JHEP11(2018)126 Springer November 3, 2018 November 21, 2018 September 13, 2018 : : : Accepted Received Published Published for SISSA by https://doi.org/10.1007/JHEP11(2018)126 . 3 1806.00024 ) supersymmetric gauge theories. We show that the space of supersym- The Authors. N Brane Dynamics in Gauge Theories, D-branes, Solitons Monopoles and In- c
In this paper, we propose a string theory description of generic ’t Hooft defects , [email protected] = 2 SU( N NHETC and Department of126 Physics Frelinghuysen and Rd., Astronomy, Piscataway, Rutgers NJ, University, 08855 U.S.A. E-mail: Open Access Article funded by SCOAP Keywords: stantons ArXiv ePrint: in metric ground states issetting, given by Kronheimer’s the correspondence modulibrane is space configuration of realized can singular be as monopoles used T-duality. and to that study We in the conjecture full this dynamics that of this monopole bubbling. Abstract: T. Daniel Brennan Monopole bubbling via string theory JHEP11(2018)126 17 14 = 4 gauge theories 43 N 9 19 46 16 5 31 6 9 13 32 23 35 50 – 1 – 38 41 15 33 45 10 45 27 8 15 42 3 14 38 20 29 19 22 2 ) irreducible monopoles N 3.4.1 Action3.4.2 of T-Duality on fields String3.4.3 theory analysis T-duality and line bundles 3.2.1 Low3.2.2 energy effective theory Vacuum equations 2.3.2 Review of Kronheimer’s correspondence 2.4.1 Summary 2.2.1 Quick2.2.2 review of bows Bow2.2.3 construction of instantons Bow2.2.4 varieties Bow variety isomorphisms: Hanany-Witten transitions 2.3.1 Taub-NUT spaces 2.1.1 ’t2.1.2 Hooft operators Singularity structure: monopole bubbling A.1 Review ofA.2 Taub-NUT spaces Kronheimer’s correspondenceA.3 for a single Generalization defect to multiple defects 4.1 SU(2) irreducible4.2 ’t Hooft defects SU( 4.3 Physical ’t4.4 Hooft charges Irreducible monopole bubbling and future directions 3.4 Kronheimer’s correspondence and T-duality 3.1 Brane configuration 3.2 SUSY vacua 3.3 Monopole bubbling 2.4 Reducible singular monopole moduli space and 3D 2.3 Kronheimer’s correspondence 2.1 Review of singular monopole moduli space 2.2 Instantons on multi-Taub-NUT and bows 1.1 Outline and summary A Kronheimer’s correspondence 4 Irreducible monopoles 3 Reducible singular monopoles in string theory 2 Singular monopoles and their moduli spaces Contents 1 Introduction JHEP11(2018)126 = 2 and = 2 (1.1) is the funda- ]. By P N N W N 33 + in which 1 cr are param- = 2 Λ theories and f cr ∈ N ∗ Λ , ∈ . v) = 2 b 3 , R a P, N 1 ; S → , m a TN ( -fiber degenerates [ 1 in the gauge group, and S P mono Z This theory can be embedded in = 2 gauge theory with v) 3 ] for a complete discussion. N P, 8 ; ) is the cocharacter lattice of the gauge group, N , m ], the expectation value of ’t Hooft opera- a 2 ( 31 -fiber of Taub-NUT: cochar -invariant instantons on Taub-NUT ) is the ’t Hooft charge, 1 S K G – 2 – ( 1-loop Z where Λ v) , cochar b fixed point where the ( Λ similarly dissolve into a ’t Hooft defect. πi /W K 2 ], this dissolving of instantons is equivalent to a process ∈ 3 ]. See upcoming work [ e to denote the line operator associated to the ’t Hooft and Wilson )) | R 35 ] P 31 P P ( P,Q Z |≤| [ X ( v L | wt fundamental hypermultiplets. Λ = × E N 2 0] P, [ ) SYM theory although similar results apply to ≤ action is translation along the cochar L ], the authors proposed a brane configuration to do so for products of N f Λ K D 6 ], and by using AGT or spectral network techniques in theories of class v) is the contribution from the degrees of freedom trapped on the ’t Hooft ]. N ]. We will specifically be interested in ’t Hooft operators in theories that lie ∈ P, 26 31 31 ] ; , , – 8 ) is the Killing form, 6 – , m , )) is the weight lattice of the centralizer of the cocharacter P,Q collectively denotes the masses of the matter in the theory. Here, the sum is over 23 6 = 2 SU( a , P ( ( 2 m In this paper, we will restrict our attention to supersymmetric ’t Hooft defects in In order to understand this contribution, we need to have some way to study monopole From general field theory considerations [ In this paper, we will be studying supersymmetric ’t Hooft defects in 4D , N Z Here the U(1) Here we use the notation There are additionally subtleties associated with SU( ( 1 2 3 1 [ mono wt charges [ Λ Weyl group. mental hypermultiplets as noted in [ 4D theories with string theory as the low energy limit as the effective world volume theory of a stack of D3- bubbling. In [ minimal ’t Hooft operators.brane In configuration this describes paper, ’t’t we Hooft Hooft provide defects operators. more and rigorous give arguments why a this brane configuration for generic eters encoding the expectation valueand of bare electric and magneticbubbling line configurations defects which respectively, are indexedZ by the effective ’t Hooftdefect charge [ v tors in these theories is generically of the form where ( ators can be computedtheories exactly by [ using localizationS in the case ofat Lagrangian the 4D intersection of these two classes. instantons dissolve into theKronheimer’s U(1) correspondence [ in which smooth monopoles in supersymmetric gauge theories. In these theories, the expectation value of ’t Hooft oper- Monopole bubbling is the processdefect in (singular which monopole). smooth monopoles Thismagnetically are non-perturbative absorbed charged phenomenon into line is a an defects ’tcharge important in Hooft of aspect four the of dimensions. defectdiscovered by It and studying decreases the traps bubbling the quantum of effective U(1) degrees magnetic of freedom on its world line. This was 1 Introduction JHEP11(2018)126 . ) ) ]. ]. 4 N N 4.4 29 ] to , 12 12 , , 6 11 ] and use = 2 SU( 35 N N D3-branes. These N = 2 theory by introducing N ]. N-1 20 are a basis of simple magnetic N-1 } p I h . Because these D1-branes have ) gauge theories, ’t Hooft defects { 2 N we will review some relevant details 2 where ], but we will extend this analysis to include all I h 12 . In the low energy limit, these D1–NS5 I = 2 SU( p 1 I 3 N P – 3 – ] and Kronheimer’s correspondence [ 3 = p P 17 , See figure 13 2 5 2 p . 4 1 1 p See figure 6 . 3 In this description, smooth monopoles take the form of finite D1-branes stretched . This figure illustrates the brane construction of a generic ’t Hooft defectin the SU( 4 In the following section, we will study the brane configuration proposed in [ We will show that in this description of Note that we are truncating the world volume theory of the D3-branesThis to an has been shown for certain ’t HooftTo defects our in knowledge, [ the identification for irreducible singular monopole moduli space is not known. We = 2 SYM theory with ’t Hooft charge 4 5 6 a sufficiently large mass deformation. ’t Hooft defects. This configuration is very similar to thecomment Chalmers-Hanany-Witten further construction on [ the case of irreducible singular monopoles and how it relates to our results in section operators are describedbetween by the adding D3-branes. spatiallyworld transverse One volume NS5-branes can theory at see by that fixedbranes performing they connecting positions Hanany-Witten the source in transitions D3- magneticDirichlet and such charge and NS5-branes. in that Neumann the there boundary See D3-brane are conditions figure D1- at opposite ends, they carry no low energy We give the explicit description offor reducible section singular monopole moduli space in preparation describe reducible ’t Hooft defectsSYM (products theory of which we minimal embed ’t in Hooft the defects) world in volume theory of a stack of 1.1 Outline and summary The outline of this paper isof as follows. singular First, monopole in moduli section instantons space. on Additionally, Taub-NUT we spacesthem will [ to review the show that bow construction singular of monopole moduli space can be realized as a bow variety [ can be realized asspatially finite transverse NS5-branes. D1-branes runningsystems will between introduce the fixed, magnetically D3-branesgiving charged and rise point to sources some that ’t auxiliary, have Hooft no defects. moduli, thus weights. branes. between the D3-branes which separate in the low energy limit [ Figure 1 N JHEP11(2018)126 v . J . 2 for some J I -v δ ). Here we also see that by ⊗ = m I p 2 p 1 ) is computed by the Witten index of the 1.1 . We then expect that this well behaved brane ( – 4 – HW Trans. HW for the case that 3 1 by identifying the moduli space of supersymmetric mono ]. Specifically, we find that the vacuum equations are v Z 3.2 21 , p 20 ] in which 6 ) SYM theory. We show that this configuration is indeed the correct N -invariant instanton configurations on Taub-NUT given by Kronheimer’s , we propose a new brane configuration to describe generic ’t Hooft defects 1 2 K 4 4 x -v = 2 SU( . This figure illustrates how Hanany-Witten transformations are realized for a brane m N 1,2,3 In section We then study the relationship between Kronheimer’s correspondence and T-duality. Then we argue that this brane configuration can be used to study monopole bubbling. We then concretely demonstrate that these NS5-branes induce ’t Hooft defects in the x sponding U(1) correspondence. Thus indence. this This setting, is one T-duality of is the identical main to results of Kronheimer’sin our 4D correspon- paper configuration can be usedlocalization to results understand of monopole [ effective SQM bubbling. of the This bubbled is D1-branes. supported by the Specifically, we find that T-duality maps singular monopole configurations to the corre- In this setting, monopole bubbling occursbranes when becomes a coincident D1-brane with stretched between anthis adjacent NS5-brane. seemingly D3- singular By configuration using cannot Hanany-Witten be intersect transformations, exchanged the for NS5-brane, one but inbrane rather which to is the one coincident D1-brane of with does a the D1-brane D3-branes. connecting See the figure NS5- vacua of the world volume theorymoduli of space the in D1-branes analogy with the withgiven appropriate [ by singular Nahm’s monopole equationsequations, which define we a identify the bowmonopole moduli moduli moduli space. space space of Then by supersymmetric using by vacua the studying with bow the reducible description bow derived singular in section degrees of freedom as we expectis from Hanany-Witten ’t dual Hooft operators. to that Note of that this figure brane configuration D3-brane world volume in section Figure 2 configuration of transverse D1-performing (red), a series D3- of (black) Hanany-Witten moves andworld the NS5-branes volume NS5-branes source theory. magnetic ( charge in the D3-brane JHEP11(2018)126 . 