JHEP09(2020)019 and 1 S Springer × 2 July 25, 2020 June 18, 2020 : : September 2, 2020 AdS : -BPS geometries. 1 2 Accepted Received 4) charges and a single , Published Published for SISSA by https://doi.org/10.1007/JHEP09(2020)019 = 4 SYM is also discussed. [email protected] , N SYM = 4 . 3 N 2006.08649 The Authors. c AdS-CFT Correspondence, D-branes, Long strings

We study classical solutions to the Nambu-Goto string on , corresponding to strings stretched between wrapped branes extending in AdS. [email protected] 1 S × 3 Department of Physics, UniversitySanta of Barbara, California CA at 93106, Santa U.S.A. Barbara, E-mail: Open Access Article funded by SCOAP Their relevance to the SL(2) sector of Keywords: ArXiv ePrint: Abstract: AdS The solutions are obtained byposition analytic of continuation the of giant branes.SO(6) magnon charge. solutions, These cut We at solutions compute the carry their one energies or and show two their SO(2 relation to David Berenstein and Adolfo Holguin gravitons in Open giant magnons suspended between dual giant JHEP09(2020)019 6 7 and (1.1) is the 5 . The S p λ × 5 AdS ], and the magnons 4  2 p is the t’ Hooft coupling and  λ 2 sin 2 λ ]. This connection makes it possible to ex- π 11 1 1 + – 1 – 4 = 4 SYM is that it admits a description as an r = N 1 6 J S − × 3 ∆ AdS remains finite. The dispersion relation follows from the symmetry J 1 1 − S ] for a textbook-level review). Relative to a ferromagentic ground state, 2 10 × 2 ] 4 , 3 AdS is a Cartan generator of the SO(6) R-charge, , while ∆ J An important feature of planar 3.1 3.2 Rotating string in = 4 SYM theory. In particular, the string excitations associated to the t’Hooft limit of → ∞ algebra of the spininfinite spin chain, chain and has itare a is in centrally an extended short supersymmetry all multipletsthe algebra of orders anomalous [ the result dimension centrally is inthe extended constrained the multiplet. by algebra. t’Hooft This the shortness kinematics coupling As of of such, the the multiplet their is shortening contribution a condition to BPS on condition for the states represented Here, quasi-momentum of the magnon impurity.J The infinite chain limit corresponds to the limit integrable spin-chain that computes theoperators spectrum (see of [ anomalous dimensionsit of is single trace known bysector of supersymmetry the spin-chain, arguments single that magnonrelation excitations in [ are characterized the by the infinite exact dispersion length limit of the SU(2) tract information about strong couplingA dynamics particularly from important calculations example inN is the the dual duality geometry. betweenthe string theory theory can in befield understood theory. in a lot of detail from the study of planar diagrams in the 1 Introduction The AdS/CFT correspondence has established atotically bridge between AdS quantum spaces gravity in and asymp- gauge field theories [ A Giant gravitons and LLM geometries 3 Open giant magnons in AdS 4 Discussion Contents 1 Introduction 2 Review of giant magnon solutions JHEP09(2020)019 ). J (1.3) (1.2) ]. These 12 giant magnons ]; the dispersion Q 8 , 7 is a discretized angular k 2 | ˜ ξ ] is that it decouples finite size  6= 0), so it becomes possible to 5 − 2 p ], rather than having an infinitely λ ξ  | 2 10 ]. The results of the central extension 2 , 9 π λ = 4 SYM spin-chain studied by Beisert 10 ]. This is in contrast to the plane wave sin 4 5 2 N λ + π 2 + 1. In the field theory dual, the coordinates 2 < – 2 – + 1) Q are (complex) coordinate positions of D-branes, | k ˜ ξ ξ r Q ( | ∆ are both finite. ξ, r = = J, 2 J = = 4 SYM, the net central charge must vanish. This is J − J N − ∆ ) from quantum effects ( ∆ ∞ J < . The variables 5 while taking the long wavelength limit on the spin chain simultaneously, S → ∞ is the Beisert central charge of the open string , and ]. The Hoffman-Maldacena limit allows for a different reliable extrapolation of ] and the open strings between