JHEP11(2008)061 can 5 S × 5 June 6, 2008 October 15, 2008 November 7, 2008 y AdS November 20, 2008 Received: Revised: ific position in the bulk nstructed. We show that iagram. Accepted: Published: lynomials can be viewed as ese dimensions emerge. We are captured very naturally he Yang-Mills theory. In this and h, South Africa Published by IOP Publishing for SISSA = 4 super Yang-Mills theory. Six of the dimensions N BPS sector of the field theory by rewriting it in terms 2 1 Brane Dynamics in Gauge Theories, AdS-CFT Correspondence. Type IIB string theory on spacetimes that are asymptoticall [email protected] National Institute for Theoretical Physics, Department of Physics andUniversity Centre for of the Theoretical Physics, Witwatersrand, Wits, 2050, South Africa, Stellenbosch Institute for Advanced Studies,E-mail: Stellenbosc SISSA 2008 c Robert de Mello Koch Abstract:

of Schur polynomials. Thea Young diagram book labeling keeping of device theseaspects which po of summarizes the how the geometryby operator of the is the Young co extra diagram. holographiccorrespond to dimensions Gravitons boxes which added are at a localized specific at location a onKeywords: spec the Young d of the string theoryarticle are we study holographically how reconstructedreorganize these in the dimensions t and dynamics local of physics the in th be defined using four dimensional Geometries from Young diagrams JHEP11(2008)061 1 8 2 5 8 1 19 20 21 23 25 25 = 4 super N -BPS sector). There rovide us with a def- 1 2 in its correlation func- rding to the AdS/CFT orrelation functions of he string theory is a 10 di- r local dynamics should be the strong ’t Hooft coupling these dimensions emerge and Thus, we will compute these these free field results can be ample, type IIB string theory ensions that are holographically this in the is given by four dimensional 5 S × 5 – 1 – = 4 super Yang-Mills theory. See [3] for detailed N 5 S in these dimensions emerge is a fascinating problem. In this × 5 local physics The dynamical content of any quantum field theory is captured 3.1 The annulus 3.2 An arbitrary3.3 number of rings Eigenvalues Yang-Mills theory (see [2] for an explicit demonstration of examples of what information is coded and how its coded. Acco correspondence, we expect alimit classical geometry of to the emerge field in theory. In this article, we focus mainly on c 3. LLM geometries 4. Beyond LLM 5. Discussion A. A review of the LLMB. geometries Correlators inition of string theoryon on spacetimes negatively that are curved asymptotically spaces; AdS for ex 1. Introduction We do not yet know what string theory is. AdS/CFT [1] seems to p article, we explore this issue. 2. Gravitons in AdS tions. The extracoded into holographic the dimensions correlation functions together of with thei Contents 1. Introduction are many aspects ofmensional this theory; definition apparently, that there are need anreconstructed to additional in be 6 dim the clarified. Yang-Mills T theory.further, Understanding how how BPS operators because thesecorrelation correlators functions at are weak protectedextrapolated, coupling [4]. with with no confidence change, that to strong coupling. JHEP11(2008)061 . - 2 n ). m S This N 1 r Young etric group o goals in mind: (i) BPS sector [5]. f a single holomorphic 1 2 esentations of SU( amics of the field theory. ur polynomials. Second, diagram and (ii) to stress indices. To compute the ) has a definite symmetry . An example of a Young nsor has been symmetrized hange of variables: we will d theory limit, are known j he Young diagram records. duct rule that can be used N gram, plays a central role. m of the + U( i able simplification of correlator e computation of correlators for − -boxes corresponds to a completely N n – 2 – and the hook of each box in the Young diagram. 3 has a weight equal to indices. A Young diagram with a single row containing j n boxes also labels an irreducible representation of the symm A Young diagram with the weight of each box displayed. n ) irreducible representation corresponding to a particula N and column i Figure 1: BPS sector of the theory is captured by the singlet dynamics o 1 2 Our approach entails performing a reorganization of the dyn Young diagrams are a useful way to label the irreducible repr We will start with a quick review of Young diagrams. We have tw These are not the Dynkin weights. They are simply an alternativeA basis Young to diagram the with trace basis. 