1 = 4 (1.2) N 1 v (c) D3-brane. See figure th 1 I . -v 4 -1 1 m 1 v , I h I p , this non-singular brane configuration can ]. While this mapping is simple at the level 3 (b) I 35 X – 5 – = D3-branes on which our theory lives to a single spa- P D1-branes connect to the I N 1 p -v -1 1 m ) is given by the Witten index of the resulting effective SQM ]. This formulation is a consequence of Kronheimer’s cor- 1.1 15 D1-branes (red lines) stretched between them and spatially transverse 1 1 m v ). (b) We move a single D1-brane down so that it is spatially coincident ⊗ (a) . This figure illustrates how Hanany-Witten transformations of our brane system can be -invariant instantons on Taub-NUT [ 1 4 x K -v We argue that as in the case of section 1 m 1,2,3 x of field configurations, itdirectly. is However, in difficult the to caseduality of derive reducible between the singular singular monopoles, data monopole we ofquiver can moduli gauge the use theories space the with corresponding semiclassical fundamental and bow mattersingular the to variety monopole make Coulomb the moduli branch exact spaces identification of of as 3D reducible a bow variety. See figure In this section webe will described review as singularhyperk¨ahlerALF bow monopole spaces varieties moduli which [ spacesrespondence are and which themselves show gives that moduli an theyU(1) spaces explicit can of map instantons between on singular 4D monopole configurations and bubbling contributions to ( of the world volume theory of the finite D1-branes. 2 Singular monopoles and their moduli spaces is given by connecting thetially stack transverse of NS5-brane where be used to studybubbling monopole bubbling can for be generic understood ’t by Hooft coincident operators D1-branes semiclassically. and Again, thus we would expect that the T-duality. In summary, we find that a ’t Hooft operator of charge NS5-branes (black with the NS5-branes.Hanany-Witten transition (c) so We that can the D1-branes pull now the end NS5-branes on the through NS5-branes. thedescription D3-branes by by making use performing of a the identification between Kronheimer’s correspondence and Figure 3 used to show thatbranes monopole (black bubbling lines) is with non-singular. (a) We consider a system of 2 parallel D3- JHEP11(2018)126 . t ), G C ∈ (2.4) (2.1) (2.2) (2.3) ], and M ∞ 42 X ) k, ~w ~ ( ). This space has the bow ∞ , . X M ; I, ∀ m ], birational equivalence [ , which carry non-trivial mag- → ∞ → ∞ γ 0 ( r 35 r X . -invariant instantons on Taub-NUT ≥ M K Ω I d 2
)
r o Const. Bow f ) i ) as 1
Birat. Equiv. ˆ r 2 ,N i / X, , m ) 1 i k, c B I − ( N ) as D r 2 ∞ 2 H K ( / S I 3, is the gauge covariant derivative and ) gauge theories with fundamental matter ( – 6 – = 3 O Z , i m − N 2 (SU( U(1) inst r π , + B 1 ( C 4 I M r X m O 2 M = 1 γ = = + i m − ˆ r m γ = 4 SU( 2 γ ∞ m r γ 2 6= 0 obeys the property X N
irrSym. Mirror Kron. Corr. ], Kronheimer’s Correspondence [ m = = ], where ] respectively. γ 29 ), and the moduli space of U(1) , ~ B ) X 15 is the asymptotic magnetic charge iff ,X – m i 12 bow , ∅ cr coupled to a real, adjoint Higgs field 13 A Λ M 11 P, γ 6= are a basis of simple coroots. G ( + [ ∈ ) is a hyperk¨ahlermanifold. ) } I ∞ ∞ X M m i γ H X X ∂ { ; ; = m m γ γ . This graph shows the dualities between the moduli space of reducible singular monopoles ( ( ). Here the abbreviations Mirror Sym., Birat. Equiv., Kron. Corr., and Bow Const., refer X i K M where M D The set of solutions to the Bogomolny equation subject to these boundary conditions ), Coulomb branch of 3D U(1) inst 2. 1. M M defines the moduli spacefollowing properties: of monopole configurations: Here is a regular element ofAdditionally, the Cartan algebra that defines a splitting of the Lie algebra of with the asymptotic boundary conditions 2.1 Review ofSmooth singular monopoles monopole are moduli static, space with finite gauge energy group field configurationsnetic of charge. a These 4D classical Yang-Mills fieldequation theory configurations are defined by solutions of the Bogomolny ( bow moduli spaces( ( to Mirror symmetry [ the bow construction [ Figure 4 JHEP11(2018)126 ∈ is : rot 3 is a G m (2.7) (2.8) (2.9) (2.5) (2.6) R 0 γ which ) = 0. ⊂ ∈ h M , T ) defines n T . ∞ ~x g ⊂ 2.7 is the group X 0 T D where SO(3) T ] for more details. × I, then we can also find ∀ respectively. 39 , 7 , such that ( 9 T 3 trans 0 t . R ∈ I , , ≥ ∈ × ˜ . 0 0 I m ∞ h I . See [ ˜ , m iX rot is called the relative magnetic I → → ) is the Killing form on Z m e 0 X , n n ). This space has the properties: m I r r γ ∞ X ) = , and X × M I D ) = 4 ; M ( under the action of the Weyl group in H ∞ m ∞ 1 I ) = 4 ) where ( X 3 trans n π X , γ ˜ m m P ; R ∞ R } ) as ) as 2 n m , γ X 2 I / × / ; P 3 ∞ X 1 , γ { m − n . These boundary conditions insert a magnetic X } ( − n – 7 – 3 cm ( r γ = n n r ( ( 2 ) is given by SO(3) π ( R ~x g P 4 M − n O ∞ { O M = which shift the asymptotic magnetic charge ( P = X + + R ; n M n n M n ~x m ˆ r n X class r γ R 2 n P E n 2 dim ( is the image of − r at P 2 − − which acts by global gauge transformation. m M n dim γ P = = G are the orbits of the = ~ B cochar X ) is non-empty, its dimension is given by ∞ , then its dimension is given by Λ m ∞ ∅ X ) is a hyperk¨ahlermanifold with singularities. It is non-empty iff ˜ γ R X ∈ ∞ 6= ; ]. Here ) X n m ; 38 P ∞ and are spatial rotation and translation symmetry respectively and , γ m X } ; , γ n 3 cm } m P . R n 3 trans { γ n ( ( P R P { ( n M M ] for a review of smooth monopoles. P is the set of global gauge transformations generated by is non-zero [ the closure of thecharge. anti-fundamental chamber and ˜ M where simply connected space called theof strongly deck centered transformations moduli which space gives and rise to the maximal torus of and 48 The only symmetries of a generic singular monopole moduli space is If The topology of the moduli space is given by The symmetry group of If Together with the asymptotic boundary conditions, the singular behavior ( If we lift the requirement that the configuration has finite energy, + Classically for smooth monopoles, 3. 2. 1. 5. 4. 3. 7 cr the moduli space of singular monopoles in a local coordinatesource system centered of at charge Λ solutions to the Bogomolny equations which have a singular behavior at points See [ JHEP11(2018)126 . G I h . rnk ∨ = ]. (2.11) (2.12) (2.10) G indexes ]. 45 ’t Hooft P ,..., 33 def ) = 1 . i ( 00 I P ,...,N 00 irreducible 0 P ) on the fields in the = 1 PP i -SUSY due to the fact 2.7 R 1 2 I. 00 ∀ . That is, ) in the path integral where P ) M i = 2 vectormultiplet. In this , ( where I 0 2.7 = h ) N 0 i . This is a disorder operator in ( ≥ P I 0] I with the imaginary part h P, ⊗ [ are defined as the coincident limit of X L P are its structure constants [ 00 0 iX , P , p I + cochar , PP h . Λ I R 0] Y , ∨ p 00 ∈ G – 8 – G P [ =1 P and Φ = L I X rnk monopole is of charge 1 00 ∨ 0 − P = G th ζ i ) PP i ( R I : h 00 ∨ P M def =1 G monopole. . i X N in space. This is a special case of the ’t Hooft line operator in ζ = th U(1). Thus, mutually supersymmetric ’t Hooft operators must i = n 0] , ~x ]. The definition of reducible ’t Hooft defects requires a notion of 0 ∈ will be unimportant and hence we will ignore it for most of our P P [ 33 such that ζ ζ L 8 )[ · ∨ 0] G P, ( [ identifies the real, adjoint Higgs field ’t Hooft defects of charge are representations of minimal ’t Hooft defects, each of charge L wt ζ 00 I Λ p P = 2 supersymmetric theories, reducible ’t Hooft operators are related to irre- , = 2 supersymmetric theories, as is our focus in this paper, there is a supersym- I ∈ 0 ’t Hooft defects. These are irreducible ’t Hooft defects with minimal charge — P P N P N , = In Reducible Often in this paper we will differentiate between irreducible, minimal, and reducible These operators impose the same boundary conditions ( In Here we use the notation where the 8 def Here P according to the charge of the Consequently, they are S-dualthe to product a of Wilson minimal representations line of with a reducibleducible representation ’t Hooft given operators by byof the the corresponding products Langlands of dual their group associated representation the ’t Hooft defects, In short, reducible ’t Hooft defects are simply the product of minimal ’t Hooft operators. minimal that is irreducible ’tThese Hooft are defects S-dual whose to ’t the Hooft Wilson charge line is with a theN simple minimal cocharacter irreducible representation of ’t Hooft operators. ’t Hooft defects. Thosedefects. defined by the These boundary areweight conditions S-dual above are to the Wilson line with irreducible representation of highest the choice of where Φ is thepaper, complex, the adjoint valued choice Higgs of discussion. field of Additionally, the for the rest of this paper, we will only discuss supersymmetric metric version of thethat ’t they Hooft break defect. translationwhich invariance is These and specified come operators by with preserve all a have choice the of same conserved choice of supercharges Now we can constructthe the quantum ’t field Hooft theory defectpath that integral operator imposes at the a boundary point which conditions the ( ’t Hooft boundary conditions are imposed along a curve in space time [ 2.1.1 ’t Hooft operators JHEP11(2018)126 ]. 42 (2.13) (2.14) (2.15) )[ ∞ X ; m , γ (v and contributes to M ) ], in the case of re- , n ) ( i ~x 42 ∞ , 6 . Each lower-dimensional , X | ; ) P m ∞ , γ X ; } | ≤ | ) i m v ( | I , γ . h ) { is inserted at i (v ( ( ) ) I s i ( ( h M I n ) where h M ~y n ~y ∞ v). As shown in [ P → ) → X n |≤ ) ( P, a X i ; v n ( – 9 – | ( ~x i m : ~x i ) is the smooth component of M , γ ) = ∞ = ) describes the degrees of freedom of the free (unbub- (v ) = lim ∞ X n ∞ ; ∞ P X ) by M X ; m X ; ∞ ; m , γ m as X m ; (v , γ n , γ P, γ m ~y ( } (v M has the special interpretation of describing monopole bubbling ) n v) is a quiver variety. s M ( P P, γ ⊂ P, { ( . We will further denote the transversal slice of each component ( ( ) M M ], the authors provided a theoretical framework to find explicit in- P M ∞ c M M X 18 + , ⊂ ; cr m ) 15 n Λ – ∞ ~y minimal ’t Hooft defect of charge ) decomposes into a collection of nested singular monopole moduli spaces of ) has the stratification , γ 13 X ∞ ∞ ∈ (v comes from how the nested components are glued together to form the total th ; ) i s X X ( m ; ; M m m , γ M (v Physically this should be thought of as follows. Singular monopole moduli space We will denote the moduli space of singular monopoles in the presence of reducible ) P, γ P, γ s ( ( ( 2.2 Instantons onRecently, multi-Taub-NUT and there bows has beenspaces. a significant In amount [ stanton of solutions work on on these spaces describing that instantons relies on on a ALF central algebraic object called a bow. As moduli space. This isically determined describes by the the moduli transversaldefect. of slice smooth of In each monopoles the component case thatgiven which of were by phys- swallowed reducible a up defects, quiverthe by the moduli the smooth transversal space. ’t monopoles slice Hooft induces is This’t particularly a indicates Hooft simple corresponding defect. that and quiver quantum is SQM mechanically, bubbling on of the world volume of the decreasing charge and dimension: component describes themonopole singular is monopole absorbed into moduli thedefect defect. and space reduces As that the we know, resultsture number this of of when reduces degrees the a of charge freedom smooth of in the the ’t bulk. Hooft The complicated struc- given by v M ducible monopoles, M where Here each component bled), smooth monopoles in the bubbling sector with effective (screened) ’t Hooft charge 2.1.2 Singularity structure:The monopole singular bubbling locus of configurations. In theM case of a single ’t Hooft defect, singular monopole moduli space the charge at We will refer to this as reducible singular monopole moduli space. ’t Hooft defects inserted at where the JHEP11(2018)126 Σ → E . The small ). A represen- Λ which are at 1 S ∈ x ]. 