JHEP09(2009)052 July 14, 2009 s, Extended : August 30, 2009 : September 9, 2009 alize, when com- : Received eneralized s-rule. We flavor symmetry can be Accepted llows to easily count the 8 Published , , 7 , r we realize these theories at 6 gauge groups. 10.1088/1126-6708/2009/09/052 b E doi: SU = 2 superconformal field theories superconformal field N Published by IOP Publishing for SISSA = 2 N and Yuji Tachikawa a [email protected] , Sergio Benvenuti a 0906.0359 Supersymmetry and Duality, Brane Dynamics in Gauge Theorie We describe configurations of 5-branes and 7-branes which re [email protected] SISSA 2009 Department of Physics, Princeton University, Washington Road, Princeton NJ 08544,School U.S.A. of Natural Sciences, Institute1 for Einstein Advanced Drive, Study, Princeton NJE-mail: 08540, U.S.A. [email protected] b a c analyzed in a uniform mannerinfinitely in strongly-coupled this limits framework; of in quiver particula theories with

Abstract: Francesco Benini, pactified on a circle,recently new constructed isolated by four-dimensional Gaiotto.dimensions of Our Coulomb diagrammatic andfurthermore method show Higgs a that branches, superconformal with field the theories help with of a g Supersymmetry, Intersecting branes models ArXiv ePrint: Keywords: Webs of five-branes and theories JHEP09(2009)052 4 6 6 7 8 9 1 4 8 16 17 17 20 21 24 12 14 14 20 25 27 29 14 = 2 N deal of non-perturbative brane, physically realizing ing a system of D4-branes ple, four-dimensional s configuration can be lifted to flavor symmetry – 1 – 8 , 7 , 6 E ] theory 1 − theory theory theory N 6 7 8 theories theories A E E E [ n n T E E theories n E = 2 = 3 and the = 4 and the = 6 and the -junction N N N N N theories -junction and n E 5.1 Formalism 5.2 Rank-1 5.3 Higher-rank 4.1 4.2 4.3 4.4 4.5 Higher-rank 3.1 Classification of3.2 punctures Generalized s-rule 3.3 Derivation of the generalized s-rule N 2.1 2.2 Coulomb branch 2.3 Higgs branch 2.4 Dualities and Seiberg-Witten curve supersymmetric quiver gauge theoriessuspended between can NS5-branes in be type IIA implementedM-theory, string in us theory. which Thi D4- and NS5-branes merge into a single M5- 1 Introduction Brane constructions in string or M-theory can tell us a great 6 Future directions A Seiberg-witten curves B information about supersymmetric gauge theories. For exam 5 S-dualities and theories with 4 Examples 3 General punctures and the s-rule Contents 1 Introduction 2 JHEP09(2009)052 . e tion ]. The 1 semi-infinite N M5-branes, and the , suspended between 6 S5-branes and horizontal h is only possible when x It is natural to combine rom this point of view, finity. All branes fill the to define a complex coor- section with another M5- ics of the system [ 6 -branes wrap the cylinder ors on the worldvolume of x uration is shown on the left multiplet charged under the . Right: the compactification of , the stack of three M5-branes is 3 , M-theory circle. 2 , 1 t = t full punctures the , say ⊙ t , with the direction – 2 – M5-branes with more general configurations of . Vertical lines stand for NS5-branes extending . It was found that marginal coupling constants 11 1 x N , and ] theory, with three full punctures. It has no obvious type 1 − as the compactification of M5-branes on a sphere with defects N 1 A [ ] that the system can also be seen as a compactification of 2 T , and the defect corresponding to intersecting ). In this particular example, when all of the vacuum expecta maximal puncture simple punctures 11 ix or , and at three values of . The example shown has two SU(3) gauge groups, each with thre t + 3 , 6 2 , full x 1 , 0 marks the x • while horizontal lines are D4-branes extending along simple puncture . In this representation, both the intersections with other 2 5 . Left: a brane configuration in type IIA. Vertical lines are N , . Left: the system in figure = exp( 4 t x The system lifts to a configuration of M5-branes in M-theory. It was recently shown in [ M5-branes on a sphere by a further change of coordinates whic one can consider compactifications of are encoded by the positions of the punctures on the sphere. F M5-branes the lines are D4-branes. Right: its lift to M-theory showing the two NS5-branes, or endingspace-time, on an NS5-brane and extending to in M5-branes corresponding to the fundamental hypermultiplets, and onetwo gauge bifundamental groups. hyper the direction along the M-theory circle, values (VEV’s) of theparameterized by adjoint scalar fields are zero, three M5 dinate Figure 2 Figure 1 intersected by one M5-brane. setup is schematically drawn in figure two infinite ends canthe be M5-branes. thought of as Webrane call conformal the defect the operat defect corresponding to the inter along N The symbol IIA realization. the Seiberg-Witten curve which governs the low energy dynam all of the gaugeof couplings figure are marginal. The resulting config JHEP09(2009)052 ]. 1] 4 , has 6 oups, ert on boxes. x ) flavor ar quiver N N ]. This last 7 ]. Our proposal 8 ]. Furthermore, it 3 1) 5-branes on [1 , boxes. Recall that D5- ]. The Coulomb branch al superconformal field full punctures, depicted 6 akes manifest the moduli y further, and it will be N , ch the different properties ], is isolated in that it has 5 s appropriate [ o not change the shape at ere do not have moduli. It onger has a realization as a aux consisting of 1 s on the sphere. Instead, we urations which realize five- -infinite 5-branes on suitable − ]. In particular it provides a gravity dual was found in [ ranes around, as was the case sional versions of the theories al superconformal field theory ,1) 5-branes in type IIB string eb. 2 on [ N y because the direction rg duality [ local 7-branes, which are known A [ he web construction, once 7-branes T ], in the sense that compactification on ]. In order to study configurations with 1] 7-branes, and (1 (1,1) 5-branes. 1 , − 11 N , N -th 7-brane. This leads to a classification in 3 ] will be crucial. We will need a generalized i A [ 6 M5-branes on a higher-genus Riemann surface T – 3 – N ] in the context of string junctions, which we will parallel 5-branes into smaller bunches, composed by ] in the framework of [ 14 N – 10 . This theory, called 12 , 2 flavor symmetry, originally discussed in [ 9 8 ]. We will see that each of the punctures corresponds to a , 1 7 NS5-branes, or of 5-branes on the , − 6 i N N k E theories using a quiver gauge theory consisting only of SU gr A ]. It arises in an infinitely strongly-coupled limit of a line 8 [ , , in fact labeled by Young tableaux with 2 7 T ] that there are more general punctures or defects one can ins flavor symmetry, because each full puncture carries an SU( N E 4 3 symmetry [ ) 8 , N 7 , 6 E D5-branes, of N ]. It then gives a uniform realization of the four-dimension It was shown in [ It is clearly important to study the properties of this theor The realization of the field theory through a web of 5-branes m 2 realizes the theory 5-branes, and then end 1 i branes terminate on D7-branes, NS5-branes on [0 only two ends, which can accountpropose at that most configurations two of special intersecting puncture theory D5-, give NS5- the and five-dimensional (1 version of 7-branes. Therefore the resulting systemto has lead mutually non- to enhanced symmetry groups when their combination i on the right hand side of figure punctures. The most fundamental one is the sphere with three S version of the s-rulereview studied in detail, in and [ its “propagation” inside the 5-brane w gauge theory, as a natural generalization of Argyres-Seibe observation will naturallydimensional lead theories us with tois, propose as 5-brane stated config above,in that [ these configurations aretheories five-dimen with new realization of the general punctures, application of the s-rule [ is the natural building blockcorresponding from to which the the compactification four-dimension of nicer to have anothercan description be of understood the easily. samebrane One theory configuration problem is in from that type whi this IIA theory string no theory. l It is basicall corresponds to normalizable deformationsinfinity. The of Higgs the branch can web7-branes: be which seen it d by then terminating corresponds allin to semi moving type the IIA endpoints with of D4-branes 5-b ending on D6-branes. the M5-brane worldvolume, naturally labeled by Young table can be constructed as a generalized quiver gauge theory. Its along the line of Argyres, Seiberg and Wittig [ space, as happens with the more familiar type IIA constructi bunch of symmetry, as shown in [ no marginal coupling constantshas because (at three least) SU( points on a sph This kind of classification arises straightforwardlyhave from been t introduced: we can group k terms of partitions of JHEP09(2009)052 . 4 ], ]. 