3 1 2 We will also exploit this connection in what follows. The matrix [5, 6]. Theuse reorganization Schur polynomials we which employ provide is a achieved complete by description a c reorganization will allow atwo significant reasons. simplification of First, th to the collapse Schur any polynomials productthe satisfy of two a Schur point polynomials nice functionsexactly into pro of [5]. a Schur sum Thus, polynomials, ofcalculations. this Sch in reorganization The label provides the of a free the consider fiel Schur polynomial, a Youngto dia recall the definition ofthat the the hooks hooks and and weights encodeor the the antisymmetrized. weights specific of way a in which Young a te under permutations of itsA indices. Young diagram It with isantisymmetric a tensor this single with symmetry column that containing t A tensor transforming in an irreducible representation of S diagram, we need to define the weight A box in row dimension of an SU( boxes corresponds to a completely symmetric tensor with diagram with the weights filled in is given in figure 1. JHEP11(2008)061 f the elbow lies. umber of boxes the en in figure 3. ing from the given box combinatorics of tensors agram. These two lines roduct of the hooks. For th the hook lengths filled , the totally antisymmetric m, and another line starting l indices take distinct values. . 4) − 2 N 3 · 1)( 3 · − 5 2 – 3 – N ( 2 N = ) irreducible representation corresponding to this Young N d A Young diagram with the hook length of each box displayed. The hook shown is associated with the box in which the corner o Figure 3: The dimension of the SU( The hook length for the given box is obtained by counting the n To obtain the hook associated to a given box, draw a line start Figure 2: diagram is given bythe the example product we of are the considering weights divided by the p The hooks and the weightswith provide a an definite efficient symmetry way to undertensor encode swapping with the indices. three For example indices, gives a non-zero result only if al elbow belonging to the boxin passes is through. given in A figure Young diagram 3. wi towards the bottom of the pagefrom until the you exit same the Young boxform diagra towards an the elbow - right what until we you call again the exit hook. the An di example of a hook is giv JHEP11(2008)061 1 − (1.1) (1.2) ) irre- N is equal N the Schur R ) gives the f 2 U N ( R χ , U provide a convenient Z roblem entails defining harpen the question we e a useful labeling of the trix e normalized by a factor s. The factor rs have a description in a the hooks corrects for the . The product rule follows , ll geodesics and hence they point functions and whose 2 e Young diagram describing , we need to define graviton ) . ol to explore the AdS/CFT alar o graviton states. This gives apping any two indices only n Z n ) R ( was given in [5], where it was from the Young diagram. We -charge of order n σ ry. The graviton annihilation i ing [7 – 9]. For a recent review i Z ⊗ R ( s dual geometry emerges. If the Z . 1 S χ R · · · RS S δ 2 (1) R R 1 σ f i 1 i R Z = g ) σ . These results have been extended to de- E values, the second index any one of ( ) S counts the number of times the SU( R 2 values. These are exactly the value of the X R Z N ( χ S – 4 – − 2 S n )= S R χ N † ∈ 1 Z X ) σ ( R 2 ! g Z 1 ( R n R . In addition to this, the Schur polynomials also satisfy χ ) χ boxes, then the Schur polynomial is built using products R D Z )= ( n 1 Z ( R R χ BPS sector of the theory. A particularly nice property of the such that each term is a product of χ appears in the tensor product 1 2 Z S in irreducible representation U is a Young diagram with Recently is has become clear that Young diagrams also provid According to the AdS/CFT correspondence [1], these operato R weights of the correspondingfact Young that diagram. not The allcosts division elements a by of sign. this tensor are independent - sw invariant variables of matrix models.shown The that first Schur such polynomials example built using a single complex sc parametrization of the If Thus, the first index can take any one of values and the third index any one of of traces of Higgs fields is that they have a diagonal two point function The Littlewood-Richardson number Schur polynomials to the product ofa the product weights of rule called the Littlewood-Richardson rule ducible representation immediately from the fact that when evaluated on a unitary ma character of fine operators in multimatrix models, that have diagonal two multipoint functions again follow fromsee group [10]. theory reason dual quantum gravity. In particular, for operators with an polynomials should be dualare to asking: new the geometries goal [2].Young of diagram this We article labeling can is of now tocorrespondence, operators s investigate the is how dual thi to geometry provide shouldwill a emerge argue useful naturally that to coherent this states is in the indeed gaugeus the theory a that case. way we to propose explore are the“know” Our dual dual about approach t geometry: the dual to gravitons geometry. movecreation this along To nu define and p the annihilation coherent states operatorsand creation in operators the that we Yang-Mills definethe