41 , 3 comes with an associated . We will additionally use . } 9 i ζ ∈ E { subject to these conditions. = large representation +1 i In order to facilitate this, we will E I , e i 10 e in between wavy line segments. We will and the . } i } i x e { { – 10 – = to be the set of marked points E i in between edges . i ∈ I ∈ I ζ ζ small representation is a directed quiver diagram where nodes are replaced by a collection is required to satisfy Nahm’s equations with certain boundary/matching bow for an example of a bow. E 5 A ) monopole moduli spaces can be written as bow varieties associated to certain to denote the set of N i I additionally use the notationthe Λ end points of the A set of marked points denoted Λ = A set of continuous wavy line segments, denoted A set of directed edges, denoted See figure Now we will give precise definitions of bows and their representations and review how The bow construction of instantons on multi-Taub-NUT relies on two such repre- Really each pair ofSingular small monopole and large configurations representations for of other the gauge bow. groups will be given by the moduli space of certain 3. 2. 1. 9 10 instanton configurations on the corresponding ALF space. of connected wavy lineline segments segments. with This marked is points specified at by: the connection of any two wavy ADHM/ADHMN construction of instantons/smooth monopoles [ they can be used tois give less explicit interested instanton in solutions technical on details Taub-NUT space. can feel TheBow free reader data. to who skip the rest of the subsection. sentations of a bow:representation encodes the the datatation of describes the the multi-Taub-NUT instanton space,give bundle. while an the These explicit two large construction representations represen- of can instantons be on put Taub-NUT together in to a manner analogous to the tation of the bow thenthat can defines change a rank vector at bundleconnection the edges (or on of really the a different sheaf)conditions intervals and with at at connection marked edges points insolution and Σ. of marked the The points. Nahm’s equations for A the representation connection of on a bow is then given by a 2.2.1 Quick review ofHeuristically, a bows bow canstructure. be thought Bows are of essentially as quiversof a with these quiver wavy wavy lines with lines instead a as ofalong intervals different nodes. the with type edges One a of of should collection representation the think of original marked quiver points to Λ form that a connect space together Σ (in our case Σ = instanton moduli space whichlar is SU( formally calledinstanton a configurations bow on variety. multi-Taub-NUT spaces. first As provide we a quick will overview show, of singu- the bow construction. in the ADHM construction, which relies on quivers, each bow JHEP11(2018)126 ) Λ e Λ, ( t and
∈ ∈ such E x x s x
E ), where → ) e ζ → ( ( , and marked respectively. h } x
i ζ ζ C E with Hermitian { > R : : x ) with generators ) I C x ∓ RL ( e ζ ( R B respectively. 9 ). And for each with coordinate R , ζ ζ ) e ( 6 ζ e and for each point e ( , segments R σ h for }
N I . x e E 3
∈ { 8 ) R ζ ) ∓ → 5 ζ ζ ∈ e ( ) of rank E e ) 7 R ( ( ζ 3 t e ζ
σ are the segments to the left and right 2 ⊂ , ν E x 2 e → : 4 6 ⊥ e ) ζ ζ , ν ζ x 1 e
. E LR e ν x bow with edges ) are the beginning and end of B ζ 5 ζ of a bow consists of the following data n ζ ( E = ( 3 i where A – 11 – e e | ~ν ) 4 ζ + ζ 1 ) and ( x ζ 2 3 R ) is the head, tail of the arrow ( ) = 0, we define a set of linear maps e , we associate a line segment ζ e o (2) representation of dimension ∆ ( x − ( ) , t su ∈ I R ) − 2 e representation ζ ζ ζ ( ( 1 h e A according to the shape of the bow. R | )) where ζ . ζ σ i ( connect along marked points and edges to form a single interval (or 1 ) = , ζ , i ζ , we assign a non-negative integer where N , we assign a vector x h and a set of linear maps , we define a vector bundle ∈I ) . σ Λ where ∆ ( e ζ Λ we define a one-dimensional complex vector space ζ x x ( R S ∈ ∈ I o ∈ ∈ E C ∈ I ∈ E x ζ e x ζ e → = ( ζ x
. σ . This gives a representation of ( } E are the segments to the right/left of } . This figure is an example of a type n i : x ρ { ζ J for each we define an irreducible { inner product we define ∆ of the point that The intervals circle) Σ = For each For each For each To each wavy interval For each For each 6. 4. 3. 1. 2. 5. Figure 5 points Bow representations. JHEP11(2018)126 , =1 3 i } ! i † (2.16) (2.18) (2.19) (2.17) , T ) 2 { LR e , RL iν e )) i B B e + ( ( )) t 1 e , ] ( ν ) ] − h ! x 15 , = = s 15 – ( − − C . + δ e )) s s 13 ( e ( ) ( δ ) δ x s h ) ) 1 ( )) . † x s − ) − 3 e 2 ( I − s R ( ν x s g ) ( t )) ∆ s ( I 3 e ( d ζ ) − ds ( , ν ν δ ( − O x 1 − † j h t † ) ) ( ( s ) − − T s x † x ( − g g ( + x e † J δ ) + ig ) I ) s g RL Q ⊗ e † e x ,B i B ) λ ( x RL LR e e ~ν j ( J ( B ( − T x ⊗ ( LR e ( ) ( B B g ! − ) j iσ Λ ⊗ x s 1 x O B s † )) )) ( T ∈ ( ( − RL ) e J Q 1 e e X + x − )) + e g g ) ) RL ( ( e e B − 0 0 0 t 1 2 B ( x LR LR e h e g ) ( B T T ( h 1 1 − ) x B B 1 − − − ( s − ⊗ 2 − ] + − ( R s and skew-Hermitian endomorphisms g − − )) + 2 g LR 1 ( e 1 ∆ s e s δ 0 ( − ) ( iT B t δ T g RL – 12 – e = = † (( Λ x e ) ∈ − ∗ B O ~ν − + − X x e − † 1 T ) LR s e s + d ( + ds ( 7→ ,T B δ ) = x RL 2 e ( j † s T − ρ ) ( · B s iT O − ( e ~ν h which have the pole structure 1 2 ∗ i 0 x x + B RL LR e e ζ , T T I ( T J ∈E i 1 σ + j e B B X and T ⊗ T [ − 1 2 j i 2 − e ) = ⊗ : σ T + s ) j B ( + g s d j ds ( 3 iσ j T T
ν ,B s ∈E + e j X 0 ∇ ! T + P † x x I . These gauge transformations act on the various field as J Λ. = Im ⊗ = 0 = E ∈ 1 µ over the interval µ x This equation can be rewritten in a more familiar form as [ ζ = = — a Hermitian connection E x T s Q where then the linear maps are required to satisfy the “Nahm equation” [ near bundle on ∇ If we reorganize these linear maps as As in the ADHM and Nahm construction, there is a gauge symmetry of the instanton 9. 8. 7. JHEP11(2018)126 ). and and ) 2.19 ∅ (2.25) (2.22) (2.24) (2.21) (2.23) (2.20) ). We e satisfy ( ζ i h ( is deter- t E R k → TN , ) e , ( ω ) t ~x ) E + ( : 2 e,i ]. V t dξ ~ d 18 LR e , = ) b + x , µ 15 0 . – 2 e, i − . s )) )) 13 ∈ E dt . ( e e RL ) δ ( ( e -intervals) in which Λ = i )[ as in the Nahm equation ( t x , a + ( k h 0 σ ,B J , µ − x − RL e ⊗ I RL e s LR s ds t ( isa , b 1 Λ ( N δ db ( H ∈ δ i † ,B − X ψ x t ) LR e } ) . First, we need to specify the data of the , b i ) µ s k − s − T ( exp RL e ∂ ( ] ··· i { b C 2 ∂x C ( -edges and = 0 P σ ν 2 TN ). Using this, we can reconstruct the self-dual ν d k i iT ψ = ⊗ 1 2 − ψ − – 13 – t i + 1 µ 2.23 T + D 1 ψ : log( LR e RL e 0 + t B ,T B LR e 3 d T RL ds db e [ LR e † ,D i Ψ = B ) B = Ψ t h LR e µ ) + b + D 2 ( ∈E iD d e X iT † 1 2 Ψ + + ] by 1 and the linear maps for each edge e ds T 15 ( X } i s – Z t . For the small representation, in the bulk of each interval the = ∇ { ) 13 e = 2 ( . Here, we will denote the triple of skew-Hermitian endomorphisms from t µ ds E 0 = A ∈ I → -type bow (a circular bow with 1 ζ ) e ∀ − ( k h A E : After solving the Nahm equations, we construct a Dirac operator Now we can construct the actual instanton configuration. This is specified by a large The metric on the multi-Taub-NUT space can then be defined by reducing the “flat” = 1, = 0 with boundary conditions defined by the i ζ RL e ds dt of the linearly independent solutionsgauge of field ( as in [ and use it to construct a matrix will denote the maps of this representation as As in the ADHM and ADHMN constructions, we find the kernel of this operator by Nahm’s equations and gaugemined symmetry. by Here, the the gauge angular invariant coordinate data of on representation which is allowed to have non-empty Λ and generic data for the b metric Now we will construct instanton solutionsmulti-Taub-NUT. on This requires specifying aof small an representation. This isR a representation condition (7.) as Note that this is simply the complexified Nahm2.2.2 equations with certain Bow boundary construction terms. of instantons JHEP11(2018)126 ]. E Σ. , 14 ∈ , , and (2.27) (2.26) ) = 0 4 G ) } s 3 ζ e ¥ ( ~ν / { R , Σ at E , Λ → , where -invariant [ I E ) K ) R( 2 ' ¤ ∈ I 2.19 ( ζ invariant will be of special is the corresponding Dirac ). ζ ω K ) R( ( 1 £ Nahm’s Equations R is the fiber of s , ,
-invariant instantons on muti-Taub- ) ) E e e ( ( K t h = E E s × dω . × E 3 ) , ) e ∗ e ( x ( t h E = E – 14 – E HW Isom. HW ) = 0 × ) and ζ . In the case of a Cheshire bow representation, we → x dV . Further, fix the specification of ( → are the intervals to the left and right of an arrow . The bow moduli space is then given by set of all π r ) , E ) e ]. This allows us to exchange an adjacent edge and R ) R ( e 2.18 R ( h t E 42 → [ , ζ E E L x ζ mod 2 × × C ) R( 3 ) 0 End( ) ¢ ], there is a special class of large bow representations, called e , that give rise to U(1) e × ( 4 ( h t ⊗ x such that ds t E E action of translation along the fiber coordinate will be determined C H : : : H K ∈ ) where ) R( + − x ∈ I e e 2
