JHEP01(2015)126 is β , if Springer der which limit. We ) is a root eformations iβ January 1, 2015 January 23, 2015 : ondence December 10, 2014 → ∞ : : N 2 YM g 10.1007/JHEP01(2015)126 Accepted Published Received doi: th energy far below the string SYM d by operators in the field theory, ) close to a root of unity in a double Published for SISSA by = 4 -deformations, to understand their dual iβ β is geometric by studying the spectrum of X N are very important. When exp( β [email protected] , is a large classical geometry in the X symmetry, known as 3 . is an orbifold. When exp( corresponds to a finite deformation of the orbifold. Finally 3 in certain double scaling limits. We study the conditions un X X 1408.3620 -deformations of The Authors. β → ∞ c D-branes, AdS-CFT Correspondence, Gauge-gravity corresp

We study a one parameter family of supersymmetric marginal d , N [email protected] geometry, where X = 4 SYM with U(1) × N Department of Physics, UniversitySanta of Barbara, California, CA 93106, U.S.A. E-mail: of unity, the space these open strings can givescale. rise to The a large number-theoretic number properties of of states wi open strings between giant gravitonsas states, as we represente take ArXiv ePrint: AdS Giant gravitons and thelimits emergence in of geometric Open Access Article funded by SCOAP Abstract: David Berenstein and Eric Dzienkowski Keywords: scaling limit sense, irrational, sporadic light states can be present. of argue that we can determine whether or not JHEP01(2015)126 ]. 1 4 7 N 3 10 14 16 20 limit 2 YM g 1. Our N is a real urbation ≪ β 2 YM = 4 SYM [ g ]. The orbifold 5 , N ) and 4 [ iβ , and 5 S n a very special set of ns of y a classical, geometric × 5 ued that a parametrically → ∞ that in the large = exp(2 large geometry arises as the N guarantee the appearance of root of unity, then the above q eld theories are equivalent to 2 YM vatures are small, gravitational − rized by two parameters, g s itational states are very massive. = 4 SYM = 4 SYM theory. This is a continuous ]. In some of these theories, at large N 1 N lead to a well defined classical geometry in β – 1 – = 4 SYM when = 1 a primitive s -deformations where N = 4 SYM, protected operators and the q q N -deformations of β orbifold with discrete torsion of AdS s Z is fixed with × s q Z ], but finding examples where the gap can be followed from pert 2 -deformed spin chain with boundaries β , which is in turn a subset of the Leigh-Strassler deformatio β This question is too hard to answer in general. It has been arg It is known that if In this paper we examine this question of emergent geometry i SU(2) sector and strong gauge coupling, the vacuum can be characterized b large gap in anomalous dimensionsa for geometric operators dual [ is enough to correspond to classical geometries. supergravity background. In thisexcitations classical can background be cur treatedIt semi-classically, is and natural non-grav to ask what is the set of such gauge field theories In some contexts these are known as N quantum theories of gravity in higher dimensions [ 1 Introduction The gauge/gravity duality conjectures that certain gauge fi 5 The 6 Geometric limit interpretation 7 Conclusion 3 Open strings between giant4 gravitons Marginal deformations of Contents 1 Introduction 2 Field theory results for giant gravitons in cases. We consider the family of family of supersymmetric conformal field theories characte limit corresponds to a this limit. effective description of the undeformed number. The classical weakly coupled string theory on a very main focus is to understand which values of theory to strong coupling is hard. and JHEP01(2015)126 . e n ∞ , but = s nuous Z N × it relative bmanifolds s 2 YM . aracterizing g Z q deformations, / ]. 5 ] for a review). β is believed to be S 15 14 5 × S 5 -duality setups is by cle. We will show in × ase of T 5 r strings, but our goal will . If we think of these as bility program has solved etries, one should be able s it in terms of a spectrum try can become infinite in , it is necessary to make a t geometries than a sphere 5 se; the spectrum of modes d theory diagnostic of this rent values of boundary limit is essentially S re connected by a continuous ] (see also [ acterized by a gap in energies system arising in field theory , which are the Kaluza-Klein be continuous as a function of × valently, by having an infrared aic equations that need to be , see however [ 13 anslate into a large set of states e us a recipe for the emergence 5 – ], but only one such geometry can 11 6 to have the appropriate periodicities 5 ]. The different possible string theory 1 S string theory on AdS . This limit can be considered as a boundary – 2 – → ∞ ] and thus in principle is solvable by a Bethe ansatz N classical 9 – integrable system [ 7 2 YM 5 g S × 5 , and the geometry depends only on integers, the volume of the quotient geometry depends only o , we think of the system as a stringy geometry, since the spher 5 S N , show that the emergence of different geometries is disconti q 1, but that does not mean that it is still continuous at s, t 2 YM approaches a root of unity in a particular double scaling lim g ]. D-branes are geometric objects sometimes described as su ≫ β 16 N . Such behavior for a physical quantity on the unit circle, ch ), for t 2 YM g πit/s , then we can get a classical geometry which is not just AdS parameter. After all, the rationals are dense on the unit cir ] for a recent review of the integrability program). For the c N β 10 The spectrum of closed strings in Naively, just like when one studies T-duality for torus geom For the purposes of this paper we will assume that the integra Notice that the different sphere quotients have very differen An alternative route to understand the origin of geometry fo 2 YM g = exp(2 , at fixed , but not on controlled by an integrable system [ the possible values of between the Kaluza-Klein scale andscale the that string is scale, very or different equi from the string scale. to understand the appearanceof of closed a strings large thatexcitations volume are in geometric much the given lighter lim geometry. than This the spectrum string can be scale char in the phenomenon for the particular theories we are studying. of parameter space. What wefractal. are saying The is purpose that this of geometric this paper is to find a very precise fiel we know that theresults string theory as is a integrable; twist the of integrable the AdS q discontinuous jump, even though theparameter. two dual In field string theories theory, a theβ spectrum of strings should give different answers. To go from one geometry to another one As such, tools relatedof to geometry. integrability Unfortunately, can this does inthat not principle are automatically giv tr easily followed;solved to the find corresponding these states twisted are algebr hard to solve for analytically (see [ geometries are connected tobecome each classical other at by a T-dualitysize given [ in time. all directions That when is, we take only one such geome s group only acts on the and they are alsocomputed different on from them each other by in restricting the states spectral on sen the this paper that if is of finite volume in string units in this case [ For finite values of to rather a supergravity deformation of it. using D-branes [ the problem of understanding the geometric origin of AdS be to understand the other geometries that can appear at diffe JHEP01(2015)126 . β ] and com- 18 ]. This provides 22 can be obtained by 5 be extended in some strings, we can check ll show that the spec- S ition of distance. So if persymmetric D-branes m of states for various rings at describing local × wist affects the boundary fied boundary conditions, erve, these giant gravitons 5 her. Indeed, the properties arge, and that a volume of xtension of the spin chain. ular position, and a radius. e volume limit in the closed ngian. With D-branes, they es with respect to the central uble scaling limit exists that dS nown energies. gma model. Basically, we are eed, any twist can be undone string scale). If an extended re sensitive to such a redefini- can also have pairs of parallel en them, testing if they can be such localized states up to two erstood recently [ r gap between light states and es can be constructed and that at they can have a well defined rane also become light (they are onstructed in [ c questions are easier to engineer. ation by having an S-matrix whose ], which remain BPS states for all 17 ]. Because these states are BPS, we will is not exactly a root of unity. 