JHEP09(2018)008 = 4 N d Springer July 5, 2018 : August 23, 2018 : September 3, 2018 : SQCD theories with Received d Accepted = 4 quiver gauge theories. Published and 6 N d ones. One further consequence d theories and promotes the Higgs d Published for SISSA by [email protected] https://doi.org/10.1007/JHEP09(2018)008 , to the status of a new operation between non-special . This paper establishes this operation formally and = 1 SQCD of integrating out a massive quark while N d nilpotent orbits to Kraft-Procesi transition . 3 special quiver subtraction 1803.11205 The Authors. Field Theories in Higher Dimensions, Field Theories in Lower Dimensions, c

We study the vacuum structure of gauge theories with eight supercharges. It , [email protected] Theoretical Physics, The Blackett Laboratory, Imperial College London, SW7 2AZ, United Kingdom E-mail: Open Access Article funded by SCOAP Keywords: Supersymmetric Gauge Theory, Solitons Monopoles and Instantons ArXiv ePrint: quivers that do not necessarilyof have two a quivers simple is an embedding extremelyvacuum in structure simple string of resource theory. gauge for theories, The the and subtraction a theoretical yet crucial its physicist power interested role is in in so the the remarkable that coming is discoveries bound of to new play and exciting physics in 5 and 6 dimensions. is the analysis ofthe the gauge effect coupling in remains 5 the infinite. different In types general, ofmassless the singularities states subtraction that within of arise quivers thebranches at sheds that Coulomb those open light branch up. singular on and This points; allows the including for a the structure systematic nature analysis of of of the mixed the branches of new 3 Higgs mechanism known as the quivers. This is the examines some immediate consequences. OneKraft-Procesi is transitions the from extension the of classicalis the to the physical the realization extension exceptional of from Lie algebras. Another result has been recently discoveredeight that supercharges, in the the new Higgsinfinity, massless can branch be states, of described arising 5 in whenThe terms of description the Coulomb of gauge branches of this couplingof 3 new is nilpotent phenomenon taken orbits draws to as from Higgs the and ideas Coulomb branches discovered of in 3 the analysis Abstract: Quiver subtractions Santiago Cabrera and Amihay Hanany JHEP09(2018)008 changes instanton g 8 E = 4 theories and the small N 13 d has the same scale as the 2 /g 6 = 4 theories in terms of the theories this is not the full story. N d d 0. and 6 In the past, the Higgs branch of theories d → , the factor 1 at infinite coupling d 1 2 d ] and the appearance of new flat directions in /g 20 – 1 – – 5 ]. 2 3 ]. However in 5 4 15 6 9 has scale of mass and new massless states arise at the limit. There 5 2 ] and the introduction of nilpotent orbits in the description of Higgs /g = 4 + 1 dimensions [ ] to the very recent relation between the moduli space of 3 d 23 [ 2 1 , 1 ]. 22 [ 2 ] for a recent application of this effect to the dynamics of M5-branes on ADE singularities. , the factor 1 d = 2 SYM [ 21 N The questions about the physical phenomena at infinite coupling could have been asked In 5 d The myriad of examples goes from the study of monopole condensationSee and [ electric-magnetic duality 4.1 Exceptional algebras 4.2 Classical algebras 1 2 non-perturbative studies of themonopole formula Coulomb branch of 3 on 4 quantum Hall effect in transition many years ago. Thetions two main about reasons the thattools unchanging prevented were this nature at progress of were the the the disposal misconcep- Higgs of branch the theoretical and physicist. the fact However, with that the not advent enough of the is also a global symmetrythe enhancement Higgs [ branch, that growstension in dimension. of a In 6 string,the and dimension new of tensionless the strings Higgs appear branch at grows infinite by coupling. 29 in In a some process cases called the with eight superchargesbranch has counterpart. not This been is duenot studied receive to quantum with the corrections understanding the [ thatThe same the Higgs classical intensity branch Higgs as branch ofwhen does the SQCD the (with Coulomb limit eight to supercharges) infinite at coupling is finite taken gauge 1 coupling 1 Introduction The analysis of moduli spaces is aquantum key field step theories in the with study eight of new supercharges. physical phenomena arising in 5 Integrating out a massive6 quark in 5 Conclusions and outlook 3 General definition 4 New Kraft-Procesi transitions on nilpotent orbits Contents 1 Introduction 2 The subtraction of two quivers JHEP09(2018)008 ] ] n d 48 D = 4 46 , (2.1) ]. [ N 47 45 theories. , d ]. In sec- nilpotent 28 bad quiver sub- , ] in unitary 40 ] it was first , also utilizing 37 27 4 44 , 37 , 25 [ ] are recast in the quarks) that constitute . By subtracting 3 . This is a new way 37 n non-special (which is the Higgs branch = 4 quivers. 