T Q B B ζ R,e ζ ( are defined as in ( R T ) = , r e , ) = ]. R ) is the harmonic function for multi-Taub-NUT and ,B ( L,e ~x 17 ) R( x ζ ( 1 , ( ¡ ∈ E Q V bow 13 bow and a small representation R( A sub-interval R e Σ for the large representation This isomorphism of representations is explicitly given by As we will see in the next section, instantons that are U(1) Hanany-Witten isomorphism 1 M • • − → k the marked point in exchange for modifying the local values of data [ 2.2.4 Bow varietyAn isomorphisms: interesting Hanany-Witten feature transitions offormulations bow of varieties the is same that bow there variety. are One often such many isomorphism different, that isomorphic will be useful for us is where This describes the moduli space of instantons on multi-Taub-NUT with fixed asymptotic A E large representations modulo gauge equivalence. This is given by transformation and therefore, the corresponding instantons will be2.2.3 U(1) Bow varieties As in the ADHM/ADHMN construction, therecorresponding is a to moduli space the of instanton set configurations of all (large) representations of a bow. Consider fixing a type This is because the U(1) by a non-trivial shift in can use gauge symmetrywhich to means eliminate that thisare Σ shift unrestricted. has since Thus, effective there any endpoints is shift on of a the which fiber the coordinate gauge can transformations be of compensated by a gauge NUT. These bows have the special properties: potential: interest to us. AsCheshire shown bow in representations [ where JHEP11(2018)126 ω + . (2.30) (2.31) (2.32) (2.28) (2.29) where 3 ) with is well dξ R 3 ω R 2.29 ) given by + is the Hopf ∈ K 3 i dξ R ~x , but it has the = 0), where the 4 ~x R . dV , R 3 ∗ in the base ∈ 2 . = S dV . , 3 2 This is a map between singular ∗ ) + 1 , k = 11 )Θ ) and certain instanton configura- 3 2 4 ζ ~x ( ( R ds 1 R − ω , dω = , dω V + 2 ) + . To our knowledge, the proof of Kronheimer’s | i 1 + ds ζ ~x A dξ ( ∗ d~x 1 − R · ~x is the Hodge dual restricted to the base – 15 – globally ill-defined, but again Θ = | Θ = , f 2 d~x 3 ) = ) 0 2 ∗ dξ ) k fibration to a 2-sphere ζ k =1 ~x ( i 1 X ( + , R S V | . Topologically, it is homeomorphic to ~x 1 | ~x,ξ 3 = ) + 2 is not a globally well defined 1-form. However, Θ = (time independent on 2 R 2 ζ 3 ) = 1 + ( ds dξ R ~x ) = ( R ( V isometry which has fixed points (NUT centers) at ~x,ξ ) = 1 + ( k ~x K ( f . Its metric can also be written in Gibbons-Hawking form ( V 3 fiber coordinate and R ). This space is also a 4D ALF hyperk¨ahlermanifold which is given by a fibration over 1 k ) obey the relation S 1 i ζ S TN ( R is the fiber degenerates. This renders ξ 1 There is also an extension of Taub-NUT to include multiple NUT centers called multi- Taub-NUT has a natural U(1) isometry (which we will refer to as U(1) Taub-NUT has a metric which can be written in Gibbons-Hawking form as S Full details and proof are presented in appendix fibration over fiber degenerates. Thus, 11 1 1 Again there is athe U(1) defined. correspondence has only beenproof for given the for case the of case a single of defect single and defects. prove the In case of the multiple appendix, defects. we both review the S the substitution This action has a single fixedS point, called the NUT centeris (in this globally case well at defined. Taub-NUT ( where translation of the fiber coordinate where Taub-NUT is a 4Drealized asymptotically as locally a flat (ALF)property hyperk¨ahlermanifold that which the can restriction of be fibration the of charge 1. 2.3 Kronheimer’s correspondence Now we will briefly review Kronheimer’smonopole correspondence. configurations in tions on Taub-NUT spaces. Thus we will first2.3.1 give a quick Taub-NUT review spaces of Taub-NUT spaces. As we will see, thisfigurations. will be intimately related to Hanany-Witten transitions of brane con- where the JHEP11(2018)126 . ] . ) ˆ 1 k F x ∗ − ( iP ξ k ψ ) are , this − TN = A (2.39) (2.34) (2.35) (2.36) (2.37) (2.38) (2.33) e action Z A ˆ , F , we can = k K invariant and 3 g . 3 K σ dx TN R ~x ( . ∧ V ψ A which fixes the → 2 cpt 2 ~x ] where Γ[ } = 1 H n ω dx − P k X + ∧ { V A 1 dξ -invariant instantons on dx K ∧ . ∧ ) = Γ[ n , Z ) , P dω k ω − 3 ∗ , + generates a gauge transformation TN ) ψ ( ) = g ω dξ irreducible singular monopoles with x dg . is equivalent to a gauge transforma- ( . + 2 ( cpt 1 that has been lifted to the full ) + k k ∧ − ψ V Dψ , H 3 n -invariant means that the U(1) ig dξ Dψ R ~x V ψ ) and TN K ( 3 )( Dψ ) = + → lim ∗ x ~x → D ( ) under the identification − → 2.31 ˆ V k ψ ) Ag = ψdω 1 P 2.1 – 16 – − 3 action near the NUT centers lim + − . These non-trivial 2-cycles in ω , − R TN 1 g . Kronheimer’s Correspondence gives a one-to-one 3 n ψdω K 3 F 3 − R P as in ( R ω k = 3 R − A F ∗ ξ A ˆ = ( 3 + A ∈ V = , and the orientation form Θ R ∗ 3 3 ]. Here, U(1) action to the gauge bundle. As shown in appendix n 3 f ∗ R F ˆ dξ A R ~x 35 K A A . Using the form of the curvature n = ( , this connection can be extended globally iff ˆ ∧ ~x A ˆ 3 F → lim A R ~x reduces to the equation . F 3 ˆ 3 F R ∗ of our gauge bundle − ∗ ˆ = → A k ˆ = F action is specified by the collection of ’t Hooft charges ˆ to describe an instanton, it must satisfy the self-duality equation: F K TN ˆ ∗ : A is a connection on the base ] inserted at positions is the curvature of π 3 n 3 is the curvature of R 30 R P , generates translations along ˆ A F F f 22 As shown in appendix Away from NUT centers, we can choose our connection to be in a U(1) This space has a non-trivial topology given by -centered (multi)-Taub-NUT [ This is the familiar Bogomolny equation ( have the limiting forms which can be rewritten as compute the dual field strength Now self-duality Now the for where where limiting behavior of the lift of the U(1) gauge where Here which defines the lift of thelift U(1) of the U(1) Consider the singular monopolecharges configuration with mapping between this singular monopolek configuration and U(1) on the connection tion [ is the root latticehomologous of to the the Lie preimage ofprojection group the lines running between any two NUT2.3.2 centers under the Review of Kronheimer’s correspondence JHEP11(2018)126 . = 4 G (2.41) (2.40) , , with the I I H H I . 1 ) and have the | − 2 I m 1 n g N I ~x n m 2.38 P − P as a bow variety, we = 4 gauge theories I ~x action. Further details X | = C 2 N K = m M − γ 2 ∞ ) monopole moduli space is − where the data of the singular ]. This identification equates ) = N N 3 x ) gauge theory with fundamen- ( R m 42 , ψ N ) . ,X x 29 I ( with a bow variety, we can use semi- , A h V ) n ]. However, while it is easy to give the n 12 ( I ] that SU( M 4 , ~x p = 4 SU( → 11 29 = 4 theory. This equates smooth monopole ~x I , ], which identifies . One such equivalence exists for the case of X N = lim 3 N 42 . To our knowledge, a similar identification for 20 = – 17 – bow , m X ) c i M ( n 11 M I ~x h → ∼ = lim n ~x ) theories [ ~x ) satisfy the Bogomolny equation ( ) of a 3D = N M i -invariant instantons on multi-Taub-NUT. By using the = 4 linear quiver gauge theory with gauge group , C invariant instanton on multi-Taub-NUT is in one-to-one X ~x ,X : K Ω 3 2 i M d N K R m n -invariant instanton and singular monopole field configurations, = A 2 P K n ]. Consider the reducible singular monopole moduli space with ) gauge theory with magnetic charge = 12 3 N R ), a U(1) F ,P n I 1 ~x H 2.40 → lim m I ~x m I ) and only bifundamental matter corresponding to the quiver X I = m It is one of the general results of [ Kronheimer’s correspondence tells us that singular monopole moduli space is equivalent m to the Coulomb branch ( U( ˜ γ I c This was extended to include minimalthe (and bubbling hence locus reducible) in singular monopoles [ away from irreducible singular monopoles is not known. We will comment ondescribed this further by in the section Coulombmoduli branch of space a for 3D SU( Coulomb branchQ of the 3D classical equivalences to identify reducible ’t Hooft defectsM in SU( tal matter. Then bycan using identify the the bow results variety of describing [ it is difficult todescribing use a Kronheimer’s given singular correspondence monopole to moduli specify space. the data2.4 of the Reducible bow singular variety monopoleNow moduli that space we have and an 3D exact equivalence between to some moduliexplicit space construction of of the U(1) modulisection, space instantons we on see multi-Taub-NUT from that thecorresponding previous singular to monopole Cheshire moduli bowexplicit representations map space between [ U(1) can be described as a bow variety limiting form ( correspondence with a singular monopole configurationmonopole on is encoded inabout data this of correspondence the NUT can centers be and found lift in of appendix U(1) correspondence, the corresponding field configuration has the limiting behavior Therefore, since the pair ( Therefore, by using this limiting form of the fields with the identification of Kronheimer’s JHEP11(2018)126 ) I I = ( i ) = ). e ∈ I ) ) with I ) n ( |E| (2.42) i j ) = ( I I i ( ζ a p ~x 1, ( ( ζ I R U( − = 4 quiver n Λ, N Q and each 1 ∈ 1 for each factor. specifies I − ), this bow mod- − → I x ) N ,...,N m g N ) specifies n , and the insertion ( I I p m I p p 2.42 edges with identical = 0 +1 with H I ) I ) = I I,n n x ) wavy segments ( I I m ( p a bow ) P I ζ n ( ( I 2 and , M ) where each hypermultiplet 2 p P Λ for R I − ) − I n i p ∼ = = ,..., N g ∈ x N ; − P p n I m U( has c M x P I = 1 ~m,~p ) . We will think of this bow as being n I Q ( indexes the points a i 6 (Γ ζ P Λ, there are = C I − where , = 0. ) with gauge couplings I F ∈ m M I = 4 quiver gauge theory associated to the ) γ G n N m ( 0 where ∼ = I ) factor in the gauge group. The hypermulti- ∈ E +1 = N is in between p I ) I to identify . = ) Q U( i i 3 m e n 3 | m ( ∞ = 4 e γ I , x – 18 – I p m Λ = I m Λ, there are 1 + 0 | X h Q x ; ) = 0. F which we identify with the mass of a fundamental n m ) ∈ = m ~x = ) G determined by the position of the singular monopoles N n = ( j i , γ +1 I,i ~x n ζ G ~ν ( I } ( : ~x n i ~ν ]. This is because Kronheimer’s correspondence explicitly R , x P Λ. = I P { 42 ) specifies 2 x ( n 2 ∈ p . Each wavy segment m N ~m I = c ] we can describe the Coulomb branch of this 3D M x indexes minimal ’t Hooft defects which are at a position 1 with n 42 ) ) couples to the U( indexes the edges ∈ E n P I − ( I i ) p p = SU( I ( i e G n,I . 1 P ) 1 are the parameters where ,...,N n p m I ( ) n p edges n ~x ( I n = 1 : ) is the Coulomb branch of the 3D ,..., . This space is equivalent to . We will additionally take p While this identification used the semiclassical equivalence ( ) n I n i ( I P ( C , p ~m,~w I ~m I,n = 1 I = h n M i Asymptotic relative magnetic charge ˜ m ’t HooftP charges is associated charge wherepoints the edge Gauge group = I P p Now we can use the dualities in figure For this bow, in between Using the results of [ This theory has a gauge group ) • • • I,i ( Remark. uli space description of singular monopolethe moduli space singularity captures structure the as full in geometry [ including hypermultiplet. In between each pair gauge theory by the modulisplit space of up the by bow marked in points figure and comes with the data of a triplet tal representation of the flavortransforming group under U( plets have mass parameters which additionally break each factor of the flavor symmetry group U( and Additionally, there are fundamental hypermultiplets transforming under the fundamen- charge where quiver Γ where ~ν JHEP11(2018)126 M and as in c M c M . . In order to = 4 theory with ] for describing c M c 3 M (2) N ζ 12 , 6 2 (2) ζ 1 (2) ζ ] for the case of instantons and 2 x +1 1 21 , p (1) ζ 20 . The data of the singular monopole bow ), which we only a priori trust away from M 3 (1) ζ 2.42 ∼ = ) can be realized as a bow variety and gave fundamental hypermultiplets coupled to each gauge – 19 – m ) = 0. n = 4 gauge theories with fundamental matter with c M ( I , γ 0 p n 2 (1) , p N n P 0 ζ in analogy with [ ( P m 1 (1) M ζ c M = 1 I and x p I +1 0 p m (0) ζ ) are realized in each of these different realizations of ) and 0 s ∞ ) = p (0) . Then we used the semiclassical identification of the Coulomb ) ζ I m ) of certain 3D ( a ,X ζ bow n U( ( bow s R -1 ,P M 0 Q p (0) M m ζ ˜ γ = ∼ = ∼ = , ) G C K N . This figure gives a the bow describing the Coulomb branch of 3D M U(1) inst . 