them are obtained by cutting the giant magnons | λ 6 [ ˜ ξ 11 is a second Cartan generator of the SO(6) symmetry (which is orthogonal to − ∞ 2 ξ J | < For a closed spin chain in The central charge for the open string is sourced by the D-branes in a similar way The scaling of the dispersion relation at strong coupling suggests that these states 2 appear naturally when addressing collective coordinate descriptions for giant graviton where momentum on the and they live onξ a disk of radius one correspond to the central extension ofwith the open planar boundaries. Excitationsexact of dispersion the analogous central to extended theory the are above, given expected by to have an than in the caseis of the D-branes source in ofgravitons flat [ the space: central the charge.for physical the In separation sigma this model of at case, a thematch location the pair of with supersymmetric of the traditional D-branes D-branes D-branes [ field are theory giant computations up to various loop orders [ related to the level matching constraint.strings This instead condition of can be closed relaxed strings.open if one strings More considers that precisely, open end one on canlong supersymmetric study spin D-branes giant chain. [ magnon solutions for where A particular feature ofeffects the of double scaling the introduced systemstudy in ( the [ regime where the charges relation of these bound states is given by limit where λ/J quantities between the strong and weakalso coupling regimes. have simple Bound dispersion states relations of thatand have in been the computed classical both string infrom the limit. spin-chain understanding Their model short contribution to representations the of anomalous dimension the also spin result chain algebra [ conformal field theory. should have simple descriptions in termswidely of classical studied. strings. They Such string require solutionscoupling a have to been particular be double a scaling finite limit and that adjustable allows parameter for [ the t’ Hooft on the spin chain dynamics, but they are not global BPS states with respect to the global JHEP09(2020)019 ˜ ξ to ξ = 4 SYM, , as single 2) has the , rather than R . N 5 k × 3 AdS S with the mass of the will both be complex ] (see also references | , which are suspended ˜ ξ ˜ ξ 19 − ξ, ξ | : the straight line from ˜ ξ . Our goal is to show how this ξ, R for both infinite strings and finite × 5 3 S ]. Also, the momentum × AdS 23 open giant magnons 5 directions. More precisely, we study such 5 1. These expressions have been also obtained AdS – 3 – ]). AdS | → ⊂ ˜ 17 ξ | n , | S ξ | × ), where the central charge now corresponds to length t ) suggests the identification of R 1.3 1.3 ] (see also [ for dual giant gravitons directly from the string sigma model. ]. The Coulomb branch arises from classical solutions of the 16 ˜ ξ – 21 , ξ, , but now as solutions in 14 20 R × 3 S ]. Recently, a combinatorial origin of this central extension in computations 18 ]). These expand into the AdS directions, rather than on the sphere. Because ]. See also the appendix for details. The boundary conditions on the spin chain 22 13 These solutions will be called in this paper We show in this paper that indeed, the dispersion relation of these giant magnons is The purpose of this paper is to find the classical solutions for the open strings that It is expected that similar expressions hold for the SL(2) sector of the spin-chain The relativistic form of ( given by a continousof limit the of open ( stringcoordinates stretched that along are the outside AdSthe the directions, string disk and solution of now that depend radiusin on one. the the complex There precise plane position is must of an not cross additional the constraint disk for of radius one. There are hints that this type being an SO(6) angular momentum, becomes an angular momentumbetween on dual giant gravitons.dual giant In gravitons the that sigmastrings extend between model, dual in these