions of the theories as well4.1.1 as the 5d 5d dualities. SU quviers obtain the following 6d quiver theory at low energies the D8-branes. Dependingare on introduced the to Youngcorresponds diagram, some various to middle fundamental the gauge hypermultiplets Young nodes diagram in [ the 6d quiver. Hence, after a Higgsing JHEP05(2019)203 . - C − , 7 23 4.22 B O . Note 23 , and set the m and the number m . N+4n-nm D7 m , we can consider either + 4 + -planes. We consider the 2 − N − 7 + 4 O N+(n-1)(4-m) D5 [1-m, 1] . Then, the splitting type of the N -planes compared to figure in order that the resulting 5-brane − 7 1] BC +1)-th gauge node in 6d quivers i O m, − . Then, we move the branch cuts of N+4-m D5 [1 -th right columns from the center. Here, 24 i X 1] N D5 − -planes. As in section – 25 – m, -plane to be − 7 − 7 [1+ . The resulting distribution is depicted in figure O O X m N+4+m D5 [1+m, -1] -th left columns and i N+(n-1)(4+m) D5 so that some of the D5-branes and D7-branes cross the branch cuts 5-brane loops as in figure are distributed into the next columns, which are left and right to 1] n m, -planes or splitting only one of the two 4.1 − − [1 7 color D5 branes corresponding to the ( N+4n+nm D7 O X i 1] − +8 -plane is determined to be . The 5-brane web diagram after splitting two m, − N 7 [1+ O X Next task is the resolution of the The we fix the splitting typelower of the upper web diagram after pulling all the 7-branesresolution creates outside the has a 5d gaugeand theory interpretation. The splitting both former case first. The conditionshould that have the a final 5d 5-brane gauge webplanes. theory after pulling interpretation The out also relative all constrains difference the the between 7-branes splitting the type two of splitting the type is not important, and hence are distributed into the we need to careall about the the 7-branes condition outside thatdetermine has the the a resulting number 5d 5-brane of gauge web D5-branesmiddle theory diagram in one. the interpretation. after By column pulling In repeating next this fact, tofor analysis, this the the it column condition ambiguity turns which out parameterized may is that by there next is to only the one degree of freedom distribution ambiguity. We label thenumber of ambiguity the by D5-branes a in non-negativeof the number the left column D5-branes to in the the middle right to column be to the middle to be that the slope of the linescharge is precisely. schematically depicted and does not represent the corresponding 5-brane node in 6d quivers the center, respectively. The number of the D5-branes in these columns can change by the Figure 24 JHEP05(2019)203 ) n 1 1) ] 1 − M | (4.3) (4.4) (4.5) (4.6) (4.7) R [ m, SU( − ) 2 M 1 gauge nodes. − n SU( . Relatively, it can be , , 1] 1 1 n-1 D7 N+2n-nm+2 D7 n-1 D7 1] 7-brane and ( m, − − − − − · · · − [1 n n ) X 1 m, ≤ ≤ n-1 D5 − n-1 D5 1] l l n N+(n-1)(2-m) D5 − [1-m, 1] ≤ ≤ M m, [1+ SU( X − for 1 for 1 1 D5 1 D5 ) N+2-m D5 n and C + 2 , N D5 m, m, B N ) ) – 26 – l l nm, nm. − − − SU(2 1 D5 1 D5 n n ( N+2+m D5 − ) + ( − + 2 + + 2 1 n n n n − 7-branes in a counterclockwise and clockwise direction re- 1] 7-branes, and the D5-branes become NS5-branes. On the n , N 1] + 2 + 2 + 2 + 2 n-1 D5 [1+m, -1] m, . This is the origin of 5d quiver theories with 2 n-1 D5 N N N N − SU( N+(n-1)(2+m) D5 [1 = = = = 26 X l l 7-branes in a counterclockwise and clockwise direction respectively. 1 1 n-1 D7 N L R 1] M n-1 D7 C − − · · · − ) m, 2 N+2n+nm+2 D7 and N [1+ . In the upper part of the diagram, D7-branes and D5-branes cross the branch B X 25 SU( . The motion of the branch cuts of − ) . The 5d gauge theory realized by the 5-brane web is then 1 ] After pulling out all the 7-branes, the 5-brane web diagram becomes the one in fig- 1 N | 27 L [ The parameters are constrainedthe such that flavors should the be rank greater of than each zero gauge at group least. or the number of where ure SU( other hand, in thecuts lower of part the of thespectively. diagram, Then, D7-branes the and D7-branes D5-branes7-brane and cross respectively. the the After D5-branes branch crossingvertical become the lines [ as branch in cuts, figure the 5-brane loops are divided by 2 realized by moving somethat amount move of are D5-branes indicated and by D7-branes. arrows in The this D5-branes figure. and theas D7-branes in figure cuts of the Then the D7-branes become [0 Figure 25 JHEP05(2019)203 1] C − , m, n B − [1 X 1] − m, ). The global [1+ 7-branes. On the 4.1 X C , B and N+2n-nm+2 D7 [1-m, 1] C , = 2. N-2m+6 D7 B n 7-branes. ) since the very first 5-brane [0, -1] 1] N+(n-1)(2-m) D5 n m, − + 8 [1-m, -1] [1 X N 7 1] N-m D5 [m, -1] − N+2-m D5 m, [m, -1] -plane. D7-branes do not change its change when it [1+ − 7 N D5 X O N D5 – 27 – [1+m, -1] 1] 7-branes denoted by the yellow circles in the upper , ) gives a different-looking 5d gauge theory. However, N+2+m D5 4.3 N+m D5 in ( -plane. m [m, -1] − 7 N+(n-1)(2+m) D5 O 7-branes. The [0 . For simplicity, we write a web diagram of . The red lines arise from D5-branes which cross the branch cuts of N+2m+6 D7 1] 25 1] 7-branes denoted by the black circles in the lower part of the diagram arise D7-branes on top of each other. [1+m, -1] 24 m, − n − [1 m, N+2n+nm+2 D7 X + 8 1) which is the sum of the number of the tensor multiplets and the number of 1] N − − . The schematic diagram after moving the branch cuts of . The 5-brane web diagram after pulling out all the 7-branes outside of the 5-brane loops n m, [1+ + 2 The different choice of The number of the Coulomb branch moduli can be easily counted and it is (2 X Note that a D7-brane is mutually local to an N 7 web gives 2 we claim that they are dual to each other, andcrosses it the is branch the cut of distribution an duality. 1)( the vector multiplets insymmetry the analysis Cartan by subalgebra 7-branes of the should 6d recover quiver SU(2 theory ( Figure 27 from the one in figure 7-branes from figure and part of the diagramother arise hand, the from [ D7-branesfrom which D7-branes cross which the cross branch the branch cuts cuts of of Figure 26 JHEP05(2019)203 . ) 1] n 1), 28 m, 29 − (4.8) − + 8 [1 n . X N +2 1] +2] − n N m, +14]. We only , the resulting +6 1)( N [1+ − 1. Therefore, the N [2 X 25 n [2 − − − n and N+3n-nm+1 D7 +8) 4) ≤ -planes. The quantum . The motion corresponds C − N − , k N-2m+7 D7 7 23 n B ≤ ) D5-branes is connected to O +6 SU(2 [0, -1] m N+(n-1)(3-m) D5 − N (4 + +2) k SU(2 N + N+2-m D5 -planes in figure N N+3-m D5 7-branes. The analysis is essentially the − −· · ·− 7 - C O - ) D5-branes for 1 O7 O7 N D5 m +8) N D5 – 28 – n and − = 2 for simplicity. This web diagram can be obtained . (4 +2 n B k ), the 7-brane analysis should give the SU(2 N+3+m D5 N 28 . + 4.8 N+2+m D5 25 N SU(2 − -plane into +4) N+(n-1)(3+m) D5 − . About the motion of the branch cuts of 7-branes, we can only n 7 O N+2m+7 D7 SU quivers +2 4.1.1 N Q − N+3n+nm+1 D7 SU(2 − ) . The schematic diagram after moving the branch cuts of . The 5-brane web diagram realizing the 5d Sp( n -plane, the column which originally has − + = 2. 7 n N After pulling out all the 7-branes, the final 5-brane web diagram is given by figure O Sp( Due to the processflavor of symmetry. obtaining The number ( of the Coulomb branch moduli is again (2 5-brane configuration is depicted in Note that in this casethe the distribution ambiguity doesthe not column matter. which originally Due has resulting to 5d the theory presence is of We then consider theresolution case splits of the splittingsame as only in one section ofconsider the the two one corresponding to the upper transformations in figure Figure 29 write the web diagramby for pulling the all case the with with 7-branes outside the 5-brane loops from the configuration depicted4.1.2 in figure 5d Sp Figure 28 7-branes in the case whento one the resolves upper one ones of depicted the in figure JHEP05(2019)203 ) ). 