1 M to pinpoint the exact isomorphism = SU( G c reducible monopoles. We willdefects confirm in that the this gauge configuration theorymetric describes living vacua reducible on is ’t a exactly Hooft given stacksmooth by of monopoles. D3-branes Then by we showing willmonopole that argue bubbling the that and supersym- this show brane that configuration it can makes be correct used predictions to for study the geometry of ( table 3 Reducible singular monopolesIn in this string section theory we will study the brane configuration suggested in [ an explicit identificationdo for this, the we case usedand of Kronheimer’s reducible correspondence singular tobranch establish monopoles ( the isomorphism between M the singular locus, to pinpoint exactly which2.4.1 bow moduli space describes Summary In this section we showed that Figure 6 gauge group group. In this bow gives us a completevariety. identification We of are the simply full using singular the monopole relation moduli ( space with a bow JHEP11(2018)126 ) I = th N m = 2 σ (3.1) (3.2) 1 ] with − N )+1) ) I 20 σ I ( v H = 4 U( I I ˆ ˆ − A A v ∞ N 1 1 ∞ ∞ +1 = X S S I and ( H H v ∞ th X ) = ( ) to specify which pair D3-branes localized σ σ Gauge Coupling 2 I ( ( g I I N -directions. This is the . 3 Asymptotic Holonomy: Asymptotic Holonomy: , 7→ I 2 by stretching a D1-brane , = 0 and fixed location in 1 σ H I with ) x 9 , I H 6 8 I v , NS5-branes (indexed by K h 7 R , I − 6 . k , p ∈ E × such that ) 5 } I I 3 e x +1 , n p ) gauge theory with two real Higgs I 1 = 0 P P ) SYM theory as derived in [ v { R N ( U( I = N v I = Number of n Edges ,...,N X Lift of U(1) 9 P =1 N , I X 1 Instanton Bundle = Fund. Hypermults. = 1 R ∞ = 2 SU( I – 20 – I , N ) H ζ I and I +1 ,X E I m I m R ) at distinct points between the ) -direction. ]. ) of I ζ 4 3 σ H m ( ∈ < v I x E ˜ . To each NS5-brane we associate γ P ( , x 44 R I I σ = U( at distinct fixed points in the , m 1 2 σ v v D3-brane, localized at ], these NS5-branes introduce minimal/reducible singular c = I I 37 I Rank of , x G 12 m th H X 1 σ for ˜ γ . We will then show that in this setting, T-duality is equivalent x Instanton Bundle i I = v 0] + 1) m = ( I P, = [ γ I ) ) ) Λ σ G L 4 ~x 1 N N N h ∈ x − =1 x N I and ( Q = SU( = SU( = SU( As argued in [ th = I Number of G G G points Gauge Group = 0 and by projecting out the center of mass degree of freedom and adding a sufficiently G 12 9 , 8 . This table shows the different realizations of the data of the 4D gauge group, relative , ) localized at K 7 , ) SYM theory. Consider flat spacetime X,Y C 6 . For our purposes, we will consider the case of a general configuration with , bow Now we will introduce ’t Hooft defects by adding Now we can introduce a smooth monopole with charge 5 3 c N U(1) inst M , Here we index the NS5-branes by M x 2 , M 12 1 M , . . . , k 1 D3-branes. of D3-branes it is sitting between in the smooth monopoles of charge standard construction of smooth monopoles in SU( fields large mass deformation as in [ between the x The low energy effectivegauge world theory. volume theory We of then these project branes to is a that 4D of 4D 3.1 Brane configuration Now we will describe theSU( brane configuration for reducibleat ’t Hooft defects in a 4D Table 1 asymptotic magnetic charge, ’t Hooft charges, and Higgs vev. expectation value of to Kronheimer’s correspondence. JHEP11(2018)126 N = v ∞ (3.3) (3.4) X 13 . 3 I v , and Higgs vev H I I H , m I , m I D3 I X D3 P = = m (right) ˜ γ 2 m 2 − (right) γ m p − D3 , I D3 h ) n I ( + (left) p – 21 – + (left) I D5 X 2 v = NS5 ) σ ( (right) I − ]. h n 50 = = (left) ~x = , relative magnetic charge ˜ I σ D5 X ~x NS5 h ], D1-branes ending on D3-branes source magnetic charge in the L : I L σ p 20 I 1 1 = m p P n P = . I P H ) I . v ]. In this T-dual configuration, we can pull NS5-branes through an adjacent 7 . This figure shows the brane configuration of a single, reducible ’t Hooft defect with − 11 4 x 1 +1 v I Our brane configuration is the T-dual to a configuration consisting of D3/D5/NS5- As we know from [ We should ask why we expect this configuration to give rise to ’t Hooft defects in v Here we use the convention of [ ( I 13 1,2,3 x a configuration. This can be seen as follows. branes [ D5-brane by performing aconnecting Hanany-Witten the transition D5-brane which creates and NS5-brane or in destroys order D3-branes to preserve the linking numbers a fixed location, 2.)theory. This it brane does configuration not canmanner. be introduce seen any to new reproduce these degrees properties of in freedom the in following world the volume low theory energy ofD1-branes the connecting D3-branes. the While NS5-branes our to the brane D3-branes, configuration it does is not Hanany-Witten have dual any to such See figure the D3-brane world volumedefect theory. in the If worldproperties: volume this 1.) theory brane of it configuration the sources gives a D3-branes, rise magnetic it to field must in a have the the ’t world following Hooft volume minimal theory of the D3-brane at magnetic charges are given by Figure 7 ’t HooftP charge monopoles and shifts the asymptotic magnetic charge so that the ’t Hooft and relative JHEP11(2018)126 ]. 6 ] in ] for (3.5) 37 21 , , 29 20 -directions. 3 , 2 , 1 x ], this choice of phase 5 ]. Additionally, since the , 1 29 . D 1 D -directions. This is clearly the 5 , 4 (right) x -plane in which to separate the D3- − (right) 5 U(1) of a ’t Hooft defect operator is x 1 − R D ∈ i 1 D ζ + 4 x R + (left) – 22 – ), we can go to a dual frame in which there are + (left) D3 1 ]. NS5 29 < v are the number of D3-branes that end on the left, right 4 σ (right) are the number of NS5/D5-branes to the left, right of the x − D3 D5 = = (left) / D3 NS5 NS5 L L , (right) ], the string theory embedding of instantons is given by D0-branes D3 21 , , (right) 20 is equivalent to the requirement that all NS5-branes are parallel to each D5 ζ / NS5 One may be curious how the phase Note that this construction is fundamentally different from that of [ As shown in [ It is clear that in this dual Hanany-Witten frame the D3- and NS5-branes impose Thus, by performing a sequence of Hanany-Witten transformations (for example send- Similarly, Hanany-Witten transitions can be realized in the D1/D3/NS5-brane system singular monopole configurations. Wethe will string take theory an description approach of similar instantons to and that monopoles. ofinside [ of D4-branes and thebranes string stretched theory between embedding D3-branes. of smooth In monopoles each is case, given this by D1- was justified by showing that the which singular monopoles are obtained by taking— an that infinite is mass by limit sending of a smooththe D3-brane monopoles utility of with this attached construction D1-branes is off that to it infinity. is especially As nice3.2 we for will studying discuss, monopole bubbling [ SUSY vacua Now we will demonstrate that this brane configuration does indeed describe reducible same choice of other and are perpendicularrequirement for to preserved the supersymmetry. D1-branes inRemark. the operator in the world volume theory ofRemark. the D3-branes. encoded in the geometry ofis this equivalent to brane a configuration. choicebranes. As of shown direction in Thus, in [ the the requirement that mutually supersymmetric ’t Hooft defects have the porting any massless degreesany new of quantum freedom degrees and ofNS5-brane freedom hence is in this the heavy low configuration compared energy tobe will theory sourced all not [ at other a introduce branes fixed location inTherefore, given this the by NS5-brane the system, position configuration this of reproduces magnetic the the NS5-brane charge “minimal” in will the properties of a ’t Hooft D1-branes connecting theis NS5-branes clear to that the theD3-brane. D3-branes. NS5-brane sources In magnetic this charge dual in configuration, theopposite world it boundary volume conditions theory on the of D1-brane. the This will prevent the D1-brane from sup- branes so as to preserve the analogous linking numbers ing the NS5-branes to positions given brane and (left) side of the given brane respectively [ by conjugating byoccurs T-duality. when In an this NS5-brane crosses brane a configuration, D3-brane, a changing Hanany-Witten the transition number of connecting D1- where (left) JHEP11(2018)126 . - 3 th σ , 2 , (3.6) 1 x + 1) σ in the and ( ) . We will also σ ( i 8 th ~x σ , ) and consequently 1 ) stretching between I n σ h ~x -direction on a circle so 2.19 = = 4 ) σ x σ X ~x ( : , which introduce localized J . See figure σ h I , . . . , m s ) = = n = 1 4 , ,P i x σ I ~ν H ( NS5-branes which we will index by n X p -direction and at points 4 = . x ) = 0. n fundamental walls 2 ( I p = , – 23 – σ 1 , p in the ) m − n ) ( I σ σ p s s ) where each factor corresponds to an interval in the n σ X − where m = +1 D1-branes (indexed by p I . Each interval is intersected by some number of D5-branes σ U( s ] for similar analysis of a T-dual configuration. σ + σ +1 σ , p m σ s Q 17 = ( I , ) which lie at distinct points ∼ m 2 σ = 2 15 g σ and ≥ G to denote the number of D3-branes in between the , which introduce localized bifundamental hypermultiplets from D1– σ I ]. ] that this frame exists if we satisfy s This brane configuration has p 6 ) gauge theory, the supersymmetric vacuum equations for the D1-branes σ ,...,N q 29 N 14 = 1 I 4) quiver gauge theory with domain walls induced by the interactions with D3- and = 2 SU( -direction bounded by NS5-branes, , 4 x Gauge coupling: FI-parameters in each interval are given by the Gauge group: N For purposes which will become clear later, we will wrap the Here the data of the brane configuration maps to the 2D SUSY gauge theory as We will consider the Hanany-Witten dual configuration in which D1-branes only end Similarly, in order for our brane configuration to describe reducible ’t Hooft defects in It was proven in [ • • • = (0 14 While this is not abubbling. necessary condition, For it the will rest make the of following this analysis paper easier when we considering will monopole specify to the case where this condition is satisfied. that the D1-branes stretch alongThus, we the will circle identify direction but do not wrap all the way around. directions. We thenthe have NS5-branes at (indexed by use the notation NS5-branes. This is summarized in the table bifundamental walls D1 strings as in [ on NS5-branes. These are localized at distinct points 3.2.1 Low energyThe effective low theory energy effectiveN world volume theoryNS5-branes. of these The D3-branes branes will willfundamental give be hypermultiplets rise a to from two-dimensional D1–D3 strings, and the NS5-branes will give rise to a 4D must be the samethe moduli as space Nahm’s of equationsmoduli supersymmetric for space. vacua singular must In be order monopolesof given to ( the by demonstrate D1-branes. reducible this, singular See we [ will monopole now analyze the world volume theory by the ADHM equations or Nahm’sspace equations of as supersymmetric appropriate. vacua Thisof for tells the us instantons that D0/D4-brane and the system moduli thatsystem is is the given given by moduli by the space smooth moduli monopole of space moduli supersymmetric space. vacua for the D1/D3-brane vacuum equations for the world volume theory of the lower dimensional branes are given JHEP11(2018)126 - 5 ∼ = 4) =p , © , 3 s , R 2 (3.7) , 1 x = (0 . =p-1 I ¥ s N m Spin(3) = is the contri- 9 — -symmetry of 4 → f R x =p-1 S ¨ R and the D3-branes s along the 8 — ) up to a choice of 2 ) +1 =p-2 σ