2 ] to study the anomalies and C N collects some conclusions d 43 (the information of mixed quivers. In [ 2 quarks is studied in [ 6 5 − N 2 Higgs branch at infinite coupling transverse slice d < ]. The second is the understanding instanton on 1 f ], new discoveries have been possible 41 : quiver subtraction n N − ◦ A 40 Kraft-Procesi transitions | – 1 2  ◦ subtraction of quivers 25 − [ orthosymplectic ) and quiver | 1 2  ◦ 3 N + 1 electrons) and the reduced moduli space of 1 ]. − ] for an extension to orthogonal and symplectic gauge – 2 – ] to n = 4 presents the formal definition of the 1 ◦ 44 40 40 3 N in terms of = ] were derived in the context of brane dynamics of : d d quiver theories [ 37 Q Higgs branch at infinite coupling can be understood in ]. d d 24 = 4 theories ) gauge group. Quiver subtraction can simplify the computations of we use the result to compute the Kraft-Procesi transitions be a 3 N N 4 or 6 Q = 4 theories [ d physical effect. The aim of this paper is to formally establish d ] Kraft-Procesi transitions between nilpotent orbits are used to characterize d N 42 SQCD with eight supercharges. Section d d SCFTs, and the nilpotent orbit data is utilized in [ d = 4 SQCD with gauge group U( we present a simple example in which the results of [ 2 (which is the Higgs branch of SQCD with gauge group SU(2) and N 2 d C ] where these transitions are introduced as a specific type of Higgs mechanism in 3 operation. In section = 4 SQED with gauge group U(1) and 37 = 4 quivers can greatly benefit from its application we use quiver subtractions to analyze the effect of integrating out a massive quark on N In section A natural consequence of these two developments, and the advances that made them 5 The study of mixed branches has become relevant in the description of infra-red physics of This is a generalization of the reduced moduli space of 1 See [ N d 5 3 4 d group factors. Note thatRG in flows [ between 6 moduli spaces of the theories. The case of 3 treats the SQCD theoriesthe with mixed Sp( branches when the theories under study are more complicated 3 of 3 instanton on the starting point of the discussion in [ quiver gauge theories with unitary nodes, and [ of interpreting the resultsType that IIB in string theory. [ Let the Higgs branch of 5 and future directions. 2 The subtraction ofThis two section quivers illustrates with an example the idea of language of quiver subtractions. Section traction between nilpotent orbits oforbits of exceptional classical algebras algebras, and extendingtion the also physical between realization presented in [ this operation and to examinehow some the of study its immediate of3 consequences. mixed In branches particular, and we differentbranches show can singularities be within collected Coulomb in branches this of way into Hasse diagrams). possible, is the promotion of theto mathematical a concept new of physical operationquivers between the quivers: change the in thea 5 remarkably simple way.quivers This and operation it can was be furtheremployed derived to extended from in describe the [ a work in 6 [ in the last year. Thein first terms one of is Coulomb the branchesof description of the of 3 small the instanton 5 transitionCoulomb in branches 6 of 3 and Coulomb branches of 3 JHEP09(2018)008 ) Q ( ) and (2.5) (2.7) G b ) and C U( , ). × (5 ) Q a ( The moduli sl F with complex 7 . ⊂ } of U( M 2 2) 1 , ) corresponds to a − ≤ (3 Q 1] ( ) , O C 0 M divides into two half-hypers, ). Its closure is called a b , section 3.3], taken from the ,..., C 5 matrices 37 [0 , × and ⊗ (5 ) (2.6) 1 a ) on the definition of sl U(1) (2.2) C 0] . , 1 ) (2.4) ... × ( of 0] = 4 theory with gauge group C (5 S , 8 ,..., sl is: 0 is the set of all partitions of the inte- = 0 and rank( N (5 U(1)) (2.3) λ , Coulomb branch ,..., ⊂ U(2) 2 } Q sl d 0 O ) × ) , 5 × ⊂ [1 (5) (1 ,M ⊗ 1) , [ , ∩ N ∈P ) 1 2 3 λ (U(1) U(2) − 2) – 3 – (2 1 , 1] S , ¯ ) = 0 = × , O (3 : 0 (2 S M , N ) = ) = 1) tr( Q ,..., , Q | for physicists can be found in [ ( transverse to the nilpotent orbit ) = ( 2 ( 5 , that can be defined as the union of all nilpotent orbits: F C Q × (2 ) = U(1) ( 5 , Q C H ( 2) , ∈ G (3 , M { 1) ) corresponds to a phase in which only scalar fields from the hyper- ) is in turn determined by the square nodes: , Q Q next to minimal nilpotent orbit = . This quiver has Coulomb branch: nilpotent orbits ], which is the standard mathematical text on the topic. In this section we follow Slodowy slice ( ( 6 (4 nilpotent cone , 1) 29 F H , 2 6 , since it can be understood as the set of all 5 (5) (2 { ¯ is the O is the 1) ) is the , 2) , 2 C , (3 (2 (5) = S (5 O P sl Higgs branch Note that a globalIn U(1) particular, factor a decouples, hypermultiplet hence connecting the two symbol nodes with labels An introduction to 7 8 6 one is in the irreduciblethe representation other with one Dynkin in labels the [1 complex conjugate [0 material in the bookthe [ same notation. where ger 5. Where N ⊂ The multiplets acquire non-zero VEVs. Ingauge a group generic is non-singular fully point broken. of The the Higgs Higgs branch branch the of where nilpotent variety entries that satisfy: phase in which only