4.2 (4.9) 4.11 (4.11) (4.12) (4.10) , + 2] ). n 4.1 , + 2 , , N = 0, which yields, [ . | + 2] | [3] [3] 6 − m n − SU(2) ) gauge node with other SU(2) ) [2 n + 2) flavor symmetry of = 2, the 5d quiver ( n − − . − n Sp n ) + 2 + 2 n 2.4 + 2 N N SU(3) SU(3) N and SU(2 SU( SU( − 2.3 ) − · · · − − · · · − n ) ) − · · · − − · · · − n n ) ) + 2 | n n ) in the SU( [1] + 2 , we again obtain a brane configuration with + 2 n N | | 1 [2] [1] + 2 N S N – 29 – + 2 SU(2 SU( N ), we have obtained two types of 5d quiver theories. SU( N SU( SU( 4.1 − ) − · · · − n − · · · − ) − · · · − n + 2 ) gauge nodes. In particular, when − · · · − n ) N SU(3) n SU(2 +2 SU(3) − ). For example, let us consider a case where gauge nodes and the other type has one SU( − 1 gauge nodes. It is in fact more interesting to see the TST-duality + 2 N − , it is possible to generate further dual 5d theories by acting the 4.3 − | N [3] . SU , it is possible to decouple some flavors from the 5d theories whose UV n + 2] | 1 gauge nodes. The S-duality or the 90 degrees rotation gives 2.4 SU(2) · · · − [3] 2 3 SU( n SU( − SU(2) 2.3 + 2 Tao theories which we will discuss in detail in section − [2 n − n n N +2 N + 2] T n ) duality on ( Z , + 2 gauge nodes. Those two theories are described by essentially the same 5-brane web As in section As in section N [ and can be done in parallel to4.1.4 the analysis in section Higgsed cases Let us then moveAfter on performing to the a T-duality circle along compactification the of the Higgsed 6d quiver theories ( two the completion is the 6dfixed point. SCFT. Suppose After we decoupling havedecoupling dual some the 5d flavors, same theories whose the flavors UVUV in 5d completion completion the is theory is both the a has same theories same 6d a leads 5d SCFT, 5d SCFT. to Working UV another on dual explicit 5d examples is theories quite whose straightforward which can be regarded as gauging SU( where it has 2 has a peculiar form where it has or the 45 degrees rotation. The resulting theory is where it has 2 SU diagrams and we claim thatclaim they in section are dual to each other. ThisSL(2 is the generalization of the of the vector multiplets in the Cartan subalgebra4.1.3 of the 6d 5d quiver dualities theoryFrom ( the same 6d quiver theoryOne of type ( has only which is consistent with the sum of the number of the tensor multiplets and the number JHEP05(2019)203 D7 l l=2 n-1 -th right i D8-branes N+n(4-m)-Σ l j is that we l . We defined − l n ). In this case, 22 k l=2 n-1 4.1.2 1. 4.2 − -planes, the splitting , n − N+(n-1)(4-m)-Σ(l-1)j D5 7 -plane. After the second ··· O − , 7 . Compared to the case in -planes, we fix the splitting O . The first step is we resolve − N = 1 7 l to a 5-brane web yielding a 5d O D7 -th left column and the for 30 i 4.1.2 n-1 j l k = N+4-m D5 and l j , to the + so that the final 5-brane web admits a 5d - - l 1, which originate from the i 4.1.1 22 7-branes, and the splitting type of the lower 1] O7 O7 D7 n − N D5 m, k C – 30 – n − [1

≤ n X and i . Due to this distribution ambiguity of D7-branes, one D7 1] B ≤ 7-branes. The second step is moving the branch cuts of − n-1 i m, N+4+m-k D5 C 4.1.1 [1+ -planes. When we split two − X and 7 such that they satisfy O B l l -plane into -planes. The difference from the cases in section , j − − l l=2 n-1 i 7 )-Σ(l-1)i 7 n O O D7-branes for 1 l − n . We choose the branch cuts of D7-branes in the columns left from the center D5 N+(n-1)(4+m-k k , we have the branch cuts of D7-branes in the columns that extend in either n k l . Type IIB brane configuration after performing a T-duality to figure yields a 5-brane web which admits a 5d gauge theory interpretation. l=2 n-1 )-Σ l i n 4.1.2 D7-branes extend in the left direction. 30 -plane is chosen to be In order to go from the brane configuration in figure The requirement that the final 5-brane web diagram admits a 5d gauge theory inter- n − k 7 N+n(4+m-k D7 type of the upper O gauge theory interpretation. Whentype we may split be only generically one7-branes of which the arise two by the quantum resolution of two or one pulling all the 7-branesfigure outside. It turns out that thegauge 5-brane theory, configuration we depicted take in twoeither steps two as or in one section of the pretation constrains the numbernumber of of D5-branes D5-branes in in eachsection the column middle except column for the isleft center. or again right always The direction. Thewhich branch gives cuts a also web affect diagram the admitting number a of 5d D5-barnes gauge in theory each column description at the final stage after column from the center column.fixed to Again, be the numberextend of D7-branes in in the the left middlethe direction, center column and extend is in the the branchof right cuts direction. of Furthermore, we D7-branes also in assume the that the columns branch right cuts from hypermultiplets attached to some gaugethere nodes is another in ambiguity the of 6dbranes distributing which quiver D7-branes we theory in saw ( in additioncan section to the allocate distribution of D5- in the Type IIA brane configuration in figure Figure 30 non-negative numbers 5-branes and two now have D7-branes in various columns, depending on the number of the fundamental JHEP05(2019)203 . ) 1 ] 31 1 M | R (4.17) (4.15) (4.16) (4.13) (4.14) 4.1.2 [ SU( when two − and D7 ) l , , , 2 30 ] l=2 n-1 1 1 1 2 M | − − − 4.1.1 R [ n n n SU( N+n(2-m)+2-Σ l j [1-m, 1] ≤ ≤ ≤ p p p l , the final 5-brane web l=2 n-1 ≤ ≤ ≤ − · · · − 31 ) [0, -1] 1 ] N+(n-1)(2-m)-Σ(l-1)j D5 − 1 n for 1 for 1 for 2 − | n M gauge nodes with various flavors R [ SU( l SU D7 i − ) n-1 j p ) N+2-m D5 n − l l ( ] + 2 j , ) n D7 | 1 n N D5 +1 p k k − p [ – 31 – X n xN + 2 = − l l

l n ( li − SU( 1 D7 1 ) +1 n-1 − i n =2 − p − l X n N+2+m-k D5 X k n = ) l 1 − ] -planes, we obtain the 5-brane configuration in figure − − 1 − ) − n − n 7 | m n N m k l O )( ) L p p [ l=2 n-1 − )-Σ(l-1)i n SU( − − m [m, -1] ( n n ( n N+(n-1)(2+m-k D5 + ( − + l − · · · − n n n ) l=2 n-1 2 ] [1+m, -1] )+2-Σ l i n + 2 + 2 + 2 2 N | , p L [ N N N i . Type IIB brane configuration after the two steps from the one in figure SU( = = = = N+n(2+m-k D7 − 1 p p p ) L L N 1 M ] When we split the two 1 -planes are resolved. N | − L [ 7 where SU( 5d quiver theory. The procedure is completely parallel to the oneafter in section the two steps.configuration gives After rise pulling to allattached a to the each 5d 7-branes node. quiver from The theory figure explicit with expression is Figure 31 O step, the 5-brane loops are divided by vertical lines and the structure eventually leads to a JHEP05(2019)203 . (4.18) (4.20) (4.21) 4.1.3 when one D7 l , 30 . l=2 n-1 1 1 − − n n , N+n(3-m)+1-Σ l j ≤ ≤ ) p l 1 p + 2] l=2 n-1 − ≤ n ≤ n | 2 N [0, -1] after the two steps. After N+(n-1)(3-m)-Σ(l-1)j D5 − 1 SU( for 2 32 k for 1 gage nodes with various flavors [ SU , D7 n-1 j +1 − · · · − l N+3-m D5 k ) ] 2 2 - N − | D7 n O7 + 1) n N D5 ) has parameters associated to the distribution k k -th right column from the center are the same p [ i – 32 – SU( + n 4.13 − 1, and the rank of each gauge group as well as the n , the final 5-brane web configuration gives rise to a ) D7 ] 1 1 − n-1 − i N 32 − | l N+3+m-k D5 + 2 (4.19) n ( l , n p k 1 [ lj − SU( gauge node and other − n 1 ··· X n = − − , =2 l l l X n ) Sp l=2 n-1 − )-Σ(l-1)i n n − = 1 p quiver theory ( ] + l n | k nm N+(n-1)(3+m-k D5 N [ + 4 SU for − n l l Sp( n k l=2 n-1 + 2 )+1-Σ l i n + 2 , ) = N l p j j N . Type IIB brane configuration after the two steps from the one in figure = 2 + = = N+n(3+m-k D7 l p 1 p i N R R -planes, we obtain the 5-brane configuration in figure Since the two types of the theories have the same UV completion as the 6d SCFT, -plane is resolved. − − 7 7 -th left column from the center and the they are dual to eachFurthermore, the other, 5d which is aof further D5-branes generalization and D7-branes. of the Again,we all claim argue the in that combinations section descend they from aretheories the dual same also to 6d give each theory, various other. 5d The dual 90 theories. degrees or 45 degrees rotation of the 5d In this case, the distributioni ambiguity of D5-branes andcolumn D7-branes due does to not the matter orientifold since action. where O pulling all the 7-branes5d from quiver figure theory withattached one to each node. The explicit expression is where ( number of the flavors should be larger than zero at least. When we split only one of the Figure 32 O JHEP05(2019)203 is + l N (4.23) (4.22) , we have ). ). 20 . 4.22 4.23  , ] n +1 l l k + 8 1 − =1 N l [ X n NS5 D6's − − l+1 lk 1] where the number of ] Σ l=1 n-1 1) , 1 | 1)) k [ − − NS5 ··· n , n . Compared to figure 1 D8's 1 D6’s N+8n D6’s N+8(n-1)- , k 2 33 + 8( , + 8( N NS5 N NS5 ···  N+8(n-1) , 2 , . We consider the generalization analogous SU( SU 3 . By considering the Higgsing specified by the D6’s ··· n , – 33 – NS5 NS5 1 4.1 , we obtain the 6d quiver gauge theory quiver gauge theories with the gauge node at the D8's − − · · · − n − · · · − N+8-k n-1 k ) -plane. The global symmetry is generically SU( , n n SU + 8 − k + 8) 8 N+8 D6’s NS5 N ··· − ] NS5 O , N 1 1 − | . The simple generalization is the 6d linear quiver = 2 D8's N D6’s n n + 8 − l k k SU( [ - N lk 4.1 − N D6’s O8 ) - NS5 =1 , n, n n l SU( N O8 NS5 P − ··· . Type IIA brane configuration for the 6d linear quiver theory ( . Type IIA brane configuration for the 6d linear quiver theory ( ] ) SU( A n, N 1 − | , A n k SU( [ [1] d Figure 34 Figure 33 6 d U(1). 6 We can again further generalize this quiver by Higgsing which is induced exactly by It is also possible to decouple some flavors from the 5d theories. Then, each 5d theory × with a condition ) l n k an extra NS5-brane on8 top of the same mechanism discussed inYoung section diagram [ generalization of what weto studied what in we section did in section The type IIA brane configuration is depicted in figure 5d SCFT. 4.2 6d SUIn quivers this with subsection, an antisymmetric weedge hypermultiplet consider having 6d one hypermultiplet in the antisymmetric tensor representation, which is a will have a5d UV fixed point.should Decoupling give exactly another the dualities same flavors between from 5d the theories dual which 5d theories has the same UV completion as a JHEP05(2019)203 , . - ) n − m 7 37 , the gives 4.22 O D7 1 . S 33 4.1.1 (n+1)N+4n-(2n+1)m introduced m . Contrary to the 5-brane loops as in 36 nN+4(n-1)-(2n-1)m D5 n -plane is determined to be − , T-duality along the 7 1 . The 5-branes depicted as red O S -th gauge node in 6d quivers ( 37 . We will study these 6d quivers i -planes attached to 5-branes. The 2N+4-3m D5 -planes create − ) on − 7 34 7 O O . 