1 N x 1 2 − N . This A [ 3 T f ] theory. When 1 -junction, where theories. − , which might be N 5 n N Higgs branches. To ], see figure E A ane configurations as [ very far away, one is 15 means that the brane T i n we will use in table th generic punctures can g on them overlap, there e 7-branes, thus realizing 7 8 9 − − − e 5-branes on the 7-branes cs apart from the decoupled rms of a dot diagram, that f the nes on appropriate 7-branes. r to as the m the center of mass motion. governing the supersymmetric by considering a junction of lized s-rule will be detailed in − 2 ation using the machinery in [ 1) 5-brane [ angle 5 6 , − 4 − − − − – 4 – flavor symmetry. In section signifies that the brane extends in the corresponding ) 5-brane webs. To get a 4d theory, the direction 8 flavor symmetry arise in the strongly-coupled limit , we review some aspects of the 7 − , 8 , p, q 6 7 B , E 6 gauge groups. We conclude with a short discussion in ] theory E 0 1 2 3 1 − − − − − −− − −− − − − − − − − SU . The most basic object in the brane-web construction is the to study how we can use 7-branes to terminate 5-brane junc- N i x 3 A we write down the Seiberg-Witten curve for the theory on the [ T A under the column labeled by a number D5 NS5 − 1) 5-brane 7-branes , 1) 5-branes, which we argue is the five-dimensional version o , (1 , which naturally leads to our identification of certain 5-br . In appendix -junction . Configuration of suspended ( 6 4 -junction and N Finally, the web construction makes it clear that a theory wi The paper is organized as follows: we start in section N direction. arise as an effective theory by moving along the Higgs branch o is compactified on a circle. The symbol extends along the direction junction between a D5-brane, an NS5-brane and a (1 two 7-branes are alignedcan in be such Higgs a way branchand that directions moving the corresponding the 5-branes toleft extra endin breaking with pieces th a apart. puncturegeneric with punctures. By multiple moving 5-branes the ending extra on the pieces sam section of quiver gauge theories with Table 1 2 by showing that the SCFTs with There, the symbol system is rigid and doesAccordingly, not it allow does any not deformation, givecenter apart rise of fro to mass. any We 5d can low consider energy a configuration, dynami which we refe multi-junction. Finally in appendix tions, realizing more generalconfigurations type of of punctures. these The systemswe s-rule will will also describe. besection Several formulated examples in illustrating te the the five-dimensional genera theories with read separately, we provide further checks of this identific D5-, NS5-, and (1 We begin by summarizing the type IIB or F-theory configuratio 2.1 We study the flavorsee symmetry the and Higgs branch, the we dimensionsWe need proceed of to then terminate Coulomb in the and section external 5-bra JHEP09(2009)052 , to at = 2 1 acts 1 (2.3) (2.2) (2.1) S S N N ,...,N Z 4 , of vector- × h N n Z D5-, NS5- and full punctures ating one vertex N and ]. It is a 4d v 4 where n , 2 N li of the 4d theory. The Z er . × rane web, and connecting kground vectors, therefore 2 heory compactified on this ry. The 5d vector multiplet . the Wilson line along N ]. We devote the rest of this nder this duality, the single al diagram are adjacent, see − Z 2 2) / 2. The quaternionic dimension Coulomb branch are 3 ] theory [ 3 N 1 2 − − C − − . 1) 5-brane. Center: multi-junction of d ) N N 2 , 2 is A theory. Right: the dual toric diagram of N [ 1)( = 1. 3 d 6 T − E = v N αβγ ( n αx,βy,γz ( -junction configuration compactified on − = M5-branes on a sphere with 3 – 5 – whereas the multi-junction of h → N 3 n N ) 1) 5-branes meet, see the diagram in the center of , C = (1 ], and is ] theory constructed in [ Coulomb 1 4 . It is known that this procedure produces the toric x,y,z N − 3 ( Higgs M N ) global symmetry. Therefore the flavor symmetry is at C A , the web of 5-branes, or equivalently M-theory on the non- M N [ 1 H T S dim 1) associated mass parameters. The complex dimension of the − dim N = 3 5-branes — this realizes the -th roots of unity such that NS5-branes and by N N 3 N C , with 3( are 3 , with a particular triangulation. ) ∈ N ) N Z . Left: single junction of a D5, an NS5 and a (1 . For a given 5-brane web, the dual diagram is formed by associ × 3 α, β, γ N When compactified on Let us first recall the salient properties of the x,y,z Z D5-branes, / 3 gives a 4d vector multiplet, and the real scalar pairs up with compact Calabi-Yau, gives rise at low energy to a 4d field theo form a complex scalar.parameters This and is moduli true of for themain both web dynamical proposal end and up of in bac this parameters and paper modu is that the low energy gives rise to the figure (1,1) 5-branes corresponds to the blow-up of the orbifold Here on ( three bunches of The scaling dimensions of the operatorsand parameterizing the the multiplicity of the operators of dimension N least SU( C Figure 3 section to perform various test of this proposal. each of which carries an SU( to each face, bothtwo compact vertices and whenever non-compact, thethe in corresponding the rightmost faces original diagram in b ofdiagram the of figure origin a non-compactthreefold Calabi-Yau threefold, is and dual thatjunction to M-t corresponds the to original the five-brane flat space construction. U isolated SCFT, obtained by wrapping Coulomb branch is of the Higgs branch can be easily found from the effective numb and hypermultiplets calculated in [ JHEP09(2009)052 ] 1 (2.7) (2.6) (2.5) (2.4) s. In the artan gen- ther it changes the ed as the number of , . eformations as well, we 1) ore general configurations 2 -junction picture. First of − ). Deformations of the web rthogonal spacetime filling orresponds to normalizable e-functions. Practically it s. A semi-infinite D5-brane , 6 − realized on the worldvolume N ts. These are background or N s in the web contributes two the edges. Then we find t two rigid translations acting , ,x ome finite distance. The same ( tablishes one relation between 2) te 5-branes terminate, without 5 N x − 2 3 + 4) ) global symmetry, each realized on N = 2 N N 1)( internal lines 2 1 non-normalizable deformations which n 1)( − − − − N ( N N ( = internal lines – 6 – ] to study some 5d conformal theories, or in [ 8 = junctions n , n Coulomb = 2 2 M N C deformations = n dim deformations semi-infinite 5-branes extending in the different direction n junctions n -junction has three copies of SU( N semi-infinite 5-branes has N ). ) global symmetry factor and are in correspondence with its C N -junction configuration, N 2.2 N The dimension of the Coulomb branch can directly be determin These are the properties we wish to reproduce from the For the break the SU( reproducing ( all notice that the closed faces in the webintroduced diagram — in this the will next be sections. true even in the2.3 m Higgs branch In order toneed see to the terminate the Higgs semi-infiniteprocedure branch, 5-branes was on which adopted, 7-branes corresponds at for s instance, to in local [ d Each bunch of which means (where D4-branes end on D6-branes)can to end, study without 4d breaking gauge theorie any further supersymmetry, on an o erators. In the toric diagram, these correspond to points on the worldvolume of are described by realdynamical scalars fields which are depending inmeans on 5d the that vector normalizability multiple a ofboundary mode the conditions is at wav background infinity.real or degrees Each of dynamical of freedom, the depending andthe single each positions on of junction of whe the the junction internalon points. 5-branes the es We system then need as to a subtrac whole. Therefore next section we will introducechanging 7-branes the on low-energy which theory. semi-infini Thenof the the global 7-branes. symmetry is 2.2 Coulomb branch Let us next count the dimensiondeformations of of the the Coulomb web branch, inside which the c two-dimensional plane ( JHEP09(2009)052 - )- er ee he 6 (2.8) ,x 5 ]. This pt was x 16 -junction l 7-branes. N me this just me way as it KK monopoles plane. The compact . 9 , 2 8 , verything becomes pure l 5-branes are coincident, 7 ] 7-brane as obtained by onal to the 5-brane is, as − er of the 5d theory, such ymmetries can be seen as x ing the previous setup, or to a point in the ( bal symmetry is unbroken, ctor multiplet. The motion can be mapped to different Doing a T-duality along the N p,q volume, and states of the 5d 2 nce, consider a configuration − 2 ver along the Coulomb branch various y whose toric diagram is the dual ed symmetry group. The simplest N = 2 theories, see for instance [ 3 In this case the central ) flavor symmetry. N = k r ). Issues related to the s-rule will be 1 i 1 − 2.3 =1 i X N -junction configuration, we have M-theory on – 7 – 1 + 3 N ]. Each of them has a U(1) gauge theory living on − 8 ) 5-brane can end on a [ , N we will discuss many specific examples. 7 = p,q 4 Higgs )-plane of the gauge field on the 5-branes pairs up with the thr position to give a Higgs hyper-K¨ahler moduli space. Moreov 6 M 9 , H ,x 8 , 5 ]. After removing the decoupled center of mass motion we get t 7 ] in a T-dual setup, the low energy 5d dynamics on a 5-brane sus x 1 )-duality of type IIB string theory. The configuration we ado . In section x 6 dim 3 Z , parallel 5-branes can fractionate on the 7-branes, in the sa D7-branes at a point, showing SU( and the 5-brane web is completely suspended between paralle separate simple junctions, free to move on the k 1 7-branes of the same type collapse to a point. In a duality fra N N k after having compactified it, we get a system of D6-branes and 4 x As analyzed in [ Once all semi-infinite 5-branes end on 7-branes, the global s The dimension of the Higgs branch is maximal when all paralle Mass deformations reduce the Higgs branch dimension. Moreo 1 corresponds to plane, gauge symmetry enhancement willtheory occur fall on naturally their under world representations ofcase this is enhanc when mixed Coulomb-Higgs branches originate, as in more familia can split into shown in table application of SL(2 D7-brane, and more generally a ( in type IIA.geometry: This a can toric be conicaldiagram Calabi-Yau further threefold to uplifted singularit the to M-theory, 5-brane where web. e For the scalars encoding the component along the ( full structure of the moduli space could be studied as well. setups of string orwithout M-theory 7-branes where by all various of themdirection dualities. has been For moved insta to infinity. discussed in section 2.4 Dualities and Seiberg-WittenThe web curve of 5-branes in type IIB string theory we consider here its worldvolume. When 7-branes of various type can collapse pended between a 5-brane and aof 7-brane does the not 7-brane contain any in ve that the the direction length of ofkept the the finite. 5-brane 5-brane On is can thebefore, not be other a a taken hand non-normalizable the to paramet deformation. motion infinity of recover the 7-brane orthog dimension of the Higgs moduli space: explicitly realized on the 7-branes [ that is when all massand deformations we are are switched off at and the the glo origin of the Coulomb branch. each bunch of Reassuringly, it agrees with the known value ( happens in type IIA [ JHEP09(2009)052 , 3 ) N -th 7- . The i y, which rises in our ]. A similar 2 ]. Since there ) flavor group. 