scalarIn fields a from generic vector non-singular multipletsmaximal point acquire torus VEVs on U(1) different the from Coulomb zero. branch the gauge group is broken to its The edges of the quiverdamental denote and hypermultiplets antifundamental of representations thespace of theory of the transforming this groups under theory the they has fun- two connect. special branches. The given by the circular nodes: The flavor group As is now conventional, this quiver represents a 3 JHEP09(2018)008 , 0 ]. is to Q D 45 (2.8) (2.9) , ) (the (2.12) (2.13) (2.11) C and 28 , , is different as a new Q λ +1 27 , n ( ) is as follows: 25 0 sl [ Q ( ⊂ 1) , ) 2 C 1 ]. In this particular (2 − ¯ n 29 O smoothly equivalent 1 . The Coulomb branch , . Notice that the quiver Q ⊂ (2 instantons when 2 ) and ¯ )     n ) is C O Q | 3 1 1  ◦ ( 1 A 9 Q , ( C − (2 ) 5 1 subtraction of quivers (1 ) (2.10) ⊂ C − ◦ ∈ O ) is the union: C 1 ◦ , | x C ) 1 1  ◦ ) with ∪ O 0 − ◦ ) ) (5 instanton on 0 D Q n, − 3 3 | λ 1 1 (  ◦ ( ( sl 1 1 n 1 , , C O 2 sl A     (2 (2 ⊂ a λ − ◦ ¯ ∈ C ) − O ≤ 1 0 3 [ 1 λ x ∪ O ∪ ,     ) = can also be realized as the Coulomb branch of − ◦ to a point 1 (2 – 4 – = ) D | 1) 1) 1) 1 1  ◦ , , , ¯ 3 ( 0 ) O 2 2 2 1 − ◦ Let the rest of the section focus on the Coulomb , C ) at λ Let us define the (2 (2 (2 | ( relation between partitions [ 1 2  ◦ minimal nilpotent orbit = − ◦ (2 Q : ¯ ¯ O O O ( O ¯ | are obtained by subtracting the ranks in 1 1 ) =  ◦ 0 O − 0 C | Q 0 1 2 = =  ◦ Q − D , generalizes the moduli space of ( Q ) = | | λ 1 1 1 1   ◦ ◦ − C transverse ¯ Q O − 1 ◦ ( − − 1) C , | 0 1 1  ◦ ◦ Q     ] to SQED with 3 electrons. It is in this sense that we said before that 2 (2 24 = = = = ]). The relationship of ¯ O : D 0 28 Q natural ordering ) = Q ( : 0 C is also the reduced moduli space of 1 Q n in a − Q ). The closure of a nilpotent orbit of ). 1 Q denotes the is defined as the closure of the = slice − ( n n C 1 D ≤ a , is also known: The closure of the smaller orbit The variety is the mirror dual in the sense of [ 9 D 2 d a This means that the singularity of 3 the set of closures offrom nilpotent (2 orbits of where notation is taken from [ The ranks of thewhile gauge the flavor nodes nodes of are taken to be the same as the flavors in The subtraction ofquiver two quivers. a quiver gauge theory This quiver is: case, this means that: branch where The closure of a nilpotent orbit. JHEP09(2018)008 ). ). Q D Q ( ( H are the ⊂ C with less Q 0 ◦ ) Q 0 Q ( . C 0 Q and the same connec- 0 the Coulomb part of the Q ) and: or with one exception: there is with equal rank to Q Q 0 ( Q Q is the difference between the rank . ⊆ C 0 ) (2.14) )). D ) Q th gauge node of 0 D i of the set of points in the Coulomb ( D Q ( ( The physical implication is that locally C C ) is the same as that at the origin of the × H . Q
0 ( is defined as: Q Q 0 interior Q , while its Higgs branch is isomorphic to are equal to the ranks of the gauge nodes of – 5 – ⊂ C C ) 0 0 th position of − ◦ i Q ) Q 0 ( Q ) = ) should be understood as: Q C are restricted to be in the subspace ( Q = have the same number of gauge nodes and the same ( 0 Q B 2.14 D and the rank of the Q ∈ C denotes the . The mixed branch is: 0 Q x ◦ ) two hypermultiplets become massless when the VEVs of Q ) 0 and Q ( − Q ( Q C Q C = . D is isomorphic to D ) are strictly lower. Q 0 ( subtraction quiver . Hence, the appearance of new massless hypermultiplets and the transition Q B 3 th gauge node of i Z / with higher rank than their counterpart in 2 , p. 9] (where C Q is a quiver with the same number of gauge nodes as 49 D tions between said gauge nodes. of the rank than their corresponding nodes in of except for a set ofranks the of gauge nodes in a connected subset of the quiver, insame which to the those attached inan the extra corresponding flavor nodes attached in to the gauge nodes adjacent to gauge nodes of The rank of the gauge node in the There is at least one flavor attached to one of the nodes in the subset of gauge nodes The flavors attached to the subset of gauge nodes in The ranks in the gauge nodes of ) = Then, the )[ D 1. 2. 3. 2. 1. D ( ( Let two unitary quivers connections between said gauge nodes, such that the branch. This generatesthe the non-zero possibility VEVs for of a mattersubtraction phase hypermultiplets quiver transition are to fully a defined mixed by branch the where Higgs branch of3 the General definition The right hand sidemixed of branch equation ( More generally, the appearancemassless of states singular along transverse the slices Coulomb indicates branch at that points there that are are new not necessarily the origin of This opens up a newH Higgs branch, which isto equivalent a to mixed the branch KleinianCoulomb at surface branch a singularity of point branch; it its important tocould restrict be to larger, the and interior, since the the singularity transverse would sliceNew not at massless be other states points andin mixed the branches. Coulombthe branch massless vector multiplets of C JHEP09(2018)008 0 ) 6 6 e in Q E C (by of (4.1) ) n, ( min . This . D slice and its sl ( )) theo- n ) n 0 is an orbit , the Higgs ∈ O Q ) ) = ( × H O is the x D ) = 4 quiver Q C (SU( 0 , is isomorphic ( ( ) 6 ρ 0 Q at C H E T N ( Q ]. These are new ) C d where − 46 Q min and the precise form ( ) and the two next orbits , ¯ Q O 6 C ) = G Q e ( = Q ( ) = Lie D 0 B ( , Q C ( Q C 1 | = 4 quiver 2 ◦  . Therefore, the 3 next to minimal nilpotent orbit − N 2 3 with Coulomb branch d C 2 − ◦ Q nilpotent orbit of | 4 ◦ ◦ can be utilized to demonstrate the physical ] (this is a consequence of the analysis of the for closures of nilpotent orbits, due to the fact that − 3 is the closure λ 3 0 ¯ 45 – 6 – . This implies that new hypermultiplets become O , Q 1 ) − ◦ | 28 D 2 ◦  to label their different nilpotent orbits. Instead of the more ( instanton on λ = 4 quiver C = of section : 6 N E , then the Coulomb branch ). Q ) ) d minimal non trivial Q D Let the closure of the ) are closures of nilpotent orbits of the same algebra ( ] used for exceptional algebras we have decided to adopt the notation ( 0 C Q 29 H are in the same position and have the same labels as the flavor . ( 6 C D . The 3 E , for the 6 G E . This means that there is a singularity on ]). It can also be used to analyse vacuum of theories whose Coulomb O ) and 37 min . min . Q n to . ( quiver subtraction . n ]: If the Coulomb branch of C Q and a new Higgs branch opens up, which is isomorphic to 39 by and n x . A transition is possible to a mixed branch of 10 G D min ], where . nodes on 32 The flavor nodes of , n ) transverse On the other hand, the closure of the minimal nilpotent orbit of This conjecture can be used to study the structure ofIn the particular, new it massless states can that be used to analyse the Coulomb branches of In this section we deviate from the notation G 3. Q 10 ( exceptional algebras do notconventional have Bala-Carter partitions labels [ min that are greater in dimensionsection respectively. is This restricted decision to is the motivated smaller by the orbits fact of that each the exceptional discussion algebra. of this to the reduced moduli space of 1 realization of Kraft-Procesi transitions between nilpotent orbitsus of start exceptional algebras. with one Let example. A transverse slice in be denoted was found in [ 4 New Kraft-Procesi transitions on4.1 nilpotent orbits Exceptional algebras The definition of and their transverse slices arebrane given dynamics by in [ [ branches are closures ofresults, nilpotent let orbits us of discuss them exceptional more Lie explicitly algebras in [ the next section. appear along the Coulomb branchof of the any new unitary Higgs 3 branches that open up. ries [ singularity is smoothly equivalentmassless to at branch of this we mean thatHiggs the part is Coulomb isomorphic part to of the mixed branch is isomorphic to Conjecture 1 under the isometry group of C JHEP09(2018)008 0 Q (4.4) (4.2) (4.3) instanton 5 A , figure 1], we 39 ), and hence their 3 = 4 quiver based on the        1 5 A N − ◦ d 2 min 1 2 − ◦ = | | 3 ◦ ◦  5 is the extended Dynkin diagram − a , since the only gauge node of 0 1 2 = 3 Q − ◦ ) there are six hypermultiplets that ]: − ◦ Given the results of [   , section 2]. The closure of a nilpotent orbit 2 1 Q ◦ 1 24 ( mirror quiver of SQED with 6 electrons 50 1 | 2        − ◦ 1 ◦  d . Let us compute the subtraction: | | 3 ◦ ◦  − 5 − ⊂ C a 1 −    6 1 2 = is computed to be a singular space, isomorphic − ◦ E , the Coulomb branch of the subtraction quiver – 7 – | 2 ◦ 1 1  6 1 is the 3 − ◦ S 3 | | E 1 1 ◦ ◦   1 ◦ − 4), and it carries exactly one more flavor than its min D − ◦ is well known. 3 − − 1 6 = ∈ < 1 1 : 2 − ◦ E 1 − ◦ 0 ) is defined as in [ | x 4 ◦ ◦ 0 | − ◦ − ◦ 1 ◦  Q O | 1 1 ◦ ◦ min − ) = 2. The quivers of this set corresponding to the groups   3 0 O − − ◦ C Q 1 1 1 − ◦ ) = 0 ht( | (the top node with rank 2) is adjacent to a node with smaller 2 − ◦ 1 1  − ◦ − ◦ ) = Q ( | | 1 1 11 ◦ ◦ D   Q    Q ( transverse to min C fit into the definition of section of the orbit (see more details in the following section). We thank Giulia Ferlito = = = = C 6 0 E Q D satisfy the properties for quiver subtractions (section height ]. According to section 8 . The resulting quiver 2 can always be realized as the Coulomb branch of a 3 min 46 . E Q [ and . ): n ≤ 6 of a nilpotent orbit ht( 5 C ) Q Z should be the singular space , A ⊆ / and O 0 2 (6 S ] or, in the new language of nilpotent orbits, the closure of the minimal nilpotent 7 Q C sl min E height 24 − [ , = 6 , with ranks equal to the index of each node [ 2 Q ) = 6 This implies that at a point 5 E The C D a E , = ( 11 4 Lie algebras with F of height ht( weighted Dynkin diagram for pointing this out. become massless and a newH Higgs branch opens up, equivalent toQuiver the subtraction Kleinian singularity in exceptionalhave a algebras. set of quivers whose Coulomb branches are closures of nilpotent orbits of exceptional orbit of Notice that with the same rankrank as (the middle nodecounterpart with in rank 3 and eight supercharges. Its Coulombon branch is the reduced moduli space of 1 The slice to D of with Coulomb branch JHEP09(2018)008 ) i for D ( G C ] and min 6 5 1 7 3 ) . 