theo add background. or remove The boxes state to th with a box added/removed needs to b JHEP11(2008)061 1 2 (1) O t is easiest to -charge of the BPS operators charge of R 1 2 ions of operators factor plays a key R background. This 5 c N S in global coordinates state/operator corre- ]. By probing the × 5 p 5 S S × ur dimensional Euclidean chur polynomials and re- e mapped to a theory on avity we will conformally 5 discuss our results. of the adjoint scalars of the s that correspond to closed ries that arise when the field f the geometry of the dual IIB ons can be identified with the se spacetime dependences will y to see that where our results in the half-BPS operators that . non-relativistic fermions in an setting. In section 3 we turn to E scription for the † N kl ; they have two point function (we Y ) of the graviton. The operator dual 5 background [1]. The ij Y S . For the operators we consider, after 5 Z,Y 3 D S S × . = 5 ) × jk Z N δ R il √ – 5 – δ Tr ( . The boundary of AdS = 3 S E = 4 super Yang-Mills theory, with † kl × Z N R ij Z 5 D S × 5 , the dynamics reduces to that of 3 S . We will make use of two scalars 3 is the weight of the added/removed box. The extra × S c R × - it is the space that the field theory is defined on; this is why i R 3 where . Under this map operators map to states and conformal dimens S 3 c N × S Chiral primary operators in To conclude this introduction, we mention the papers [11, 12 In the next section we start by discussing gravitons in the Ad = 4 super Yang-Mills theory is a conformal field theory. By the p R × external potential [5, 6,moduli 13]. space The of phase the space dual of geometries these [2]. fermi mapping to to a graviton with one unit of angular momentum is are dual to Kaluza-Klein gravitons on the AdS interpret the field theory after mapping to operator maps into the angular momentum (on the Our computations will mostlyspace, be where performed we in can thestricted theory use Schur on the polynomials. fo knownmap To results our make for results contact to correlators the with of theory the on S dual gr suppress the spacetime dependenceplay of no this role correlator in as this the article) serves to illustrate the mainconsider ideas LLM in geometries; the section simplest 4theory possible is background concerned is with built geomet from two matrices. In section 5 we of 2. Gravitons in AdS role in ensuring that the correct dual geometry emerges. BPS operators in thestrings gauge these theory papers have with already singlesupergravity. understood These trace some articles aspects operator used o anin eigenvalue density the de gauge theory. Usingand the those results of of [11, section 12] 3.3 overlap, it they is are eas consistent. spondence, the theory onR 4 dimensional Euclidean space can b N map into energies of states.can We be will constructed primarily from be thetheory interested s-waves on of complex combinations is JHEP11(2008)061 . 2 ,x (2.1) (2.3) (2.4) (2.2) 1 s hence y,x (extracted 3 S . ) z ! . -graviton state is m = 1 z ¯ m z n O ow !  n z inates of [2]. In these n z Z . The graviton that is = 0 and at some fixed r mics of coherent states =  Tr y  m z . d malized  dZ O † he dual quantum gravity on , ) m,n Tr Z N  δ n dynamics. Define an operator !  √ n d -wave on the second dZ Tr ( 1 s N = and a radius   . a . φ =  n n , Tr m   . The remaining two coordinates of the ↔ †  to make three new coordinates, ) ) 5  a N ) 1 5  S ) Z Z N N Z √ N Z ↔ is the complex conjugate of on the vacuum in the super Yang-Mills the- d N  √ √ dZ √ ) Tr ( Tr ( z √ Tr (  , – 6 – N Z Tr ( . . It is also an N  3 ) √    ! √ , n Z Tr / S ( ! n N 2 ) Tr ( n d | direction. It is localized at 1  =0 dZ n √ N ∞  z Z z 1 √ X | Tr m ) φ )  √ z † 2 | e Z z =0 =0 Z N N ∞ ∞ 2 | | Tr X X n n e z plane using an angle √ √ | 2 2 2 Tr ( 2 N 1 2 Tr ( 1 , −  √ extracted from the ,x |N | e |N | |N |  1 3  x = = = 1 = = S exp ↔ z E is easily determined (¯ N z O ensures that this operator has a unit two point function and i O = N † z ] = 1 1 2 z † O − O D -wave on the field theory N a, a [ s is a graviton creation operator. It is useful to use the coord . We will write connections like this using a double sided arr 5 ) and is smeared along the † S 5 a S × 5 As a check of these identifications, we will now study the dyna is combined with the radial direction of AdS 5 . It is this localization that we are trying to describe. A nor dual to a normalized state. It is straight forward to compute This suggests that the action of Tr ( We can also define the graviton annihilation operator as The factor of Note that coordinates, there is an ory matches the actionAdS of the graviton creation operator in t where The normalization from of gravitons and verify that wedual recover to the expected a gravito graviton coherent state as dual to the operator created