giant the gravitons are that strings respect stretching the between same half of the supersymmetry. sigma model of analytic continuation works inthe detail analogous for variables these solutionsSome and classical to solutions that also seem explainthe to the dual address origin some giant of of gravitons, these have issues, been but that studied do in not [ include 3,4 of [ of this, the magnons,need rather to than become looking extendedsame like in (complexified) a the point-like Lie AdS particlestrings algebra directions at should as as arise the SO(6), from well. center an one of Naively, analytic AdS, expects since continuation of that SO(4 those the solutions corresponding already found classical for the charge that correspond to thedual classical Coulomb giant branch gravitons of the [ SYM field equations theory of are motion the in so flatthe called space. D-branes Associated can field be theoryeigenvalue configurations also solutions that of treated describe the semiclassically classical BPS for equations the of field motion for theory the on dynamics (see sections of anomalous dimensionstherein). in field The theory correspondingconditions has analysis is not been of understood the explored yet. open in SL(2) [ spinare chain associated and to the their SL(2) boundary sector. In this new case, the D-branes that source the central from studying open stringsstrings. on with open boundaries.extension that In determines the this massesas of case argued W-bosons in the [ in the central Coulomb extension branch of is actually the same central follow from the works [ BPS string stretched betweenof the a branes; string. this Beisert’s is resultthe a is positions quantity recovered to from that the the edge is open of analogous spin the to chain disk a when we length take the limit of states [ JHEP09(2020)019 . ], 5 N 2 S 18 g (2.4) (2.2) (2.3) (2.1) = λ ]. We are we study 5 3 ψ. 2 . 2 dψ + cos we review the solutions for giant + 2 0 2 2 clear. We can choose a parametrization ∞ ∞ ψψ 2 t ψdθ 2 → ∞ 2 −  < ∆ sin t θ AdS . These are a particular analytic continuation = to be large. Then we seek for the solution with 1 q – 4 – = = = 0 . The simplest of such configuration is given by λ , p < S λ J is the momentum of the excitation, and J, + cos p ˙ τ σ ψ , and their generalizations to two-charge magnons × 2 p while its endpoints rotate along the equator of − 5 3 dτdσ dt 5 S ∆ − × Z AdS 5 = 2 AdS which we describe. A particular simplification is possible λ π 2, where we have used a cosine of the angle, rather the sine 2 S 2 n √ π/ and S ds AdS × = 1 × R S S R × 2 and 2 π/ − AdS for a fixed momentum =  = 1 arises from the fact that the string becomes asymptotically a ferromag- ψ ˙ θ is one of the SO(6) R-charges, J Upon substitution of the rigid ansatz, we get the action The paper is organized as follows. In section net for the SU(2)velocity chain. for In the the motion of presence the of giant giant gravitons gravitons, themselves. this is the correct coordinate The condition Note that the spatial part ofsingularities the at metric is that ofof a the sphere of angle. radius We one haveorder with chosen two to coordinate a make slightly the different analyticfor convention continuation for a into the rigidly metric rotating of worldsheet the coordinates sphere for in the Nambu-Goto string given by the least energy a string that sits atThe the motion origin takes place of on Where is the t’ Hooftshould coupling. also consider the In t’ order Hooft for coupling the semi-classical string description to be valid we Let us recall the scalinginterested limit in of the the following giant scaling magnons limit: of Hoffman and Maldacena [ classical strings on of the string solutions on due to the boundarythese conditions solutions in necessary terms on of the the giant LLM plane gravitons. is2 also The discussed. interpretation of Review of giant magnon solutions rather than the SL(2) sector. We will explain themagnons origin in the of sigma this model. constraintdisk in inside We also this the interpret paper. LLM those planeand solutions of we in discuss terms their of open a string geodesic versions on ending a on giant gravitons. In section of condition arises from perturbative calculations from studying the SU(2) sector in [ JHEP09(2020)019  , as = 0, p (2.8) (2.9) (2.5) (2.6) (2.7) θ diverges 1, this is 2 0 ≤ ψ r and the angle ) takes the form ψ . This is not the 2 2.4 dθ 2 = cos r r + 2 dr 2 ψ ψ 2 r 2 2 r = 1 , the expression for + 2 + − ψ σ 1 ) 2 ψ 2 2