4 semi-infinite 5- n , -junction theory 2 N N sider the 5d theory of them on the i 5-branes, we can group iven by the type of the k of the nctures [ e wrapping a holomorphic N brane, we can group some hree points on a sphere, it anes wrapping vanishing 2- ] to understand the possible bunch of e type IIA description. We lassification in [ hree bunches corresponds to ing to the fact that the flavor -branes, of which we will see 17 ssified by partitions. Partitions which requires the 5-branes to 0) theory are classified by Young , 2 , and end (2 N 1 − = of 5-branes carries a U( ] theory. The flavor symmetry SU( N i k 1 k A 5-branes is not realized on the low energy − i N N P A [ – 8 – T . This chain of dualities is closely related to the singularity over the curve thus obtained describes with ∗ 1 M5-branes on the sphere with 3 generic punctures C N A × N of , and thus confirm that it indeed gives the SW curve of C } A i 2 there are more than one possible kind of punctures in . Type IIB string theory on a circle is dual to M-theory on ] in the suitable limit. 4 k 2 { x , are still expected to be mapped to pure geometry in M-theory = 4 super Yang-Mills theory. N > 4 N in appendix ] when 1 boxes. We will see below that such classification naturally a . We can then send the IIB circle to zero to obtain the 4d theory 4 ∗ S , C N 2 bunches made of the same number 0) theory. Wrapping . Our proposal was that at low energy this gives a 5d field theor , × n ] theory found in [ N ∗ 1 (2 Z − C 1 N × − A N [ N A set of Another chain of dualities which we use is the following. Con Generically, instead of ending each 5-brane on a different 7- A T Z We thank D. Gaiotto for suggesting this possibility to us. / 2 3 gives rise to an SCFT, up to some restrictions on the type of pu C after further reduction to 4d flows to the brane. Thus the possible kindcan of then punctures be are naturally represented cla by Young tableaux, reproducing the c tableaux with overlap, and reduces the number ofsymmetry mass carried deformations by accord the bunch gets reduced. Given a bunch of construction involving D3- and D5-branesboundary was conditions employed in of [ the 3 General punctures and3.1 the s-rule Classification of punctures According to [ are no marginal parametersis associated an to isolated a SCFT,punctures. configuration but The of with possible t type a of more punctures in general the global symmetry g the however not to a toric geometry. compactified on a circle along a torus. The web of 5-branes is then mapped to a single M5-bran curve on construction. For that purpose, it is sufficient to consider a M5-brane now wraps a curve on branes extending in theeach same of direction, the because punctures. each of the t 5-branes and end them together on the same 7-brane, them according to a partition one described above: the fibration of However a diagonal U(1) for the whole set of is then realized oncycles. the Webs homology of of 5-branes themany which examples singularity, in require by section mutually M2-br non-local 7 compactified on the type IIB mirrorwill of use the these toric well-developed Calabi-Yau techniques singularity to in find th the SW curve JHEP09(2009)052 a ). 2.8 (3.1) is the k n es, which ]. In simple 5 . } . N } } ]. 1 , where itrary choice of 3 { This question was
21 ) . Then the counting 3 {z – k times y that the parallel 5- n b . In order to count the ...J 18 A,...,A| s necessary to correctly 4 hey effectively decouple . { U( more than one D3-brane =1 multiple 5-branes ending f 7-branes. We saw that i i . [ k } at a theory with punctures at is, we cannot have more ding of the s-rule, to which gree of the punctures. This ion i e extra 5-brane pieces. e 7-branes and move the cut i k ows that the effective theory ely disconnected. One is left ee for instance [ ces are taken very far away, Q ed to the puncture of “lower k pplies to general intersections f a theory with only punctures { ixed Coulomb-Higgs branches J =1 S i X on )-dual version of this statement is ] in the construction of 3d gauge 6 − Z 1 (decoupling the center of mass). : , p H − NJ M N = we get one of the three terms in ( i } k we mean the partition ) N i } 1 b { − A – 9 – { J ( J =1 i X = p H M H 5-branes. This agrees with the flavor symmetry associated to dim k ]. The full puncture corresponds to the partition 2 ] in the context of string junctions in the presence of 7-bran , which was originally introduced in [ 14 – 12 s-rule ]. Then the flavor symmetry of the puncture is 8 ) 5-branes suspended between different numbers of 7-branes. It is easy to count the dimension of the Higgs branch for an arb The s-rule states that there are no supersymmetric states if It is worth stressing how this construction makes it clear th p,q Here and in the following, with the notation 3 It is easy to check that for the partition punctures. The internal web always contributes theory [ cases it would be enoughthan for one D5-brane us between to a D7 use and a an T-dual NS5. version Any of SL(2 it, th Many examples and comparisons withdimension known results of are the in Coulomb sect branch,we we devote need the a next precise subsection. understan of a lower type is effectively embeddedof into the the Higgs maximal branch o type,when i.e. two obtained or using morebranes the 7-branes ending maximal on of them number a overlap, we o pieces puncture can break around, are the to 5-branes aligned on realize in th i.e. Higgs such when branches. a one When wa gives thefrom large the extra rest VEV’s pie of and thewith web, goes and a to some puncture 7-branes the composed can Higgson of be the left branch, a same effectiv smaller t 7-branes, number thatalong of is a the 7-branes, more Higgs and generic branch puncture. undertype”, This consideration plus sh is some the decoupled SCFT modes relat describing the motion3.2 of th Generalized s-rule In general the Coulomb branchis gets due reduced to by lowering the thetheories de in order to correctlywhen account D3-branes for end the on dimension D5’s of and m NS5’s, and lateris studied suspended in between e.g a givendescribe pair the of dynamics of D5 D4-branes and between NS5. D6’s The and NS5’s, same s rule i also an s-rule. However we needof a version ( of the s-ruleanswered which a in [ Each puncture contributes according to its defining partiti number of bunches of puncture, found in [ of legs gives for the Higgs branch at the puncture JHEP09(2009)052 and 2 n e number, say ot represent a mal polygon. A separated by black , in the sense that lygons. Consecutive ndary conditions in s in the dot diagram. t must satisfy an extra or not, respectively; 5- , depending on whether changing 5-branes with It is always possible to such that the four edges ne a tessellation of such -branes, the dot diagram ny derivation to the next 2 are adjacent. The lines in in the dot diagram, which toric diagram. Moreover it encodes the parameters and ual to the web: we call it a parallel 5-branes end on the ost in the dot diagram. The s on space-filling 5-brane and h face (even non-compact) in to understand how the s-rule integers If it is an triangle, the three edges must be composed by the sam – – In the dot diagram, we can represent the fact that to a black one. These same 7-brane by separating defined momentarily. We say that these segments supersymmetric configurations arethe now web constrained. determine a convexConsecutive The polygon, segments can bou made be of separated the bythe white external corresponding or line black parallel dots 5-branesbranes end which on do the not same enddots 7-brane on in the the same dot 7-brane diagram. turn into segments Then we proceed tosegments tessellate the separated dot by diagram aminimal with polygon white minimal can po dot be act eitherconstraint: a as triangle a or a single trapezium, edge and i of a mini For the sake of clarity, we prefer to state the rule, leaving a In the presence of 7-branes, we proceed as follows. • • subsection. The s-ruledot is diagram better visualized on the diagram d toric geometry. Given ais web constructed of on 5-branes a which squarethe do lattice web, and not by a associating require line athe 7 connecting dot dot two to diagram dots eac must whenevergo the be back two orthogonal and faces to forth from theorthogonal the 5-branes lines web and in diagram vice the to versa.moduli the web. Notice dot of that diagram, the the by 5d ex webboundary field diagram conditions theory, in whilst all theform this web a information determine convex is the polygon,polygon. l external whereas lines In the this detailsis particular of completely case, triangulated the the by web dot minimal determi triangles diagram of is area really 1 a string junctions are essentiallypropagates the inside same. complicated We 5-brane willbe webs. also understood need using Both the issues, brane creation c effect [ junctions are trivalent. we can directly borrow because the supersymmetry condition explained in section JHEP09(2009)052 , 1 n , Lower 2 n 5-branes simply go le edge of minimal ane web. White dots 1 n − same 7-brane, they have nal consecutive segments 2 n the dot diagram. When und, we can go back to the ns such that each 5-brane n te triangulation of the dot rresponding 5-branes have to trivalent junctions can occur. s are mapped to intersections terminate on another one and 2 triangles are mapped to the l s-rule when applicable. raint. Notice that tessellations as no SUSY vacuum and some / n he unconstrained case: different “flop transition”. along the Coulomb branch. One agation of the s-rule inside. In the ey represent s-rules, and come from 2 square, given the same constraints the other one, even though there is × inside the dot diagram, and has to be is mapped to the intersection of three bunches jumps over – 11 – n junctions can occur, and 1 n there are some examples of minimal polygons with the is mapped to the intersection of four bunches of 4 1 n another 5-brane, meaning that they cross it without ending. 