21 q , 20 – 3 – ]. An all-loop conjecture was also suggested in this work, 19 ] for open strings rather than closed strings. 2 2) spin chain [ | (2 su , namely half BPS giant gravitons [ 5 S × 5 In the third set of calculations we will explore this spectru In the second calculation we will examine precisely how the t This is the route we will take in this paper. There is a set of su For practical purposes, D-branes are usually better than st In this paper we will do three main calculations. First, we wi truncating classical solutions of the infinite open string c we will still haveConsequently, we BPS will states derive a relative rich to spectrum the of central states with chargeconfigurations k e and under whichstill conditions leads an to appropriate a do large geometry, including when conditions for the spin chainlocally induced by by a these field D-branes. redefinition,tion. but Ind the This boundary means conditions that a if we understand the details of the modi in AdS information. Closed strings usually describelow local energy inform limit canare be already eventually localized converted in to some directions, a so local local lagra geometri they are indeed BPS. addressing the gap problem [ trum of open strings stretching between giant gravitons indirect A evidence at strong coupling that the open string stat putations of the spectrum ofloop strings order stretching were between two performed inwhere it [ was argued that thesecharge open extension string of states the are BPSassume stat there exist dual strings with the same energy in the si Once we have chosen whichare half characterized of by the coordinates supersymmetriesThe which to dual pres live states on for a these disk: localized giant an gravitons ang were und the D-brane is becoming large. stringy states characterizes the emergenceD-branes and of study geometry. the We spectrumconsidered of strings to stretching be betwe closeof to the each other, spectrum or ofwe far such have apart open such from strings D-branes eachboth can ot and that be can a used follow transverse to the direction infer spectrum to a of the defin such D-brane open is becoming l directions, and sharply localizedposition in that others (in is the sensitiveD-brane sense becomes to th infinitely shorter large length instring scales theory, volume a as than collection we of the take openKaluza-Klein the strings excitations larg ending on of the gauge D-b fields, etc.) and a simila of a given geometry where strings are allowed to end. They can JHEP01(2015)126 : . s 5 5 ars S S . This ]. Giant Y 25 theory was coordinates PS configu- and 2 . ] it was shown 5 X , x 1 26 S x entum. In a quantum nent of a single chiral o go beyond operators SO(6) of the R-charge. xactly computable. ⊂ field theory computations g that they do not carry on. The tay at fixed radius. ], equivalently described by lar variable. To define ob- states can also be described 4) solutions. 3 irs of coordinates on the droplet. In [ 24 y equal to one (this is always ~x S [ trings between branes. Such a antum hall system [ = 4 SYM y the role of a coherent back- eds a superposition of angular e set of states to have more than ], where it was argued that they N 23 only. We think of each component = 1 (2.1) 2 6 X x ]. These states are spherical D3-branes + 17 is bounded below by zero. On this disk, the 3 ··· S – 4 – . Afterwards, it was understood that any gauge + X 2 2 ]. x 30 + – 2 1 ], where the spectrum of strings was described in terms 28 x is half-BPS to all orders and that a complete orthogonal coordinates in a polar decomposition, with a radius and a 19 6 , , X 5 , 4 18 , as the collection of unit vectors . These states are parametrized by a radius and an angle. In 3 5 5 x ] is that an open string is represented as a product of matrice S 27 with fixed volume and rotate with angular momentum in the 5 S . A general recipe to attach strings to giants in the dual field = 4 SYM. We will label the lowest components of our chiral scal ⊂ , with no trace taken, thought of as a word in the letters X ]. Strings are attached by adding boxes with labels to a half B 3 N and construct our BPS states from S 27 , etc. This identifies the SO(6) R-charge as rotations of the 2 ...Y 1 ix n + 1 YX x Half BPS states are constructed solely from the lowest compo The dual half BPS states were understood first in [ The basic idea in [ The states described by Young tableaux have fixed angular mom To add strings that stretch between giant gravitons we have t X,Y,Z = ∼ X as corresponding to a complex combination of consecutive pa by three sphere. This is the three sphere we choose to wrap the belong to a disk, because the radius of the half BPS giant gravitons movemeasured with relative constant to angular the velocit scale of the AdS geometry)scalar and field they of s we can choose to write the These states preserve half of the supersymmetries and an SO( a parametrization of the They are mappedangular to momentum chiral in AdS primary scalar operators, implyin invariant operator built out of ground for field theory computationsformalism in was setups developed that in stretch [ s of an open chain with boundary conditions and wasW found to be e ration made of Youngin tableaux. these Techniques states to were deal developed with in basic [ system, this means thatjects they that are are delocalized localizedmomentum in in states. the the dual dual These angular angu geometrically variable, localized one states ne pla gravitons that wrap the spherethat are general hole fermion states droplets in can the be fermion mapped tobuilt supergravity only out of Young tableaux and free fermions. Thisone lets giant us graviton. generalize th The nextin observation terms made of was droplets that of these free fermions in a two-dimensional qu were sub determinant operators built of wrapping an Giant gravitons were originally discussed in [ described in [ basis of these states can be obtained from Schur polynomials 2 Field theory results for giant gravitons in JHEP01(2015)126 . 2 ]. Y ix 32 (2.4) (2.5) (2.6) (2.2) + 1 − ins can are the 1 k (2) spin s lead to x π (2) sector su ∝ ... su is restricted y spin chain λ z ) + 2 a amics of the giant itons − ]. It is also expected 1 omagnetic a 31 ( sites is a tensor product  k † 2 respectively, and they end e subject to a Gauss’ Law. ricting to the SU(2) sector a Y ˜ ξ coherent state is defined by k − lattice sites, there are in a simplified setting in [ spin chain of Cuntz oscillators n giants. pace in terms of an occupation ξ, † 1 k a .  ˜ !# ξ expansion [ ...X k + m 2 a +1 n . This linear system of equations is dependent rescaling can be put into a i  /N + − z 1 N ξ 1. The ground state of the spin chain is N a ) − YX ∗ √ N 1 i at each location of the chain. Because the ˜ . The conventions on factors of λ / − m n + 1 z √ ∗ λ | ≥ . Thus, we label the states as = 1 (2.3) ˜ λ k z ∗ − N = | ] † Y λ i YX ]. The one loop Hamiltonian in the – 5 – √ ! = z 19 aa † k ≃ | ↑i + 1 = 19 ˜ a ξ − k i   Y ( 1 k − † 1 = ], and by an − a i = z N 18 ˜ λ − [ and +1 m √ k , . . . n z z 2 N N