2 0 r ds − 2 σ + cos + cos 2 ) r r θ r sec 2 σ 2 variables: 2 + ψ ψ ∆ r − p 2 a + 2 cos r 2 2 0 = 1 both of these expressions scale with + cos 2 2 − r tan 0 0 (1 dσ = sec r sin 2 r 2 0 sin sin σ √ ψ

2 (1 r r 2 2 , minimizing the action ( ψ 2 Z √ 0 0 a 2 2 λ ) ) ψ π 0 0 ψ ψ λ π √ − remains finite and it is the same as the on-shell r r sec 1 in the solution. Notice that by contrast 2 – 5 – ( ( √ p p 1 2 J = a → = dσ dσ dσ dσ − J = cos r = J r Z Z − Z Z 2 0 − λ λ λ λ π π ψ ∆ π π 2 2 √ √ 2 2 √ √ = ∆ = =  , but it is a flat auxiliary geometry. By virtue of J J ∆ = ∆ = 5 S × ] by a matrix model ansatz. For closed string solutions we can sew 5 24 1 in order for the solutions to make sense, as AdS = 1: | ≤ r σ | = 0. sec . Substituting the explicit solution a 1 − total ) p 2 r The case of open strings stretched between giant gravitons is entirely analogous except Note that − is a real coordinate of the sigma model. Near forming a closed polygon.string This is equivalent to the levelthat matching one condition must of impose amust that closed be the attached endpoints to of the location the of string the lie giant inside gravitons. the These disk. giant gravitons The preserve end the points The angle between the twooriginally end-points noticed is in identified [ withtogether the various momentum of of the these defect straight line solutions on the disk such that in the end ∆ so the density of thecan conserved have charges a becomes smooth infinite endingeffective near on energy such an of points. infinite the configuration, This spinaction, ∆ is chain which that how is one has clearly not thethe been length edge closed. of of However a the the straight auxiliary line disk segment geometry: connecting the two points on ψ (1 whenever is non-singular and stays finite.is the More same to as the minimizing point, the for geodesic these problem simple we strings, found extremizing above. These develop a singularity whenever It is convenient to write them also in terms of the of a simple geodesicoriginal problem metric with of an effectivea metric flat metric on a disk. The conserved charges are given by Using the coordinate transformation JHEP09(2020)019 ]. (3.4) (3.2) (3.1) 10 remain direction. ,J = 1 has to and we will φ ˙ φ ) (3.3) σ ρ ( = 1. These are + 2 ˙ φ τ , have a fixed value = φ φ + cosh , 2 τ ], as they have motion in ) 0 ρ = 27 t ]. That is, )( ! ρ . . ( 2 2 . The problem again simplifies 2 21 ) , 0 iρ dφ ρ while rotating at angular velocity 20 . This condition with [ 2 + sinh τ 5 → r 2 ρ q ψ + = 1, both of the charges ∆ dρ 2 ), so that AdS 0 r ρ r + can then be seen to be the coordinates for + 1 1 dτdσ 2 ]. p 2 ρ on which the branes are wrapped shrinks to Z will evolve in time in the same way that the 26 2 1 1 + ( ρdt 3 dσ dψ inside – 6 – λ 2 2 S π φ r 3 = cosh( 2 Z √ cosh S r τ + − cosh − 2 geometry. The metric is given by ∝ =