σ R, ( i 5 m 0 0 7 ~y ~x , s — R σ s SU(2) = 6 — 4 4) theory. The D3- and NS5- × x , = diag( 1 ) 5 R, σ — bf ( ) at ∞ = (4 S ~v N-1 ⊗ ] + are FI-deformations, N is contribution of bifundamental walls +1 SU(2) f σ 1 FI I S σ s 4 ∼ = s s bf S S , s + S σ R s [ FI =3 S ¤ m – 24 – + 3 — Spin(4) =3 ) § bulk s 3 σ σ → ( 2 S ~x i — ~ν =1 R ) factor is given by
~x R σ =
t 0 s =2 0 £ — — — x R m = (0 =2 N ¦ s D1 σ D3 NS5 =1 m ¢ D1-branes (red) that end on the NS5-branes ( is the bulk theory of the D1-branes, m Coordinates σ . This figure illustrates the Hanany-Witten frame of the brane configuration in which m 4 bulk . This table specifies the brane configuration whose moduli space of supersymmetric vacuua x =1 S ¡ s The Higgs vevs for theordering. U( This theory has • 1,2,3 x the D1-branes with our truncationbranes is then described impose by a boundarytheory. conditions that This break canthe the be D1-brane supersymmetry theory deduced to from by a directions. Spin(8) noting Then that the the introduction truncation of D3- breaks and the NS5-branes breaks Spin(4) where bution of fundamental walls(NS5-branes). (D3-branes), and The action of this gauge theory is of the form Figure 8 we are studying thethere space are of supersymmetric vacua(black) of give the rise wold to volume fundamental theory domain of walls D1-branes. at Here the intersection with D1-branes Table 2 is described by singular monopole moduli space. JHEP11(2018)126 ] R 4) 0) 46 , , are (3.8) (3.9) 4) has B (3.10) (3.11) (3.12) A , ) = (1 See [ = (4 a 4) SYM σ , = (0 15 N N ] N . , 10 , = (4 i and ( 4) SYM theory. B ˜ Φ) combine into ψ , o 1 , N i γ Ψ ˜ B Φ 0 A ) ˜ Φ). Here the vector 4) SUSY. ) , ∂φ − = (4 iγ , a ∂E ) forms a real SU(2) σ D + ( = N ¯ , θ a ¯ ˜ c Φ( ,D = (0 + ] γ M θ , F ) N (Φ)) + 2 symmetry whereas , Ψ ¯ ) + )Φ + a G Im[ 2 3 φ , − , , are a doublet of Dirac fermions X ] ( R, 2 2 µ D ¯ Ψ F ψ , A , D ,F,E E with superfield strength Σ, a Fermi ¯ χ Φ( 2 2 U(1) B + ¯ ˜ Φ). λ φ = 1 ¯ ] for a review on 2D D, θ V × , iV , B + 1 . ] A 10 − ¯ θ = (Re[ − ) 2 R, 2 ¯ a + ¯ ΨΨ + 46 1 φ , φ a ˜ Φ = ( Ψ = ( λ θ ) ) ∂ σ ) 1 1 ( M 1 + µ − D D . See [ 4) vector multiplet and the (Φ γ D 1 0 , a , – 25 – R ¯ , ΣΣ + λ ) where the ∂ + + ˜ + + B Φ] from (Φ ) n + , 0 0 , 5 0 ¯ = θ ) i D D B = (0 1 ,D X Tr D ( ( A − 1 ( µ ) ,F,D − θ + + D a + [Φ ,G + N 2 ¯ ¯ A Spin(4) θ θ 1 D , λ , ¯ 1 θ σ s ¯ λ + + A ( + A ⊂ , χ , 2) SUSY have U(1) ... 2) vector-superfield + , ψ A i doublets and , v iθ iθ , µ , 1 1 iθ ¯ µ 0 A χ iθ γ = v φ γ R − − χ R, dt ds d c ,X − c − A s − 1 2 Ψ = (2 γ iγ ¯ ¯ χ = (0 Z ) ψ ψ F = ( B E i , v A 1 N + + − 2 + ¯ 0 χ v 4) SUSY. Φ = ( -directions. Thus the total theory has V θ θ θ g v 1 − N , 3 A 5) are real scalar bosons that encode the fluctuations of the D1-brane B 8 − , A − 4) SYM theory which is given by dimensionally reduced 6D , A 2 + + + ) , 4) SUSY, these fields reorganize themselves into a single , 3 0 A χ 1 a = ( , 1 2 2 = µν are SU(2) , v µ x ¯ = (0 c σ λ φ φ 2 F ( γ V γ , A (Φ) is a holomorphic function of all chiral superfields of the theory which = (4 A A ]. The bulk theory of the D1-branes is described by = ( A φ N µν bulk Ψ ¯ ¯ = (4 i χ γ i S ˜ 10 E = 1 Φ = Φ = Ψ = V N i = = ¯ = = N 4) twisted hypermultiplet. These decompose into the component fields 5 a µ A , and direction. The SUSY transformations of these fields are given by [ , are the gamma matrices for Dirac fermions in 2D with along the A i (for δχ δX δA symmetry. δX λ x µ i R γ R = (0 X Under The bulk contribution of the action is given by Due to our truncation of the full string theory, the bulk theory of the D1-branes is Note that theories with N 15 Here the Pauli matrices for SU(2) SU(2) in the vector-multiplet and where and the superfields are written explicitly as in [ where triplet. Here receives the contribution multiplet Ψ, and two chiral multipletsand in Fermi the multiplets adjoint combine representation as (Φ a a for a review of described by SYM theory [ This is composed of a SU(2) JHEP11(2018)126 (3.16) (3.17) (3.18) (3.13) (3.14) (3.15) 4) bifun- 4) vector , , , ) 4) hypermul- I , = (0 s = (0 − , N N s ). These couple to = (0 ( δ , +1 I 2 σ N 4) (twisted) hypermul- I , Q , s 2 I σ 1 Q s Ψ + c.c. t Q ), with constituent bosonic . . = (0 4) hypermultiplet describing I D + , I 2 I 1 , which encodes the supersym- =1 2 N -directions, this boundary the- 2 ) . N I dθ X 3 Q s ] ¯ , FI Q , Q ( c 2 I , I S = Z ν = (0 1 1 1 + ) x 1 f ,X Q s Q I = b ) N ( 1 I δF δS ν X s [ Q FI t + ¯ δF − D δS abc I s 1 ( , ˜ θ V Φ), has a non-trivial coupling to the F- and ¯ δ + ) Q 2 , I a d s , =1 N – 26 – θ I X M 1 2 Z , is given by fundamental − ) 1 2 d ) f s s s ( ( S = 2 ( + 3 Z δ 3 ) ν ν a I dt ... 1 are constant on the interval ( bulk = ¯ Q Z δM = C I δS Tr FI 1 ) respectively. These domain walls contribute the to the full -symmetries. Take the Ψ =1 N ∈ I δD Q I X δS 2 R E ) 1 2 − s 1 ,J dt ds ( I I 2 R, 2 2 = , which are dependent on the interaction of the Z iν a 4) vector multiplet (Φ Q Q f as appropriate to the representation. These fields additionally con- , I S = 2 M ¯ ) + Q iV FI s ( ( S = (4 1 =1 N ) and ( ν 0 I X I N ∂ 1 1 2 ,J = ) = symmetry associated to the rotations of the I = s 1 t ( fundamental domain wall theory to be described by a doublet of fundamental chi- R f Q D ν δD th δS Similarly the contribution of bifundamental domain walls is that of Now consider the contribution to the action from the domain walls. The contribution Now we will consider the contribution to the action In order to determine the vacuum equations of this theory, we will need to eliminate I which have the effect of addingvacuum boundary equations. terms to the supersymmetry transformations and damental hypermultiplets on a domain wall preserving the same supersymmetry. This can These fields couple to the F- and D-terms as where tribute to the E-term for the Fermi superfield Ψ as ral superfields infields conjugate ( gauge representations,action ( as from the fundamental domaintiplets walls restricted to theSpin(3) world volume ofory the preserves domain the walls. SU(2) the By nature of preserving the where the F- and D-terms: metric FI-deformations to the theory. This is given by the auxiliary fields multiplet with all hypermultiplets intiplet the in theory. the HereD-fields. the As is usual for hypermultiplets, this coupling is given by JHEP11(2018)126 (3.19) (3.20) (3.24) (3.22) (3.23) (3.21) ) with σ 2 B , σ 1 . B ) s ( . symmetry. These ) , 1 . L σ ds ν R s ) σ L σ +1 , 2 ) 4), we only impose half − s σ , ) L σ s B σ s 1 t s s ( − ˜ − Z δ D s R σ − σ σ ( = (0 s 1 − 2 s δ ¯ ( B 2 σ B − δ . N 1 σ ) 2 s + B σ iX ( B 3 σ σ δ 2 ν 1 ) + σ ) = 0 B B 1 3 − t ) + a ν 1 ˜ ). These are described by the action X σ D . M − σ 2 σ ) + + 2 R σ ¯ 1 A 1 B 7→ s + ¯ σ B − σ 2 ,L µ 2 a 2 R σ − σ γ s B = 0, the stationary vacuum equations become 2 X B s θ σ iX 1 ( 0 2 2 = B − – 27 – − δ d ¯ D A + B s σ A ( σ ( 2 1 1 δ Z − B B σ B σ σ 2 σ 1 as appropriate to the representation. Here we use the B Tr 1 ) and ( 1 A ], shifts the bulk dependence of the FI-parameters to B ¯ B ) σ ¯ B ,X σ B 1 a ) dt σ 1 ( ) 1 p L σ =1 σ ) for any superfield Λ. These fields additionally contribute 17 s B , ( s X σ ,L s ( p =1 Z B ( 3 σ X σ 1 2 15 1 p =1 Λ( V p =1 1 2 X σ + ( B X σ + σ ds ν − s = = 2 1 ) → +1 ... s R σ σ bf bf = s s σ ( = s δF δD bf δS δS Z V Ψ S i E − ). Since the domain walls break SUSY to ) = lim 3 t X L,R σ ∂ 3.12 s 7→ = 3 t ˜ X D This transformation, as inboundary dependence [ at thecontinuous. bifundamental By domain choosing walls the where axial gauge the FI-parameter is dis- For these transformations, the bulkby contributions making from the the FI-parameters shift can be absorbed fields as in ( of supersymmetries of theare bulk generated by theory: those which preserve SU(2) These also add boundary termstions. to the supersymmetry transformations and vacuum equa- 3.2.2 Vacuum equations Now we can determine the vacuum equations by examining the SUSY variations of the bulk These couple to the F- and D-terms as where notation Λ( to the E-term for the Fermi superfield Ψ as constituent bosonic fields ( be written in terms of two chiral superfields in conjugate representations ( JHEP11(2018)126 , ) 1 − which (3.25) (3.26) R σ s } . a − T s { bow ( δ = ) M 1 } − a σ 3 X ν { + σ 2 , , B I σ 2 σ 2 , 2 ) B Q ¯ 1 B − = = R σ − , s x ) σ . This identification is given by I Λ and similarly the bifundamental RL 1 }| e − s ¯ n } B ∈ s ) ~x σ ( − ζ 1 x δ ( s = ) are identical to the Nahm’s equations ) , with (one plus) the rank of the gauge B ( R σ ) δ ,B ,J N { ν ) , I σ I I,σ 3.25 1 1 } ( 1 = − ) + ( e ¯ | B Q ~ν Q L σ ~ν σ ), we can determine the data of the correspond- I 1 s { Λ ) | 1 = = , – 28 – B I − I Q x s σ , 2 3.26 s LR ∈ E | e ( − − Λ B ) δ , I s I ) 2 ( σ ( E , with the number NUT centers on multi-Taub-NUT: σ 3 e δ p Q ν I ) + ( I with the Chern classes of the instanton bundle (note that |{ 2 2 ,I = L σ − ,B ) can be read off from the ranks of the σ ¯ Q s a = Q ζ ). Therefore, we can identify the moduli space of supersym- σ σ I ( ( ) m 2 1 X ~ν − |E| n ¯ R B ( I =1 Q N s = = I X p σ ( 2.20 2 ) = δ e a 2 1 =1 ) N ) i ~ν B I X T ( σ σ ) = 0), ζ ν ), ) − − . These are the complex Nahm’s equations with boundary terms for ( i ] + 2 ( ] p σ where with a moduli space of instantons on multi-Taub-NUT N ¯ R − ζ 1 X ( ) σ iX B n ,X 2 . Therefore, the number of fundamental walls correspond to the rank of R ( I vac 3 σ X, + p [ B 1 X i : SU( σ ¯ 2 1 [ M B 1 ∈ E i ( I,n G X + B e + ( p =1 P 3 = X σ -directions. p =1 3 X X , = 2 1 X σ 1 1 X 2 , The total number of markedgroup points, The numbers one of the The number of edges, p 1 D D + + In summary, we can match the data of the brane configuration to that of instantons Now by studying the identification ( Under the identifications x • • • 0 = 0 = on multi-Taub-NUT by specifying Further, we can identify the fundamentalwalls walls with with the 4D gauge groupTaub-NUT and centers. the In number this of identification,the bifundamental the positions walls FI of correspond parameters are the to mappedthe NUT the to centers number the which of position corresponds of to the positions of the NS5-branes in for the bow construction ( metric vacua ing bow variety. The ranks correspond to the ranks of the gauge group of the 2D theory in the different chambers. it is clear that these SUSY vacuum equations ( where singular monopoles. which can be written as a real and complex equation: By integrating out the auxiliary fields we see that this reduces to a triplet of equations JHEP11(2018)126 I x . In 1 (3.28) (3.27) ) with σ q < v ], one can 4 σ 6 x 1 + is the radius R ,..., , are identified as } σ = 1 i , where m ( {