2 triangles and only black dots. / propagation of the s-rule and 2 n jump over 5-branes in which only 1 . Upper line: examples of minimal polygons with the dual 5-br n , 1 respected. In general the tessellation of the dot diagram leads to inter which are separated bypolygons. white dots, This and is a they again act as a sing n parallel 5-branes in which, because of the s-rule, only Once a consistent tessellation of the dot diagram has been fo Let us stress that when multiple parallel 5-branes end on the • − n 2 If no consistent tessellation exists, it means that the web h 7-branes have to beends added. on its Notice own thatdiagram 7-brane, for in the terms boundary prescription of conditio area gives 1 back a comple web diagram mapping lines to orthogonal 5-branes. Area 1 usual junction of three 5-branes.of 5-branes The in other which, minimal because polygon has of to the s-rule, cross a it. 5-brane cannot We say that the 5-brane line: two examples of different allowed tessellations of a 2 no real displacement. In figure separate consecutive segments whicheither act multiple as 5-branes a ending singleweb, edge. on some 5-branes the Th same 7-brane, or prop on the external edges. The webs of five-branes are related by a Figure 4 dual web. For instance, a triangle of edge of A trapezium of edges n straight crossing everything. to be coincident. Asthe we s-rule saw propagates this and isbe there represented coincident are as by white well, white dots simply dotsrespecting inside, because the i the of s-rule are the co not geometric unique,tessellations const just are as related they by are changing not parameterscan in or t finally moving check that this prescription agrees with the usua JHEP09(2009)052 o for (3.2) (3.3) . works p,q 16 X , 15 , 14 . It is easy to 5 , e 12 , NS5 which becomes ) monodromy osses it, it can happen 10 Charge conservation at Z of the NS5, all but one ve a modulus related to , annihilation mechanism, 7 to the other side of the l junction between a D5, iagrams we show. This is esult of 5-branes jumping ranch cut, it is generically . Suppose we cook up such y counted from the web of m: it gives mass to scalars 6 ave one modulus controlling )-strings, and we shall soon configuration, no more than rved geometry, in which the Y must be broken; moreover ! ee many examples in the next s we choose the cut in such a their size: they are completely is transformed as faces, and so effectively freezes us subsection. Mostly the same p,q n figures 2 pq τ p − − 1 pq 2 ) 5-branes. We will also emphasize the b d q + + 1+ p,q

cτ aτ ], which used the brane creation/annihilation = – 12 – → 14 ] 7-brane creates an SL(2 ! τ ], is measured counterclockwise. We represent the p,q 7 c d a b

= ] in the case of the web of ( p,q 14 X – which was not clearly mentioned in the previous literature. 12 1) 5-brane, and suppose the D5 ends on a D7. Let us take the cut t given by , ]. Let us start by reviewing how the brane creation mechanism τ 6 1) 5-brane when crossing the cut. The process is shown in figur We follow the derivation given by [ When a 5-brane cannot end on another one and therefore just cr , run away without crossing theNS5: 5-branes. We when can they then move crosshowever the the the D D5 NS5 disappears nowa by crosses (1 the the brane branch creation/ cut. We are left with an the axiodilaton monodromy as a cut originating from the 7-brane. Then see that the samepropagation rule of applies the s-rule to the web of ( when a 7-brane crosses a 5-brane. A [ understand the s-rule in itsone D5-brane standard can formulation: stretch in between a aforbidden D7 SUSY configuration. and an NS5. WhenD5’s See moving figure remain the as D7 5-branes tothe stretched the junction between other requires the them side D7 to and be anti-D5’s, the showing NS5. that SUS mechanism of [ that a face inits the size. 5-brane web This getsin visually frozen vector shows and multiplets the does which effect parametrizesome no the of moduli. longer size the ha Indeed of Higgs it internal over can mechanis other be ones, geometrically checked do then notfixed open have by faces, moduli the as related structure to the oftheir r the web. size. Only The (and dimension all)5-branes: closed of it faces the is h Coulomb thesection. branch number of is Example closed then of internal consistent easil faces. tessellations We can will be s 3.3 found i Derivation ofLet the us generalized now s-rule derive the generalizedrule s-rule was stated formulated in the by previo [ which, following the conventions of [ when we cross the cut.transformed by Accordingly, the when monodromy a and 5-branea crosses it schematic changes a way its b to slope depict5-brane in the just the correct follows d situation a inway geodesic. the that true In they cu all doan our not NS5 web intersect construction and the web. a (1 Consider first the usua JHEP09(2009)052 s propa- same D7, 1) 5-brane , tion has been ame from the . comes a (1 ven though the s-rule is e D5 ends on the NS5 USY breaking is apparent 1) 5-branes meet the same , sign. The dotted line shows ing to non-supersymmetric ⊗ ygon is acceptable. The 5-brane sappeared because of the brane ation is still supersymmetric ve. The generalization to more both (1 t was shown as if it jumps over the n does not respect the s-rule either. let us show that the s-rule ond figure. The polygon violates the onding polygon. ions are the same as before because of 1] 7-brane. Each D5 can end on a different , – 13 – 1. 1) 5-branes. This is as in the previous example. The novelty i − , τ → τ . Consider a configuration where two D5-branes, ending on the 1) 5-branes still carry a constraint: they behave as if they c 7 , we also showed the dual dot diagrams, whose precise construc 6 . propagation of the s-rule. Left: non-SUSY configuration. E . Left: non-SUSY configuration, which violates the s-rule. S . brane creation mechanism for 5-branes. On the left an NS5 be 1] 7-brane — in particular they cannot end on the same 5-brane , , see figure In figure in the second figures,Right: where an SUSY configuration. anti-D5 is Onlywhich one present. D5 cannot The ends end on polygo on theother NS5. the one. other The pol one and therefore just crosses i 5-brane. The propagation ofs-rule the as well. s-rule is Right: manifest a in SUSY the configuration, with sec the corresp tensions do not balancewhile anymore. all other On onesafter the cross pulling other the without hand, D7 terminating, to if the the only configur other on side. because it meets a D5, which comes from a D7-brane shown by a respected where the D5’s meet the NS5’s, it is violated where the cut associated tocreation/annihilation the mechanism, monodromy. but On thethe the boundary cut, right condit along the which D5 has di same [1 Figure 7 Figure 6 Figure 5 meet two NS5-branes, endingNS5, on resulting the in same [0 twothat (1 the two (1 given above. It isconfigurations, easy do to not check respect thatinvolved the configurations dot s-rule is diagrams prescription straightforward. we correspond ga In particular gates JHEP09(2009)052 3 = C = 11. M we also H C 7 . M 8 H emoving one 1) 5-branes when global symmetry, ) (4.1) , 3 global symmetry is kn 5’s. We are left with δ 3 aligning the 7-branes branch. In the web of there is a single Higgs e junctions. oup is broken to U(1) global symmetry, three jm 3 heck of the s-rule. The ǫ up. 3 ns, see figure = 1 and dim m ld theories will be made il ǫ lexity, where all rules pre- 3 C and one not. Generalizing ± ious subsection. 2 S-dual frame. Namely, there is from the s-rule: m 1] 7-brane. In figure M f the flavor group, dim m , + C ± . If the two masses further satisfy . It has SU(3) 8 kn 3 = ǫ 1 jm m δ -th SU(2) flavor symmetry. The masses il i ǫ 2 . The dimension of the Higgs branch is the = 3 the visible SU(3) 3 m ]; in fact, it is given by 8 free chiral superfields m + 1 N ± A – 14 – [ kn 2 ǫ T 1] 7-brane, which become two (1 m , jm ǫ ± = 2 multi-junction has SU(2) il 1 of one SU(2). δ 1 ] that for m N 2 2 ± m ( lmn theory Q . We see here that even the Higgs branch dimension works out 6 6 ijk E E Q = there is a two-dimensional Higgs branch, corresponding to r = 4. This is the theory W 3 H m and the ± M is the mass parameter associated to the H = = 3 = 2 i 2 = 3 multi-junction was already shown in figure m N N piece of 5-brane and splitting the remaining web in two simpl If all masses are zero, all four Higgs branch directions open m If one mass is zero and the other two generic, there is no Higgs 5-branes this is guaranteed by the s-rule, see figure branch direction. Insuch the that web the of multi-junction 5-branes can split this in corresponds two to simple junctio If the three masses are generically non-zero and the flavor gr there is no Higgs branch. If the masses satisfy N , where each index is in the In order to see why, let us pull the D7 to the other side of the NS Even though trivial, this system allows us to perform a nice c • • • ijk 0 and dim mass deformations corresponding to the Cartan generators o Q 6 mass deformations and then no marginal couplings, dim It has already been noticed in [ The 4d low energy theory on the actually enhanced to 4.2 two NS5-branes, ending on the same [0 showed the dual dotthese diagrams examples, one with gets tessellation, the one set of allowed rules we4 stated in the prev Examples In this section weviously consider stated many will exampleswhen of become possible. increasing clear. comp Comparisons with known4.1 fie The where superpotential for general mass deformations is of the hypermultiplets are then crossing the cut. Nowcannot the be usual two NS5-branes s-rule stretching applies, between in a a D5 and different a [0 number of the massless hypermultiplets. Let us reproduce th JHEP09(2009)052 3 . m 6 ± E = 0 (4.2) (4.3) = 1 e meet . For 2 4 gether, m , whose = 0 and 6 C m as ] 7-brane 1 E in m p,q and one Higgs = 0, N ce of D5 can be t 3 1 m m = ± subgroup. at a particular point 2 e over its projection. 