− dr 1 S ) is independent of , the two dual giants must preserve the same half of the g = S ]. This description of the physics in terms of a disk also = 1 σ = − ( S 2 × ρ 25 2 √ [ 2 ind 2 × ds ds 5 2 ds S . As in the previous section we may consider the Nambu-Goto dτdσ AdS × 5 AdS 5 S Z λ π ⊂ 2 √ AdS 1 S − 1 = S = 1, these do not exit the disk. The analysis has been carried out in [ S r × 2 AdS at each time and are located at fixed values of This way we find that The induced metric on the string worldsheet for these solutions is given by We choose to parametrize the worldsheet coordinates by As a result of the end-points not reaching φ = 1 along an which can be seen toof be an the analytic giant continuation magnons ofby the of sigma introducing Hoffman model the and action change of Maldacena, of the variables solutions supersymmetry. one extra dimension. be looking for solutionsdo where with the motion ofTo have the a dual reduction giant to gravitons that move at speed one in the We will be interested in solutionswith of the the equations dual of motion giantcoordinates where gravitons. of the string the rigidly dual As rotates giants such, of do. These rotate at constanta velocity different in type of solution to the rigidly rotating GKP string [ between two D3 branesω that wrap an action for a string on an 3 Open giant magnons in3.1 AdS Now we proceed to study the analogous solutions for the case of open strings stretching zero size at finite. The disk coordinates the LLM plane of appears directly from the field theory dual [ same half of the supersymmetry. As the JHEP09(2020)019 3 1 ), ,J S , > 2.4 (3.8) (3.6) (3.7) (3.5) | r 1)) | − 2 ). This is r σ ( ( where the 2 0 r ], which starts 3 ϕ ) S 28 + × 2 3.1 0 r which corresponds to R ( 2 shrinks to zero size at 1 β 2 S 5 into − is the angle in the plane of × dφ ) is the radius of the LLM in the Nambu-Goto string ˜ 2 ρ ρ 3 0 i ρ + AdS σ 1) 2 = 1 somewhere have a similar → ⊂ − AdS r 2 ρ ρdθ 3 r 2 ( S = cosh( = cosh 2 0 r ) ) , in a parametrization g r 0 2 σ ) ( σ σ remains finite. dσ + sinh ( ϕ − 2 g 1) + 2 a J r + σ ) + − dρ τ − τ + σ 2 t φ ( – 7 – + r 2 ω r βτ ω cos( ( 2 0 dr = = = = ϕ = 0 t r θ ρdt 1 φ ) which cross the unit circle are not physical, as they r = = 1. 2 βg S 2 φ 3.3 × ˜ s ω + 2 d 3 cosh can be easily seen to be the length of a curve on a flat 2 = 0 − 2 t ϕ r ω AdS = + + 2 2 0 2 0 r ds r = 1, yet the energy ∆ p √ ) can be analytically continued via r dσ 3.6 R dτdσ Z λ π 2 √ is the distance of closest approach to the origin and − a = The action for the rigid string in these coordinates is given by: The metric ( The introduction of a third charge is straightforward. We will do that analysis in the The coordinates of this auxiliary plane geometry span the region outside the disk in It is important to note that although superficially similar to the solutions of ( S = 1. In particular, solutions to ( Where we are interested in the form: Here again it isaction. convenient As to noticed use inplane the the coordinate. coordinate appendix, the coordinate is expressed in Hopf coordinates. We make an ansatz for the embedding coordinates of 3.2 Rotating stringNow in we consider the sigmaa model two-spin of magnon a solution. rotating The string metric on is a simple generalization of ( behavior to thebecomes Hofmann infinite Maldacena at solution, in that thediscussion density of of the following thefrom section. the charges Nambu-Goto We ∆ string, mostly to follow make the the discussion analysis. in [ would require the radial coordinateprinciple of solutions AdS of to minimal becomenot length stable complex. when in Even that one though they removes there should the are receive in inside quantum of correctionsthe since the LLM they disk, plane, are these no rather are longer than BPS. the inside. Solutions with straight line in this auxiliary geometry from the startingfor point the to change the of endeverywhere point. variables on to the make worldsheet,r sense the in reason this case being we the must also impose that geometry in polarminimized coordinates by a straight line, where where closest approach. The energy computed this way is also proportional to the length of the The expression JHEP09(2020)019 ) R are × (3.9) 3.10 θ 3 (3.12) (3.11) (3.10) in this ) is of = 0 S g σ 3.8 J σ ∂ ). The other ) implies that = ! 3.10 3.10 1)) − 2 r ( 2 0 ϕ might seem daunting as ( = 0. Because the D-brane we + g 2 t 0 by using its equation of motion: σ r f i ∂ ( g σ σ 2 is free to vary.