0 -directions are identified ) . This is exactly the bow i 3 ( ∞ , σ ) with the positions of the 2 πR ζ ) , X 2 , 1 σ , 2.4 ( 3 I x H , ν 0 ) I σ ∞ ( 2 m is Hanany-Witten dual to that of πR X = exp , ν 2 8 I ) o X σ ( 1 A ν = 1 ∞ m S = ( ˜ γ H = exp σ π 1 ~ν 2 ) , – 29 – n I A . h . ) 1 ∞ 1 ∞ n S ( I S /R I p π 1 and the numbers of D1-branes I 2 = 1 X 0 σ , ( ] to show that the ’t Hooft charge is appropriately screened ~x R σ = 16 ~x n = exp P σ with NS5-branes, and the wavy lines ~ν is isomorphic to the one described in σ e 8 is the dual radius of at infinity and 1 . Thus, in this setting, Hanany-Witten isomorphisms are literally Hanany- /R σ S m = 1 0 , the bow variety describing the moduli space of supersymmetric vacua of the brane R ) = different NUT centers: The holonomy of the gauge fieldof exp the The hyperk¨ahlermoment parameters 7 ) i Additionally, although bubbling involves an intersection of a D1-brane with an NS5- In this setup, monopole bubbling occurs when a D1-brane becomes spatially coincident Therefore, since the brane configuration of figure Note that this is simply the bow variety specified by identifying marked points ( σ • • ζ ( during monopole bubbling. brane, the bubbling configurations areHanany-Witten actually frame non-singular. in Specifically, which we allthis can of go case, to the bubbling the NS5-branes D1-branes are will localized at at most distinct make them coincident with another D1-brane with and intersects anbranes NS5-brane. may indicate One that mayHowever, this be there brane worried description are that breaks several thisreproduces down reasons intersection the for with that correct bubbling NS5- effect suggest configurations. adapt on the the the computation opposite. bulk from [ dynamics. First, Specifically, the as bubbling argued in locus [ Thus far we have presentedvacua analysis of that the shows brane thatmonopoles. configuration the However, moduli since matches there space that is very ofto of little supersymmetric see known the about that monopole moduli this bubbling, it space analysis is extends of difficult to reducible include singular bubbling configurations. where 3.3 Monopole bubbling singular monopole moduli space with the data where the Higgs vev is defined by the holonomy figure configuration of figure variety describing reducible singularspace monopole of moduli supersymmetric space. vacua of Consequently, this the brane moduli configuration is given exactly by reducible with D3-branes, edges D1-branes. Further, the positionswith of the the FI NS5-branes parameters inR the Witten transformations of the brane configuration. JHEP11(2018)126 in 1 (3.29) 2 N 3 dual magnetic frame . Further, notice that in . Now to study monopole N-1 1 9 p ≤ -1 1 1 k . k 10 1 1 , k 1 h 1 3 p See figure ] for a proof. = ) of reducible singular monopole moduli space – 30 – 6 16 P 1 1 ] it is shown that this brane configuration reproduces k 2.15 -directions. Here, one can see that in this frame, bubbling 6 3 . See [ , 1 2 , 2 p 1 = 2 SU(2) SYM theory. This can be described by the x ≤ -1 1 1 N k k bubbled D1-branes such that 2 the bubbled D1-branes in addition to the D3- and NS5-branes. We 1 m 1 D3 D3 D3 D3 D3 3 ≤ only 1 k NS5 v): 2 P, . This figure describes a Hanany-Witten dual frame of the brane configuration in which ]. This can be seen as follows. Now the D1-brane world volume theory is given by a quiver SQM described by the Consider the case of the In fact, this brane configuration has actually been shown to reproduce some key data of 42 1 [ Note that this exists because 2 16 c can now perform a sequencewhich of D1-branes Hanany-Witten only moves end to on go NS5-banes. to the quiver Γ( consider adding a reducible ’t Hooft defect localized at the origin with charge where there are bubbling, consider singular monopole moduli spaces. Inthe [ structure of theM bubbling locus ( above brane configuration as explained above by adding a large mass deformation. Now created by pulling NS5-branesstudying through the a supersymmetric vacua, D3-brane. there isof See no monopole figure obstruction moduli to describing space. theconfiguration Therefore, singular gives it locus a is good not description unreasonable for to monopole conjecture bubbling. that this brane monopole bubbling appears tothe finite be D1-branes a (blue) singularassociated occur D1-branes process. when in they red) In become inis this spatially the non-singular coincident figure as with it we the corresponds can NS5-brane to see (and at that most bubbling coincident D1-branes. of Figure 9 JHEP11(2018)126 ], is 6 (3.30) (3.31) 2 1 v)) defines P, (Γ( = 2 SYM theory), 2 ’t Hooft defects in SQM 1 N , k ≥ 1 ) M v p v) N ]. The main result of [ P, ( 31 (b) ). . -invariant instantons on Taub- ) mono K 1 v) Z k 2.15 1 ). Now “resolve” the configuration P, -v v) Γ( 3.6 P, ( v) in ( P, +1-times. This SQM has a moduli space of (SQM ( 1-loop 1 W 3 k Z v)). Thus, this brane configuration shows that I ) 2 M b , – 31 – − P, (v 1 ) are glued into the full moduli space. Specifically, p πi v) = theory (and hence for SU( (Γ( 2 ∗ e P, 2.15 | ( P action to the gauge bundle around any NUT center is 1 2 SQM = 2 |≤| X K M v mono | N Z ) = is repeated i N 1 0] k ], the expectation value of a ’t Hooft operator is of the form 1 P, v [ 31 L h v) is a contribution from monopole bubbling. This has been computed for 1 p (a) P, ) SYM theory subject to the constraint ( ( N . This figure shows the two Hanany-Witten frames of our brane configuration that we mono 4 x Z 1 -v By Kronheimer’s correspondence this is dual to U(1) Additionally, this construction has also been shown to reproduce exact quantum in- 1 = 2 SU( 1 k -k 1 1,2,3 x m Consider a general reducible singularN monopole configuration with by pulling apart all of the defects into minimalNUT ’t where Hooft the defects. lift of the U(1) This provides a powerfulmonopole verification bubbling that more this generally.itself brane Further, a this configuration semiclassical also can effect. suggests be that used monopole to bubbling3.4 study is Kronheimer’s correspondenceNow and we T-duality will study the relationship between Kronheimer’s correspondence and T-duality. the monopole bubbling contributionSQM is derived exactly from given this by brane the configuration: Witten index of the bubbling where several key examples by using exactthat techniques in such the as case AGT [ of the SU( the moduli space of supersymmetricthe transversal vacua of slice this of 1D each quiver singular SQM strata formation about monopole bubblingtheory by considerations using [ localization. Recall that from general field where the node of degree supersymmetric vacua given by there is a SQM ofindicates bubbled how the monopoles singular living strata on in the ( world line of the ’t Hooft defect which Figure 10 are considering: (a) thethe standard unbubbled frame monopoles and removed). (b) the Hanany-Witten “dual magnetic” frame (with JHEP11(2018)126 - K (3.38) (3.32) (3.33) (3.35) (3.36) (3.37) (3.39) (3.34) in the metric . 2 ) | , under T-duality, ω σ ~x V + . 1 , √ − 0 / ~x dξ | . 2 )( , Xdξ σ ~x cochar 2 ( P , ) 1 Λ d = , the theory will be described 3 ω − − 0 p ∈ ∗ 0 V + ) = σ ∞ = TN ( , V dξ ψ πR I dξ ) σ X ) 2 σ ~x ~x )( ω ~x , ( ~x I -invariant gauge → ψ lim ( + , h ~x 1 K H fiber has radius 1 to be a periodic scalar field, which in decompact- I − K 7→ dξ 1 ( V ∞ m = exp S ψ X , + dV , dω I V U(1) = 2 SYM theory. As before, near each NUT ) 3 + σ X dξ ˆ ∗ – 32 – ~x √ 3 A ∈ d~x . ω R = · N = 1 ∞ 1 ∞ σ A S m S V P -valued scalar. d~x I ˜ γ t ) = √ π = ~x ) 1 ˆ ( 2 A , α 3 ~x , dω ( V R α ( | ) ψ σ A σ = ( ~x in the 4D σ I = 2 ~x exp ih 1 p − e → lim ds dξ ~x ~x ) | TN 2 7→ ~x ( we allow to be a α ψ σ R X → . Further, the first Chern class of the gauge bundle is given by is the dual radius of 1 ) σ S . By T-dualizing along the circle fiber of ( I p /R h ) = 1 + ~x TN = = 1 ( -directions. Additionally, since the 0 3 σ , V 17 2 R P , 1 Let us embed this gauge theory configuration in the world volume theory of D4-branes x Here this equation is only strictly true if we take 17 the one form roughly transforms as ifying the T-dual In T-dualizing, Buschergenerates duality a tells non-trivial B-field usthe source that existence at of the the NS5-branes positions term the in of the the transverse NUT space centers. at the This position indicates of the NUT centers in Again we will use the notation where where 3.4.1 Action ofLet T-Duality us on consider fields theinvariant action instanton of on T-dualitycenter, on the gauge gauge field field configuration can be describing written a in U(1) the U(1) wrapping as the world volumeby theory taking of the D3-branes coincidentexactly in limit produce the (that the presence brane is configuration of reconstructingproceeding for D1- the with reducible singular the and reducible monopoles. technical monopoles), NS5-branes. details However, this before of Then will this calculation, we will first motivate this result. infinity where where and the Higgs vev is given in terms of the holonomy of the gauge field around the circle at given by JHEP11(2018)126 - p 3 18 to , 2 , p Φ = 1 1 given x (3.41) (3.40) − p TN ζ is a gauge ) and the TN NS5-brane 1 -direction S 5 th 3.36 x (where + 1) NS5-branes which ,Y ) in ( , p 0 σ | , ψ A σ 3 ~x R σ = 