(2) one loop open spin chain with λ √ √ , n + su 1 ≃ | ↓i and n  | X N N 2 π √ . In this setup, the one loop Hamiltonian for the open spin cha . Attaching various such words to a single giant graviton doe 8 / 2 YM ]. a coordinate in the complex unit disk. The domain of ∗ g Y W λ 34 z = = ]. These are described by the deformed oscillator algebra and letters between the different ξ 33 loop with X X are the complex collective coordinates of the two giant grav − i 1 = ˜ = 0. For general giant gravitons, the one and two loop boundar λ z | H 0 ] where we take i z z 7 λ, 0 | = a The ground state of the The usual description of the SU(2) sector is in terms of a ferr i letters can jump in and out of the word when we consider the dyn z | They live on a disk of radius up being rescaled complex conjugates of the same as those in [ where of Cuntz oscillator coherenta states, one for each site. Each where chain [ X disk of radius one. The normalized coordinates are called for the spin chain Hamiltonian is and graviton, it is betternumber to for describe the states of the Hilbert s That is, the sites are linearly interpolated between the two Hamiltonian was computed in detail in [ letters in the word for the more general caseThe where counting of the these states restricted in string the states Fock space was ar produced an approximate Fock space of open strings and a 1 completely solvable with so they are described by a ‘boson’ at each lattice site. For spanned by word is then attached to a configuration of giants. We are rest be described in terms ofat raising each and site lowering operators [ for a because the states are not normalizable for characterized by the following equations [ JHEP01(2015)126 , on X p (2.8) (2.9) (2.7) , just (2.10) (2.11) ), which n ip ethe ansatz ground state exp( is the half-BPS ted from a square ≃ . The relative cost .... J 1 1 ates in terms of the − + − n he giant gravitons are n 3 finite, this background i 4 | to do is check that our (2) spin chain are BPS. dS radius units. Their on from a Bethe ansatz ]. The central charge is r the centrally extended ˜ k /z ξ n +1 su 21 z + 1) − t en them are short massive , 2 ξ k | | finity by taking the effective ( ˜ metry algebra can be sourced ξ 20 tral extension is characterized → | fixed in position), the result is , with 2 −
1 for parallel branes in flat space. ... 4 − ξ defect to the left, we need to take N π | n + i → ∞ defect in the background of the i t Y N 2 | i 2 YM 128 spins can jump to the left or to the π g n N Y | 4 2 loop YM n -th g X − ˜ z ξ n + 2 -th + 2 − H n | h 2 ˜ X ξ ξ z + + 1) – 6 – − ], bound states in the N i k + 1) ξ ( | Ξ = 36 = k ( loop i N 2 − z r 1 | π . When we take 8 2 YM ]. Acting on an impurity at fixed quasi-momentum H = g h X ). In the field theory this energy is the dimension of an J to a given infinite spin chain, first on the left and then on the X,Y 2.8 − [ from the right of the X ∆ , where ∆ is the operator dimension, and → + 1) + + 1) + 1. For closed strings the central charge is confined. However J k k X Y − − number of ) = ( = ( ip N J − and at the same time, we need to take ∆ n ]. At this order, we have that i 1 has a unique well defined momentum in the coherent state. In a B is a normalization factor. Now, when we look at the spin chain 35 ] it was argued that the all-loop energy of such a string resul . To jump an − z Y Y s 19 N 2) symmetry of the (infinite) spin chain discovered in [ → | In [ There is a subtlety when computing the ground state energy. T One may also try to understand the central charge constructi To do this, we first need to expand the coherent state ground st | n (2 i s and specifies the energy ofby the adding correpsonding or removing state. one This cen consistent with their result ( su exactly given by operator. On theargument was gravity that side, this this open is string appears an to energy be measured a in BPS A state fo root expression right; basically it sends multiplets. The shortening of representations is standard perspective. As discussed by Dorey in [ like in flat space, theby central D-branes, charge extension so of that a the supersym open string states stretching betwe a ferromagnetic ground stateproportional (keeping to all exp( other impurities This is derived fromground a state Bethe has a ansatz similar calculation. local form What to we a want Bethe ansatz. can be interpreted as the momentum of the so each in amplitude for these two components of the wave function is built from an order R-charge [ | right of in the interior (away from the boundaries) the contributes infinite energy.Hamiltonian It is to standard be to ∆ remove this in occupation number basis where JHEP01(2015)126 Y with (3.2) 5 S × . Like their 5 tions for the 5 S correspond to the i ] for conventions and J happen if the relative solutions of the string atz sense. This should 11 ] called giant magnons. satz to be these classical ching between two giant isk in the to confirm the conjecture 36 is the R-charge of the half- [ 5 The corresponding classical 1 S pect these strings to be BPS in chains are also BPS, their J etry), they should correspond lutions are cut at the location the giant gravitons. ons, albeit modified to include al BPS strings satisfy a simpler 2) (3.1) ) = 2 (see [ p/ ( 2 2 ]). There, the ik = 0 (2.13) sin n 38 = 2 (2.12) z 2 is the geometric angle subtended by the λ 2 π p n +1 − z n + the Cartan generators of the SO(6) R-charge, ) + exp( z 2 i 1 +1 and J – 7 – n Q + ik z 5 1 − r Q S n + − ] (see also [ z n = = 1 z 37 − 1 2 n , ). This shows that the local structure of the one loop J J z -matrix between defects. The scattering matrix of two 22 − 2.5 S ∆ = 4 SYM with large angular momentum in the infinite chain limit with fixed angular momentum on the N has a pole at exp( 5 S 2 × , k 5 ], we can conjecture that the corresponding open string solu 1 ]. To do so, we need to solve the string sigma model on AdS k 22 19 the number of magnon excitations, the quasi momentum of the bound state. In particular, Q p Since these bound states have been identified with classical The BPS states in BPS ground state whichstring is infinite solutions in were the found infinite chain in limit. [ spin chain dual, these stringsset have infinite of length. equations These that du dual the string full states set of shouldthe states. also effects satisfy Since of a our the simple openstring new set sp solutions boundary of but conditions. equati truncated Thus so we that take the our endpoints an fall on proposed in [ sigma model [ are bound states of elementary excitations as shown by Dorey which is equivalent to and which is exactly the equation ( angular momenta of the string on the Each state is characterized by the quantum numbers 3 Open strings betweenIn giant this gravitons section wegravitons want in AdS to compute the energy of a string stret open strings are locallyof the the same, giant but gravitons. where the We will classical show so this in the next section. wave function is thatpersist of to a all bound orders. state of magnons in a Bethe ans details). This translates to the equations string stretched between two points on the edge of a maximal d with momenta boundary conditions given by the(they giant are graviton. expected Since to be weto ex short solutions representations of of the supersymm sigma model that have this property. momenta are at a zero for the state, we would expect that such ordering of momenta can only with JHEP01(2015)126 = 1. (3.6) (3.4) (3.3) (3.7) (3.5) ], and z assical i ¯ z 25 i , . They are P 18 α, θ , and boundary to the collective 5 S ]. In the dual CFT, by 17 p [ en spin chain. Then we d to make sure that the he conjugate variable to spin chains in three parts. e radius one [ rotate with angular speed coordinate), then it moves e at the edge of the disk. in the vitons. ve with angular velocity one 1 erent states of the dual field the ends follow the positions ry variables z L/N herwise we could not compare 2 − om the giant graviton). Lastly, | ˜ 3 with the constraint 1 ξ ,  .  2 p − 2 p