3 dy the monodromy of 6 m xyz ± . , known as the affine 1 , 5 = 1 1 , nnot be removed. d) A 1 1 X 2 ional SCFT’s are collected in omology realizes the lattice of generic d) ) m X = 3 global symmetry in 5d. , = tanton moduli space of 2 CB 6 m E ) monodromy around a [ t the monodromy matrices. n two simple junctions. c) Z showing the SU(3) , C , has = 0, 6 , R 1 6 E 1 esponding monodromies have to be multiplied in 1 − , , 0 1 m X X c) = non-perturbatively. The combined monodromy = – 15 – . Our configuration shows instead that we can 6 3 E m CB is described by the equation cone over dP ± N , Q 2 3 , B = 2 multi-junction for various values of the mass parameters Z , which is the del Pezzo surface dP 0 0 m , , 3 1 , which is conjugate to ( 1 N × P ± 3 ], the 7-branes are an ordered system according to the order w X X 7 P N = 3 = = Z 1 Q / 3 3 P A m C R . Extended Dynkin diagram of It is known that eight out of nine 7-branes can be collapsed to ). Let us denote the monodromy matrices of the 7-branes we use , now one D5 can jump the NS5, the s-rule is satisfied and the pie 4 3 3.2 . In fact, m Figure 9 B ± . Higgs branches of the = 2 a) generic b) Another way to understand the symmetry enhancement is to stu m Following the conventions of [ = 3 it is an homogeneous cubic equation, and thus a complex con , and thus M-theory on the CY 4 6 Its projection gives a cubicof its in complex structure moduliE space where it is toric. Its h the system of 7-branes. Let us recall that the SL(2 and removed, as well as the junction can be split. is given by ( no Higgs branch due to the s-rule. The piece of D5 on the left ca direction corresponding to splitting the multi-junction i N From left to right. a) Generic masses and no Higgs branch. b) of our system is then Figure 8 quaternionic dimension is 11.appendix Some properties of the except the opposite order. Here and in the following we always repor making the F-theory 7-brane of type their cuts circling counterclockwise. Obviously, the corr In the literature, the basis is usually given instead by correctly: indeed, the Higgs branch is the centered one-ins configuration. so that a single O7-plane splits into JHEP09(2009)052 . . 9 11 = 3 C , each ij = 1and M where the Q C C 4 ollapsed to Z M , see figure 6 that is known . × C and giving rise 5 subgroup. E 4 6 } Z 2 A 2 / . White dots on the 2 , 3 = 9. The Coulomb } ) 1 2 C { , H 2 SU(4) CB { M × ion . Ithasdim nditions. In fact the 7-th H on is non-toric, because we he three punctures, we get } moduli space. Again, more and subgroup of 2 2 } 3 ree hypermultiplets 1 { SU(2) subgroup, see figure nted in section , (3) 1 eb with jumps corresponding to the × , 2 1 and single line. In red the only closed dual , } 1 4 { 1 = 0 and dim { . In our description we instead grouped ns . Let us study the monodromy as we did 7 C 7 showing the SU(2) E E . Indeed, the Higgs branch has the correct 7 M 7 ]. It is particularly interesting to realize the E C puncture by moving away 2 pieces of 5-brane 3 E SU(2). The only closed face is well visible. A , which is conjugate to ( } – 16 – [ 4 × 1 T , 2 P 1 4 , global symmetry, 9 mass deformations, dim theory. On the left dot diagram of 1 Q 3 2 { 7 R E for its 5-brane web and application of the s-rule. The rank-7 10 theory 7 U(1) symmetry, dim E × of one SU(3). 2 3 . extended Dynkin diagram of configuration of the 7-brane. Nine out of ten 7-branes can be c = 21. This is the theory 7 and the E H = 17, see figure . web of 5-branes for the M SU(2) theory, using punctures of partition = 4 H H = 4 multi-junction has SU(4) × Figure 11 . The total monodromy is N 2 M 6 N We can realize the other puncture, represented by the partit H E four, four and two 7-branes together, showing an SU(4) one point, making an F-theory 7-brane of type as the affine for SU(4) to U(1) flavor symmetry, from the Figure 10 and dim index being in the tessellation realizes, on the external edges, the partitio More arguments supporting this identification will be prese Notice that the dual geometrycannot we have remove to the compactify 7-branes M-theory del without Pezzo changing does the not boundary have co toric points in its complex structure global symmetry is believed to enhance to along the Higgs branch.a Performing theory this with operation SU(3) on one of t dim 4.3 dimension as the one-instanton moduli space of branch is lifted by the s-rule. This theory corresponds to 9 f The collapse three bunches of three 7-branes, displaying the SU edges separate collinear lines thatpolygon, have which to is be a thought closed oftessellation. face as The in a the visible web. symmetry is On the SU(4) right, dual w JHEP09(2009)052 . . } ], 8 } = 3 1 6 2 E − 1 H { { N where 3 6 M , which A Z 6 [ H and ] contains × P T 5 6 3 flavor sym- } Z A 2 Q 8 [ / , 3 2 3 7 T , { R 6 C , E . These numbers } SU(6) subgroup. 3 ow 2 × 12 { description as the web = 10 and dim C SU(3) theory 7-brane of type hich have M × . We can construct all these C lation. The visible symmetry is B theory, we start from 6 . In red the only closed dual polygon. } = 29, see figure 2 N E 3 H { configuration. It is known that ten out . SU(6) flavor symmetry is manifest, see 5 8 M × E and H } 5-branes end on the same 7-brane on each side theory. On the left dot diagram of , showing the SU(2) 3 8 2 8 – 17 – { E N E SU(3) SU(2). Inspection of partitions shows that , global symmetry, dim } × 6 × 3 1 { = 1 and dim SU(3) C × ]. M 5 theory known as the affine C A theories 8 realizes SU(2). Thus one suspects that the theory [ 7 theory. E T n } A SU(6) 2 2 8 E ) 3 E { ⊃ 8 CB SU(2). The only closed face is well visible. E × and the . extended Dynkin diagram of . web of 5-branes representing the = 6 . SU(3) theory in its Higgs branch, specified by the three partitions = 6 multi-junction has SU(6) 13 N × 8 N Let us study the monodromy produced by the 7-branes. This is n Note that E Figure 13 figure Instead in our description an SU(2) The Coulomb and Higgs branchof dimensions 5-branes, can and be found are from dim the the tessellation realizes the partitions the and move on the Higgs branch such that Figure 12 is conjugate to ( of eleven 7-branes can be collapsed to a point, giving us an F- theories with the multi-junctions. To get the rank- 4.4 match those of the 4.5 Higher-rank 50. This is the theory The realizes SU(3) and On the right,SU(6) dual web with jumps corresponding to the tessel There are higher-rank versions of the theories we saw above w metry, and whose properties are summarized in appendix support for this identification is given in section JHEP09(2009)052 , N nal hree (4.5) (4.4) ber and and one } 4 N { ]. the punctures are 1 ting the dot diagram − , and the Higgs branch N . ] theory and substituting d count the dimension of 6 . 1 l one, such that the flavor 15 inal theory, the dimension 1 . A − ed faces in the dual web of [
− 14 T Nk : dual web of 5-branes. The two + 1 N e are needed. The dimension of A he s-rule. In red the two polygons [ T old H theories. More generally, given some 1. in the new one. The Higgs branch M 8 1 = 12 , see figure ]; the punctures are two − H E 1 − N − Nk N . The Coulomb branch has dimension N 5-branes end on the same 7-brane in the dim N } 4 and web, see figure A 7 N m [ N 6 + 3 3 T E – 18 – N N, N E 3 3 · { . The web of 5-branes turns out to be, due to the +1= theory is embedded in N 8 = 3 new H = H N E M C and one M H copies of the ] theory, we can construct a new theory with the same global M H } 1 C − N theory, in the example rank = 2. On the left: dot diagram with t N dim k 2 dim 6 A [ N, T 5-branes end on it in the new theory. The total number of exter 5-branes ending on the same 7-brane. More precisely, the num 2 theory is embedded in N E 7 N 1. The rank- N, in the original theory, and is 3 2 Nm − { k N E N , and the Higgs branch dimension 30 . the rank 16 , one . The Coulomb branch has dimension } } 6 N partitions on the external edges, tessellated respecting t N 2 in the new theory. The rank- In the same way, we can get the rank { } 3 N N, 2 N The Coulomb branch dimensionaccording to has the to s-rule,5-branes. be and It worked then turns counting out out theis by that number tessella when of the clos dimension is 1 in the orig dimension is easily determined: type of 7-branes insymmetry the new is theory the is same.original the same theory, as However in whenever 5-branes the origina was 3 see figure symmetry but larger Coulomb branch starting with the each 5-brane with { set of punctures in a one which become closed facesconcentric in closed the faces are web well of visible. 5-branes. On theof right the multi-junction; therefore threethe 7-branes Higgs on branch each is edg { Figure 14 dimension 18 We can work out thethe web Coulomb of branch: 5-branes that dim respects the s-rule an jumps, a superposition of JHEP09(2009)052 two (4.6) . 