4.22 N-m D5 35 - mentioned previously. Then, from figure - O7 O7 m -th right column from the center. By imposing – 34 – i m D5 . -N+4+3m D5 . The resolution of the 1] , 4.1.1 +1 -plane was attached. If we fix the splitting type of the upper -th left column and i m − and consider their 5d descriptions as well as the 5d dualities. 2 7 1 . Then, the splitting type of the lower − 1) color D5 branes originated from the O . Type IIB brane configuration after performing a T-duality to figure S , the number of D5-branes at upper left and lower right is one more than C N color branes are added to the two columns at the center since all the [ − D5 , i n X 25 B , 1] +8( , -(n-1)N+4(n-1)+(2n-1)m 1 . Then, we move the branch cut of these four 7-branes in such a way that some of N − Figure 35 m 36 2 After this motion, we obtain the diagram in figure Next, we consider the resolution of the two − D7 N [ -nN+4n+(2n+1)m cut. These 5-branes becomes partout of that the the walls splitting newly each generated gaugeto 5-brane node. loop the Moreover, gives tuned it extra distribution turns colorwe parametrized D5-branes by for see each that column due 2 5-branes loops contribute as color D5-branes to these. When we move to the next gauge the D5-branes and D7-branes go acrosscase the in cut figure as dictated inthe the ones arrow at upper rightquite and parallel lower to left. section Apart from this small asymmetry,lines the are procedure the is ones coming from the D5-branes which went across the 7-brane monodromy above due to the conditionwhich that originally one of suchplane 7-branes should to be be attached toX the 5-brane to figure the condition that weout should that obtain the a distribution 5dThe ambiguity gauge is resulting theory again distribution interpretation labeled is in depicted by the in the end, figure single it non-negative turns integer charges of the split 7-branes are determined by the non-negative integer First, we start with thetype 6d IIB quiver brane gauge configuration. theorydistribution Analogous ( of to the the D5-branes brane should setupand be the discussed considered. in section There isare no distributed center into column in this case The brane setupcompactified for on this theory is given4.2.1 in figure 5d SU quivers JHEP05(2019)203 and C , B D7 D7 (n+1)N+(2n+1)-(2n+1)m (n+1)N+(2n+1)-(2n+1)m n D7 n-1 D7 . D5 D5 n D5 n-1 D5 [0, 1] 36 nN+(2n-3)-(2n-1)m nN+(2n-3)-(2n-1)m [N-2m+1, 1] . The branch cuts of 35 1 D5 2 D5 2N+1-3m D5 2N+1-3m D5 B 1 D5 N-1-m D5 N-1-m D5 planes in figure [N-2m+1, 1] − B 7 – 35 – C C O 1 D5 m-1 D5 m-1 D5 [N-2m-1, 1] 2 D5 1 D5 -N+1+3m D5 -N+1+3m D5 . Diagram after the motion in figure [N-2m-1, 1] are moved in such a way that the D5-branes and the D7-branes indicated 1] Figure 37 , D5 D5 n D5 n-1 D5 [N-2m, 1] +1 -(n-1)N+(2n-3)+(2n-1)m -(n-1)N+(2n-3)+(2n-1)m m n D7 n-1 D7 +2 N [ D7 D7 X 1] . The diagram after resolving two , -nN+(2n+1)+(2n+1)m -nN+(2n+1)+(2n+1)m 1 − m +2 N [ Figure 36 X by the arrows go across the cut. JHEP05(2019)203 1 S (4.25) ) on 4.22 D7 n, 2 (n+1)N+3n-(2n+1)m , 7-branes so that -th right column gauge nodes. i ··· C ] (4.24) , 7-branes, we consider -planes is resolved. +1 n − -planes in the diagram n SU [0, 1] = 1 2 and nN+(3n-4)-(2n-1)m D5 7 C − N O 7 [ B O − and ) , while we keep the lower half n B 2 m, ` N 36 + 1) SU( 2N+2-3m D5 ` 2 . ). B − 1) N-1-m D5 -plane into - n , − O7 − − · · · − 4.24 7 ) m 2 – 36 – C O + (2 m D5 + 1) N N m ) ` + 1)( SU( n + − (2 + 1)( n -N+3+3m D5 ) 1 − n − N . In this case, the distribution ambiguity does not matter as N + (2 38 1 + ( SU( − + 1) − nN ] n n 0 D5 − N [ = 2 = ( = -(n-1)N+(3n-3)+(2n-1)m ` 0 +1 N N . The diagram for the case where only one of the two n 2 D7 . After we resolve the upper N -nN+(3n+1)+(2n+1)m 35 -th left column from the center is actually the same column as the Figure 38 Therefore, after moving all the 7-branes outside, we interpret this diagram as the i performed for the upperpart as part it of is. thediagram After diagram depicted moving in the in figure branch figure cuts,the it is straightforward tofrom see the that center we by obtain the the orientifold action. In other words, we have Here, we consider thein case figure of resolvingthe only same one procedure, of namelysome the we of two the move D7-branes the andor branch D5-branes equivalently cuts cross the of them. motion the The of explicit D7-branes motion and of D5-branes the is branch cuts exactly the same one that we The parameters are constrainedthe such that flavors should the be rank positive. ofis each described In gauge by summary, group we the see or 5d that the quiver gauge the number theory 6d of ( quiver4.2.2 theory ( 5d SU quivers with an antisymmetric hypermultiplet following 5d quiver gauge theory: where node, the extra color D5-branesbranes reduce at by the two columns and at there the are both only edges. two additional color D5 JHEP05(2019)203 . , we (4.26) 4.1.4 and in , n, , n, . 3.6 ··· ··· + 1] , , ) (4.27) 4.2.1 n 1 ] = 1 = 1 M | ). Diagrammatic R + 6 [ N SU( [ 4.22 − − , p l ) j ). Since their type IIB 2 , p 5) ) l ] p i 2 M | ) − j 4.26 [ p − n l SU( ( − l + 6 ( +1 n p X +1 N = l ) and ( n p X = l − − · · · − SU( ) 4.24 − m n ] ]. . m l n M | j 39 lj [ + 1) , p − · · · − + 1) SU( l n =2 2 – 37 – l X li p − 2 − − ) n =2 + 3) n n l -plane [ − X ] 1) n − ), which is obtained by Higgsing ( N (2 n | n − 7 i , we have discussed two different types of 5d description [ − − O + 2 SU( 4.23 m + (2 planes, we obtain the 5d quivers N + 1) N 4.2.2 − N ) m 7 p + 1) + 1)( SU( O p n + − · · · − − ), which are 5d quivers ( ) − (2 + 1)( n 2 1) ] − n n − N 2 | 4.22 − i [ N n . l + (2 SU( 1 + ( 1 + ( k and in section + 2 − − − + 1) = ) nN . The different point is that some of flavor D7-branes exist at internal columns. n n l n 1 N j − ] N 4.2.1 | L + = 2 = ( = = 2 [ SU( l 4.2.2 p p i L R SU( − N M When we resolve two When we combine mass deformation and S-duality as discussed in section After moving all the 7-branes outside, we obtain a 5-brane web diagram. It is possible A [1] where where section The effect of such D7-branesHere, are we again summarize quite the parallel resulting to 5d what description. we discussed in section theories which have an identicalwould 5d be UV interesting future fixed problem. point. Classifying all4.2.4 these dual 5d Higgsed theories cases Now, we go on to thederivation 6d of theory ( the 5d description is quite parallel to what we did in section two types of 5d6d theory UV are fixed dual point. to each other, which meanswill that be they able have todecoupling the limit, obtain identical all various these dual dualities 5d can be theories. also reduced Furthermore, to by the considering dualities the for flavor another set of 4.2.3 5d dualities In section for the 6ddiagrams theory come ( from the identical type IIA diagram for the 6d theory, we claim that these to can read off from the diagram that this theory is the following 5dNote quiver gauge that theory: we haveattached an to anti-symmetric the hypermultiplet lower at unresolved the left end node as a 5-brane is JHEP05(2019)203 ] N [ − plane (4.29) ) − N 7 O SU( give the set of ) plane. These l 8’s configura- ) (4.28) j + 1] p, q n D − · · · − n N | ) 2 N + 8 , and − SU( l i 1 − SU( . k , 8 [ s D8 − − m O NS5 − N ) n ) N 1 − +1+ ] l n D6 2 SU( | lk N k N [ − 1 NS5 − =1 planes, we obtain the 5d quivers l 8] s X SU( , − A − 7 s O ) transformation on the ( NS5 Z , − · · · − – 38 – 5 + 4 ) ] D6 3 2 − N N − | n n NS5 k [ SU( + 2 − N D6 ) with any possible value for ) ] 2 N = 1 NS5 8 D8 N − s | 4.27 n - N k [ O8 + SU( − ] ) 1 A plane together with eight D8 branes, leading to indefinite sequence of 1 N | , − n 8 k [ SU( O . 6d brane configuration for the quiver [1 We claim that these two types of the theories have the same 6d UV fixed point. Also, When we resolve only one out of the two tion 5d, many D7 branes are convertedthe to other other hand, 7-branes resulting we in discussS-dual lack below frame of that flavor or 7-branes. one by On canspecial implementing still cases the have also SL(2 an give Lagrangianinto intriguing description a dual in pair pictures the of depending 7-brane. the resolution of the In this section, weposed consider the of 6d an SCFTquiver configurations type in made tensor of branch D6principle, which branes obtained are stretched through the between com- Higgsing twoprevious of NS5 general section. branes. 