2 p , 2 ξ  |  | = ξ 2 | = 1 and the positions of the giant gravitons. π λ 1 cos sin i 4 − | z ˜ ξ | . If we have a giant graviton with angular 2 2 , i 1 i r r + z J r r ξ, and quasi momentum 2 2 2 − N Q 1 + 1 – 8 – Q (it is a point in the = r Q 5 ], we recall the classical infinite string solution. The L ) = ) = . The time on the world sheet is identified with the = = S θ α 22 1 2 = J J t, x E tan( cot( − for the truncated solutions and get the answer conjectured inside . Furthermore, both notions of the angle direction between ∆ 2 | 3 1 J z S | = and | 1 plane. This follows because the ends of the infinite string cl ξ | J 1 z − , wrapping an L are coordinates on a unit disc. ˜ ξ is a function of the energy, angular momenta, quasi momentum , we see that L r ξ, ], namely First we need to find a relation between In order to impose the truncated boundary conditions, we nee We verify that these truncated solutions are dual to the open Using the variable definitions of [ 19 related to the magnon excitation number in [ we need to relate ∆ coordinates of the giant gravitonsthe in energy the open and spin angular chain.need momentum to Ot of check the that string theof infinite to string the that can giant of be gravitons the truncated (this op so is, that the string ends don’t fall fr Let us label the coordinates on the sphere by where external time variable. The solution depends on two auxilia string solutions can endequal on to the one giant in gravitons the and that they worldsheet coordinates are called in terms of the descriptiontheory, based these on also fermion sit dropletshave on and energy a coh disk which can be normalized to hav First we need to relate the positions of the giants gravitons The corresponding angular momenta are the where field theory and gravityangular momentum. are the same; the angle is related to t with angular velocity equal to one and sits at Fixing momentum solution travel at theTraveling at speed the speed of ofand light light in if and this we case moreover cut means the they that solutions, they resid they mo actually end on the giant gra JHEP01(2015)126 are (3.11) (3.13) (3.16) (3.17) (3.10) (3.14) (3.15) (3.12) (3.20) u, x to have the ˜ ξ is constant. 1 ξ, z ) (3.18) tricting the range u ( 2) 2 ) and then solving for 2) . p/ it p/ )) sech he D-brane configuration. 0 | . We also have the relation ˜ cos( u  ξ 1 exp( dx ) sin(

˜ . Consequently we have u ξ − 1 − i ˜ ξ . ξ ˜ ξ Z , u ˜ Z ξ ξ ) (3.8) ) (3.9) | 0 t ) ) (3.19) tanh( = 0, we choose α α ∂ u or α u s then given by
) = 1 ) = ) = ( t 2 tanh( − ¯ 1 0 ξ ) = 2 , t Z Z 0 ) 0 u u )) =  t = 0 for convenience, where 1 u 0 = 0 the real part of u ∂ 2 p ) cos( ( u )) sin( )) cos( u 2 t t sech 1  θ θ θ ¯ Z z − ℑ . tanh( tanh( du 1 1 sin tanh( ℑ   J i tanh( sinh( sinh( (tanh( cosh( Z 2 2 p p t − + , dx dx − x   constant at    ) and – 9 – ) 2 2 2 p p p  − it 1 2) − is infinite. For our case, we want to cut the string 1 Z Z ) 2 p z   
) sin sin 2) ) u u and the values of p/ u 2 p θ i i λ λ  θ π π ), the string moves at constant angular velocity equal p/ p 2 2 exp( sin sin sin √ √ + + ξ 2 2 2 cos cos(   sin r r r = = cosh(  sin( cosh( 3.10 r r r 2 2 p p cosh( (tanh( − i 2 1 2 2 2 it iv t x    ) = J J ξ e e 2 p 1 + 1 + 1 + , t −  = = ( = = ( 1 for that matter) is finite. λ λ λ cos cos π π π u 1 2 ) = 2 2 2 v u ∆ √ √ √ u ( z z 1 sin 1 u is found from the real part of z = = = p (or J x − tanh( ∆ are enough to determine are auxiliary functions. Notice that at ˜ ξ ξ, u, v . Since we need the real part of 1 As we can see from equation ( The energy and the angular momentum are given by After some lengthy algebraic computation, one shows that To find our string, we need to fix the boundary conditions by res , u 0 where The quasi momentum to one. In these solutions, the range of over the appropriate range. This is evaluated at so that conditions. The classical solution for the infinite string i which will be imporant when computing ∆ same real part, whichThe can boundary be conditions achieved now by read a global rotation of t u of the variables so that proportional to each other. so that the range of JHEP01(2015)126 - N × (4.2) (4.1) . (3.26) (3.27) (3.23) (3.25) (3.24) r N = 4 SYM ant is de- are N ˜ Z  3
˜ Z ˜ 2 − Y, ions that preserves A + 3 ˜ − X, s to cyclically replace ˜ Y . 2 + dimension of the fields ) (3.21) + )) A u . 0 3 ( ) u 2 2 2 ˜

x X without tildes for the lowest ) ) ( are essentially identical. Call  ential 0 0 X 2 g manner: u u sech 2 ) (3.22) F J | Tr tanh( 2 ˜ ), and then u 3 ξ ( θ dx λ − 2 N − ) tanh tanh , this determines the value of and + Z 1 ξ 1 ) + | = 1 SUSY notation, the 1 ) u − − X,Y,Z  x 2 sech J α 1 1 ˜ ( Z , u π λ N u u 0 − 4 X du o ˜ + Y θψ Z (tanh( ) cos( 2 2 p, u θ   ˜ (tanh (tanh X, √ Q 2 2 p p – 10 – = 1. It follows that   . = 4 SYM, protected operators and the   r 2 2 p p n 2 2 − | ) + cosh(   ˜ A ξ = x sin sin Tr  ]. If we use N ( 1 2 2 2 3 2 p − − sin sin J λ X r r

 ξ r r 2 + | dependence, ∆ 2 2 − 2 2 + − − in the adjoint representation of the gauge group. We will A angular momentum is given by r  ! ! sin ) = ˜ 1 1 Z ∆ ˜ 2 2 Z λ λ π π λ λ λ i π J λ π π π 2 2 √ √ 4 . Since we know 2 2 2 ˜ x, θ √ √ √ Y by requiring that the beta function of the gauge coupling con ( ˜ Y,



Q ˜ 3 ˜ X = = = X, = = = = h ˜ 2 X, , λ 2 2 J 2 2 J Tr J 1 are arbitrary complex parameters. The gauge coupling const − λ , λ 1 3 2 ) = λ J , λ W 2 − . One then sees that 2 , λ r . Because the superpotential has enough discrete symmetrie r (∆ 1 2 ± = 4 SYM admits a three parameter (complex) family of deformat λ 1 N Notice that apart from the SU(2) sector = = 1 super conformal invariance [ ± stant vanish. ThisX,Y,Z amounts to the vanishing of the anomalous termined from where The three parameter family is characterized by the superpot So we find that The as expected. 4 Marginal deformations of A We now want to fix Similarly, one finds that the N has three chiral fields be interested in the case where the gauge group is SU( matrices of superfields. We will reserve the letters component of the corresponding superfields, in the followin JHEP01(2015)126 , ¯ θ ]. + h 35 = 4 θ, F (4.4) (4.3) sym- ( θ nation N 3 -charge + orbifolds ]. At one R α = 4 theory 39 ψ has N ), and in this an not control = 4 SYM [ ≃ ˜ X iβ α = 1 corresponds N h W e case where we turn he dimension ∆. We SU(2) sector. This is umbers (this necessarily = exp(2 charge. The h form i are identical and they need original spin chain of eory and that moreover has which for R ds we use (this convention requires the alous dimension is enough to ˜ charges. This is depicted in Y ˜ ent paper. This is because, as 1 Z equal to one. In that case the 1) ˜ Z 3 ]. In that case, planar diagrams eory. ˜ ˜ = 1 were obtained in [ YM 5 Z X ˜ g | 6 Y, , 2 respectively, and  h 4 0. / alone, | ˜ = Y, 3 Tr ˜ . X, 1 ∓ h  ˜ Y 3 λ = 1, we call ˜ J > X, = 0, the superpotential has a U(1) − ˜ be equal to zero. The study of general X . This is easy to obtain by requiring that h ˜ X,Y − 3  X, ∗  λ . This SU(2) is part of the SU(4) R-charge ˜ X h Z 2 YM Y Tr ˜ Y G θ ˜ – 11 – 2 X d  into -charge of Z . When ) = 2 Tr R 3 2 . When X h | charge equal to h h | , λ , R YM 2 G = 4 SYM theory itself [ YM , operators with these quantum numbers can only mix with symmetry charges of the theory. In our conventions the , λ g (1 + h 1 = 3 N in analogy with the corresponding charge for = 0, and in that case λ 2 YM J W 3 g λ = 4 SYM. This can be conveniently understood by noticing that can be independently rotated from each other, a linear combi = ˜ in the field theory have ∆ Z N , the anomalous dimensions of 2 ˜ . It is convenient to rewrite the superpotential as follows λ X . For the particular case of roots of unity, these result from X ˜ 2 Y, λ . We can classify our fields by the U(1) ]. Results to various loop orders when in terms of YM → α ˜ g X,