is an S We can also describe the r JHEP11(2008)061 is is R 3 d boxes; S × m ) is equal ich is the Z R E ) result vanish. . i Z z ( | N S d χ dt † i ) | ula can be obtained z  Z h  2 ( 1 R m , = N χ  product rule and the two ) his large  ,  D er of boxes) ) ) . R z  S a derivation of it using Schur 2 coherent state on Z O . Z N ( 2 O d r ( n 1 √ R ) d R n N dt − χ d is mapped to the coherent state n ight forward to verify that ( 1+ Z i χ R -charge tion. The operator Tr ( 2  N n † z is a Young diagram with  f z  R N 1 R ˙ √ n d 2 2 O O N φr O ! S Tr ( ) )   n n R R R = † S z X d d ) 1+ X i ( ( n Z z N  = | = 0, which are by now, familiar results for ( ! R R R √ , =0 boxes; ∞ r X X X n H n 2 X n | χ is equal to the order of the group. From our r )) n 2 z | 2 z Z m,n m,n m,n m,n – 7 – | = ) ( δ δ δ δ  1 2 R − i − h χ d zz = = = = = e , the operator z ( | = 1 and ˙ 3 R = ¯  ˙ S = ( d φ dt i ]= m P i n z z | × |  z )) O ) h H R , | Z Z ). A product of Schur polynomials is easily computed using N z = R h [ Z √ ( L Tr ( = (Tr ( χ irreducible representation labeled by  E ] n n [14]. z )  S 5 O iφ ) , S † − is a Young diagram with BPS operators, the conformal dimension of our operators (wh × R Z N [ re 5 R † 1 2 z √ = O Tr ( D z  Before moving on we will give a discussion of (2.1). This form , so that ( i z the sums run over all possible diagrams with the correct numb to the Schur polynomial and hence that ( graviton’s energy in the dual description) equals their the dimension of the Since we study more elementary techniques we know that all corrections to t the Littlewood-Richardson rule. Using this rule, it is stra Note that this multipointpoint function correlator for was Schur computed polynomials. using the After conformally mapping to where we have used the fact that | using elementary (just Wick contract)technology methods. since We this will will give generalize nicely in the next sec gravitons in AdS The Lagrangian governing the low energy excitations of this The equations of motion determine JHEP11(2008)061 n ), N M M p An Z (2.5) (2.6) ≡ 4 µ ) so that . . ) n N lized state dual to from the “bottom ( Z next to B, and the N ( O  R B R 2 R . f + 1 N χ √ χ an expression for Tr (  ) boxes. R to be first term corresponding to (1) boxes. We will use the 2 − d O the super Yang-Mills theory, is a Young diagram with ) O N M tuations about this state. R ( Z , X is ted. This is the case since we take B ( = ) 4 we have (in general the right O 1+ , Z = n B 3  ( , χ ) χ B Z m,n ( f . )+ to the vacuum of the super Yang-Mills δ = 2 ! ) B √ Z )+ n p χ B d Z ( , dZ ( f Z ) ) n p ( = p B Z √ boxes; we take  ( Z χ  χ ) pN χ m N Z – 8 – χ N − √ ( Tr ( −  and Tr ) i↔ √ − ) ) p χ p Z ) units of angular momentum. The result (2.5) can B Z ( Z . In this section we will study gravitons propagating | Z  ( Z 5 p ( pN S = √ × Tr ( ) 5 χ χ χ is a Young diagram with  B Z ( f n B be a Young diagram with B  √ ) = ) = ) = χ 2 4 3 ) R † p Z Z Z n p  Z pN ) graviton state in the background Tr ( Tr ( Tr ( ) where Z √ N n Tr ( Z as shown in figure 4 below. ( √ Tr ( (1). Let B B  χ O  to denote the Young diagram obtained by removing to denote the Young diagram obtained by stacking R R − (1). O Our background is given by taking the “vacuum” to be the norma Finally, consider By small we mean small enough that backreaction can be neglec 4 hand side is a sumthe over symmetric all representation hooks is with always an positive) alternating sign; the which has been given in [15]. For example, for notation + notation This suggests that the operator to be Gravitons on this background willoperator correspond to dual (small) to fluc an again be obtained by using Schur technology. To do so, we need 3.1 The annulus A simple warm upcolumns example and each is column provided contains exactly by the case that right corner” of 3. LLM geometries In the previous section, we applied Tr creates gravitons which each carry to the operator on different backgrounds. These backgrounds correspond, in theory to study gravitons in AdS is a number of JHEP11(2008)061 , is (3.1) (3.2) N , leaving . R ! + after multi- n f n B ) ative of a Schur µ ; these boxes all . R B , . i +1 n and = (1+ + 1 O R n . n | ) R rule for Schur polynomials. N + 1 , to the leading order in the + B Z B f ( f n + 1 f B √ √ 2 to produce + . From (3.2) we can read off its eated in section 2, since in that n χ ) µ B √ R n B µ d in terms of  ( cancel with weights in 1+ ) R R Z B 1+ p N X − ( † f -graviton state, to leading order in a √ p = = n χ and = ↔   +1 (1) number depends on exactly which box is ) i ! ) # n R O n ) B Z Z – 9 – O ( N f B Z √ + 1 B ( f 2 √ √ = 1. We can consider the action of a derivative on n χ + 1 B Tr ( | √ n/ 1 χ ) c n n N µ n  + 1 √ ) ! c n N ) Z N √ (1 + Z r √ N ( c (1) where the N Tr ( = √ = O  χ , we have n r The definition of + n n + O