6d These brane It cases configurationdescribed is discussed can in in be, however the the in non-trivial previous to sections. get It is 5d partial description because, following to the have Lagrangian procedure description in 5 Special cases: the 6d quiver gauge theories of the 5d SU quiverdual theory theories. ( S-dualitydecoupling of limit these will give 5d other theories set of will dual also theories which give have the various same dual 5d theories. UV fixed point. Flavor where Figure 39 which has an antisymmetricgroup hypermultiplet of the and quiver eight also fundamental coupled flavors to of tensor multiplets. the leftmost gauge JHEP05(2019)203 1) − (5.1) , ) invariant. For ) 7-brane pair. Z ] , 1 , 2 N N - [ . ] N N X O7 B B N , [ ) transformation on the C A is attached to a (1 N 7-branes). We will show a − Z which will be converted into N A , ) } . 7-brane is attached to 5-brane. 2 A B N C N B 39 ) or ( C SU( CBN , 2 where B N planes, we restrict ourselves to two 2 ] quiver − N 7 into N − · · · − N [ We will discuss various dualities along the O ) 2 B {z nodes BCA B N 2 n A A BCA A – 39 – SU( with eight D8 branes and a NS5 brane being on 2 − · · · − A − − 8 ) ) O N N plane and four D7 branes would be an obvious example . It is straightforward to show that it can be expressed − SU( | 7 SU( planes, which is either ( − O 40 A B − − - 8] A 7 ) A , C ) 7-branes with four D7 branes (four A O A O7 N ) invariant 7-brane combination where B B X Z plane, together with four D7 branes, is thus SL(2 , , planes. In order to see the dual picture, it is useful to consider A A ) invariant 7-brane combinations, as the total monodromy of them is N − A 6d [1 A SU( Z − 7 , 7 O − O plane, ) or ( 8] planes. Regarding resolution of − , C 8 , plane. In particular, the cases where one of the 7-branes is attached to a 5- − A 7 O B . (i) An SL(2 again via an S-dual action. − O 7 2 ) invariant 7-brane combinations. O Z Consider the 7-brane configuration , As discussed earlier, when going from 6d to 5d, one can distribute D7 branes and D5 ) 5-brane web plane. An BCA 2 p, q branes with a suitable Wilsonclose line, to which enables usdistinctive to resolutions allocate of any number of D7(i). branes 5-brane as depicted in figure of the brane of the samegauge charge will theories. give rise Wewhen to consider reducing fruitful such a dualities 6d combinations among theory frequently the to appearing resulting 5d. through 5d quiver T-duality among many SL(2 proportional to minus identity matrix.resolution of It the follows immediately thatinstance, ( a pair of 7-branesfew distinctive as examples a of such combinations involving 7-branes from different resolutions SL(2 reduction of the 6dor theories without to the 5dcombinations theories of by 7-branes analyzing which( the are 7-brane invariant monodromies under with the SL(2 top of the The corresponding 6d brane configuration is given in figure A sequence of monodromyA analysis makes 5.1 6d [1 We first consider the case where an Figure 40 JHEP05(2019)203 ) Z , (5.6) (5.2) (5.3) (5.5) (See ) in- (5.7) (5.8) (5.4) Z 8 , ) transfor- 1) 5-brane, 1) 5-brane. Z , , .) Using the . , is attached to , is an SL(2 3 2 N 3 42 N N ), one sees that . A BN 2 ] N - , 1 , = q, p 2 B 2 [ NCBN − B O7 C ( 2 , ¯ , BN BCA 3 2 A 2 N C into 2 → = A 3 ANX N 2 ) A ] = ¯ BCA , 1 . , 2 2 CA -action of the SL(2 p, q ¯ 3 ABCA 2 [ . (See also figure T dual A = NC 2 −→ ¯ − BN S C 2 = 7-brane is attached to (0 2 leads that ANX dual N NCBN is still attached to the 5-brane CA N 9 2 −→ − dual BCA AN = S 2 −→ = − 7-brane being attached to (0 B = or S 2 B B A 3 A N 3 , CBN CA or 2 A 2 CA ¯ BBN 2 A BN 2 – 40 – , N = with ¯ A A BN A = 3 = − 3 AC is S-dual invariant C = ¯ A NCBN ] 2 B 1 A 2 = AB , B ] C O7 2 2 = 1) 5-brane. A pictorial version of this monodromies is given in N 2 1 [ 2 , = A − ] A 3 2 [ , 1 = A 2 , ¯ BA A 2 [ 2 ] 2 BCA NA 1 ANX 2 , X A CA 2 [ A , = ANX = ) invariant 7-brane combination where the 7-brane 2 ¯ BCA Z 2 A ¯ NC 2 NX , A = A A ¯ - BCA A 2 are related by a successive application BCA 7-brane configuration where is attached to (1 AN [2,1] A 7-brane is attached to a D5 brane. This procedure plays an important O7 2 X ¯ 2 B = A N A BC A CA . Again a little monodromy calculation and is an S-dual invariant combination CBN . (ii) An SL(2 ] 41 2 ). By performing an S-duality transformation ( 2 1 , N . When no 5-brane is attached to 7-branes, it is easier for one to see SL(2 2 [ Another example is 40 40 The relevant monodromy relation for this case is Using the 7-brane monodromies, one finds that BCA NX 9 8 2 yielding which yield where the 7-brane figure variance. For instance, consider monodromy, one finds that mation. Notice that before takingbut an after S-duality, the the S-duality andtransition, a the manipulation of monodromiesrole as in well showing as a the dual Hanany-Witten description of(iii). 5d quiver theories which we discuss below. See figure invariant 7-brane combination as where we used a bar “ ¯ ”(ii). to denote the 7-brane is attached to a 5-brane of the same charge. (0,1) 5-brane. A sequence of monodromy analysis makes as the figure A Figure 41 JHEP05(2019)203 [1]. = 1. = 8 (5.9) − a f . N N = 1 SU(1) a X N N = 8 and B f − · · · − N N C N - = 1 hypermultiplets: = 8 & B a f C O7 SU(1) N B ) with N N − C n A N . 2 SU(1) = 8 and X − N CBN f 2 8] , N N A 7-brane along the direction of the charge ) theory with n B B [1] giving 5d Sp( into A − 2 quiver nodes, we claim that the 6d brane A n A 5d Sp( – 41 – BCA A SU(1) [1] quiver and 5d Sp theory with 2 C ⇒ A , one moves [1] − · · · − 43 − ) gauge theory with } n − · · · − B X SU(1) = 1 case with N D5 SU(1) - − - ) invariant 7-brane combinations, (i), (ii), and (iii), we study C A N Z ) invariant 7-brane combination where no 7-brane is attached to 5-brane. - A O7 O7 , SU(1) A Z plane gives 5-brane loops which are denoted as circular loops. C , {z nodes SU(1) − O7 A − B n brane comes a freely floating 7-brane. This procedure gives a configuration 7 − − · · · − A O 8] B For the , A A SU(1) SU(1) | − − . Left: 5d brane configuration for the quiver [1 = 1. . 6d [1 . (iii) An SL(2 8] 8] , a , A A N 1) so that the With all these SL(2 − , 6d [1 6d [1 and configuration gives rise to 5d Sp( special brane configurations which yield various 5d quiver theories. Figure 44 From the figure on(1 the right of figure on the left here.figure on Using the the right. 7-brane monodromy, one can put all the 7-brane on one side like the Figure 43 Right: splitting the Figure 42 A sequence of monodromy analysis makes JHEP05(2019)203 × ) n (5.13) (5.12) (5.11) quiver quiver , yields ] n n 1 . Taking , 2 [ = 1 flavors, ’s separated 44 ) 7-branes (a X and − a , C 7 N 1]) 7-branes for m , N O , B is even, which we = 2 = 2 and Sp( → = 2 case of ]. By pulling out all N a − N 7 39 N N is now a brane junction . O = 8 and ] is resolved differently, for 1] and [2 into two For − , f m − 8 7 − N [2 [2] (5.10) O 8 . O 1]-brane. Hence the remaining − − O , 4] . ) } , } m A [4] 1] 7-brane which does not exit for + 1) with , [1 plane, e.g., − SU(2) n ) − − SU(2 n 7 plane, and ([0 O {z Sp( + 1) − nodes {z 7 nodes n n = 1 flavors, given in [ − − · · · − O n ) − · · · − a n ) ) invariant combination with ( – 42 – N Z ) gauge theory with SU(2 m , n Sp( SU(2) | − − − 4] 7 and SU(2 | antisymmetric hypermultiplets , s as well as a D5 brane. Because of this junction, when 8] A , − ≤ − D7 branes in the original setup, and as a consequence of 7- A 7 f two 5d [4] 8] ] as a limit taking masses of D7 branes to infinite. Notice O , N 5d [1 41 A ] which is made out of eight D7 branes, and its flavor decoupling ) theory only with fundamental hypermultiplets nine 6d [1 n 30 1]) 7-branes for an , . From 7-brane monodromies for this configuration, one finds that 6d [1 plane) and four D7 brane, one finds that the Sp( 43 − × 7 ) planes into two pairs of two 7-branes, each O n − nodes: duality between 5d SU(2 1] and [1 7 , O n 1 − Using an aforementioned SL(2 ) gauge theory with ) invariant way that the 7-branes are rearranging themselves to respect the original n + 1) gauge theory with Z ). n , n = even: duality between Sp quiver and SU quiver. = 2 and the same UV fixed points. N nodes, 6d theory is On the other hand,the a quiver different of resolution Sp( of the This suggests that these two 5d theories are dual to each other in the sense that they have 7-brane structure. As thiswill case discuss can in what be follows.SU(2 understood We as state the that case the when resulting 5d theories. This case leads to 5d nodes gives rise to twoSL(2 5d theories. Firstly, reducing it to 5d, we can take the S-duality in a for Sp( the 7-branes, it is not so difficultN for one to findSp( that it makes aresolution Tao of web the diagram. brane monodromy, one of D7 branesD7 is now branes converted to account a eight [0 flavors,the but web as configuration we for haveassociated the the massless with [0 antisymmetric a hypermultiplet, non-zero thisthe mass flavor 7-brane decoupling of is limit antisymmetric for thus the hypermultiplet. fundamental flavors, See one would figure find 5d web configuration it leads to thewhich configuration describes for a 5d circle Sp( the compactification web of diagram 6d for higher higherwas rank rank already E-string E-string discussed theory in theory. [ of We masslesslimit note antisymmetric was that hypermultiplet also discussedhowever in that we [ now have maximally along the T-dualof circle. NS5 NS5 brane brane connecting stuckresolving two the on the instance, ([ the other. See figure A circle compactification followed by T-duality makes an JHEP05(2019)203 - − − − O7 4] . , (5.15) A [3] Sp(3) − SU(4) − ) } - − n O7 Sp( - − SU(4) O7 − +1) 8] | [4] yielding 5d [1 n , - [1] A − O7 - [4] yielding 5d [3] 7-branes. The second case 3 O7 SU(2 A − 4] (5.14) , 4] (5.16) − C A A , SU(4) A A [1 − SU(4) [1 - +1) − and 3 2 B A A } n C O7 − − 2 B A 7-branes. +1) SU(4) ] n 1 SU(2 , − into SU(4) 2 [ SU(7) 2 − − X A 2 − 7 SU(2 {z C A B O +1 nodes SU(4) −···− – 43 – and m B A C − SU(4) 3 A SU(7) N − +1) 8] −···− , n − 8] A , plane out of two suitable 7-branes, one finds that the 4] A {z into +1) nodes , ) invariant 7-brane combination including the resolution 3 A A n − A C B SU(2 m − 7-branes. Performing an S-duality and also taking a weak Monodromies Monodromies S-dual and Z 7 , 7 − O C O , SU(2 5d [1 B = 3) yielding +1) − 2 A | n n ) invariant 7-brane combinations introduced earlier and S-duality . As there are eight D7 branes, one can relocate and arrange D7 [1] C Z [3] implementing S-dual invariant 7-brane combinations. 