=1 t ) β ) =1 1 ω 0 σ φ

δθ g ϕ J − ω t − σ

, so one may substitute that condition di- L 2 1) 2 ∂ r φ ∂ δθ ∂ω r L 1) ( L − ∂ β ∂β ∂ω ∂ 2 − dσ dτ r − 2 – 8 – dσ dσ r Z = Z ( + ( 2 − Z Z 0 0 g ∝ g σ direction, the correct boundary conditions for = = ∂ βϕ vanishes at the end-points of the string in order for the 1 2 θ bdy ∆ = J J σ S 1) + 1)( J − − 2 2 r r ( ( 0 ϕ 0 ]), but their physical interpretation was not made clear. Here the βg to be constant. At first solving for 28 , is precisely the quantity inside the parentheses of ( σ + 2 23 J ) θ 2 0 σ L ∂ ϕ ∂ = 0 and by the choice of static gauge in terms of the other coordinates: ( ∂ + g δφ 2 0 = r = σ direction from one D-brane to the other. The giant magnon carries that angular p J θ δr

. A great simplification is possible since in the end we are interested in describing We must then conclude that The boundary term that arises from taking the variation of the action ( σ σ ∂ J meaning of the condition is clear:in the the giant magnon doesmomentum, not but transport it angular momentum does notcase does transfer not it affect to the therectly variational in D-branes. principle order for Eliminating to the express the variable conserved quantities in terms of the on-shell action multiplied the function Solutions of this typesigma have model been (see [ previously considered for infinite strings in the are considering is extendedNeumann in boundary the conditions, which means that variational principle to bethis well-defined. quantity vanishes In everywhere addition along to the this, string. equation This ( allows us to solve implicitly for Where boundary terms are settions to vanish by imposing the appropriate Dirichlet boundary condi- of strings ending on acorrect boundary pair conditions of at giant the gravitons, string end so points. onethe must form: be careful about imposing the We have chosen to write ittion, in which terms forces of anreduces expression to that a is complicated implied equation depending by a on an current integrating conserva- constant, which is the value As we will see, one can eliminate the angular variable can be easily evaluated via the formulas: where we have set the angular frequencies to one for simplicity. The conserved quantities JHEP09(2020)019 . ). 1 ,J that 1.3 (3.14) (3.15) (3.13) Z , with the β = 1. We can see r at fixed J ) ) 2 2 ) ) − dr dr dϕ dϕ ). Although the boundary 4 2 r 2 r ) 1.3 + ( + ( = ∆ 2 2 1 + 2  dr r r dϕ 1 + − )( )( |Z| − 2 2 2 + ( 2 2 β β β = 1 (that is, if the string touches the 2 π λ β r 4 − − 2 2 r + + β β corresponds (up to a factor) to the length 2 s + (1 (1 ) 2 Z ) − − 2 Z 2 Z dϕ q q dr dϕ β J 1 1 dr dϕ f ( 1) 1) i 2 ( ϕ – 9 – r p p yields the dispersion relation r ϕ − − Z λ λ π π = 2 2 2 2 √ √ − 1 |Z| r r ( ( J = = = − dφ dφ 1 and J Z J ∆ 2 Z Z − J λ λ π π ∆ 2 2 √ √ is the length of the segment in the LLM plane connecting the | = 2 1 ξ ) can be used to express the answer in terms similar to those of ( ∆ = J − iϕ ]. It would be very interesting if a class of solutions of the sigma model 1 in terms of ξ 17 | β exp( r = ξ It is also instructive to give explicit expressions for the conserved quantities ∆ Eliminating It’s clear that theseorigin quantities of AdS), diverge even whenever though the length of the string on the auxiliary plane geometry is finite careful that the radial coordinatethis doesn’t from touch the the expressions: unit circle given by giant gravitons in [ of this type do lead to integrable boundary conditionsSince for we the are dual considering spin openare chain strings finite. description. of finite size, However, one for would similar expect that reasoning these to quantities that of the previous section, one must be conditions considered here lead tosigma solvable model, equations it of is motion notcan for expected the that be these ground fully solutions state (and described ofbranes their by the many are an magnon expected counterparts) integrable spin tonotion of chain. destroy a that simple The Bethe property. boundary Ansatz effects point This of sourced view, has by for been the the open argued, spin at chains attached least to from regular the Which is precisely of the form expected from equation ( of the string onbe an identified auxiliary with flat thecoordinate 2D central geometry charge of with the aHere SL(2) disk we sector see removed, of that andtwo the as giant spin gravitons. such chain. should The complex To do the variation,endpoints we on want the to dualresults giant minimize in gravitons. a the straight This energy, line. is Clearly a the straightforward variable minimization of by some kinematic factors: JHEP09(2020)019 = 4 ]. It 4 N = 4 SYM = 4 SYM 1. This is N N > | r | BPS geometries, where 1 2 ]. It is expected that the = 4 SYM spin chain [ 30 N ]. 