2 f the moduli ) and the di- 4.5 k > on with three generic for eral formula ( 18 -junction itself, that corresponds k blue. − N + 20  2 N on the external edges. On the right: dual web of – 19 – 3) } N − 3 2 k N, 3 { 1) + ( 1 − and − on the external edges. On the right: dual web of 5-branes. The theory, in the example rank = 2. On the left: dot diagram with theory, in the example rank = 2. On the left: dot diagram with } ) N } k 7 8 N N 2 9)( − 2 2 2 N, − N E N E N, k 2 k 2 ( (3 {  N, 2 N k { and , = = ] SCFT, to obtain a higher rank version of it. The dimensions o } } 1 6 4 C H − N k N . the rank . the rank { { M M A [ T dim dim The process can be applied to any multi-junction configurati 5-branes. The two concentric closed faces are emphasized in punctures. In particular itto can the be applied to the basic space are Figure 16 Figure 15 partitions partitions The Higgs branch dimension is directly obtained from the gen concentric closed faces are emphasized in blue. JHEP09(2009)052 , 1 is n × − as − ) E l a N a d d (5.1) +1 (5.2) (5.3) d a A d = 2 6d − a = 0. Since a k d 1 − ]. It was then n . Therefore we 1 d r Σ of holomorphic ∗ = ≥ theory. For each T 0 t flavor symmetry are a -theory construction 1 d . ng the formalism by − and to the left of s of 5-brane junctions , n N r ) d A , n 0 means that ). We require flavor symmetry originally > d d = 2 superconformal linear y found in [ a 8 N SU(2) , ≥ d for each 7 φ utive gauge groups SU( , . a · · · N × 6 SU( 2 k + 3 SU(2) E +1 > × a × d ) · · · 1 +1 2 r − − SU(3) + SU(3) n a 3 d × d φ > d 3 r SU(4) − flavor symmetry d SU( N 8 , x = SU(6) 7 , manifest flavor symmetry + 6 – 20 – · · · } } } 2 E 3 2 2 φ ; thus we can associate naturally a Young tableau ×···× 1 2 3 = 2 ) l − 2 } { } { } { N d 3 4 3 ). The Coulomb branch dimension has to be worked out x < d 1 1 2 a > r + 2.8 SU( · · · } { } { } { punctures N 3 4 6 × ; we refer to the parts to the right of x 1 1 1 < ) r { { { 1 2 d symmetry via compactifications of the 6d d 3 4 6 8 N , 0= = 7 < d , ] theory ( 6 7 8 n 6 SU( 1 1 E E E · · · E E − d k . Equivalently, they correspond to compactifications of the = 1 extra fundamental hypermultiplets for SU( A [ l S a d T k = of the M5-branes, the types of the punctures, and the manifes to make every gauge coupling marginal, where we defined N ] that they can be realized as a subspace of the total bundle N ]. 2 2 +1 ), and a . SCFTs with d +1 a − In this section we perform further checks of our proposal usi The Seiberg-Witten curves for these quivers were originall d 1 is non-negative, we have − a a differentials on a Riemann surface Σ, given by the equation monotonically non-decreasing for to the tail by requiring that it has a row of width a bifundamental hypermultiplet between each pair of consec SU( Table 2 listed. mension for the the number shown in [ for these theories. 5 S-dualities and theories with Gaiotto [ 5.1 Formalism Let us start byquiver gauge briefly theories recalling with the a chain formalism. of SU Consider groups from the dot diagram and the web of 5-branes. We do not know an F two tails of this superconformal quiver. Requirement that can naturally label a puncture by a Young tableau. k We denote In the previous sections, we found that the 4d SCFTs with d found by Minahan andcompactified Nemeschansky on can be constructed by mean theory on a sphere with three specific punctures, see table JHEP09(2009)052 , . 1 } 3 k + 1 1 (5.4) n { and the , and two } i • . differentials ≥ ··· ≥ k { d J 1) k ), one has from the Young − arameterizes the differentials with 1) group inside a d d 5.1 ) we can show that d − p (2 eight ) is the height of the of the punctured Rie- omb branch operators N i 5.1 the order of the pole − ( another puncture h d p rm ( and denote by . In the zero coupling limit nformation on the tails. l quivers ( presentation. The limit where f the form above arising non- nds to the degeneration of the of the system, to distinguish it SU(2) (5.5) ), where i ( h ]. The construction starts from the 2 come together and develop a neck. A . We call this set of a Riemann surface − at the puncture) ×···× d i G-curve • d 2) p = ( i − p 1 simple punctures can also occur in the very of the allowed poles of the degree- N – 21 – d ] we identify this situation as having a weakly- X − p 3 boxes from left to right and then from bottom up, , whose G-curve is also shown there. It is an SU(3) SU( punctures differential with poles at the punctures, encoding , N 17 2 N × = d 1) theory as a limit of a field theory with Lagrangian was d 6 − boxes, when the coupling of the SU( E N is allowed to have a simple pole. At a more general puncture, N d ]. At a given puncture specified by the partition 1 simple punctures splits off and leaves a puncture labeled by φ SU( 2 is a degree- − d φ N ]; we follow the presentation in [ theories 3 n is then given by the dimension of the space of degree- E d 1 # of operators of dim. is allowed to have at the puncture; the method to obtain is a holomorphic differential on the Riemann surface Σ which p d φ x At a simple puncture The marginal couplings of the theory are encoded in the shape is allowed to have a pole of higher order. We denote by are determined as follows. The Young tableau has columns of h d d -th box, and the bottom row has height 1. The number of the Coul aligned at the bottom. We order the φ extra punctures labeled by Young tableaux which encode the i the prescribed singularities. The formula is starting from the bottom left corner. Then superconformal tail with the gauge groups punctures of the same type, which we call ‘simple punctures’ i tableau was detailed in [ the tableau with one row of where strongly-coupled regime. Following [ of dimension first obtained in [ φ coupled S-dual description, withperturbatively. the superconformal tail o 5.2 Rank- quiver shown in the first linegauge of theory figure with six hypermultiplets in the fundamental re a sphere containing the coupling constant ofG-curve SU(3) such is that infinitely two strong simple correspo punctures of type becomes weak. Now, this splitting of The way to obtain the rank-1 dual weakly-coupled SU(2) gauge groupof with this one new flavor appear SU(2) gauge group, the neck pinches off and produce fiber direction, and mann surface Σ. For example, by studying the quivers of the fo which VEV’s of Coulomb branch operators of dimension Σ and punctures marked by Young tableaux the from the Seiberg-Witten curve. One finds that, for the genera corresponding Young tableau, the orders JHEP09(2009)052 . 6 } E 3 1 { . 6 stands for a SU(2) with E n × a USp(4) gauge ]. Here instead, we ctures of type 22 between a flavor symmetry factors, and there are in theory with six flavors, it ⊃ enhance to SU(6), which is ries are on the same footing; lely of SU groups. We start ecting two objects stands for a ). Its G-curve has three simple the Minahan-Nemeschansky’s SCFT. the subgroup of the flavor symmetry 7 nt of the enhancement to e symbol E 5 ], which was also directly realized in the . The gauge group is SU(4) 3 18 . We can go to a limit where a sphere with } 2 – 22 – 2 theory. The triangle with two SU(4) and one SU(2) { theory. A circle or a box with a number 7 6 E E flavor symmetry. 6 E and one } 4 1 ⇓ ⇓ ⇓ ⇓ enhances to SU(6). On the other hand, in the description with { 2 ]. Another realization was recently found in [ 2 theory was found in the infinitely strongly-coupled limit of 7 . Construction of the rank-1 . Construction of the rank-1 E ) gauge group or flavor symmetry, respectively. The line conn The n The authors thank Davide Gaiotto for explaining this argume 5 theory with six fundamental hypermultiplets in [ SCFT. The G-curve is shown on the right. flavor symmetries represents the Minahan-Nemeschansky’s the bifundamental hypermultiplets chargedaddition six under fundamental the hypermultiplets for twopunctures, the SU node one SU(4 puncture Figure 18 Figure 17 quiver language in [ present a method tofrom construct the it quiver shown using in a the quiver first consisting line of so figure SU( therefore any pair of twopossible out only of if the this three theory SU(3) has groups should the G-curve, it is manifest that the three SU(3) flavor symmet bifundamental hypermultiplet charged under two groups.and a Th gauge symmetry signifies thatspecified. the gauge fields The coupled to triangle with three SU(3) flavor symmetries is We end up with a theory whose G-curve is a sphere with three pun On the one hand,was in manifest the that original SU(3) description as an SU(3) gauge JHEP09(2009)052 × SU(2) . theory. 7 × 8 E theory of E ), that the 8 several kinds E 5.4 SU(6). In the and another of symmetry while ]). × } 5 3 23 2 { has only one Coulomb SCFT. U(1) ormula ( -curve is a sphere with his is the SU(3) } 8 , leaving a theory whose × 2 × E 3 nown fact of the { e, the two SU(4) cannot be e has five simple punctures, roups , quiver shown in the first line . This description shows the } mmetry enhances to } 3 ve SU factors; one has in addi- ith gauge groups SU(3) 4 nsky’s 2 1 { , because if it did, the generalized { } , SU(2) (5.6) SU(2) (5.7) 6 , one of type } 1 } 6 × × { 6 1 1 { . The only possibility is that SU(5) and { 4 SU(4) SU(3) would have SU(5) U(1) × × SU(2) enhances to SU(5). This SU(5) does not 19 × – 23 – × theory. The triangle with one SU(6), one SU(3) and 8 SU(6) SU(4) E × × ]; the realization we present here does not seem to be . In the original description, it is clear that SU(2) . We can go to a limit where a sphere with five simple 2 11 } 2 3 SU(3) SU(5) { and two punctures of type (we refer the reader to tables 14 and 15 in [ SU(4) ⇓ ⇓ } 8 2 × E 2 { and one theory was found in the infinitely strongly-coupled limit of 8 } 3 E 2 { . The original quiver has the gauge group . Construction of the rank-1 . This description shows the flavor symmetry SU(2) 19 } 2 3 { The rank-1 Indeed, the structure of the Coulomb branch indicates that t commute with the SU(6) associated to the puncture one SU(2) flavor symmetries stands for the Minahan-Nemescha of Lagrangian field theories in [ original description, it is clear that SU(3) flavor symmetry SU(2) quiver drawn in the second line of figure Minahan and Nemeschansky. It can be easily found, using the f SU(4) enhances to SU(6).distinguished. In This the is description only using possible the when G-curv the total flavor sy branch operator, whose dimension is 6. This agrees with the k three simple punctures splits. A dual superconformal tail w Figure 19 appears. After the neckone is puncture pinched of off, type we have a theory whose G theory whose G-curve has three punctures of type directly related to theof cases figure listed there. We start from the with bifundamental hypermultiplets between thetion consecuti five fundamental hypermultipletsone for puncture SU(6). Its G-curv punctures splits off. A dual superconformal tail with gauge g type SU(6) combine to form appears. We tune theG-curve gauge is coupling of a the sphere SU(5) with group to one zero puncture of type the original quiver clearly has only SU(5) JHEP09(2009)052 } d 3 and N (5.8) (5.9) { a (5.14) (5.11) (5.12) (5.13) er, we have = 3. . ) N , N 2 ermined; the tableau ups . Here theory. In this section ..., unctures of type SU(3) } 3 6 × 6; nsion 3 and 4 respectively, ees with the known fact of , the poles of the degree- E N , { 5 , k , 5.1 ne the spectrum of Coulomb es. The central charges . own data. , or equivalently the effective . c SU(6) 4; 4 6 31 , × 3 , and ) = 40 (5.10) 6) 8 theories treated above. The result is )= a 8 E − N 7 ( 2; 2 (total) = 80 (tail) = 40 E , 3 h h h E ( N have degrees and , ..., ) = (1 } SU(3 ]. 