45 i , C E A A D n constitutes D ( 28 4 C min min min min min F . 1 1 +1 2 i | 1 1 ◦  | − ◦ Q 1 contains the result ◦  1 1 2 1 − ◦ − − 1 | | ] during the analysis of 2 1 ◦ ◦  |  ]. Therefore, the Kraft- 1 2 ◦  − ◦ 51 3 − − 46 − ◦ − i 2 1 1 2 2 − ◦ D ). Similarly, we use | 1 4 ◦ ◦ − ◦ 1 − ◦ − ◦ ⇒ ◦ G | 2 ◦ ◦ | 1 2 ( ◦ ◦ − 1 ◦ 3 − 1 − ◦ − Lie 1 ◦ | 1 1 1 ◦ ◦  − ◦ | 2 ◦  ]). Notice that the case of 1 is computed and its Coulomb branch 2 2 | 2 ◦  for the next orbit in dimension after the next 40 +1 | , 1 2 3 1 ◦  i − ◦ − G | 1 Q 1 2 4 ◦ 1 37  − ◦ − ◦ 3 − and a point in the interior of | 1 2 ◦ |  – 8 – 2 4 − 4 ◦  − ◦ − ◦ − i min − ◦ i . ]) has a physical realization also for exceptional 2 − ◦ 6 3 − Q − ◦ 4 n − − ◦ Q . 2 3 3 46 5 5 1 2 2 n − ◦ − ◦ − ◦ − ◦ i = instantons. : | | | 8 3 4 2 ◦ ◦ ◦ ◦  3 3 − ◦ 5 − ◦ − ◦ ⇒ ◦ Q ⇒ ◦ i n | | | 4 ◦ ◦ 6 6 ◦ ◦ ◦ ◦ 3 − ◦ − 4 5 − 3 D − ◦ C 3 2 | | − ◦ ◦ 6 1 − − ◦ ◦ 10 − ◦ − ◦ 3 4 4 | 1 2 − ◦ 2 ◦ − ◦  ◦ − − and | 1 2 1 ◦ ◦  − ◦ 4 − ◦ − ◦ 7 n | 2 2 2 ◦  ◦ ◦ 1 − ◦ − ◦ B 2 | ◦ , 4 ◦  2 G 7 , 6 7 8 4 E 4 6 7 8 ) 4 F E E E i F F E E E Q min ( . min min min min . . . . C min min min min n . n n n n n correspond indeed with the expected transverse slices [ i . Kraft-Procesi transitions between closures of nilpotent orbits of exceptional Lie algebras. i 9 7 5 8 6 2 4 3 1 D denotes the closure or the minimal nilpotent orbit of G extended to exceptional algebrasalgebras (beyond by the [ classical realizationthe in [ first example oftypes a of quivers Kraft-Procesi and transition theirmoduli between Coulomb spaces branches non were of simply introduced in laced [ quivers. These Coulomb branches can beof studied subtracting with quivers whose this Coulomba new branches link procedure. are in closures their of Table quiver corresponding nilpotent orbits Hasse related diagrams. by Procesi The transition Coulomb branches (that of in the the subtraction mathematical literature was introduced in [ the next to minimalto nilpotent minimal. orbit The and subtractioncorresponds quiver to the transverse slice between min Table 1 JHEP09(2018)008 . . 6 +1 D i . (4.5) (4.6) Q (1) where , with ) to be min O 7 − i . i E D n − ◦ Q Q ( 13 C = min . is and i predicts that n USp(2) 7 . D 1 C E n − ◦ ,    1 B  (12) O − ◦ ). If one restricts the . This singularity is ◦ 2 ) +1 i − ◦ 6 +1 Q i 3 D ( Q C 2 ( − ◦ are used). Let us compute ] to be the closure of the next | min 4 C ◦ ◦ . 1 n 36 transverse to min − 7 3 = E 1 − ◦    2 | 2 ◦  min | 2 . ◦     n . − − n 2 = 4 orthosymplectic quiver ] for Lie algebras of type is isomorphic to the Coulomb branch of the is isomorphic to    ⊆ 2 7 − ◦ N i 40 – 9 – E | S 2 3 d ◦ 2 Q  | 2 1 ◦  − ◦ − | 2 ◦ ◦ 4 2 − | 2 2 ◦  − − ◦ 1 ◦ 5 , with complex dimension 34. Conjecture (12), with complex dimension 20: − ◦ − and their embedding in Type IIB string theory, introduced 7 , the slice    ), the VEVs of the remaining massless vector multiplets 2 2 . In this way, the computation of transverse slices allows 3 − ◦ so E i i 1 | 6 ◦ ◦ 1 Q − ◦ − ◦ D It is worth noting that the quiver subtractions can be utilized (where the labels from table of ). The fact that there are VEVs sitting at the singular point ( | ) = 2 2 ◦ ◦ − min 5 5 can be seen from the work in [ +1 D 12 . 4 1 i − − ◦ ( Q Q 7 transverse to min D | ⊂ C Q 1 2 ◦ ◦ C − ◦ − E − 7 ◦ 2 ◦ − 3 − ◦ − E ) 3 i 0 1 ◦ ◦ Q    Q +1 Q i min = = = = . Q = ( n D i . C n D ( ⊆ C S orthosymplectic quivers ) implies that new massless states appear (including massless hypermultiplets), and i D The physical implications are as follows. Let us consider the subtraction ( This is the same as theThe Higgs authors would branch like of to the thank Paul 3 Levy for confirming that this is indeed the correct transverse slice. C 12 13 4.2 Classical algebras The Kraft-Procesi transitions described inbased on [ Hence, according to Conjecture The Coulomb branch of to minimal nilpotent orbit the slice quiver subtraction the subtraction: A new prediction in to predict a newthat result, are namely not linked thecomplex together transverse dimension in slice 54, their and between Hasse closures diagram. of This nilpotent is orbits the case for of with them the possibilityis of equivalent to transitioning the to transitiontheory a that defined mixed takes by branch place the at infor quiver the the a origin moduli global of view space, the onget which Coulomb mixed a branch phase branches. in with the For mixed each Higgs case and there Coulomb is branches. a singular transverse slice, we Then, a subset of thevacuum Coulomb expectation branch values of of thein corresponding the massless subset vector multipletssit in at a singulardefined by point in the directions perpendicular to JHEP09(2018)008 ): 14 C , ), or ], let (4.7) . The (4.11) (4.12) (4.13) C (9 ) , 6 51 1 so , , (3 + 1 36 ¯ n O ] dual map (2 52 so and 1) , 4 (2   orthosymplectic quiver ¯ 1 O ] are restricted to the set ): ⇒ ◦ 40 C 2 , ) (4.8) 1 − ◦ )) (4.9) (4.10) (9 C | 2 ◦  1 C C . There are two nilpotent orbits , so 1 , , 4 | 1 2 ◦  ). In this section we show how the b − (9 (9 (9 ]. 1 C ◦ ⇒ ◦ so = ⇒ ◦ ⇒ 53 so so 2   n, 2