N ). We can write O =  ! ) † Z i n M ) ( Z n N | √ Z N † + χ 2 √ a √ Tr ( n/ N 1 Tr ( ) Figure 4: . It is straight forward to verify that (the rule for the deriv )= graviton state µ n  Z n † O ) B Z (1 + f ( , is the weight of the box added to the background Young diagram limit a graviton creation operator in the background B √ c χ † d dZ N N a  √ Tr " dual to the operator considered. Given this correlator, a normalized To see the last equality note that all weights in Thus, in this new background the operator action on an only the weights correspondinghave to a the weight boxes of added to polynomial was conjectured in [16] and proved in [17]) This formula is exact;We to will get now this compute result, thelarge two we point used correlator the of product this operator with This is perfectly consistentcase, with to the leading trivial order background in tr where plying by Tr ( JHEP11(2008)061 5 (3.3) coun- atives. charge R , Indeed, we -charge of the erpretation. 6 R · · · + . !   ) n B Z n ( f µ B . √ . χ ) i = n µ, n-zero result. 1 1 Z BPS sector. N ( − + 2 1 − B n R ) operator, to the leading order ). The term that we have written tion, which we will call f µ, n − ! B hich lowers the total Z | Z screpancy, note that in contrast ( ) χ √ f ( n e identity. i f the ]=1+ n, we can now act on the operator B B Z √ after subtracting off the † c √ N ( χ χ µ d √ i n =1+ dZ Y a, a χ [ Z !  would be incorrect - antigravitons are created ) ) 1+ -countergraviton state, to leading order

R d n ↔ Z d a, p N dZ N = Tr  1)! R √ ) √ ↔ = X Q Tr ( − Tr ( i Z , N d 1 = n added, which is the expected action of an annihilation dZ

N ( – 10 –  √ ) † n − antigraviton Tr ( √ ) B p # d Z Tr † , n dZ ( 2 -charge f ) a | / 1  B R n B  Z √ 1) ( f χ d √ − B dZ n √ n Tr c ( χ N  ) ! · · · N n ) 1 µ r + √ ! Tr d dZ N 1 )  = N − d 1 √ i (1 + dZ n N √ n Tr (   | O √  a

µ µ . However, this can’t be correct: the identification Tr ( is the product of the weights of the boxes removed by the deriv µ i c 1+ 1+ i *" *" p p Q limit, is n n An operator dual to a normalized √ √ ]=1+ N 7 † . represents an extra term that does not yet have an obvious int = = a, a · · · ; including any antigravitons in the state would take us out o † In the previous section the operator dual to the vacuum was th We call a state with positive (negative) This extra term arises from the action of the derivative on Z 6 7 5 operator. In the last line, by vacuum a graviton (countergraviton). The term This formula is again exact.in The the two large point correlator of this A natural interpretation is that the derivative operator (w Clearly [ Ignoring this term, we can identify, explicitely removes some of the boxes that ( of the state) createstergravitons gravitons moving in the opposite direc implies that where the graviton annihilation operator acts as where which is in stark disagreement withto (2.2). the trivial To background resolve considered this in di dual the previous to sectio the vacuum with the derivative operator to obtain a no find JHEP11(2008)061 . a µ after (3.4) ]= † B b, b gram . . In this case g to the action -charge (string b , R can add boxes to 1 b g to the action of − ) = 1 ) and n µ corresponding to the Z † ( − eory b O − ) . operator creates counter- ; Tr ( µ . 8 † . + 1 † b b s above, we have [ ) a n B  Z , + ( f √ ions i ators by d a µ B . 1 ) creates gravitons and destroys √ dZ b Indeed, χ ) √ Z −  , but only in the upper right hand ↔ ]=(1+ n Z b n = ) B ! 1 d +
e extra term in equation (3.3): the extra term dZ N − † |− + Tr d n dZ √ N + Tr ( a − , a Tr ( † a  + 1 √ O b ) n Tr d Z + dZ √ + 1

a – 11 –  µ [ ! n n b , √ √ ↔ no longer have a local action on the Young diagram √ + 1 c = 2 N ) = Tr ( 
† = Tr i ) n/ a Z d n r dZ  Z µ N ↔ = d Tr ( √ = dZ |− ) Tr ( †  n b  , n Z N − −  √ O Tr is that it can add boxes to O gravitons and 1 counter graviton. Tr ( d , a dZ n ) and Tr ) ) Z  d Z dZ N Tr √ Tr ( N 1  ) becomes √ removes boxes from the lower right hand corner correspondin  d dZ b
. Similarly, split removes boxes from the upper right hand corner correspondin is the weight of the box removed from the background Young dia b d dZ a c