10 ]. More concretely, these lead to operators ]. It is unclear what the precise structure of 18 29 – 10 – = 4 SYM spin chain with open boundary conditions. N . We find a nice description of the solutions in terms of an analytic λ It would also be interesting to understand how such strings suspended between dual We also found that the LLM coordinates arise naturally from studies of the string sigma The description in the SL(2) sector is expected to be qualitatively different than the of the gauge theory,operators and can that be one matched should with make valid sure open that string the solutionsgiant corresponding on gravitons gauge the that LLM theory preserve plane. differentwould halves allow of us the to supersymmetry better would explore work out. how the This physics of the Coulomb branch of quadratic fluctuations around these solutionsSYM and spin could chain in . principle comparethe This to states analysis the can dual also tocharges be closed of extended open string to strings magnons general stretched betweenthe have giants been regions generically diverge studied where as [ the theirone branes should corresponding approach expect 3-spheres instabilities shrink that to are not zero captured size. by the This naive suggests perturbative expansion that bound and scattering states is without further study ofmodel the in field the theory SL(2) side. of sector. locality This in might allow theshould us also radial to expect direction that understand given of better these AdS how solutions, certain that arise aspects one from could the also compute dual the field spectrum of theory directly. One SU(2) magnons due toalready the clear nontrivial from boundaryscattering the methods condition sigma for imposed model the by SU(2) description, SL(2) sigma as model, model even constructing seems thoughation solutions to their of solutions require from the should solutions different inverse be in Bethe related the roots by SU(2) an than sector analytic the [ continu- in a continuum limit.the string It sigma would model alsoAfter corresponding be all, the to of open scattering interestthe strings to and Bethe suspended consider bound ansatz between of more states ordinary the general of giant SU(2) solutions gravitons giant spin have to magnons. chain, a as relation noted to in [ continuation of the SU(2)these sigma states model arise from andfrom a the shortening BPS conditions solution condition of haswould the on be a central the nice form extension string to that oftheory worldsheet, understand the in suggests this as more that better would detail. from be This the spin expected chain planar should SL(2) precisely sector realize of the open the string sigma model 4 Discussion In this paper we havetions provided of evidence the for SL(2) theThe all-loop sector calculation dispersion of is relation the done fort’Hooft in the coupling excita- the sigma model and it gives rise to a series expansion in the quantity is becoming divergentstates, should in become a way unphysical similarthat and would to inject lead the an infinite discussion to amountbecomes in of non-normalizable energy [ complex. into the bulk and where the radial coordinate as in the Hoffman-Maldacena string. Solutions like these, where a physical worldsheet JHEP09(2020)019 3 r P C (A.3) (A.1) × 4 AdS , which is the N = are the energy and 2 ]. The droplet is a = 1, and considering 2 i 26 , which is one of the 3 κR x E,L ], we find that the giant Z . A vacancy in the center 2 11 P κR , so that corresponds to a coordinate on a ), where 2 N θ r ) (A.2) 2 = 1 superfield notation. The ground − R 2 N L L N N R − ( 2 − κ r ( 1 + 1 ]. The latter should be quite interesting, as = κ r r 31 L = – 11 – = = = L r r R R E = E is a Sasaki-Einstein manifold, as they share the AdS part together with a phase 5 r X = 4 SYM can be described in terms of a quantum hall droplet = 4 SYM theory in an N N where 5 ] for more details). ) that results from writing a five sphere as X 10 iθ × 5 exp( (see [ AdS 2 ix r/R + = 1 ξ For dual