1 theory by projecting out A − P which together satisfy the N ~x to the ( | 3 2 R th − A σ . Therefore, from the field per- σ ) = ~x , This can be measured by its period ~x ( NS5-branes. Now consider wrapping π B 2 ψ 20 -direction then maps the collection of In this case we have ) th 0 4 ~x x ( σσ 19 . C -field, T-dualizing them give rise to a NUT V + 1) Z 8 -field. σ B = 2 SYM theory to a -invariant instanton configurations on ~x σ B = -invariant instanton configuration on K and connection ) so that there exists a magnetic Hanany-Witten duality N → lim – 33 – 0 ], the stack of D3-branes to a stack of D4-branes ~x K σ 3.6 θ . X 3 and ( 43 , − R ]. ), we can see that these fields have the limiting form th σ 28 [ θ σ 35 ω , p σ 3.38 P TN D1-branes running from the = σ 3 R m A σ -directions in analogy with before. ~x 9 , → with lim 8 , ~x 7 ], instantons in the world volume theory of a stack of D4-branes wrapping , and the D1-branes to some instanton configuration of the gauge bundle , σ p 6 x 50 TN -direction and the 5D Higgs field describing the D4-branes in the 0 x is the standard complex Higgs field) come from the five-dimensional gauge field D3-branes in between the -direction on a circle. T-duality along the σ are T-dual to a brane configuration described by D1-, D3-, and NS5-branes. Here we 4 iX q In order to specify the T-dual brane configuration we need to specify how the numbers Consider our brane configuration in the magnetic Hanany-Witten duality frame where Note that the other bosonic fields of the four-dimensional theory x p Note that we are truncating the standard 5D Recall that we are imposing the condition ( Note that this is the relative positions as the absolute positions along the T-duality + 18 19 20 fluctuations in the frame. dependent. directions. This means that theis relative encoded positions of by the the NS5-branesintegrals cohomology on the class T-duality of circle the living on the D4-branes. and positions of the branesand are positions encoded in of the theSince instanton the brane NS5-branes configuration. NS5-branes are are The encoded chargedcenter number under in due the the to B-field Buscher of duality the at T-dual the configuration. previous location of the NS5-brane in the we will index by and the NS5-branes into a transverse wrapping the will use this analysis toabove show is that T-dual to the the brane corresponding configurationby U(1) of Kronheimer’s D1/D3/NS5-branes correspondence proposed [ D1-branes only end on NS5-branes as in figure and hence do not play a role in3.4.2 T-duality. String theoryAs analysis described in [ TN spective, it clear that T-dualitysingular maps monopole U(1) configurations on Y along the Bogomolny equations. Additionally, using theform limiting of forms the of harmonic ( function ( which is exactly the ’t Hooft boundary conditions at This leads to the standard Higgs field JHEP11(2018)126 ) Z ; p fiber (3.43) (3.44) (3.45) (3.46) (3.42) 1 -brane -brane = 0. 1 σ S TN numbers ( 0 2 p m H ]. Here we have ) as ] for more details. Z 43 . 50 where , ; . Here we identify p 3 } )) 28 R TN /R ( N ). These elements can be is defined as the preimage -brane and the NS5 , . . . , p 2 Z cpt is p ; +1 H p = 1 σσ exp( -direction [ . σ C TN 4 ). Using this, we can identify a ( : . x Z +1 ; σ 2 cpt , σ p q ,..., , σσ ` ) H b 0 { ) above, we can identify the homology +1 + where ) ) is naturally isomorphic to TN Z m 1 σ /R ρρ } ( ; Z 2 f − p -brane in the base C ; σ +1 2 is cpt p 1 ) determined by the sequence of − σ ) = +1 m − σσ Z = H 0 TN σ to an element of V of the line running between the NUT centers X ρ ; ( σ +1 ( C TN − p 2 – 34 – σ exp( } ( 2 { 3 of , σ f = H ) R ( ch 2 0 cpt TN m NS5-branes: +1 ( σ H σσ /R Z → X p = 1 σσ as follows. Given an ordering of the NS5-branes, there C , . . . , p 2 cpt b p σ NS5-brane along the 0 is = ` H as the homology cycle σσ = 1 th ]. In order to specify the class of the instanton bundle B TN 0 ) given by C σ . This is topologically equivalent to the cycle defined by the σ 0 : 50 Z σσ ; : -brane and NS5 p π C σ σ f σ > σ TN { ( 2 = diag (exp( ] determined that the first Chern class of the instanton bundle is given H ∞ 50 U The first Chern class is valued in -branes. 21 numbers, with basis elements 0 σ p +1 is the number of D1-branes running between the NS5 . We then define is the position of the p p 0 σσ σ C θ m + ], the author also computed the 2nd Chern character of the instanton bundle TN σ In order to completely specify the instanton bundle, we also need to specify the holon- In this setup, [ The rest of the data of the brane configuration is encoded in the gauge bundle through 50 That is to say, we specify the data of the relevant instanton moduli space. See [ → ∼ 21 Given this data of the instantonmine bundle the and T-dual B-field brane configuration, configuration we of can completely D1/D3/NS5-branes. deter- Now by taking the coincident where (recall that the NS5-branes are separated along a circle).omy of In the our gauge case, connection.at we have infinity In encodes the the 5D positions gauge of theory, the the D3-branes: monodromy along the In [ V by the corresponding elementgiven of by the linking numbers of the understood in the followingby fashion. Poincar´eduality. Using the basiscycles of sequence of and the NS5 the instanton configurationof [ the T-dual branethe holonomy. configuration, one must specify first and second Chern classes and where we have assumed preimage of the line running between the NUT centers corresponding to the NS5 identified the homology cycles is a natural basis of under the projection map corresponding to theσ NS5 where JHEP11(2018)126 . I on = 2 m I K which (3.48) (3.49) (3.50) (3.47) N P ) . Due to ∞ } N -direction. splits as a . i 4 p ~x to x ,..., { 22 σ TN ~x . = 1 . i in a SU( | ) | σ ]. from i n ~x ( I ~x 32 p σ 1 1 − where NUT centers located at − L 3 -direction, any such gauge ~x n,I ~x | 4 p | ) to be the preimage of this R x σ P , ∈ L d d ( = 3 i Λ 3 1 ∗ ~x d -field gauge transformation as ∗ -action in the singular monopole − K π B = + = σ i = B ~x ~x , . . . , p σ . 7→ C I = 1 R σ , dω , dω i σ =1 N ~x ~x . Now choose a line I M for ω ω σ – 35 – ,B 1 2 1 2 σ ~x = Λ ~x + + T -invariant instanton configuration on ) ) ~x ~x K ( ( V ω V ω | | i σ + ~x + ~x T 7→ T ⊗ L − dξ dξ − ~x | ~x | 2 2 − − invariant instantons that are far separated at positions = = i K ~x σ ~x Λ ], these line bundles transform under a Λ 50 σ ~x -invariant instanton solution given by Kronheimer’s correspondence. Consider is the line bundle with connection given by Λ. 6= and U(1) K Λ i ~x L =1 p σ Choose the NUT center at position In order to complete this discussion, we need to understand the action of U(1) } Here by asymptotically we mean at distances sufficiently far from any instantons. Specifically, are σ 22 ~x where interested in the behaviorperspective at of infinity and singular arbitrarily monopoleHiggs close configurations vev to and because the at the NUT the gauge centers. ’t symmetry Hooft This is defects can broken by be at their seen non-trivial infinity from boundary by the conditions the [ As shown in [ This family of line bundlespoints can be extended to include connections associated to arbitrary individually gauge equivalent tofollows. a canonical set of line bundles whichdoes not can intersect be any other definedline. NUT as centers. To this Define infinite cigar we can identify a complex line bundle with connections the holonomy of thedirect sum gauge of bundle, line the bundles Chan-Paton bundle asymptotically These line bundles can be decomposed as a tensor product of line bundles that are each the U(1) the case of reducible ’ttheory Hooft with defects far at separated smoothThis monopoles is at positions T-dual{ to a gauge theory on multi-Taub-NUT with However, since the branes do nottransformation wrap can all the be way undone aroundconfiguration the will by be a dual trivial to gauge a U(1) transformation.3.4.3 Therefore, this T-duality brane andWe line will now bundles show explicitly that T-duality exchanges singular monopole configurations with reducible ’t Hooft defects. the T-dual instanton configuration.to the Since instantons the on D1/D3/NS5-branebrane Taub-NUT configuration by configuration is T-duality, acts the related as U(1) a non-trivial abelian gauge transformation in the limit of the appropriate NUT centers, we arrive at the T-dual brane configuration for JHEP11(2018)126 is i ~x circle (3.51) (3.52) (3.53) (3.54) (3.55) (3.56) of the 4 x ∞ X . This tells , i σ ~x ~x as above. This ω ω , with connection σ 2 1 2 1 ~x t ∗ , L 1 k − ~x → − → − L . σ σ ~x ~x Z +1 Λ Λ π I σ 2 ~x )= D3-brane along the k →∞ O lim → ( lim R ~r . ~x I th : i ∈ k j + 1 where it can be smoothly continued ~x t , = 0. Note that this reproduces the we can also define 3 L , I , 0 R ∼ σρ index smooth monopoles with magnetic 23 i σ , t m )= ~x ~x C ∈ j ( ω ω i ) L +1 O I = I = 2 1 ~x x : ω , + ( j k =1 ρ 1 2 N πt , t σ ~x O 1 V σ π is the Dirac potential centered at dξ − ~x Λ – 36 – 2 m t → − → = i d L . = 2 ~x i i ∗ 1 , . . . , m I = σ ~x ~x ) ω − t L C Λ Λ ( ∗ σ p )= Z i ~x σ = 1 Λ A ( ~x Λ O I is the position of the k : 1 ∞ → →∞ lim lim ). Additionally, these asymptotic forms tell us that Λ ~r ~r S I σ p =1 s I X σ t and πR 2 3.61 I / = I s ∗ ) t , ( ∗ L ) Λ ω respectively where = , . . . , m I + +1 R I , = 1 dψ ) where again H 0 ( j → → has non-trivial holonomy along the asymptotic circle fiber 3.46 ∗ ∗ is the Cartan matrix of and is a line bundle with connection which is gauge equivalent to Λ ) t Λ Λ ∗ ( I σ i σρ ~x ~x H C L → →∞ lim lim ~r ~r Using this, the factors of the Chan-Paton (gauge) bundle of the T-dual brane config- Since this is a topologically trivial line bundle, We can additionally define the topologically trivial line bundle These connections have the property that This is trivial in the sense that the canonical pairing of the curvature with any closed 2-cycle is trivial. 23 where here the charge expression ( before decompactifying. uration are given by Therefore, this component ofT-dual the brane Chan-Paton configuration bundle ( describesan asymptotically the flat Higgs connection vev exceptin near exchange for inducing a source for the first Chern class. where all other limitsus are that finite. Λ Here These connections have the limiting forms the Cartan matrix. where line bundle is topologically trivial because its periods are trivial due to the properties of where JHEP11(2018)126 . } ). ∗ 1 σ L − ~x 6= 0 to its L = (3.59) (3.60) (3.61) (3.62) (3.57) (3.58) σ 1 1 y σ , ]. There- − ~x ) = L σ r m 35 ~x s 2 σ γ { L N − , j ∞ , the Chan-Paton ~x σ which requires scaling X Λ y 17 +1 = I ) = s x )= ( j X ( -field such that the Chan- ψ I ) : B x j ( + V j . ~x , , 1 ) Λ σ ) →∞ lim r − ~x ω I N L Λ )= + . j , (