d , is now given by dZ These identifications also provide an interpretation for th N 8 , but only in the lower right hand corner of the Young diagram, . We will define new operators that are local: split countergravitons. It is now easy to argue for the identificat The resolution to ourgravitons and paradox destroys is gravitons; now multiplication in by sight: Tr ( the derivative theory angular momentum) to label states. Arguing exactly a Thus, we see that Tr ( and Tr corresponds to a state with Denote the countergraviton creation and annihilation oper The definition of Tr ( action of in B corner of the Young diagram, corresponding to the action of (where B These identifications are perfectly consistent with (2.2). Tr in the dual quantum gravity. Note that we are using the field th acting with JHEP11(2008)061 (3.8) (3.7) (3.5) (3.6) into an . B . It is for this χ B 1) , − on coherent state and N χ e any correlators using he background . = = ( kground ) of the new local operators avitons are missing boxes, operator dual to a coherent . w. This is most easily done B Z tion of these local operators miltonian is bounded below. ( ! f r combination of Schur poly- rs are indeed well enough de- le and known two point func- ation n . The Hamiltonian is B state and then translating this √ χ i χ b χ √ z n ) | 2 b n n . ) Z O  where  z µ n a . O d z ) dZ b Tr ( z , † Z N µ  b = (1 + ! √ = 1 n − # Tr ( Tr =0 z µ ∞ 2 √ | X n  µ n z O 2 a n | 1+ ! ) , † ) 2 O µ e | n a a z µ 2 – 12 – n 1+ ,
√ z −| 2 2(1+ d dZ N = e |N | n/ 1 (1 + ) √ χ H = = µ , Tr =0 z ∞ E X n " z O + 4) (1 + O N † z N χ O = = D z n = ( = O O χ χ a ) a Z  d dZ Tr (  Tr . Diagrammatically we have To compare results to the dual gravity, we need to translate t We can again use the operators we have defined to build a gravit † b LLM geometry, using the resultsby of first [2] translating which the we Schur polynomial will into now a revie free fermion reason that the secondsimilar to term holes above in is the Dirac negative: sea. the Since countergr the sea is finite the Ha This Hamiltonian measures the energy with respect to the bac of This operator satisfies Denote the coherent state that this operator maps to by We are computing correlation functionstions using for the Schur product polynomials. ru with We the have usual just Schur defined polynomials.nomials the whose This product product two point givesthe a function new linea is local known. operators whichfined Thus shows for we that our can these purposes. comput localbelow. We operato will give a more constructive defini check that we reproduce thegraviton expected state graviton is dynamics. The Requiring a unit two point function determines the normaliz JHEP11(2008)061 or and 10 integers 1. These occupied are rings in − N 2 N columns, so i x of the Young + + i M i 2 r p = times the weight of is precisely equal to E he LLM plane 1 µ N re proportional to the . The area of the black tors, we are producing hus, the total area of the To get a correspondence rows and e localized on the Young N considered. We have seen . The Schur polynomials etime and locality on the xes in row 9 ite. This black and white n example, we have shown ve an interpretation in the N hese ripples are localized in r which the area divided by e. tor of le in the center of the annulus = 1 and the row closest to the ndition is specified the LLM plane. i . Note that µ √ = has a total of unoccupied states followed by N M B M q potential so that fixed energies – 13 – 2 . 2 x ,x 1 x in figure 5. background itself corresponds to a black disc of radius 5 B is shown on the left. The corresponding boundary condition f S × B 5 . The energies of the occupied levels are given by N = i . The AdS states and hence a radius of fermion energy eigenstate [5, 6], which is labeled by a set of π = 0 and has coordinates y M ; the row closest to the bottom of the page has N i r The Young diagram BPS geometries, identify the free fermion phase space with t plane. 2 1 is exactly equal to the number of states in the black region. T By acting with the graviton creation and annihilation opera Recall that our fermions are in an external We call the two dimensional plane on which the LLM boundary co x,p 9 = 1, in a sea of white. The Young diagram 10 states giving a phase space thathas looks a like an total annulus. of The ho telling us which levels are occupied. Denote the number of bo diagram by the LLM geometry is shown on the right. fermion state into amap boundary into condition an for the LLM geometry small “ripples” on the innerthe and radial outer direction edges of ofdiagram. the the annulus. LLM We T plane seeYoung in a diagram. spacetime direct This and is connection they athat between ar feature the locality of oscillators all in in of the spac the new examples background we are have scaled by a fac top of the page has occupied states map into rings in the free fermion phase spac color occupied regions inpattern is black the and boundary unoccupiedthe condition regions boundary for condition in the dual wh dual to geometry. As a that in the free fermion description, there are Figure 5: the weight of the box in the lower right hand corner divided by black regions is This plane is at the with the box removed/added bydual the geometry? oscillator. The Does areanumber this of of factor states the ha in black theπ/N regions region. on We the will LLM use plane a a normalization fo R JHEP11(2008)061 . µ ). N µ charge) (1 + . In the B N R . Note that 1 + dius on the LLM µ background 1+ e LLM result: we can us, i.e. those created by . √ tion value of the number and corner divided by  n the LLM geometry (we and the energy of a single = 2 s have an energy r momentum (= r e true vacuum of the theory. , ase space, this energy is just tion value of the energy of a ). The Lagrangian governing M . ! . s are ripples on the inner edge µ − 2 al to the value of radius in the N + n µ 2 N n 2 √ r Nr r 2 n O (since together they correspond to q (1 + ) 1+ dτ dφ n = µ N N  = n z zz . In this case, the Hamiltonian becomes i + N t 2 . This tells us that gravitons are small † z ) a. = ¯ (1 + † O µ N M = i ] Nr z z i =0 ∞ | z X n Na = O | a ) † (1+ 2 µ = | H a – 14 – | | z N, dτ dφ N [ z z ∂L h −| ∂ H h 2(1+ = e = = τ i i − h i z z τ z ]= | | | z d N dt H | O | i z | z h h z N, h [ = L quanta is † a states). The outer radius of the annulus is thus N We will now focus on excitations of the outer edge of the annul What is the energy of a state of gravitons created at a given ra + . The energy measured by (3.8) is measured with respect to the † To see that this rescalingcoherent does state of the job, note that the expecta The number operator acts as This is the energy ofThere the is graviton measured a with natural respectintroduce a to way rescaled th to time coordinate modify (3.8) to get agreement with th LLM picture, gravitons exciting the outer edge of the annulu of the annulus. ripples on the outer edge of the annulus and countergraviton a plane? Given the interpretationthe in terms number of of a gravitonsgraviton free times fermion is the ph given energy by of the a radius single squared graviton times Thus, the factor by whichLLM the plane oscillators at are which rescaled the is dual equ graviton is created. M is precisely equal to the weight of the box in the upper right h This is indeed thehave reproduced Lagrangian (3.41) for of the [18]). dynamics of Notice gravitons also i that the angula annulus and its white center is proportional to computed using the rescaled time so that the expected number of gravitons, is given by Comparing the expectation valueoperator of we see the that energy each to graviton has the an expecta energy of the low energy excitations of this coherent state is JHEP11(2008)061 lead . f the (3.9) ) Z units of ( b b p χ . Using the . − b ) Z Z m,n ( + δ b a n a ] χ p Z ) ittlewood-Richardson − . µ red with respect to the . ) 2 ovide a natural way to → µ r µt there are various “local” Z ( Z er right hand region of the ity ‘a’ or ‘b’ and summing b s in the Schur polynomial = a d region of the background = ed time coordinate is given vel of the Schur polynomial, χ rgument for a coherent state τ ![(1 + re. The giant graviton probe n rule defines what we mean i . This matches the energy of his characterization of typical s which each carry n z − ion, the analysis of [21] focuses | pond to Young diagrams whose y measured with respect to the ) is interpretation. = energy measured with respect to Nµ N | us sectors of the field theory. The Z ) have a local decomposition? We z  ( rdinate p h We hope to return to this issue. a a ) Z χ B Z ( 12 f − B √ ) , χ 2 Z ( m b Nr  b ) χ − matrix replacement p p – 15 – Z = pN )+ i z √ Z | Tr ( ( b . We have checked that this recipe reproduces (3.9). a † b  b b | Z n χ z  h + ) † a p )+ = p Z Z i ( Z z pN b | ) creates a localized graviton, localized at the outer edge o a → p a 11 √ H χ Tr ( | Z Z z  h † )+ units of angular momentum. ) Z B p the weight of the boxes that are added or removed. Thus, we are Z ( f ( a c a B √ χ χ  with )= 2 ct Z = Using Schur technology it is simple to verify that It is not surprising that we need to work with the energy measu Tr ( In this case, we have introduced a different rescaled time coo We would like to thank the anonymous referee who suggested th τ 11 12 will now define and give evidence for the results of [19, 17,with boxes 20], in we the know Youngthe diagram. that replacement Using we amounts this to can insight, allowingover associate at each all the matrice box possible le to identities. have Thus, an for ident example angular momentum. Does the graviton operator Tr ( of countergravitons gives is equal to the energy, as expected for a BPS state. A similar a When we now take arule product except of that Young we diagrams, are webackground only Young use allowed diagram the to and usual add ‘b’ L ‘a’ boxesYoung to boxes diagram. the to This lower the refinement right upp han ofby the the Littlewood-Richardso decomposition This suggests that each countergraviton has an energy of Thus, for example, Tr ( Thus, we can again identify operators which create graviton annulus, carrying a graviton exciting the inner edge of the annulus. true vacuum, to obtain agreementof with [18] the is dual identified gravity with pictu true a vacuum highly is precisely excited the fermion;on energy the very