giant gravitons we can use the same intuition. Now, a dual giant at radius In this picture giant gravitons are holes (vacancies) in the droplet, and dual giant Similar simplifications as the ones found here should also be possible on backgrounds x , an eigenvalue at the edge of the droplet will have energy = will have energy given by so that One can check that thedisk radius ξ other radius, a holeangular will momentum have of energy the giant gravitonThis (they are allows the us same because tothe of set the energies up BPS of condition). a giantgraviton gravitons. correspondence is between By at coordinates comparison a with of radius the the work droplet [ plane and gravitons are particles thatR are away from theof droplet. the droplet If is theamount a circular of maximal droplet energy giant has that graviton the radius with eigenvalue energy at the center gets when moved to the edge. For any The half BPS statespicture of in two dimensions,condensate of with eigenvalues a ofchiral a confining multiplets complex quadratic of chiral potential the scalarstate matrix [ of field the field theory is described by a circular droplet. The work of D.B.0011702. is supported in part by the Department of EnergyA under grant DE-SC Giant gravitons and LLM geometries with fluxes, related tothe the exact ABJM characteristic of modelof the [ the allowed field brane theory configurations set-up. depend greatly on the details Acknowledgments gravitons that preserve different supersymmetries has not been studiedof either. the form of the sigma model,allow and solutions the of Sasaki-Einstein a geometries similar have kind. a U(1) It would fibration also that be should of use to study the case of determines the dynamics of the theory in more detail. The analogous problem with giant JHEP09(2020)019 (A.4) (A.5) (A.6) Int. J. , 99 is the ground ]. 0 D (2008) 945 SPIRE Yang-Mills operators IN 12 ][ = 4 Lett. Math. Phys. N 2 3 ) (A.7) , 2 2 Ω x d ) + ρ ( ) in the bulk. Indeed, the coor- 2 1 geometry. These are a complete 2 ρ ]. x ( 5 ]. The LLM supergravity solutions x L N 2 25 d hep-th/0206079 AdS SPIRE + sinh 0 [ geometry. 1 + L D 2 N IN 5 Z r ][ dρ r S is the area covered by the droplet, and it is − + Adv. Theor. Math. Phys. ) × = x ) = 2 2 , 2 5 2 ρ ρ x – 12 – d dt (2002) 425 ) + D that is covered by particles, where ]. ρ R AdS 2 1 ( sinh 2 x D 545 ( ∝ x = cosh( Exact anomalous dimensions of 2 N SPIRE r limit of superconformal field theories and supergravity cosh d hep-th/9711200 of the droplet with cosh( ), which permits any use, distribution and reproduction in IN D − [ N r ]. ][ Z = ∝ dynamic S-matrix 2 E ds 2) SPIRE | Phys. Lett. B IN , CC-BY 4.0 ][ The Large Review of AdS/CFT Integrability: An Overview (1999) 1113 SU(2 This article is distributed under the terms of the Creative Commons 38 The arXiv:1012.3982 ]. For global coordinates, where geometry and the other one from the [ 5 32 S ]) hep-th/0511082 (2012) 3 with large R charge [ Theor. Phys. The excess energy of the solutions relative to the ground state is given by an integral A. Santambrogio and D. Zanon, N. Beisert, J.M. Maldacena, N. Beisert et al., 20 [3] [4] [1] [2] References quadratic in the coordinates of the LLMOpen plane. Access. Attribution License ( any medium, provided the original author(s) and source are credited. where we do an integral overstate the area droplet. We alsoindependent have of that the shapeWe of see the that in droplet. the A LLM geometry single the giant energy graviton of has a a particle removed fixed from unit the of droplet area. is also classification of half BPS states in the over the droplet area This identifies the coordinate dinates that arise this wayare are given the also LLM by coordinates dropletstry. [ on The a LLM plane, plane which isfrom can the the be degeneration completed locus to of a two ten types dimensions of geome- spheres. One of those arises This is equivalent to long rows [ a giant graviton will bein at [ a radius of the three sphere characterized by (see the discussion In the field theory dual, these are given by certain Young Tableaux states characterized by JHEP09(2020)019 ] , , , ]. , , JHEP , 05 (2008) (2006) SPIRE ]. (2013) 04 IN 39 87 ][ JHEP (2015) 036 , SPIRE JHEP (2006) 024 IN 12 hep-th/0610037 ]. , [ (2000) 040 ][ 09 ]. 08 J. Phys. A (2006) 13095 , JHEP SPIRE , IN Phys. Rev. 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