2 B , +1) A Monodromies Monodromies S-dual and 2 45 4] implementing S-dual invariant 7-brane combinations. − n A SU(2 , B A C − 2 ) [1 = 2 and A | SU(2 n Sp(3) − 3 m − A − A plane into the | Sp( N 4] . A T-dual version of 6d [1 X . A T-dual version of 6d [1 X , − − [2]) [4] ( N 7 A SU(7) 3 A A − O − − plane so to make an SL(2 As a representative example, the brane configuration for 6d [1 5d [3] 5d [1 − 7 is depicted in figure branes with suitable WilsonO lines, such that fourof D7 the branes are locatedcoupling close limit to to each obtain the the is realized when we split the SU(4) The first case is realized when we split the to obtain the following 5d theories: Figure 46 (SU(7) Again, one uses SL(2 Figure 45 SU(7) JHEP05(2019)203 - . ] − = 1 O7 , 2 4] [ N , (5.18) (5.19) A X , [5] quiver N A planes, 6d cases: ) invariant − − Z → m , . 7 − 4] O - 7 , = 2 O7 O A SU(5) gauge groups. See N [1 − − [5] yielding 5d [1 } SU − plane, - . + 1) O7 SU(5) − and [3] n 7 − [3] (5.17) O − SU(5) Sp − − SU(2 - SU(5) O7 Sp(3) − brane is different, one does get dual Sp(3) − SU(5) 8] − − − · · · − , − plane in the weak coupling limit. One can 7 | A [2] O − | 7 SU(7) [2] + 1) O SU(7) SU(5) − n B C – 44 – − − {z + 1 nodes 8] , SU(2 m SU(7) A − Sp(3) − C In a similar fashion, one can obtain 5d theory for B − 4] , + 1) , yielding [3] implementing S-dual invariant 7-brane combinations. A | n . Like the previous case, one has another SL(2 [1] − 5d [3] 47 46 5d [1 SU(2 Sp(3) Monodromies Monodromies S-dual and − − ) SU quiver. n [1]) − Sp( | − X − N . A T-dual version of 6d [1 B . Therefore, we claim that two different quiver theories have the same 6d origin (SU(7) C 5d [3] A − 46 + 1. In this case, the resolution of each = odd: Sp m As an example, 5d branetheory configuration is for depicted 6d in [1 figure description, rather one finds the resulting 5d theory is a hybrid of the at UV and thusdecoupling they limit, are this two duality dual would description still at holdN IR. for less We note flavor cases. also2 that taking the flavor 7-brane pair which can be convertedfurther into move an this D7 braneone across has 5-brane junction, freely and floatingresulting as the configuration D7 Hanany-Witten yields branes, transition, a whichfigure quiver play theory the made role out of of the fundamental flavors. The is also depicted in7-brane figure combination except forWhen a different performing resolution an of S-duality,the the the previous web one configuration as becomes a completely D7 different brane from is being attached to a D5 brane which makes a floating resulting web configuration giveshypermultiplet rise at the to edge the gaugebrane quiver node. configuration gauge yields In theory a a with seemingly different way different an of 5d antisymmetric resolving quiver the theory Figure 47 SU(7) JHEP05(2019)203 ] N [2 (5.23) (5.21) (5.22) (5.20) − ) [4] quiver N − planes, one − SU(2 7 ] quiver , SU(4) ] O N − N . [2 [2 − · · · − [4] plane, and D8 branes, ) − B C − − ) } SU(4) N − ) ) } 8 N − n B N C O . . D8 SU(2 Sp( [4] NS5 [4] 2N − SU(2 Sp(2) ) − − − SU(2 ), one sees that S-duality leads to − planes, D7 branes and D5 branes ) N n D6 − 5.8 . By resolving two 7 2N NS5 SU(2 Sp(3) O − · · · − SU(4) 49 SU(2 ) − − − ) − · · · − N . As discussed in the beginning of this sec- ) N , consisting of an {z 49 N {z NS5 nodes Sp( + 1 nodes SU(2 SU(6) SU(4) − · · · − 48 n − – 45 – N ) − − − n D6 ) D7 SU(2 . N 2N − NS5 SU(2 50 Sp(3) Sp(2) ) − − SU(2 − N D5 ) D6 − n ) 2N 8 D8’s - - N Sp( | 5d [4] 6d [8] SU(2 - O7 O7 − O8 + − Sp( | ) ) invariant 7-brane combination ( − Z N , D5 5d [4] Sp( 6d [8] − . 6d brane configuration for the [8] . T-dual version of 6d brane configuration for the [8] As an example, consider obtains the configuration on the righttion, of using figure the SL(2 The procedure is depicted in figure Reducing it to 5d, onegiven gets in a the brane web configuration of configuration on the left of figure D6 branes and NS5 branes.obtains the We claim 5d that theory from given by a circle compactification and T-dual, one Next we consider another special case: whose brane configuration is given in figure Figure 49 theory. 5.2 6d [8] Figure 48 quiver theory. JHEP05(2019)203 . − − ] ... 51 (6.1) N − (5.24) T SU(4) ) − [4]. ... m 3 − − ]. By turning - − . Sp(2) T ]. An example ) 42 − [3] N O7 30 m Sp(3) 3 − − − SU(2 N − for 6d [8] SU(2) SU(6) C B T 49 ) − − ]. It is possible to construct C SU(2 B m theory is depicted in figure − 46 2 , 5 T Sp(3) − ) T SU(3) N 45 , − m N 2 32 - 0: − O7 ]. The last SU(1) node can be understood [2] [ N A SU(2 − · · · − 46 − m > − 2) – T SU(2 ) − 43 theory flows to a 5d linear quiver theory [ – 46 – − m N S-dual SU(2) N T − ) T SU( m N SU(1) [ − − Tao theory − , we consider − · · · − 1) C B N N SU(2 2) − T 5.2 B C SU(2) − − N T SU(2 N ) − B C SU( theory can be realized by a 5-brane web diagram [ N T N −· · ·− SU( . The third one is the result of S-dual action of the second one. The last ) − N A B 2) C − A N T Sp( 5.1 − 1) − + 2] ] Sp( N [4] quiver. The 7-branes are distributed and combined to form an S-dual invariant − N m − − [ ] N A SU( m − . The first figure is a simplified version of the right of figure SU( 1) SU(4) − [8 + − quiver − ] N N d 6 as [ a Tao web diagram byof adding a flavors Tao to web the diagram twoThe arising end by 5d adding nodes quiver flavors of theory to the realized the quiver by 5d [ the Tao diagram is then There is another importantknown class that the of 5d 5don theories certain obtained mass by deformations,SU( Tao the diagrams. 5d Itas has two flavors been coupled to the SU(2) gauge node, and hence the quiver is equivalently written As a generalization of section 6 6d description of 5d explained in section one is the corresponding brane configuration showing 5d [4] 5.3 6d [8+ Figure 50 SU(4) 7-brane combination. The second figure is the rearrangement of these 7-branes via monodromies JHEP05(2019)203 (6.2) ], and 27 ]. Namely, Tao diagram 37 N T ] 7-branes mutually Tao theory is realized p, q N + 4 flavors in [ T n || 2 − = 2 N f ), the branch cuts are assumed N ) realized by the 6.2 6.1 CANC 1] 7-branes respectively. We will also + 8). We repeat the same procedure to || , 2 n − N Tao theory. N – 47 – T Tao theory ( Tao theory. Since the 5d 1] and [0 , N N NCAN T T || 2 ) gauge theory with − n 0], [1 N , . In this expression of ( . The 7-branes in each chamber are schematically summarized B 52 theory. ANCA 5 T represent [1 ) 5-branes, it is possible to collect the 7-branes into three chambers, the N p, q and 1] 7-brane by − , . The Tao web diagram which arises by adding flavors at the end nodes of the 5d quiver C , A We first pull all the 7-branes inside the 5-brane loops. Since [ (3rd chamber) as in figure as where express [1 the global symmetry was determineddetermine to the be global SO(4 symmetry of the 5d commute with ( top-left one (1st chamber), the bottom-left one (2nd chamber) and the bottom-right one 7-branes on top ofpart each of other. the flavor The symmetry of symmetrynumber the realized of 5d theory. on parameters Abelian 7-branes of partshould gives the can the agree be theory recovered with non-Abelian since by the counting the the Tao rank number diagram of of realizing the the total the parameters global 5d of symmetry. the SU( The 5d method theory was applied to the 6.1 Global symmetryThe from global 7-branes symmetry ofthe a symmetry on 5d 7-branes theory attached7-branes realized to can external by determine 5-branes a in the 5-branesee the global web web the symmetry diagram can flavor [ realized symmetry, be at we understood the pull from all UV the fixed 7-branes point. inside In the order 5-brane to loops and try to put We will call the 5d quiver theoryby as the 5d Tao diagram, thewe propose theory a is 6d expected descriptionby to of examining have the the a 5d global 6d symmetry UV as completion. well as In the this number section, of the Coulomb branch moduli. Figure 51 corresponding to the 5d JHEP05(2019)203 1 S (6.4) (6.5) (6.6) (6.3) . As I U(1) 1) 5-branes , × ) Tao theory. The N + 5. The number N T N 2. The total number − || N 0) 5-branes, (0 3. Also, the number of the , − || CAN 1] N , || [2 2 || − X 1] 2 N , ! ) symmetry. [2 N N pq || X 2 ] 7-brane counterclockwise. Also, the first, ) represent (1 || B 2 B 1 + p, q 6.2 − NCAN N 2 N 3 2 pq p 2 – 48 – − q A N A − || − || 1 1] A , N || [2