11 ], the model fails to be integrable. For us this means that we c is a function of ˜ ¯ Z ψ YM and = 11 / g ¯ ∂ 1 fields to be canonically normalized). We are interested in th → are the lowest components of the superfields YM λ 2 θ . For the conjugate fields we reverse all the charges, except t YM G ˜ Y 1 = 4 SYM. For general G + Let us identify the U(1) In this paper we will be mostly interested in the generalized X,Y → α N 1 ) ˜ SYM theory [ X,Y,Z is obtained when case metry, where with discrete torsion of the of these rotations, combined with spinor rotations is the U( beta function vanishes. Thisdetermine requirement of vanishing anom has net R-charge equal to zero. Also, for the vector superfiel equal to one. Thus, a superpotential term in the action of the variables of superspace have U(1) where table D each other as thereincludes are the no bare other dimension states of with the the operator). same quantum n the one loop planar anomalous dimension of to be set to zero. This requires the on both the corresponding spin chain can be obtained by twisting the of although rather interesting, isargued beyond in the [ scope of the pres also introduce the charge Fields other than the results of calculations to all orders in perturbationspanned th by gauge invariant operators made of loop order one has that match those of the to a sector ofan operators SU(2) that symmetry is of closed rotations under of perturbation th X JHEP01(2015)126 . . J X − . Any . X 2 , and the Q n 3) fields, which / se of the half- superfield can for some other Y U + (2 2. There are not 1 ¯ DU Q m/ = 0. These equations = 3) ) /
the single field
J theory limit by studying SU(2) sector. These are o such ˜ ˜ X = 0, of the same class as r all, these operators are X 1 1 0 1 0 − nside a fermion droplet of = 1. This means that this + (4 is a primary field, invariant 0). We also have that these make up the operator. This J uch giant graviton or many. | f ion is entirely controlled by R
available: the operator has a ¯ (∆ h Df race structure of the operator. | − ˜ Q ates as analogous to dual giant e fermions on a magnetic field). . These are multitraces of X space variables and requiring that J > 3) X / f , so − 1 1 1 = 0. Thus, this subsector does not ∆
3/2 ion of the collective coordinate, it can also -1/2 ˜ J X = (1 − J f symmetry and the dimensions of the fields. 2 2 ] show that the collection of these objects acts without changing the value of ∆ 2 / 3 will count the number of 0 0 0 / ] can be adapted to this problem without 1 25 ¯ 1 D − J U(1) 18 − 1 – 12 – 2 2 / 0 0 0 / 1 1 − U(1) , not only those where , then the R-charge of such an operator is h n R 2 charge of this operator is equal to is a descendant of the form 2 / 2 1 1 1 /
3 3 ˜ − X U(1) f . The operator ∆ θ ψ Z Y X Field ], and then the arguments in [ X,Y 2. All of these operators have ∆ = 4 SYM that would be built out of that is gauge invariant is identical in algebraic form to tho 24 n/ N X . Also, the U(1) that is not chiral. Remember that m 2 charge, or that . U N R . List of charges of the fields under the U(1) √ charge is 1 A particular subclass of operators is the set that are made of The set of half BPS operators can be classified in the free field As described before, we are interested in the analogue of the This is the radius of the droplet for a particular normalizat 2 we will call The right hand sidethe would ones need to we be are an studying. operator with But ∆ these operators are chiral, so n is also true for arbitrary values of give a non-zero rightprotected hand dimension. side. This Only is the independent first of possibility how many is traces entire sector is protected by supersymmetry, no matter the t If the degree of the polynomial is operators built of We can also interpret the correspondinggravitons particle and and giant hole gravitons. st Also, Configurations the can collective have coordinate oneany formalism s modifications. of [ Holeradius states will be associated to a point i can also be visualized in terms of a quantum hall droplet (fre under half of thethe supersymmetries, U(1) and therefore its dimens be rescaled to be of order 1. U(1) BPS sector of superfield Table 1 For other fields theysuperfields are have determined definite by charges. the We charge have of also defined the super operators can not havethe lowest any component anomalous of dimension a scalar either. chiral superfield Afte mix with any other (all other sectors have fields with ∆ polynomial of imply that either the lowest component of the multiplet Young tableaux [ JHEP01(2015)126 ) ]. − Y ≫ iβ 35 i = 1. (1 (4.6) n chain defect − h tion in by one. P Y described J )), so that ≫ . The only k = exp(2 2 − ]. The main n = 4 SYM, to h N k ) + 2 = q 41 √ N , iφ + − . A single 40 ··· J , and U(1) 2 , namely, its R-charge . We are interested in − i 21 n J Y 2 n mixing to occur). This , and not just for . In the planar limit, at q i h − ic limit where we take ation invariance of the q icle excitation in the spin ) + exp( + ct can be computed locally y conditions of the various ) (4.5) 1 − iφ ) for arbitrary ) (4.7) n contributes to ∆ s of Y φ . β 1 e translation invariance. Thus, k ( +1 q i n Y ( − 2) E q i . Such a field contributes also one 2 0, φ/ Z = = exp( charge than ( ¯ φ/ i 2 ψ ...X ) to the energy. exp( ( α → R φ 2 2 ], so one can guarantee that the planar steps to the left or to the right, so the φ is − n ( sin ≃ s N 2 12 ] sin E i N n 2 [ YX 2 YM N 1 a π 2) = 2 G 2 n YM π g 2 YM φ/ X ( g – 13 – = 1 in a variety of ways [ 2 labels matter and serve to measure the distance . We expect therefore to have a dispersion relation i i 1 + Tr( h 3 for example to ensure that we can stick to planar ] q n i 1 + r n and when it exists. This requires that [ − ]. One should also notice that this superpotential can r a a > O ] 11 +1 i i ) = , the n q [ φ X ) = , essentially because of the cyclic property of the trace [ can move at most ( Y defects are dilute, and we can treat them independently of φ = with X ( E Y i Y E φ /a O ≃ 1 fixed and very high occupation numbers . However, it has a smaller 2 k N , the . This is in accordance with the fact that the associated spin s ], the quantity 4sin β ∼ )) is determined by the equations of motion. The same calcula 2 i ) is a ferromagnetic ground state with minimal ∆ 40 iφ n − n for the operator -deformed theory coincide with the planar diagrams of leads to − β φ ), where each defect contributes X J P β 2 to U(1) φ / − ( 2, rather than one, so it can not mix with the operators we have → exp( / E φ − P and 1 . This has been answered for = J ))(1 YM − J g iφ We will first consider single trace operators in the asymptot An important question for us is what is the form of − . We can take in the planar limit with the value of ∆ is just a twist of the original one [ to ∆ each other on arelative first to the approximation. position of The the spin chain has transl is an impurity. In the free field theory limit each defect carries some quasi momentum equal to candidate that also carries positive values of U(1) diagrams. In this limit the each given loop order between the defects. At