3 ⊂ ( ) ⊂ ⊂ 1 − ◦ 5 − ) , can be used to overcome this obstacle, at − ◦ 1 − ◦ ) 5 sp | , 2 ]. ◦  6 1) 1 1 2 2 , , 2   1 4 , 2 1 40 (2 − . 2 ) − ◦ (2 (3 (2 − ◦ | 2 n ◦ 1 5  Q 1 ◦ ¯ ¯ ¯ | 1 c 2 1 ◦ ◦  O O O , | 1 ◦  2 – 10 – ⇒ C ======(2 : : : 3 ⇒


and Q ) ) = 4 quivers based on Dynkin diagrams [ ) ) 5 6 1) 1 5 6 − ◦ 1) , n 1 1 such that , − 1 , 1 1 4 , b , 4 , 2 2 − ◦ ) N ]. However, due to the Barbasch-Vogan [ 2 (2 (3 | 1 ◦ 6  (2 (3 1) (2 , 0 1 (2 − ◦ in the Hasse diagram of d Q Q , 4 34 Q Q Q 1 Q ) ◦ – ⇒ (3 (2 − ◦ 5 ] we write 1 C C 0 1 Q ◦ ◦ Q   , 32 C 2 , the closure of the minimal nilpotent orbit of 28 = = = = n (2 b ¯ and O 1) , 4 1) Utilizing 3 , (2 ]. See also the discussion in [ 4 D (2 34 . Due to this, the Kraft-Procesi transitions in [ 4 Q b 15 . special be the quiver: ) = 4 quivers 5 ] and further refined in [ 1 such that its Coulomb branch is the closure of the minimal nilpotent orbit of , See a particularly clearThis exposition is of a this result duality of map [ in [ 2 31 N , the closure of the minimal nilpotent orbit of 14 15 (2 d n We can perform different subtractions for each quiver, obtaining: are: Employing the notation ofthat [ sit directly above of 3 subtraction of quivers, together with Conjecture least for nilpotent orbits ofof height Kraft-Procesi lower or transitions equal of than type 2, andTransitions provide a to physical realization Q Higgs or Coulomb branches ofthat this are type of quiversof can special only nilpotent be orbits. closureswhose of In Coulomb nilpotent branch particular, orbits is it hasc not been found an in [ JHEP09(2018)008
) are 6 1 , (3 (4.17) (4.18) (4.14) (4.15) (4.16) D with the ]. C Q 28 and
is always formed The nodes in the 1) , is the closure of the 4 O 1 (2 16 type algebra and one a = 4 quiver D   1 C weighted Dinkin diagram N C , ⇒ ◦ d ) and 2 1 C ) (4.19) , ) (4.20) 1 − ◦ C be: C , (5 | 2 ◦  ) , 2 1 6 (4 so 1 − (4 , sp ⇐ ◦ ⇐ ◦ 1 ◦ (2 sp 1 2 1 2 Q   ⊂ b a ⊂ 1 ] the labels of the ) − ◦ − ◦ − 4 ) = = | 6 1 1 of nilpotent orbits of classical Lie algebras coincide 2 39 ◦  , 1 and

2 ,   ) ) 1 − ◦ 1 ) − 5 (2 (2 6 1) 4 , 1 | 1 1 1 , ¯ ¯ ◦ ◦  1 4 , 0 , O O 2 ⇒ ◦ 2 – 11 – (2 (3 (2 = = = = 2 (2 : : ⇒ ◦ D D ] of the corresponding orbit. Q ) )

Q 0 4 6 − ◦ ) ) C C 29 1 1 4 6 − , , [ 2 1 1 − ◦ ) 2 , , (2 6 2 of any Lie algebra. The 3 2 0 2 (2 (2 − ◦ 1 , ). According to Conjecture Q (2 | Q 2 Q ≤ ◦ 2 2  − ◦ (3 C Q , appendix B]. In [ ) , | | 1 1 ◦ ◦   Q   C 36 being the closure of the nilpotent orbit O C (2 = = = = sl ¯ ) O weighted Dynkin diagram 6 1 , Let the quivers (3 of the orbit. ) = D Q This physical realization can be generalized to transverse slices between ( tabulated in [ C type. weighted Dynkin diagram is the closure of the minimal nilpotent orbit of Characteristics Root Maps 2 D ). This is indeed the case, as it was computed by Kraft and Procesi in [ b C Let us illustrate the result with one further example from a , Note that the labels in the (9 16 with the called the Their Coulomb branches are: from a C type transition. weighted Dynkin diagram becomeas gauge the nodes Dynkin in diagram), theflavor the nodes quiver labels (with attached in to the thedetermined each same weighted by different connections Dynkin imposing gauge diagram that node, become allnode and the of is the ranks them equal ranks of should to of be twice the its balanced gauge rank). (i.e. nodes number of are flavors of the so Generalization. nilpotent orbits of heightCoulomb ht( branch from the where minimal nilpotent orbit of should be the transverse slices between the corresponding orbits and the minimal orbit of The Coulomb branches of the subtraction quivers are: and JHEP09(2018)008 is 3 c (4.23) (4.24) (4.27) (4.21) (4.22) ). C , (3 sp      1 1 1 − ◦ ⇐ ◦ | 2 ◦ ◦ 1 3 )) (4.25) (4.26) c 1 − be: − ◦ C C | 2 , , ◦ =  1 ) 1 6 | 2 1 1 ◦  − 1 ,   − ◦ (10 (10 2 1 1 ◦ | 1 1 1 2 ◦ ].  − − ◦ (2 so so | | | 2    3 ◦ ◦ ◦ ◦    Q 28 ⇐ ◦ ⊂ ⊂ 1 − 1 − − − ) ) | 2 6 2 2 ◦  ) 1 − ◦ 1 1 1        and   , , 6 ) 4 2 | 1 2 1 | ◦ − ◦ − 6  ) 2 , ◦ 1  1 2 2 (2 (2 , 1 1 ◦ ◦ 1 1 |   1 ◦  , 1 ¯ ¯ 2 − (2 ⇐ ◦ (2 O O 4 – 12 – C | | = = 1 3 Q ◦ ◦ 1  1 2 − Q : : (2 ⇐ ◦ = = 1 | | | 1 ) ) = ◦ 1  ◦ ◦  1 Q 1 − − − ◦ − 2 6


1 1 ) ) ) 1 ) 2 | , , 2 − ◦ 1 )  − ⇐ ◦ 2 6 2 − ◦ 4 4 2 4 1 1 1 | 1 1 1 ◦ , , ,  0 | 1 1 1 , ◦ − ◦  (2 (2 , − 4 2 4 2 2 1 1 1 ◦ 1 1 Q Q − − ◦ 1 ◦ (2 (2 (2 − ◦ − ◦ (2 − (2 | | | 1 1 0 | ◦ ◦ ◦ Q Q ◦   Q        1 0 ◦ ◦  Q   D ======C C ) is the closure of the minimal nilpotent orbit of C ) ) 2 C 4 1 , 1 , , 4 2 (3 Let the quivers (2 (2 sp D D ⊂ ) 4 1 , (2 ¯ O = 3 c The subtraction can be computed: Their Coulomb branches are: where indeed the transverse slice between the twoD orbits type [ transition. Its Coulomb branch is The subtraction can be computed: JHEP09(2018)008 d is : 8 3 d Q (5.2) (5.4) (5.5) (5.1) (4.28) for 3 ). i C D , = 1 (eight (6 so and N i d Q : 8 = SU(3), nine quarks H 2 G 1 − ◦ is the gauge coupling. Let 3 5 g − ◦ d = SU(3), and a Chern-Simons theories and 2 − ◦ = | 4 8 ◦ ◦ G d − ◦ this will result in a new effective   4 − 1 9 7 ) (5.6) ) (5.3) ]. Therefore, the physical realization | − ◦ 1 ◦  H 8 9 0, where 2 | at infinite coupling 3 6 − ◦ ◦ ◦ 28 1 / Q Q − 1 6 ( ( from the previous section and define the , d − 1 d d , → i | | 3 3 8 9   ◦ ◦ 5 − ◦ 2 Q 1 C C − ◦ 5 0 is isomorphic to the Coulomb branch of a − ◦ | SU(3) 1 ◦  /g SU(3) 4 − ◦ ) = ) =   → 4 = – 13 – 9 8 − ◦ : = C 2 : H H 3 − ◦ 8 2: ( ( 9 In order to answer this let us write the quiver = /g / 3 H theories with gauge group ∞ ∞ − ◦ H