A X 1] , [2 ANC Tao theory is SU(3 2 X symmetry associated to the Kaluza-Klein modes from the − N I T N C A-branes on top of each other means that the non-Abelian part of the Tao theory as an example. 5 N T . The 5-branes web after pulling all the 7-branes inside the 5-brane loops. We write an 1) 5-branes respectively. Hence, 7-branes with the same charge as the 5-brane , ), and obtain . Therefore, the full global symmetry of the 5d theory is SU(3 It is possible to count the number of the parameters of the 5d N 6.2 of mass parameters ofgauge the couplings bi-fundamental or multiplets theis is mass 3 parameters forexpected, the we instantons have the is U(1) compactification of a 6d theory. The presence of 3 global symmetry of the 5d number of mass parameters of the fundamental hypermultiplets is By rearrangement of 7-branes inside each chamber, the 7-brane configuration becomes By moving 7-branes between thewe chambers finally with obtain the rearrangemnt in the second chamber, the second and theand third (1 partition linescorresponding in ( to the partitionin line ( can crosse its line. Therefore, we can move 7-branes to extend in thematrix lower direction, and the charge of a 7-brane change by the monodromy when the 7-brane crosses the branch cut of a [ Figure 52 example of the 5d JHEP05(2019)203 ) − 6.7 (6.7) global ’s. Each 1 P Tao theory. N and yielding a T 1 P ). It is important symmetry. Hence, I 6.7 Tao theory. We first focus N T . , there is a bunch of 2) B − N . 2 1)( − [11] N ( − ]. Then it is possible to find a candidate for its . In the base , which agrees with the flavor symmetry of the 5 I – B – 49 – SU(3) 3 , − 1 U(1) Tao theory. In this case, it is not included in the A × 5 [1] T theory. 4 T Tao theories Tao theory is a very special case in the examples considered in 6 corresponds to a vev of a scalar in a tensor multiplet. The elliptic T 4 can be singular, meaning that 7-branes wrap the 1 T P 1 theories, which will be basic examples for the general 5d P 6 and ,T 5 5 ,T ,T 4 4 , and does not appear as a global symmetry of the 6d theory. Since the third- T T . The 6d description in the tensor branch is 3 Tao theory. 3 gauge 4 In order to realize a 6d SCFT from F-theory, we consider an F-theory compactification Let us then consider the 5d After the circle compactification, the KK mode provides a U(1) There can be a potential U(1) flavor symmetry that may come from the overall part of Let us then determine the flavor symmetry of the 6d theory ( In fact, the 5d The dimension of the Coulomb branch moduli space can be determined easily by T on a Calabi-Yau threefoldcomplex which two dimensional is K¨ahlermanifold given bysize an of the elliptically compact fibrationfibration over over a the non-compact examples we have discussed so far.in Type This means IIA that string it theory. mayrealized However, not by it F-theory admit has compactifications a [ been brane configuration conjectured6d that description all in the the 6d 6dthe SCFTs SCFTs may global classified be symmetry by the as F-theory well compactifications, as by the comparing dimension of the Coulomb branch moduli space. if there is onlysymmetry. one representation. Therefore, the 6dthe theory total does flavor symmetry not is5d have SU(12) an Abelian From this point of view, we have at leastU(12). an SU(12) However, we flavor symmetry. argue thatSU(3) the U(1) symmetry isorder broken Casimir by invariant the for anomaly U(1) SU(3) is non-trivial, it is impossible to cancel the anomaly fundamental representation of theWeyl spinor, SU(3). the complex Although conjugationTherefore, the of a hypermultiplet the in 6d the 6d anti-symmetric hypermultiplet Weyla representation spinor of contains hypermultiplet SU(3) does in a is not the equivalent to 6d change fundamentalis representation its equivalent of to chirality. SU(3). a 6d Namely, SU(3) the gauge 6d theory theory with ( 12 flavors with one tensor multiplet coupled. theory has one tensorhence multiplet the and sum two of vectorCartan multiplets them subalgebra in reproduces of the the the Cartan 5d correct subalgebra, number and of theto 5d vector note multiplets that in the the anti-symmetric representation of the SU(3) is equivalent to the anti- section We also have one tensor multiplet corresponding to the number of the gauge nodes. The counting the rank of the gauge groups, and6.2 it is 5d Let us move on toon a the candidate 5d for a 6d description of the 5d JHEP05(2019)203 1 ’s 11 P 1 P next (6.9) (6.8) 1 by the P . The theory SU(12) C with a singular , then one cannot SU(15) 10 1 C P next to ’s that satisfies the 6d SU(15) 1 1 SO(16). C P P ⊃ 8 E Tao theory is given by the 6d Tao theory. 5 gauge group should be 5. Then, T 5 is attached to T , . Sp 1 P or [15] [17] ’s attached to 1 − − P SU . may give a U(15) flavor symmetry at least. 3 SU(6) SU(9) ’s vanish simultaneously, the 6d theory is at the – 50 – 1 − − P 3 Tao theory as well as the SU(15) flavor symmetry. Λ SU(6) 5 SU(15)  T 2 1 C −  SU(1) whose size is taken to be infinity. reproduces the correct number of the 5d vector multiplets Tao theory. This is also a very special case in the examples 1 ]. 1 6 P 5 T S [ as a . The 6d description in the tensor branch is Sp 1 P Tao theory, the dimension of the Coulomb branch moduli space is 6. or 4.2 5 T ]. SU 5 , ). 1 stands for one half-hypermultiplet in the rank three anti-symmetric represen- ’s, the total rank of the gauge groups should be less than or equal to 4. Then, 3 6.8 1 , then the 15 flavors on Λ P  1 2 . Due to the strong constraint of the shape of the base geometry, only pairwise  SU(15) The last case is the 5d Hence, we propose that the 6d description of the 5d When one increases the number of compact Let us then consider the case where one compact As for the 5d We call non-compact For example, the E-string theory on the tensor branch consists of 8 hypermultiplets and one tensor C 10 11 SU(15) Furthermore, the SU(1) does noteffectively a described dynamical by vector the multiplet. SU(9) gauge Therefore, theory the with 6d 18 theory hypermultiplets is in the fundamental multiplet, The flavor symmetry is SO(16) which is further enhanced to considered in section Note that there is no hypermultiplet in the anti-symmetric representation for the SU(1). 6d anomaly cancellation condition.to If there is no gaugeTherefore, symmetry it on contradicts the the compact flavor symmetry of the 5d quiver of ( find a candidate forgauge a anomaly cancellation non-trivial condition gauge withcompact the group 15 on flavors. the Ifthere compact we increase is the no number possible of the choice for the gauge group on the compact where tation. This 6din theory the on Cartan subalgebraThe of potential the U(1) 5d flavor symmetryagain which broken may by come the from anomaly the U(1) overall part of the U(15) is has one tensor multiplet. Hence,the the 6d rank anomaly of cancellation the condition with the 15 flavors gives us a unique possibility expected 6d flavor symmetryelliptic is fiber SU(15), giving weC the need SU(15) one non-compact algebraintersections on are it. allowed. Furthermore,should the We be non-trivial denote either gauge the algebra non-compact on 2-cycle the adjacent by superconformal fixed point. Thissignificantly [ in fact constrains the shape of theTherefore, sequence the of sum the ofmultiplets the in number the of Cartan the subalgebra tensor of multiplets the and corresponding the 6d number theory of must the be vector 6. Since the gauge symmetry. When all the compact JHEP05(2019)203 . ) 3 × or 3 6.11 (6.11) (6.10) ) after theory. SU(9) 7 − SU(3) T 6.10 − Tao theory with compactification is +1 1 N 3 S T Tao theory. Given the . Tao theory. The potential Tao theory. The potential N ] anomalies. Hence, the 6d 6 T 3 + 3] +1 U(1) flavor symmetries. One T + 3)] N N × 3 N T [9 (9 . su [21] 6) [ − − · · · − − ) is a natural candidate for the 5d 2 N (9 SU(12) 6.10 SU(21) – 51 – su vector multiplets in the Cartan subalgebra. The , which also agrees with the global symmetry of the − − I 5) ··· ··· − Tao theory is described by the 6d SU(3) gauge theory 2 N Tao theory, there is a very natural generalization to a 4 U(1) (9 SU(3) T 6 × (12) Tao theory. Furthermore, the flavor symmetry is SU(18) N SU(12) ,T 6 su − 5 T ,T 4 (3) 1 2 , which exactly agrees with the number of the Coulomb branch T Tao theory. The flavor symmetry after the su 