large separationit we is have natural approximat that asymptotically we have that all orders, up to the point where we care about the periodicit chain from so that one effectively shifts the quasi momentum of each part k ∆ is equal to 1 energy (this is the sameand should as depend the only anomalous on dimension) of a defe be written in terms ofdiagrams a of generalized star the product [ result is that When there is more than one The vacuum Tr( In the notation of [ exp( (all three conserved quantummeans numbers that would the SU(2) need sector to survives match as a for sector for all value many other operators that carry those quantum numbers for ∆ and the presence of JHEP01(2015)126 ], ), (2) 11 N (5.4) . (5.3) ation su . ) -charge Y ) (5.2) by these ∂ 1-loop +1 X H n ∂ ]. The way ∗ a is a factor of q it corresponds ) 19 − h Z ) and not U( iβ , . . The answer is X h − N ∂ )) 18 Y Y ∂ ∂ n [ g energy of the cations to exp( X ∂ )Tr( − ∗ ows up. In this form the q n ivergent anomalous dimen- alous dimension in the pla- a s based on the sigma model or arbitrary ) − rability for very short closed qY X r arbitrary iβ − X mounts to adding phases to the ∂ e assumed captures both sides of Y XY ∂ (exp( )( Tr( ) (5.1)  − , for an open string whose ends attach +1 +1 )) n † n qY X X Y a a ). Additionally it was shown that this state ∂ ) ∗ − h X 2) symmetry survives that would protect the iβ | ∂ ∗ − XY in the equations of motion of (2 q – 14 – XY units of momenta. n − a exp( su ). This is directly derived from the superpotential. = 1, there is no integrability at one loop level [ n ( X ), in which case the anomalous dimension vanishes | 6 ∂ − iβ YX  Tr(( h | Y † n ∂ +1 a XY → → n † ) and arbitrary )( ]) -deformed theory with gauge group SU( iβ ] it is easy to write down the closed spin chain Hamiltonian ha . This energy is interpreted as a dispersion relation for a Y X β − = exp(2 11 XY − qY X X , ∂ h − n † Y a ∂ exp( ) is obtained by replacing ordinary commutators in the dilat  ][  XY 5.2 ]. For general n n copies of X X 43 Tr(( ]. Later on we will only be considering operators with large R X,Y n , N 2 → N 45 2 42 π ]) π Tr([ 2 YM ] it was argued that the only state affected in the SU(2) sector )[ 8 8 2 YM X G g -deformed commutators. That is 44 , ∂ q -deformed spin chain with boundaries XY Y = = ∂ β ][ loop − 1 X,Y The Hamiltonian ( To add the boundary conditions, we follow the calculations i Our goal in this paper will be to understand the correspondin H . At this loop order, only the square of the superpotential sh Tr([ The added term yieldsnar an additional limit contribution only to for the the anom operator Tr( there is an additional term that needs to be added. Because we are working in the to the XXZ chain. the second line is specific to Hamiltonian is also a sum of squares, and in the SU(2) sector f sion for Tr( and so these finitestrings size are corrections not unaccounted a concern for to by us. integ We dothis not works need with to the make giant any graviton modifi collective coordinates a finite size effects, called prewrapping,is is protected Tr( to all loopthe orders. AdS/CFT Integrability, correspondence which we in hav the planar limit, predicts a d operator by for the one loop SU(2) sector in the Cuntz oscillator basis fo and one can not argue that a deformed h result on the right hand side. Roughly, the cost to switch can be found in [ fields on the spin chain. A calculation for those giant magnon fluctuation between the D-branes with to a giant graviton made of entirely. In [ 5 The Following the calculations of [ ground state with JHEP01(2015)126 X (5.5) (5.8) (5.6) (5.7) . ... ]. These . + !# ) with the 12 to get past X k  1 a 2.9 ) Z a ) , and the corre- iβ iβ ¯ Y − exp( d state with angular defects against an into − . exp( Z 2 ∗ N Z | jump past an − 2 ˜ ˜ λ is enough. er to explain in the open ξ redicts planar equivalence . We can then easily check ℓ √ lar field theories have the . To include the boundary 2 † s ℓ ) Y nd do not require additional † n . ∗ observations: one based on N a + and eory arguments. The two ob- λ 2 )˜ 1 iβ | √ is replacement. This should be ℓ ˜ and , a ¯ ξ ) and make the replacement ) Y,Z Z − ˜ λ . − -deformed version. This is the n n ] isβ q . The equations of motion of the -product deformation [ a β iβ Y − β 2 2.4 ∗ q ℓ − 1 exp( into − ℓ ξ − 1 | are related to each other by complex ℓ exp( + 1) ξ Y q | ξ defects. In the twisted theory, the net N k ! 2 = exp(2 ( † k π i N   2 . Notice that this is an automorphism of † s a Y 4 † 1 2 s YM ) π a a g a and 4 2 YM ) )˜ . Thus the cost in phase for a iβ g + λ iβ − 2 – 15 – isβ + ) = exp[2 2 2 2 ℓ − ˜ n λ exp( exp( + r − +1 − X,Y,Z k 1 ℓ q = N ( N λ ˜ λ ) is correct (due to the central charge extension symmetry J = exp( √ √ → r ) ) s − ˜ λ 2.9 a = arises because iβ iβ ∆ J − 1 = 4 SYM spin chain for the − − for the boundary contributions. We get that k exp( − N exp( λ ∆

q = 4 SYM at one loop order, these operators can be obtained by an +  + 1 to all orders. We just copy the result in equation ( = k n N N 2 − π = q 8 2 YM g n = ends up being opposite in phase to the cost of having a loop For this more general case we find that Consider the operators of the SU(3) sector with From here, it is easy to find the energy of the open string groun − 1 X H conditions, we just need to take the result in equation ( an that the phases cancel inthought the spin of chain as hamiltonian a after local th field redefinition of the local fields the Cuntz algebra, if at the same time we take Leigh-Strassler theory are cyclic in The twist to turnspin the chain. theory We to just the replace previous spin chain is easi twist of the boundary condition is proportional to usual statement thatsame noncommutative planar field diagrams, theories which in and this case regu results from a conjugation. This resultintegrability of follows the from spin putting chain,servations and are together one that two equation based ( on just field th background. In SO(4) rotation of the ground state with only We can also understand similar SO(4) rotations of statements can be made entirelyinsight within quantum from field string theory theory. a of the all loop(with spin chain) a and twist) that of the the field theory dynamics p appropriate substitutions. We find that sponding twists. For our purposes, the difference between collective coordinates momentum The power of JHEP01(2015)126 , up (6.4) (6.1) (6.2) (6.3) → ∞ n ). The 5 N S = 4 SYM 2 YM × makes sense g 5 N ]. Notice that s 1 ℓ . This fibration t this condition 2 ates if to the AdS radius. rmed CP he string scale, this SU(3) decomposition. → se states is to consider nterpreted as a highest 4 5 e string scale. This is a ⊂ / n the gravity theory. As S 1
acena limit [ bration structure of the 5- n we separate two D-branes d more modes are available this second term in the sum → N U(1) . This shows that the volume per angular momentum 1 S × 2 YM is below the typical string scale 2 ns of D-branes, choosing suitable g / 1 → ∞ ) < ) for the giant graviton ground state, , which means that in the geometric N N 1 2 2 N | / S ( ˜ ξ 1 YM = 0. The appearance of 2 YM s
g O g − ( . Equivalently we are considering the case J -charge. BPS chiral ring states have their N (they are null geodesic in AdS ˜ ξ ξ ≃ < ℓ | R < 2 − | 2 YM ˜ 2 ξ N | 2 g ,J ˜ CP ξ π – 16 – − 4 2 YM − ξ < g | ξ 2 | + n N 2 2 n π 4 2 YM ). The main question we will ask is which states remain r g = 1 superspace = 5.8 J N . Even though ∆ , and another one at when we take the limit 4 − ξ / 1 n ∆