9 , n 6 3 , c – 24 – , n , n N , × 3 24 E 6 N 24 95 , { , 3) 3 11 ) = 11 − ...,p )= 8 8 ; N E 9 E ( ( (tail) = 50 v ,p a v (total) = 61 n 8 n v SU(3 ,p n 7 × p theories a ; ) 6 n N ,p E 5 . a linear quiver whose G-curve has a tableau of hyper- and vector multiplets. In the original linear quiv ,p h 4 SU(3 n p theory. at the puncture of type ; 6 3 d and φ ,p ), we find that this theory has operators of dimension 2 Figure 20 v N E p n ( 5.4 appears for instance in the G-curve of a quiver with gauge gro The same procedure works for the The central charge of the SU(3) flavor symmetry can also be det } 3 can also be easily calculated, correctly reproducing the kn N differential number They agree with what was found in [ One can also easily calculate the central charges Using ( { and the number of operatorsthe of rank- each dimension is one. This agr we provide further piecesbranch of operators. evidence. Using the First, algorithm let explained us in determi section that they have only onewhich Coulomb again branch agrees operator with each, the of known dime properties of these theori whereas the tail contains c 5.3 Higher-rank In the previous section we argued that the theory with three p or equivalently has the right properties to be identified with the higher-ran We conclude JHEP09(2009)052 of and h n a lently (5.15) (5.16) (5.18) (5.19) (5.17) for the theory, 7 and 20 E v of D4-branes n , and the flavor . and one of type , 2 ) theory. The case ) } N 4 7 r the leftmost and N 2 theory. 3 N 6 { SU(4) of the superconformal . . . N E . . . × = 2 SCFT’s constructed c t corners of its marginal tely it is not known how describe linear or elliptic 4; 6; ype N E , N , 5 ies further, and to determine the resulting low energy 5d U(2) , and hypermultiplets e IIB string theory suspended ts that these theories can be 3; 3 4 an RG flow, along which , , onal a 2; 2 3; 3 , , for the rank- is N , 2 , theories are similar. For the } for the rank- N 4 8 , = 8 N 7 N , N N 4 { = 8 E ], in which the class of theories of this type ) = (1 ) = (1; 1 2 = 6 7 = 6 N N E SU(4) 4 4 6 k k – 25 – , ..., E 8 k = SU(3) , k ...p ...p 4 ; ; 8 hypermultiplets transforming under the SU(3) flavor 8 SU(2) ,p ,p k N 7 7 is p ) above, one concludes that this theory has Coulomb branch ,p } ; 6 6 N 5.4 ,p 2 ,p 5 5 p p N, ; ; 2 4 4 { M5-branes on a sphere with 3 generic punctures, can be equiva ) gauge groups to make them superconformal. See figure ,p ,p 3 3 N N p ,p . This alternative construction plays the role that systems ; 2 1 2 p = 3. There are 3 S p ( ( N and 3 hypermultiplets in the fundamental representation fo . The pole structure at the puncture } = 3, is left as an exercise to the reader. N N 2 8 a ] by wrapping We would also like to compute the central charges The analysis for our candidate higher-rank E 2 N, generically vary. It would be worthwhile to study such theor 2 the rightmost SU(3 the candidate is a theory whose G-curve has two punctures of t this theory, and compareto them construct to this the theory known in values. the framework Unfortuna of [ { with 3 symmetry, therefore we have which is consistent with the known fact obtained starting with a webbetween configuration of parallel 5-branes in 7-branes, typ field and theory then on further compactifying operators of dimension found along the Higgscoupling parameter branch space. of However a thisc procedure parent involves theory, rather then a was called ‘unconstructible.’ Our web construction sugges case while at the puncture Combined with the formula ( suspended between NS5-branes and D6-branes play in order to their central charges. symmetry central charges are easily found to be each with multiplicity one. The manifest flavor symmetry is S current, or equivalently the effective numbers of vector and which is consistent with the known result in [ 6 Future directions In this paper we proposed that all the isolated four-dimensi for JHEP09(2009)052 theory groups. branch, ] theory, N ] 7-brane 1 ]. Another − D 1 ble’ theories N p,q USp − A 0) N [ , ], that is those A T 2 he canonical line [ and T place in which the ure geometry in M- SO rated in this chain of , with a [ lomorphic curve in the 2 ed in [ al, that is each external es have generically more ]. It could be interesting R nsional (2 actified 5d theory, and in lly those arising from the 4d dynamics. This allows rgy dynamics of M-theory 27 ulti-junctions by attaching e as well as mixed branches fied 5d theory. It would be e for the theory with three – with respect to the pure 4d toric. For the particular case he duality from type IIB on ld theories. In particular, the 25 n though we have not discussed [ . , the 7-branes could be actually ]. In our construction, isolated cription of the 7-brane. The web } ], however it would be interesting 2 N 22 1 { fibration over 2 T we get the 4d isolated SCFT – 26 – 1 S is described by a 5d version of the global symmetries should arise after the introduction N ) one-cycle of the torus shrinks. This space is essentially Z × gauge groups with extra fundamental matter, or systems of theories. USp p,q 8 N , . It would be interesting to understand other examples, for 7 , Z n SU 6 / 3 E dP C semi-infinite 5-branes. They correspond to the ‘constructi and possibly N = 2 theories have the property that, moving along the Coulomb SO N singularity 3 theories, the dual geometry is known to be the total space of t n E ]. However some care is required because the parent 5d theori One could also be interested in realizing all SCFT’s present On the other hand, another chain of dualities maps the web to p Finally, The first interesting question is how can 7-branes be incorpo In the particular case in which all three punctures are maxim Another question is the extension of the web construction to 2 of orientifold planes. theory. When 7-branes are present,of the the dual geometry is non- mapped to a point wherethe the elliptically ( fibered 2-fold whichof gives 5-branes the F-theory is des thus2-fold, mapped and to this curve ainteresting is single to exactly M5-brane work the wrapping out SW this an curve curve of ho explicitly. the compacti they can be deformedranks in of the such non-abelian a gauge way groups that progressively reduce a cascading RG flow takes on a Riemann surface, the problem was analyzed in [ quivers of four-dimensional to study if similar phenomena take place in the present case. SCFT’s with with more than threecompactification punctures of on M5-branes on the higher sphere, genusthis surfaces. and more in Eve genera the paper,two it bunches of is indeed possible to gluerich together dynamics the at m particulartheories. points Such of problems their do not parameter arise space for the multi-junction to rederive it froma the circle web and construction. M-theory In on particular, a in torus, t we find a dualities. For instance, onegeneric would punctures. like We already to know obtain what the the SW result is curv [ chain of dualities leads toa write suitable down scaling the limit SW the curve SW of curve the of comp the 4d theory. bundle of the del Pezzo D3-branes suspended between 5-branes play todimension describe and structure 3d of fie thebecome Coulomb and manifest. Higgs moduli spac 5-brane ends on its own 7-brane and thus the type is in [ while after further compactification on From the point of view of the compactification of the six-dime removed without changing the lowus energy to 5d use and known consequently on string the dualities, CY and to argue that the low ene instance the higher rank JHEP09(2009)052 a g (A.3) (A.1) (A.2) . The . The 2 β R , couplings, α × , then type , 2 B F L and M-theory T o. DE-FG02- 1 + , in 4 S ′ x Y-0503584, and in α π t 2 ≃ L ed, we take the web of 4 theory at low energies. x = decompactifies. On the equal to the axiodilaton s is: Foundation under Grant 3 p SW differential is defined τ l for Advanced Study. A each us the content of his 4d dynamics is captured by L es are readily obtained from is essentially independent of s by rescaling tion at an early stage of this Center for Physics during the , j β , i . α w β ) are the integer coordinates of the ′ L i,j t α 2 i, j log 2 ) C L B ) ) π d L π . This is the SW curve for the 5d theory i,j (2 ∧ ∗ (2 α C X = = dots ( ′ × w log A α – 27 – ∗ L d L C B sB )= g L ∈ ) α, β = ( d =Ω= 2 4 F α, β g ′ 0, which means that the circle SW α ]. Summarizing, the 4d limit corresponds to decompactifyin 0= 2 are the lengths of the two sides of the M-theory torus (the are dλ ) → 31 π B τ B L (2 m ]. If we compactify the 5d theory on a circle L I sB t 30 g L – = 28 ≡ t are either Coulomb branch moduli or parameters like masses, is the 11d Planck length. In the duality between IIB on A L p L l i,j C is dual to M-theory on a torus of modular parameter are parameters, and ( is the characteristic size of the web. To keep the coupling fix and 1 t w ) and i,j S L L A , the web of 5-branes is mapped to an M5-brane wrapping a curve C 2 L F. B. is supported by the US Department of Energy under grant N Once the 5d theory is compactified on a circle, we obtain the 4d t of type IIB. The relation between IIB and M-theory quantitie T L , and we can send it to zero. In this way, the non-perturbative B p paper. We also thankproject. for Diego collabora F. Rodr´ıguez-G´omez B. acknowledges thecompletion kind of ospitality this of work. the Aspen 91ER40671. S.B. isNo. supported in PHY-0756966. part Y. bypart by the T. the National is Marvin Science supported L. in Goldberger membership part at by the theA Institute NSF grant PH Seiberg-witten curves The Seiberg-Witten curves for theour 5d construction and [ 4d low energy theori Acknowledgments We thank Davide Gaiotto for his patience and willingness to t the same size as thel M-theory circle. However the field theory where weakly coupled M-theory [ IIB on where τ is on curve is obtained from the toric diagram through: where we sum over the dots of the toric diagram, ( parameters dots, compactified on a circle. We could eliminate three parameter etc. . . The number ofby moduli the is holomorphic given 2-form by the internal dots. The The 4d limit arises as other hand the (classical) 4d coupling is JHEP09(2009)052 (A.4) (A.5) (A.6) (A.8) (A.7) linear . This l , where , but in at some l − N − t , therefore : this time /ǫ N 0 N 1 N − /ǫ N, = 1, and shift , N,...,l a N = . 