1) SU(3) − +1 N 2 theory. Furthermore, the flavor symmetry of the 6d theory ( N , which also agrees with that of the 5d (3 ) is a candidate for the 6d description of the 5d 3 I 7 N T T 3 6.11 . Therefore, the 6d theory ( 3 U(1) × . tensor multiplets and N +3) flavor symmetries are all broken by the [global - gauge SU(12) compactification is SU(21) N Tao theory. In this case, there can be potential U(1) , which also agrees with the flavor symmetry of the 5d 1 1. M − N I 7 -plane in Type IIA string theory. However, it may admit a F-theory realization, and It is interesting to note that the rank of the gauge group of the 6d quiver ( It is easy to generalize this consideration by gauging the SU(21) flavor symmetry. The The 6d description of the 5d S T − ≥ 8 increases by 9.O Hence, this 6da theory possible configuration is may not be realized by a brane configuration with an SU(9 U(1) quiver theory ( N It has total number is moduli of the 5d and the other maylinear be combination of associated the toU(1) two the U(1)’s 21 has fundamental non-zero hypermultiplets. anomaly either However, from any U(1) general quiver theory by the successive gauging is number is 15, andmoduli of this the number 5d inthe fact agrees with5d the number ofmay the be Coulomb associated branch to the one bi-fundamental hypermultiplet between SU(3) and SU(12), symmetry. Due to theto have 6d 12 anomaly more cancellation flavors.flavor condition, symmetry Namely, the is a SU(12) 6d anomaly gauge free node quiver needs theory by gaugingIt the has SU(12) two tensor multiplets and 13 vector multiplets in the Cartan subalgebra. The total We move on to6d the description of analysis the forgeneral 5d a 6d description of the 5d with 12 flavors, and it has the SU(12) flavor symmetry. We then gauge the SU(12) flavor tion, it gives 2 +the 8 correct = number 10 for 5d the 5d vectorU(1) multiplets in the CartanU(1) subalgebras, symmetry which of reproduces the overall part of U(18)6.3 is again broken by Generalization the anomaly U(1) representation with two tensor multiplets coupled. Therefore, after the circle compactifica- JHEP05(2019)203 , . . ) I B +1) 6.12 N , and ), one (6.14) (6.13) (6.12) U(1) I 2 × 6.8 +2)(3 U(1) N . The lower theory. Due 1 + 6) (3 × P in the base +3 N 1 N . 1. The potential P 3 T + 9) . ≥ + 6] 1. It is interesting to N . N + 9] N ≥ Tao theories also admit B ] anomaly of some gauge [9 N Tao theory. The possible 3 6 N − T [9 Tao theory. + 6)] theory by gauging the flavor +3 − 3) vector multiplets in the Cartan N inside ) 6 N 3 +2 and − T 1 T global symmetries are all broken. N + 9)] N (9 P 5 +1) 3 N T N su 2 T N N (9 (9 SU(9 , which is exactly the dimension of the N 3) [ . SU(9 Tao theory with su . The sum of the numbers is +1) − [9] )[ N 2 2 +2 Tao theory with +7) − N (3 N N 2 N 2 − · · · − 3 (9 N compactification is SU(9 (9 3 +3 (9 T − · · · − – 52 – su 1 N N su 3 Tao theory. Therefore, the 6d quiver theory ( S SU(1) T ) does not have a Type IIA bran realization, but it ··· ··· +2 ··· ··· SU(18) N SU(15) 3 6.12 − (9) T (15) − su su and the last one stands for a non-compact ∅ 1 2 tensor multiplets and 1 SU(9) = 0. This case admits a Type IIA brane configuration that was (6) P 1 2 SU(6) N − su N − ] anomalies, the potential U(1) . The 6d quiver description is 3 3 5 Λ  can support the half-hypermultiplet in the rank three anti-symmetric SU(0) does not support a gauge algebra. 1 2 1 ].  1 +1 tensor multiplets and P 5 P N global symmetries are all broken by the [global - gauge N So far we have considered the 5d Let us finally turn to the generalization of the 5d Again, the 6d quiver theory ( In fact, the proposed 6d descriptions of the 5d where the first think of the case where considered in section F-theory realization of the 6d theory is The first SU(0) indicates an6d additional theory has tensor multiplet without anywhich gauge agrees dynamics. with the The numberto of the the Coulomb [global branch - moduliHence, gauge of the the 5d flavorit symmetry correctly after reproduces the the global symmetry of the 5d representation [ symmetry. The 6d theory is a quiver theory described by where the first group. Hence, the flavor symmetryand after it the agrees circle with compactification that isis of SU(9 a the natural 5d candidate of the 6d description ofmay the have a 5d F-theory realization. The possible configuration is The 6d quiver theory has subalgebra. The sum ofCoulomb the numbers branch give moduli spaceU(1) of the 5d The upper row indicatesrow the gives singular the type self-intersection of number the of elliptic each fibration compact overthe the same generalization. Byarrives the at successive gauging of the flavor symmetries of ( Each column represents a JHEP05(2019)203 ) ]. Z , 30 -plane. − 8 O = 0 into the 5d N ) web diagram with the critical number of p, q – 53 – 0) superconformal field theories whose Type IIA brane , Tao theory is nothing but the rank 1 E-string theory [ 3 T = (1 N = 1 gauge theories that are dual to each other. This diversity of = 8 flavors, which is understood as a circle compactification of 6d f N N . This is perfect agreement with the formal substitution of 5 Tao theory since the 5d planes and in the distribution of D5 and D7-branes as well as a part of the SL(2 While we provided the argument for the duality between 5d gauge theories by starting +3 − N 7 3 A A brief introductionA of Tao Tao diagram diagram isflavors a that generalization gives the ofconformal UV 5d theory completion ( of as 6d theory. A typical example is 5d SU(2) super- supported in part by Perimeterwas Institute done. for Theoretical Research Physicsthrough where at Industry a Perimeter Canada part Institute of andDevelopment the by is & work Innovation. the supported The Province work by of ofFoundation K.L. the Ontario of is Government supported Korea through in of (NRF) Ministry part Grants by Canada of the No. Economic National Research 2006-0093850. application” (2015) where agrant part FPA2012-32828 of from work the is304023 MINECO, done. from the The the REA workGrant People grant of SPLE Programme agreement H.H. under is PCIG10-GA-2011- of contract supported0249 FP7 ERC-2012-ADG-20120216-320421 by of (Marie and the the Curie the “Centro Action), grant de SEV-2012- the Excelencia ERC Severo Advanced Ochoa” Programme. The work of HH is also We thank Kang-Sin Choi, Hee-Cheoland Kim, Piljin Seok Yi Kim, for Xiao useful Liu, discussions.and Soo-Jong We discussions. especially Rey, Ashoke thank We are Masato Sen thankful Taki for toTheories early the collaboration Workshop at on Geometric SISSA. Correspondences of S.K. Gauge is grateful to “APCTP focus program on holography and its perconformal field theories whoseare brane also picture some is 6d not superconformalinteresting in field to the theories consider class with their considered no 5d here. obvious counterparts There brane and picture. the corresponding It 5d wouldAcknowledgments dualities. be O duality. A decoupling of thebetween same flavors 5d from gauge the theories dual which 5d theories would leads have to the another identical duality with 5d superconformal the field brane theory. picture of their 6d mother theory, one can have different class of 6d su- We have started with 6d representations are made ofOn their NS5-branes, tensor D6-branes, branch, they D8-branesmatter are and in hypermultiplets. general a written After single asvery a a rich linear circle group quiver of compactification diagram 5d the with and various 5d T-duality, theories we for have a found given a 6d theory originates from the choice in the resolution of two section T 7 Discussions In fact, this theory has been shown to be equivalent to the rank 1 E-string theory in JHEP05(2019)203 ] while 47 ] was done and repro- ) web configuration has 30 1) 5-brane charges, which ]. This confirms that the , p, q 10 = 8. There are twelve 7-branes f N 1) and (1 , (0 , 7 have the 5d UV fixed point [ 0) ≤ , f 7. Salient features are that the web diagram N ≤ – 54 – f N = 8 flavors, an explicit computation [ f N ) web plane. As 7-branes cross the branch cuts, the charges of the p, q . When the number of flavors is critical, all the pulled-out 7-branes inevitably . An example of Tao diagram for 5d SU(2) theory of 53 = 8 case has 6d UV fixed point. The corresponding ( f To see how infinite spiral shape arises, one first chooses some of the 7-brane branch N period. The resulting diagram is a web diagram of infinite spiral. of 7-brane in the7-branes ( change according to the monodromysee associated figure with the branch cut.cross For all instance, the chosensame branch after cuts one so revolution that making the a charge spiral of shape a of pulled-out a 7-brane constant becomes separation, the or a constant SU(2) gauge theory of duced the result ofTao diagram the indeed E-string describes elliptic a genus circle computation compactification of [ the 6dcuts, E-string and theory. then pulls out other 7-branes along the geodesic direction given by the charge factor (squared) of thecompactified 6d theory theory. which