) and ( ), and ask what happens to the spectrum in the limit ˜ ξ . N | , 5.7 ˜ ξ ξ 2 YM − g (or ξ | ˜ = λ 1 . This is usually what we mean by a low energy limit. Notice tha − s λ, 1 can be thought of as a dimensionless ratio of the string length = 1. The spectrum of states between them will contain light st − s s q ℓ The simplest way to understand the geometric location of the The second term tells us essentially that As a warm up to analyze this problem, let us start in the undefo The first term tells us that the momentum of the state is below t J < ℓ n < ℓ − of squares can be thoughtby of a as the distance Higgs mass that results whe to the degeneracy expected from supersymmetry). Moreover, so that the twostandard D-branes way have to to extract the be low closer energy to field theory each in other the than Mald th where of the five spherebelow is the becoming string infinite scale in as string we units; take more the an geometric limit. optics limit they are at a fixed position in little group of such fixedThe position state determines a carries fixed no SU(2) SU(2) quantum numbers, so that it can be i is satisfied for all fixed energy equal to the angular momentum along the determines a choice of an unpacking the inequalities, we need that both maximal giant gravitons first,sphere and as a to circle bundle consider over the the complex standard projective plane, fi using expressions ( and that is with a giant graviton at in this case all field theory modes survive (there is one state light in this limit.∆ We will call states light if their energy because the term in the square root is a sum of squares. In this section we considervalues various space-time of configuratio 6 Geometric limit interpretation it always remains light in this sense since ∆ because we are measuringsuch, energies in units of the AdS radius i JHEP01(2015)126 , 3 1 2 S S . ξ is no CP (6.5) (6.6) ween k q q to pick ⊂ because 2 1 1 C ]. A giant coordinates CP 5 CP is a multiple 6 , rane. To have survive, which 4 n | on determines a base. The other 2 ℓ that shares its s = 1 is a primitive , . . . x + 3 Z 1 3 s ℓ / x S q 1 − and its images l (we think of it as the q e torsion, where ξ s quotient [ CP − for a monopole spherical what happens at generic . These end up moving at . This spectrum of states say Z 1 s s tural interpretation is that | / ink of it by the method of action that can act on the , that is, we try to locate 3 Z ξ is rational. These orbifolds . ular momentum is along the s . Seen from the point of view S ween 2 n. This is one way to think of m × Z hat / β q 1 s
Z = N / units of angular momentum on 5 ˜ ξ S n 2 YM g 0 , or equivalently, that and the Landau level classical orbits are under the 1 < by a multiple of ξ n | → − s 2 | n m n and act by the corresponding quotient group − − q 5 – 17 – q n < ℓ S − − 1 | 1 | 1. This implies that 2 | . This is exactly what we expect from the optical limit , and the string is moving along with them. It also moves ξ s | 5 . 1 differs from S s > N . Indeed, it results from looking at the ]). n 3 CP units of angular momentum will survive the low energy limit. 2 YM S 46 , an object that carries g s 3 S fiber of , the allowed positions for such an open string lie in a 5 1 . What we find in that case is that a light spectrum of states bet S S on the base. These highest weight states are localized on will correspond to a brane wrapping a X n is a multiple of in the covering space ξ 1, we will need that , and then we choose the standard complex structure on this 2 2 3 ℓ S C ; we are considering the spectrum of fluctuations of a single b − in terms of coherent states. | ≃ ξ , and the SU(2) chiral symmetry preserved by the giant gravit ξ 1 space, which decomposes as a Hopf vibration with a 2 | 3 ℓ = s S as a ˜ ξ CP Z 5 / 3 S S The next case we want to look at are the orbifolds with discret A similar statement can be made for the other giant gravitons Similarly, we can ask what happens if we take In the presence of a maximal giant graviton, which is a maxima There are two interesting questions to ask. First, let us ask ; only fluctuations with s the SU(2) we need. of the similar Hopf fibration of this are interesting because the geometry is given by a root of unity for some integer longer equal to one, but instead is a fixed root of unity; let us inside the along the Hopf fiberharmonic can of be charge thought of as a highest weight state constant speed on the the effective magnetic fieldcircles is centered proportional around to some positiondirection on of the the sphere. point on The the ang and similarly can be interpretedof as the a Hopf highest weight vibration of of SU(2) heavy states can be thought of as long strings stretching bet that maps the positionimages of a la the Douglas-Moore brane [ to itself, otherwise we th a second brane in one of the image points of on a has fixed differences in momenta, plus a shift from zero. The na complex coordinate fiber with that of the the branes survive, so long as shows that corresponding values of graviton at fixed of a light state will then require that and that For generic More precisely, if we go to the SU(3) sector, we will require t building weight of SU(3) with respect to this geometric decompositio JHEP01(2015)126 = 1. (6.8) (6.9) (6.7) q ], where 48 to have such around . Hence they s of states. It is s β ξ < ℓ on lines, along the | to be smaller. ξ e of n n . Again, we are asking ˜ ξ − q The position is uniquely ction of S-duality on the choice of discrete electric tation for these states is ve fixed values of the R- viton. We can think of it = − e local interactions; if they ξ 0. In this case all of these es. See however [ ˜ and its images. It is natural r expand in es that are sub-stringy. The ξ estrict string making an appearance present paper. | ξ . ect that the BPS central charge te Wilson line does not change. by a discrete Fourier transform. | → 2 . / ξ 1, and | ξ . 1
, and more generally, to the square in a ground state where there is a
2
n ]. This means that the coordinate s 1 and we get a construction where the n → N − Z s 47 q s / O 3 2 YM finite. It can even be made very large in g S < ℓ +
| ]. For us, the D3-branes with magnetic flux , . . . q ξ N | 2 < 52 2 iβsn 2 YM , and that n . This is the dispersion relation of a squashed – 18 – 1 2 g 2 , q, q − s 1 β )-strings have the same world sheet sigma model as 2 2 | , ℓ 1 + 2 β 1 ξ | ≃ ℓ ) theory on p, q ≃ s n < ℓ N n W q 2 YM g . We will not answer this question here. s Z to be close to zero, that is / = 1 as a matrix. Both of these give the same spectrum of states. β 3 s S W . Obviously, we can also take limits where and are related to the ones with fixed s s ξ , or on 3 S ]. This would be important to understand S-duality on the set world volume, or as a U( 47 3 S What we find is that the energy is proportional to Now let us consider Now let us ask also about the special limit where A natural question is to ask how we can deal with magnetic Wils contains both the position and the Wilson line information. a double scaling limit sense so long as we allow ourselves to r Again, just as before, we ask that One simple way to do this is to take Which is more appropriate willare then local depend on on the nature of th about fluctuations of a single brane. In this case we can Taylo discrete nonabelian Wilson line given by All we need is that root of a quadratic form involving spectrum is then theas same an as that of a single maximal giant gra images of a brane are separated from a brane itself by distanc states can survive. In that case, we ask that charge modulo to assume that such branes with magnetic Wilson lines will ha should be linear combinations of the branes at the values of Understanding S-duality in detail is beyond the scope of the lines of [ setups, and the interpretation in terms of a relative discre determined by need to be at the same location, so they must have the same valu it is argued that the D-strings and ( ξ beyond what can beLeigh-Strassler done deformations with is complicated perturbation [ theory. Also, the a the ordinary strings, except for theirargument tension. is extended We would to exp theseinside as well, the with square the tension root of the formula. However, a field theory compu Wilson lines between them, in a similar vein to [ the two D-branes are on top of each other, but they differ in the hard to understand the S-dual magnetic strings between bran JHEP01(2015)126 - s