) 2 − ◦ H H d 2 = 1 1 2 , − ◦ 4 k 1 − ◦ ◦ (2 what is the effect of integrating out this massive quark on is reserved for quivers of 5 = 8, the same gauge group 1 ◦ D = ) is the closure of the minimal nilpotent orbit of i : f H C 8 C = N : , Q = 1. Let the resulting theory’s quiver be 9 (6 k Q so 2, ]. Let us recycle the labels ⊂ / ) 41 2 1 , 2 (2 as: ]: ¯ 9 O 41 Q = 3 d = 4 quiver [ One can ask the question = 9, and Chern-Simons level be the quiver of one of these 5 quiver N f 9 d d such that [ the Higgs branch at infinite coupling? In the following, thequivers. label If atheory massive with quark one less is flavor level integrated increased out by from 1 The relation can formally be written as: Its Higgs branch at3 infinite coupling 1 3 supercharges) SQCD theories, atH the limit 1 N an embedding in Type IIB string theory. 5 Integrating out aThis massive section quark analyses in the 5 physics of integrating out massive quarks in 5 where indeed the transverse slice betweenof the Kraft-Procesi two transitions orbits is [ extended in this way to quivers that do not necessarily have Its Coulomb branch is JHEP09(2018)008 8 Q (5.8) (5.9) (5.10) (5.11) and 9 Q with label 1 still quiver subtraction 8 is that with label zero and Q 9 9 D Q     | 1 2  ◦ | 2 1 2      ◦ ). Let us denote the quiver 2 − 9 | 1 2  ◦ − ◦ − 4 0 Q − ◦ 5 4 ( , and add one extra zero node − 5 d 9 − ◦ − ◦ 3 4 − ◦ − ◦ | Q 3 6 1 1 ◦ ◦ − ◦ | | 4 8 3 6 ◦ ◦ ◦ ◦ − ◦ | 4 8 ◦ ◦ − ) (5.7) − ◦ − ◦ | ⊂ C 3 6 ◦ ◦ − − 9 | | 5 1 1 2 2 ◦ ◦ ◦ ◦ ) − 1. Instead, one can use the decoupling 7 5 H 8 8 − ( 7 − ◦ − − − 5 Q Q − ◦ − ◦ ∞ 4 ( 2 2 − ◦ 6 4 d − ◦ − 3 6 − ◦ − ◦ − ◦ 9 4 C − ◦ − ◦ 3 2 2 ⊂ H − ◦ Q 5 3 − ◦ = 4 quiver with no flavor nodes to rewrite: ) 5 − ◦ − ◦ − ◦ – 14 – 8 3 = − ◦ − ◦ 2 2 2 − ◦ H N 9 4 2 − ◦ ( 4 − ◦ D − ◦ − ◦ d quivers satisfy the conditions for 2 ∞ − ◦ − ◦ 1 2 2 − ◦ 8 1 3 d H − ◦ 3 . Note that the rightmost node of − ◦ − ◦ − ◦ Q 1 8 ◦ description − ◦ − ◦ 0 2 2 ◦ − ◦ | Q − 0 1 2  ◦ ◦ d = | 1 2      ◦ 9 − ◦ − ◦ 8 = = | | 1 2 1 2   ◦ ◦ Q     − Q 8 9 = = = = Q Q 9 D . Then, the two 3 8 Q ); they are: 3 Now the difference can be taken: Let us also decoupleto a the U(1) left from of (section the leftmost node in constitutes a problem, sincethen adding subtracting an would extra result node inof a to a negative U(1) the label from right the of gauge group of a 3 The first thing todo notice not when have one thegauge tries nodes same to with number perform label of the zero gauge difference to nodes. However, this can be solved by adding and the effect ofcan a be massless identified quark utilizing becoming theresulting massive, from 3 while the the quiver gauge subtraction coupling by: is infinite, Then, we have JHEP09(2018)008 in 4 (5.14) (5.15) (5.16) (5.12) (5.13) quiver and 3 to a point in ) perpendicular 9 ) are always freely Q i ( 32 d D , the VEVs that are 3 ( ◦ C C d ) . This implies that the 3 8 n = C Q “transverse” H (     d ) 1 Kraft-Procesi transitions 3 9 . This idea is a direct con- C H − ◦ ( | 1 2 ◦ ◦ and ∞ − 2 ⊆ H − ◦ 2 S / 16 2 between nilpotent orbits, but the 0 0 , , 5 SQCD with eight supercharges does not H , | | − ◦ | 8 6 5   ◦ ◦  ◦ d 2 = ) to its subset SU(3) SU(3) 9 − ◦ SU(3) 32 “slice” Q quiver subtractions 2 C = = ( – 15 – = : : d : − ◦ 3 transverse slices 8 6 = 5 I J C 2 H S − ◦ | 1 2  transverse slice ◦ is not singular. In this case, if the VEVs of some vector     . Note that the transverse spaces is known to be freely generated by twisted hypermultiplets: 16 d 3 9 2 H C D ) = ) = 9 9 . This means that there are no new massless states arising as a D D ◦ ) as: ( ( ) 8 d d 8 3 3 H Q C ( C ( are tuned to restrict d ∞ = 3 9 is found: H C theory, this means that the process of integrating out a massive quark, while S Q 5 d can extend the notion to quivers whose Coulomb branches are not closures H The same analysis can be done with different starting points: The effects can be computed by employing quiver differences in a way analogous to One can proceed integrating out consecutive massive quarks from the theory until the One can use this result to describe the The Coulomb branch of , and are presented in table 9 respectively. 