Infinite is spiralsthe correspond in to compactification turns an on infinite identified KK a as spectrumis circle. coming the named from KK to The mode be shapediagrams, of Tao resembles one diagram. the of a circle which Taoism Suitableenables can symbol 7-brane one be which to monodromies made is compute give out the why various of partition it yet (1 function via equivalent the topological vertex technique. As for E-string theory. Note thatthe the theories of different features than theform cases as with periodic web configuration in a spiral whose period is identified as the instanton the Hanany-Witten transition creates 5-branessuch along that the the direction that newlyTao the created diagram 7-branes 5-branes which are possesses are pulled acorresponding out bound constant to by period infinite one KK giving 7-brane. spectrumof rise 6d so to The SCFT. that KK resulting the mode, diagram Tao and diagram makes infinite describes a web a diagram circle compactification Figure 53 of various charges which are7-branes the and end different point colors of meansother 5-branes different of remaining charges. the 7-branes Middle: same are charge. six7-branes supposed branch are The cuts altered colored to are dots as cross chosen denote theyweb. as such crossed that being the Right: branch pulled cut. to the A infinity. last repeated step Right: process is generates the an to charges infinite pull of spiral out 7-branes associated with the branch cuts. In so doing, JHEP05(2019)203 , , H D 89 12 02 JHEP → JHEP SCFTs , (2017) Fortsch. , E , D + 6 09 JHEP JHEP E , , Phys. Rev. SCFTs , ]. JHEP d Commun. Math. 6 , string chains , SCFTs and (2015) 002 d (2015) 73 D SPIRE 6 11 (2015) 017] IN ][ 06 M-strings B 747 ]. conformal matter JHEP ]. , d Atomic classification of SPIRE SPIRE IN Strings of minimal Orbifolds of M-strings F-theory, spinning black holes and theories in six dimensions IN ]. ][ Phys. Lett. ]. 0) ][ Erratum ibid. , , [ (1 Elliptic genus of E-strings gauge theories ]. arXiv:1509.00455 SPIRE ]. On the classification of ]. SPIRE Anomaly polynomial of E-string theories [ IN 0) SCFTs IN Fusing E-strings to heterotic strings: , ][ – 55 – ]. ]. ][ D SPIRE 6 (2014) 028 SPIRE SPIRE = (1 IN IN IN ][ 05 SPIRE SPIRE ][ ][ ]. arXiv:1406.0850 (2016) 009 IN IN d N [ Holography for 6 arXiv:1502.05405 ]. ][ ][ ), which permits any use, distribution and reproduction in 01 [ ]. ]. JHEP SPIRE , IN SPIRE arXiv:1305.6322 ][ SPIRE SPIRE JHEP IN arXiv:1412.3152 [ , IN IN [ ][ ][ ][ (2015) 468 CC-BY 4.0 (2014) 126012 arXiv:1310.1185 arXiv:1504.04614 arXiv:1404.3887 63 More on the matter of This article is distributed under the terms of the Creative Commons Classification of [ [ [ arXiv:1404.0711 arXiv:1407.6359 D 90 (2015) 779 [ [ (2015) 294 63 334 arXiv:1411.2324 [ (2018) 143 (2014) 002 arXiv:1502.06594 arXiv:1408.0006 arXiv:1312.5746 02 multi-string branches 08 098 Phys. Phys. (2014) 046003 Phys. Rev. Fortsch. Phys. [ (2014) 003 (2015) 054 [ generalized ADE orbifolds [ Tao web diagrams also give a computational tool using topological vertex methods. For higher rank gauge theory or quiver theories, the configuration is basically same as The constant period is expressed in terms of the instanton factor of the 5d theory. It B. Haghighat, S. Murthy, C. Vafa and S. Vandoren, K. Ohmori, H. Shimizu and Y. Tachikawa, J. Kim, S. Kim, K. Lee, J. Park and C.B. Vafa, Haghighat, A. Klemm, G. Lockhart and C. Vafa, A. Gadde, B. Haghighat, J. Kim, S. Kim, G. Lockhart and C. Vafa, 6 B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, B. Haghighat, G. Lockhart and C. Vafa, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, L. Bhardwaj, B. Haghighat, A. Iqbal, C. Koz¸caz,G. Lockhart and C. Vafa, M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa,J.J. 6 Heckman, J.J. Heckman, D.R. Morrison and C. Vafa, D. Gaiotto and A. Tomasiello, [8] [9] [5] [6] [7] [3] [4] [1] [2] [13] [14] [10] [11] [12] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. is expected for aidentified circle with the compactification KK ofcircle, mode a more of 6d precisely, the the theory, radius compactification. and of the the Hence, constant T-dual it circle. period possessesshown in can a the be compactified Tao diagrams in the main text. JHEP05(2019)203 d 6 2 SO and , (2017) /T , 2 1 (2015) JHEP ]. T S Nucl. , 01 (2015) 218 , USp D-type ]. (2015) ]. d 09 D 92 6 (2014) 162 SPIRE ]. JHEP IN SPIRE 2015 , 10 RG flows ][ SPIRE IN JHEP theories on theories on IN D ][ , ]. 6 supersymmetric gauge SPIRE ][ (2015) 073 supersymmetric quiver 0) 0) (2015) 080 d , , ]. IN ]. Phys. Rev. PTEP JHEP d 5 theories 5 ]. , , ][ 07 , supersymmetric 12 SPIRE SCFTs ]. description of = (1 = (1 (2016) 080 d IN = 1 5 d SPIRE D SPIRE 5 ][ 6 10 IN SPIRE IN JHEP d N d N JHEP IN d N SPIRE ][ Anomaly polynomial of general territory , ][ Geometry of 5 , k IN ][ k S S arXiv:1503.08136 S ][ A new JHEP ]. [ arXiv:1508.00915 , [ Anomalies, renormalization group flows arXiv:1505.04439 ]. [ arXiv:1503.06217 SPIRE [ ]. IN theories SPIRE ][ (2015) 087 arXiv:1504.08348 IN Supersymmetric Casimir energy and the anomaly 0) – 56 – [ , (2015) 131 ][ Charting class 07 SPIRE arXiv:1501.01031 ]. (2015) 097 theories of class (1 arXiv:1505.04743 ]. arXiv:1408.5572 [ IN 12 [ [ Geometric engineering, mirror symmetry and 08 ][ (2015) 014 arXiv:1507.08553 ]. = 1 [ Tao probing the end of the world Evidence for C-theorems in SPIRE JHEP SPIRE Chiral theories of class 07 N Coulomb branch anomaly matching , ]. ]. ]. ]. IN IN Branes and six-dimensional supersymmetric theories (2015) 123 JHEP ]. , ][ SPIRE Duality walls and defects in JHEP 0) ][ , , Branes at orbifolds versus Hanany Witten in six-dimensions IN 11 (2015) 167 hep-th/9712145 [ JHEP ][ SPIRE SPIRE SPIRE SPIRE , SPIRE (2015) 142 07 (2015) 043B06 hep-th/9712143 = (1 IN IN IN IN (2014) 103B07 [ IN ][ ][ ][ ][ N 09 JHEP ][ , , d 2015 2014 JHEP arXiv:1505.00009 Instanton operators and symmetry enhancement in =2) , Instanton operators and symmetry enhancement in [ (1998) 180 N ( JHEP d , (1998) 003 Instanton operators and symmetry enhancement in 4 PTEP arXiv:1504.03672 PTEP arXiv:1504.05988 , [ 03 , → [ B 529 arXiv:1506.03871 0) , [ (2015) 052 (1 d arXiv:1505.05053 arXiv:1503.05159 arXiv:1506.06753 arXiv:1506.03807 arXiv:1408.6745 theories and exceptional gauge theories 019 083B02 gauge theories minimal conformal matter JHEP Phys. 045004 [ 6 and class S theories: part II [ polynomial class S theories: part I [ and the a-theorem in[ six-dimensional SCFTs [ 09 G. Zafrir, D. Gaiotto and H.-C. Kim, S.-S. Kim, M. Taki and F. Yagi, K. Yonekura, Y. Tachikawa, I. Brunner and A. Karch, A. Hanany and A. Zaffaroni, S. Franco, H. Hayashi and A. Uranga, A. Hanany and K. Maruyoshi, H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6 D. Gaiotto and S.S. Razamat, N. Bobev, M. Bullimore and H.-C. Kim, K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6 M. Del Zotto, C. Vafa and D. Xie, J.J. Heckman and T. Rudelius, C. Cordova, T.T. Dumitrescu and K. Intriligator, K. Intriligator, 6 J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, [33] [34] [30] [31] [32] [28] [29] [25] [26] [27] [23] [24] [20] [21] [22] [18] [19] [16] [17] [15] JHEP05(2019)203 , 02 (2015) (2015) ]. JHEP ] 12 , 06 ]. (2015) 157 SPIRE IN 12 ]. JHEP SPIRE superconformal JHEP , ][ , IN partition functions [ = 2 N SPIRE (1997) 56 JHEP ]. N IN , -planes d T hep-th/9605150 7 5 ][ as a linear quiver [ ]. (2015) 141 O SPIRE N B 497 T 04 IN SCFT’s ][ SPIRE hep-th/9902179 0) IN , [ ][ (1996) 562 arXiv:1403.3657 JHEP , , = (1 d ]. 5 Five-dimensional supersymmetric gauge Nucl. Phys. hep-th/9611230 , Five-branes, seven-branes and [ d N B 475 6 Mass-deformed (1999) 006 -triality SPIRE n Webs of five-branes and IN Topological strings and A hep-th/9608111 03 – 57 – ][ [ ]. dualities to (1997) 152 arXiv:0906.0359 d 4 [ JHEP Nucl. Phys. ]. SPIRE , ]. , IN fixed points from brane webs and Seiberg-Witten curve via toric-like diagram ]. ][ (1996) 753 B 492 ]. d n SPIRE ]. Lifting Type IIB superstrings, BPS monopoles and three-dimensional SPIRE gauge theories and IN IN d E (2009) 052 d ][ SPIRE 5 ][ SPIRE arXiv:1310.3854 B 388 IN SPIRE 09 [ field theories IN ][ IN ][ n ][ Nucl. Phys. E , JHEP , arXiv:1410.6868 Five-dimensional SUSY field theories, nontrivial fixed points and string Phys. Lett. [ (2014) 014 Brane webs, , ]. F theory and orientifolds 06 arXiv:1411.7903 arXiv:1507.03860 [ [ SPIRE arXiv:1410.2806 hep-th/9702198 IN arXiv:1509.02016 dynamics [ (2015) 089 field theories JHEP gauge dynamics 082 theories and degenerations of Calabi-Yau[ spaces 163 [ [ five-dimensional H. Hayashi, Y. Tachikawa and K. Yonekura, N. Seiberg, H. Hayashi, H.-C. Kim and T. Nishinaka, M. Aganagic, N. Haouzi and S.O. Shakirov, Bergman and G. Zafrir, A. Hanany and E. Witten, S.-S. Kim and F. Yagi, 5 F. Benini, S. Benvenuti and Y. Tachikawa, K.A. Intriligator, D.R. Morrison and N. Seiberg, O. Bergman and G. Zafrir, 5 G. Zafrir, O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, A. Sen, [46] [47] [43] [44] [45] [40] [41] [42] [38] [39] [36] [37] [35]