], ℓ N 12 .A β plus a 2 YM g 2 is still a as being , we can 2 ˜ ξ /t , ℓ β 1 β/π = 0. In this ℓ ]. | will survive to 50 ξ t | e torsion. These , we notice that ˜ ξ . At least naively, β ]). These should end 6= can be larger than l btained from a stringy | . Such a linear term is 49 ξ , even if they are larger ξ n | | ξ them in more detail [ nes correspond to cutting | s one in AdS units, with theory properties of instantons [ he distance from the origin rresponding giant graviton of questions could be asked -deformed theories will also l structure of states is very ently. Keeping form involving Momentum versus wrapping One can even fix e exponential in 1 β le. In our case we are dealing ], where the different rational ith the closed string spectrum. ble to the AdS radius. n-Maldacena backgrounds and e s very interesting. es of ed to be interpreted as having a 51 soning to [ will typically never be close enough close to zero, we can have states for n | − ξ q | cancel for some − q flux is known. This problem of reading the , it should be possible to produce these from . We then find that 3 with 2 S H ξ /t – 19 – 1 = ] the light states came from wrapped strings on a < ˜ ξ they look similar to orbifolds with discrete torsion 51 | t r/t − , the states always survive. This means that we should 2 ℓ changes as we go away from the fixed point β/π | = t . If we use a continuous fraction approximation to 1 ], even in more general deformations where they become unsta s 2 ℓ . Then the states whose momenta are multiples of 54 t , < ℓ 2 53 | . However, if we let such that n n − q r, t ]. These should also be applied to the orbifolds with discret − by a phase such that the factors of . The value of 1 ξ t 58 | 2 Z | ξ | ]. Seeing as our classical solutions stretching between bra × t 55 A natural question to ask is how much of this picture could be o The next question we need to ask is what happens for irrationa In this case, if we also separate the branes slightly, taking Notice also that if Z = 0. Even if the full result is not geometric, certain classes / | 5 ξ reasonable approximation. Thus, for sufficiently small valu solutions in the TsT transformationsthe that work generate [ the Luni have a list of sporadic light states that depend on the number nothing survives. This is because the numbers 1 well known solutions of the sigma model on details are beyond the scope of the present paper. small circle, while in ourare case T-dual they to carry each angular other, momentum. so exploring these issuescomputation. further i Given thestates Lunin-Maldacena are geometries, known the [ co approximations to an irrational numberwith play open an strings important stretched ro between D-branes,It rather is than natural w to imagine that the closed string sector in thes up matching the Lunin-Maldacenawhere geometries the when squashing we of explore the sphere and the low energies, so long as the first order Taylor expansion of th natural difference is that in the work [ than the string scaleS by a factor of to zero at finite gives a finite squashing; the sphere is still of a size compara by a factor of roughly flux can also be analyzed using other techniques with D-brane sphere, where different directions have been squashed differ the dispersion relation becomes a square root of a quadratic find integers constant term, and more crucially, a linear term will arise. like a position dependent relativenon-trivial Wilson H-flux line. in the This geometry will (this ne follows similar rea related to which | case we jump between geometric duality frames depending on t similar to what happens in the study of Melvin models [ think of the branethe other as directions having forming at a least stringy one geometry. large The circle genera of radiu ble [ in the corresponding orbifold with discrete torsion. JHEP01(2015)126 ] 20 ], in ]. In 19 51 ) which ], where = 4 SYM = 4 SYM. y of giant 5.8 57 , N N 56 ]. It would also ) and ( 12 in models [ 5.7 epartment of Energy e possible in part by very strongly on the owever [ k supported in part in on, the giant gravitons = 4 SYM and which of tween giant gravitons in derstood as a set of BPS hat need to be done are l charge argument [ y simply. We argue that at these states locally look method [ r orbifolds of ntum numbers of the string N tring scale) that survives. ggests interesting questions f -deformations of hat the action of S-duality is ling limit if there is a rich set quations ( ween giant gravitons with the β hain discovered by Beisert [ the states, but where we don’t for all toric quivers, since their -deformed theories, this suggests β β = 2 theories, where we also expect a ] for 52 N -deformed theories because the spin chain β – 20 – , and the notion of geometry jumps discontinuously as we β = 4 SYM, except for a twist which only affects the boundary N ]. 60 ] and it would be interesting to see how this can affect the stud = 4 SYM theory at strong coupling and in the at infinite coupling, in a way that is very reminiscent of Melv 59 N β We also discovered that in the orbifolds with discrete torsi It would be interesting to explore this issue further in othe Apart from previous two loop results suggesting this centra These results can be ported over to the This strong coupling limit with a large set of states depends graviton states. It’s also interesting to explore dual geometries can also bebe deformed interesting by to the Lunin-Maldacena understand other marginal deformationsexamples o with dual geometries are expected). of open string states with low energies (low comparednumber to theoretic the properties s of move in general, we expect a similar structure as a function of makes it possiblephysics to can take be limits interpreted and geometrically understand in the geometry strong ver coup this paper we alsocorresponding provided energy classical and open quantum numbers. stringvery We similar states also to bet argued those th that appear in the Bethe ansatz. that determines their energy. Acknowledgments D.B. would like topart thank by Greg DOE Moore under grant forOffice DE-SC0011702. various of comments. E. Science D. Graduate Wor supported Fellowship by Program the (DOE D SCGF), mad that resolving these issues might be very non-trivial. In this paper weboth have the analyzed thetheory. spectrum of We argued open that stringsstates this be for set the of central open charge extension string of states the can infinite be spin un c 7 Conclusion them are geometric. A lot less is known about such cases (see h This again resultsdifferent in [ the same spin chain, but the twistings t carried electric Wilson linesregarding how on S-duality their acts worldcomplicated on volume. when those looking states. at This different Considering su questions t [ is almost the same as in conditions of the open stringsstates. in a The simple energies way of based these on strings the are qua encoded simply in e central charge extension toexpect control the integrability [ allowed energies of JHEP01(2015)126 , l field ] , SYM ] = 4 ommons , (2000) 038 N = 4 ]. 01 N , superstring 5 global symmetry SPIRE (2000) 196 ]. S ns, Orbifolds and ]. 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