0 t ) 0. The curve in survives but the t P k p = . ( − i → p j N ǫw − . To decompactify a + ǫ P i 1 0 we are left with the t dt i, p − + to set C  ∧ N → , for is the sum of t w j k in such a way to keep the w to diverge as 1 ǫ  l , 1 ) l i , ǫw t  t . dw e. 1 ( i,j . However i,j k n: 1 he flavor currents. j − − − ǫw 1 C C e − i,j β 1) − p p N 0 limit: we get king i C a N 1) −  α 1) ( P → -junction (see the toric diagram in − ], keeping the SW differential fixed l − , we will take i,j j =0 =0 2 ǫ t i + X N t β k X ( C 1 to reorganize the summation in powers N − · · · = ( ǫw ≤ j pǫw ) j and ǫt + j e β + + l + 1 α i i = − ( , i − X p 0 e N = appearing in front of two exponential functions ≥ 1) in the first summation): it will be some power SW w i,j p – 28 – i,j ). We can then rescale . Now consider the coefficient of − i,j ) C ( t dλ . Eventually, we get the most general polynomial N l w appears, multiplying the whole series expansion of C ( l . Notice that the total number of parameters is , so that the term N 1 t = j N A.4 and )= ≤ p = ) X P /ǫ i,j j 1. We can set the coefficients in such a way that the 0 k : l l + C l, ( + t − a − N α, β 0 allowing the coefficients , i X ( N ǫw j 0 N N =0 + t l F ) = 0. As remarked in [ X ≥ w + → t e t 0 i ( i,j ǫ · · · 1 N, N P = )= = P 0= + α l multiplying exponential functions of = )= l )= p w,t − ( i,j N in such a way that all linear combinations diverge as 1 w,t C F to set ( t, w w ( , without powers of l i,j i F F does not diverge. In general the coefficient of − reads: − C /N 0= 2, as in the 5d curve ( 1 ) l,N / t w i,N − 0= ( a 0= 1 N C whose coefficients are linear functions of the t P with combined degree 0 + 2) and are polynomials of degree , =0 N i − 1 ǫw t w N j − ) is w P P N terms ( 3 a l = . Such single combination must be finite in the → , corresponding to To be concrete, the curve for the compactified 5d Consider the coefficient of and : + 1)( 0 t w − t w N, Nǫw N figure curve finite. one circle of the M-theory torus and scaling the parameters under such a shift corresponds to a harmless redefinition of t To get the 4d limit, we first of all do the following redefinitio circle, which will come from a combination of in terms of which the SW differential is it terms of We can change indices to only one linear combination ofe the of power, as long as this does not lead to divergences in the curv the power series of the coefficient all divergences cancel. Ta N in where series in there are two linear combinations of the Now we take a scaling limit allows to set the combinations of the ( a of term two linear combinations diverge as 1 JHEP09(2009)052 . ∞ , dual (A.9) es. It 1 flavor flavor defor- (A.10) (A.11) (A.12) , n n 8-th del , heterotic E E 7 8 = 0 , E t × . 8 at N E global symmetry, k ] would be a good ) 1) 8 t ( 32 E − N t in terms of which the ( P -brane can be absorbed ]. g a stack of an O8-plane N 2 w t vitational supersymmetric 1 anates, which describes the the end of the world’ of the , = 2 SCFT’s with sition of the Calabi-Yau. − + ntifold. The Coulomb branch pactification of F-theory with e paper [ taining vanishing 6 N 1) x N φ 1 k − component of which measures the − + dt -instanton of the t N 8 ( x ) k 1 0)-theory with t 1 E 1) ( , − ) 1) − 1 t t − ( N − − k t φ N = t ( P ( 1 P + k x dt . x − -instanton. On the purely geometric side, it t 8 N , = gives five-dimensional theories with t · · · 7 , and in fact = , -instanton which describes the Higgs branch of 6 – 29 – 1 8 k + dt + E S E SW 2 dt ∧ λ ]. Let us discuss them using branes. T-duality along − · · · k N φ gives four-dimensional 10 dx + x , 1 ≡ 2 2 9 = S − φ k N Φ = SW x 2 N dλ x 1) ) t ( − ] theory. We can rewrite it in a more inspiring way as 2 1 t P ( − ) encode all parameters, i.e. Coulomb branch moduli and mass 2 t gives us a D3-brane probing a system of O7-planes and D7-bran t N ( 1 j A flavor symmetry. In the main part of our paper we provided new, [ P = S ) and the differential agree with what found in [ T n N E x in the base. The total space contains the ninth del Pezzo. A.10 2 holomorphic differentials on the sphere with poles of order S k theories n E Compactification of this theory on The most basic theory is the six-dimensional (1 Finally, we introduce a new coordinate Further compactification on mations, of the The curve ( The polynomials curve reads: where are rank string. This system has avanishing dual geometric realization as a com the compactified B starting point to the huge literature on this subject. which is realized on anheterotic M5-brane M-theory. very It has close one to tensordistance the multiplet, between the 9-brane scalar the ‘at M5-brane andinto the the end end of of the the world. world, The becoming M5 an Here we provide a brief review of what was known about non-gra theories with The holomorphic 2-form is the theory; therefore this theory arises on a point-like realization of these theories in four and five dimensions. Th Pezzo. The Higgs branch is realized here by the extremal tran symmetry. On the sideand using a branes, few D8-branes we such have thatis a the real D4-brane dilaton one diverges probin dimensional, at the andprocess orie at where the a origin D4-brane the turns Higgs into branch an em is given by compactification of M-theory on Calabi-Yau’s con symmetry, originally discussed in [ JHEP09(2009)052 - 6 8 , , 4 7 N , , 3 6 z (B.4) (B.5) (B.2) E gauge = 8 , 7 u , 6 E ) contributes is completely N 9 , 8 , 7 , gauge theory lives. M5-branes probing 6 . The worldvolume nto the latter as an n 3 theory, and put a 7- x , 2 N E , e is a codimension-two , whose dimensions are 1 , i 0 anes as instantons. Dual, u . 7-brane with x . h is the so-called ‘centered -instanton moduli space of ental of SU( cribes the non-perturbative able, therefore the Coulomb the 7-brane is then N iven by more detail. nch emanates from the origin ls of = 6 = 30 ]. In particular, the two-point . These theories have real D3-branes, we have parameters 8 8 33 E ∨ E N ns have not been well understood. parametrizes the Coulomb branch , ∆ D4-branes or -instanton moduli space of the gauge 24 C on which an 8d N versions of the five-dimensional N ∆ (B.1) 9 , of , , h 8 , ∆ (B.3) , N 1 z 7 , N 6 − x = 4 n = 18 7 = 2 theory. ,...,N ∨ E 7 E – 30 – n n ∨ E and E 2∆ ∆ Nh 3 k , , . The scaling dimensions of these Coulomb branch 2 5 , , 6. When there are ∆ N E 1 4 , , D3-branes, extending along 0 x 4 , , h , x N theory, by putting complex dimensions, parameterized by the positions of the 8 = 3 . The coordinate E = 12 N 6 6 , E 6 4 , ∨ E 3 ∆ h ). Z / C B.1 currents are characterized by the number extending along n E n E . is the dual Coxeter number of the respective group, given by ) . The center-of-mass of the instanton configuration along n N n ∨ E E h which describes the process of branes being absorbed into br SU( k 8 Its Coulomb branch has Central charges of these theories was found in [ When a D3-brane hits the 7-brane, the former can be absorbed i We start from a flat 10-dimensional spacetime of type IIB or F- In a similar manner we can consider rank- One way to understand this result is to recall that the 7-bran , 7 D3-branes along the directions for each D3-brane. However the D3-branes are indistinguish , 6 i purely geometric realizations of these higher-rank versio E group operators are and is of scaling dimension 1. The natural coordinate around dimensional Coulomb branch, and the Higgs branch is the 2 to We probe this background with N instanton. In the four-dimensionalof the language, Coulomb the branch and Higgs it bra is identified with the decoupled from the restmoduli of space.’ the system, The so quaternionic the dimension of true this Higgs space branc is g is then better to use the F-theory language which clearly des properties of 7-branes, which we provide below in a slightly brane of type where where ∆ is the dimension of the lowest dimension operator, the O8-plane or the ‘end-of-the world’ brane, respectively group is of the form theory on the D3-branes is the rank- function of the object and produces a deficit angle. The transverse space to a which is normalized so that one hypermultiplet in the fundam u respectively, whose dimension is 3 branch is parametrized invariantly by symmetric polynomia theories and six-dimensional exactly as shown in ( JHEP09(2009)052 , ]. l =2 , ]. ) N , =2 ]. global p, q ]. 6 SPIRES ( N , E ] [ , SPIRES ] [ SPIRES ]. super Yang-Mills global symmetry SPIRES ] [ (1998) 261 (1999) 1785 n , ] [ SUSY QCD and E 3 =4 ]. ]. , N SPIRES =2 ]. ] [ B 534 N gauge theories and new rank hep-th/9603042 [ SPIRES SPIRES ] [ ]. hep-th/9712086 =2 ]. [ superconformal field theories duality in string theory ]. SPIRES ] [ ]. hep-th/9902179 ]. N ]. [ ] [ =2 arXiv:0712.2028 =1 [ Nucl. Phys. quantum mechanics and symmetry , N SPIRES N (1996) 159 Constraints on the BPS spectrum of SPIRES Uncovering the symmetries on ]. 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