6 Conclusions and outlook We have formally definedsequence the of concept the of physical realization of The results of integrating out massive quarks one by one are depicted on tables include the appearance of new massless states. D generated by twisted hypermultiplets, andphysics hence of have integrating the out form a massive quark in 5 view of the 5 the gauge coupling remains infinite, does not involve thelast creation quiver of new massless states. moduli space multiplets of tuned are not into a the singular point subset withconsequence respect of to this tuning, the and directions there in are no new branches opening up. From the point of the interior of Note that in this casesubtraction this is not a of nilpotent orbits. Another difference with the previous cases studied here is that the JHEP09(2018)008 d d ) i ) i D 8 6 5 16 ( D 4 5 8 are 5 are 5 H H H d H ( 3 i i H H H d I C 3 H C | 0 0 0 1 ◦  1 −◦ −◦ −◦ 1 − 1 1 | 0 1 1 ◦  − ◦ 1 1 − ◦ 1 1 − ◦ − ◦ − 1 | −◦ 1 2 ◦ ◦ | | 1 1 1 ◦ ◦  1 −◦ − ◦ − − ◦ − − ◦ 1 1 − ◦ 2 1 1 1 1 − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ 1 1 i 1 1 − ◦ 2 1 1 1 i D | | 1 1 ◦ ◦  − ◦ − ◦ D − ◦ − ◦ − ◦ − ◦ | 1 1 ◦  2 1 1 1 ◦ − ◦ − ◦ − ◦ − ◦ − 1 2 0 1 1 ◦ theories at infinite coupling. theories at infinite coupling. − ◦ ]. On the right, the quiver subtractions − ◦ − ◦ ]. On the right, the quiver subtractions − ◦ − ◦ − ◦− d 1 d 2 0 0 41 1 41 ◦ ◦ − ◦ − ◦− 1 − ◦ 1 − ◦ − ◦ − ◦ )[ − )[ i | 1 ◦ i |  2 0 0 0 ◦  ◦ ◦ ◦ Q Q ( ( d 2 d | | 3 1 1 1 1 1 2  ◦ ◦  3 C 1 C − ◦ −◦ −◦ − 1 1 − − ◦ – 16 – | 5 3 2 3 4 ◦  1 2 − ◦ ) = ) = i 4 1 1 3 2 3 2 − ◦ − ◦ − ◦ 1 2 − ◦ − i − ◦ − ◦ I | | | | | H | | 8 5 3 ◦ ◦ 6 4 ◦ ◦ ◦ ◦ ( ◦ ◦  ◦ ◦  2 3 1 ( 1 − ◦ − ◦ ∞ | | − ∞ 2 3 − − − − ◦ ◦ 1  − ◦ − ◦ H 7 5 4 3 2 . . H | 4 3 ◦ ◦ 1 2 − ◦ − 1 1 i − ◦ | | − − − ◦ − ◦ − ◦ − ◦ 2 4 1 2 ◦ ◦  − − ◦ i i i Q 6 4 3 2 1 | | 5 3 ◦ ◦ Q Q ◦ ◦ Q 1  − ◦ − − ◦ | − ◦ − ◦ − ◦ − 2 3 − − ◦  − − 0 5 i i 3 2 1 4 ◦ 2 − ◦ − Q Q − ◦ − ◦ − ◦ − ◦ − 2 1 ◦ − ◦ − ◦ 0 0 = = 4 2 1 3 ◦ 1 ◦ i i − ◦ − ◦ − ◦ − ◦ − ◦ − D D 1 1 ◦ − ◦ 0 0 0 3 1 | quivers such that 2 ◦ ◦  quivers such that − ◦ − ◦ − ◦ 1 − ◦ d − d 0 0 0 0 ◦ ◦ ◦ ◦ | 2 ◦  2 2 / 1 / 0 3 1 , , 2 2 2 are 3 are 3 , , / 1 / 2 / i | | | | 5 6 7 8     ◦ ◦ ◦ ◦ I i 1 3 5 i , , i , , , Q Q . Quiver differences between Higgs branches of 5 . Quiver differences between Higgs branches of 5 | | | | | 9 8 7 6 5      ◦ ◦ ◦ ◦ ◦ SU(3) SU(3) H SU(3) SU(3) SU(3) SU(3) SU(3) SU(3) SU(3) i 5 6 7 8 i 9 8 7 6 5 Table 3 quivers. have been computed Table 2 quivers. have been computed JHEP09(2018)008 ] d 40 = 1 = 4 are 5 N N i theories J d d d quiver sub- 0) physics, i.e., and 5 ) , i d D 4 ( quiver subtractions H d = 1 SQCD theories. = (1 3 C N N extends the physical re- d d 1 −◦ 1 − ◦ i | 1 1 ◦ ◦ D − − theories at infinite coupling. 1  ]. On the right, the quiver subtractions 0 d 41 − ◦ 0 ◦ )[ i Q Superconformal Field Theories in Four or 1 (  1 quiver subtraction d 3 − ◦ 1 C 1 ◦ − 2 – 17 – − ◦ 1 ◦ | ) = − ◦ 1 2 ◦− ◦ i i J | 2 3 ◦ − ◦ ( − Q ]. In this note we provide an example in which it can | 1 2  ◦ ∞ − , were some of this project was further developed. S.C. − 44 1  H . − 2 1 1 ◦ − − ◦ i 1 ◦ Q − i 2 ]. However, it has recently been discovered to be a more general 0 / Q 1 , ]. 37 , i = | | 5 6   ◦ ◦ J i 44 D SU(3) where the ideas for this work where kindled. The two authors are also SU(3) quivers such that ] to exceptional Lie algebras and to nilpotent orbits of classical algebras d i 5 6 instanton transition [ 40 , 8 are 3 E 37 i are specially suited, is the transverse slice between the Higgs branch of 5 Q . Quiver differences between Higgs branches of 5 We have also shown explicitly how the More Dimensions thankful for the hospitalityPhysics and of Geometry the of IFT F-Theory is supported in by Madrid an EPSRC andConsolidated DTP Grant the studentship ST/J0003533/1, EP/M507878/1. organizers and A.H. of EPSRC is Programme supported the Grant by conference EP/K034456/1. STFC Acknowledgments We would like to thankClaudio Giulia Procesi, Ferlito, Rudolph Travis Kalveks, Schedler Paul andvelopment Levy, of Gabi Noppadol this Mekareeya, Zafrir project. forand A.H. helpful is conversations the thankful during organizers for the the of de- hospitality the of Aspen Winter Center Conference for Physics tractions SQCD at finite anddescription at of quiver infinite subtractions for coupling. orthosymplecticand quivers, also It derived from utilized is the in work also [ in desirable [ to produce a similar formal that are non-special.remarkably We simple. want We to hopecomputational mention that power. once the We more examples believea that shown that very these are the useful two sufficiently tool tool features in illuminating willwith the presented of set make eight herein its of of supercharges. resources is One available to computation study that the remains physics to of 6 be done, for which In particular we show howthe to Higgs characterize branch the of effect the of theory integrating at out infinite a coupling. massivealization quark of on transverse slicesalgebras and [ Kraft-Procesi transitions in nilpotent orbits of classical terms of D-branes dynamicsquiver in gauge theories Type [ IIBconstruction, string that plays theory an and importantthe moduli role small on spaces the of description ofbe 3 6 utilized to analyze the physics at infinite gauge coupling of 5 Table 4 quivers. have been computed JHEP09(2018)008 ] SU(2) ]. ] = 2 ]. 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