of energ heavy this half-BPS fermion. states. Inshapes Almost addit all are such small states fluctuations corres around astates fixed is limiting done curve. using statistical T the ensembles true which assign vacuum, the to eachby state. In both cases above the rescal rather naturally to rescaling timeLLM differently for geometries the have vario atimes time coordinate as that well is asunderstand warped, these the so time global that rescalings time in coordinate. the field theory. Our work might pr JHEP11(2008)061 - R (3.10) ) boxes . Using and the p N p µ ( µ O (1), we have + charge (with − O µ µ R is (1) boxes. Thus, is i O p N i + is 1+ p till it has . P B B ples at the edges of the N ct of expectation values, +  y s added/eroded to produce m m eed to remove boxes from a ided by ation values, we are dealing   ) i ) + i ons and their i i p p ) p p state have  B Z N N Z Z ) i i ( f ) boxes and p p 2 B Z B √ ( f √ χ √ Tr ( Tr ( N B ( √   m χ O n n  O ) i   † i p ) ) ) p i has i † † B p p N Z i i Z ) creates a new background. Indeed, whenever f i ( p p B N p N Z B √ i i Z Z – 16 – ( p √ p χ Tr ( B ) boxes eroded from the Young diagram. These two √ √   χ Tr ( Tr ( n N ≡  ( , when   ) O B i N i † i (1). We can also consider excitations with a large Y p i i p O O * Y N h i Z . When we build our coherent state below, we will assume p ≡ = p N √ Tr ( The Young diagrams corresponding to giant gravitons.  = i p Y µ † ) B Z f ( ). These excitations correspond to giant gravitons. Figure 6: B . Set √ p χ N ( * O are all distinct) i p We have considered excitations that correspond to small rip To create a giant graviton, we need to build up a single column Note that, to leading order in respect to the background) is annulus. These excitations correspond to point like gravit the expectation value ofwith the a product classical is limit. the product of expect charge, of stacked in the column; tosingle create column till a we counter have giant apossibilities scar graviton of are we n shown inthe figure 6. giant by Denote the number of boxe is in fact a consequence of the fact that The fact that thewhere expecation we define value expectations of by the product is the produ that all giant graviton states contributing to the coherent (the weight of the last box eroded to produce the countergiant div the weight of the last box added to produce the giant divided b JHEP11(2008)061 s, ) and (3.12) (3.11) N cillator. , ( ) i O µ as a constant = r + . p z µ , . ) O † | z 0 ate B . ih (1 + = ) is given by + 0 | n 0 # =1 ; n i Y z A ) − | ih . ry [18]. O 0 B Z = + ,  ( a closed string of the form f † 2 n
)(1 ergy B b − | r ) √ p y giant graviton coherent state d χ µ dZ µ + N − p )(1 a , − √ µ. p removes one unit of energy from  ) dτ dφ µ µ Tr L A (1 + ↔ 2 ) B n a " ]= + r ) . N † Z  µ d 2 1 √ to a constant we have treated Y Z dZ N b, b with a string. The same computation, # N r [ Tr ( = (1+ = ( √ , n · · · 1 )  = B Tr ( A N − B B Y Z i † p † )! 2 ( f z )! p )! | n a O B n √ p µ, χ − H )! – 17 – M | µ − Y Z n z + M 1 B , M −  n , A a M p + µ A + N + ) µ ( i − h + † ]=1+ Y Z N z , B N Z N † | A − ( p ( 1+ N √ p ( µ d µ dt + Tr ( p a, a Tr ( 2 i N [ b | +   z = are not normal bosonic operators - they are Cuntz oscillator µ h + s n n # ) † ) B p =1+ = = = µ a n µ | n O p z L c c z , | N N ↔ O
and ) (1 + ; since the equations of motion set (1 + is again the weight of the box which is added/removed by the os = = d dZ A Z N  c N N † † adds one unit of energy to the giant; =0 ∞ √ √ X n =0 † ∞ log Tr ( X n Tr AA BB A 2 " N d " dr = = small enough that the expected number of gravitons in this st are the operators introduced above which satisfy Nτ z z 2 † N O − , b 1. It is straight forward to verify that † = Finally, consider probing the background t ≪ 13 13 p using different methods, has been given in [12]. We probe with µ the methods we introduced above, we can write in performing this rescaling. We take which is the correct Lagrangian for D3 branes in an LLM geomet The Lagrangian governing the lowis energy excitations of this satisfying a, b, a an operator dual to a giant graviton coherent state is given b the counter giant. The operator In these formulas, An operator dual to a normalized giant graviton state with en JHEP11(2008)061 ) . µ  i L 1+ ng a lattice √ , z = -term. See [11, 12] · · · o , F 2 R , z 1 s. In the background b z . | limit, where our Cuntz , ero. One can define a
,  , ! H d , ) | N dZ ¯ n | z nnulus L 2 ) L 0 σ . We can put each site of n b r tate parameter; denote the † z omalous dimension, at large n for a string moving in an nnulus. These loops are not -loop anomalous dimension. ih 2 o , z  α . escaled by the (now familiar) nnulus, or z∂ ) − omes from the 0 ( O ) R σ | large 2 z o d n e loops is outlined in appendix ∂ 0 dZ · · · +1 ) l − | R s and , 2 µ ih  a 2 α λ n he dimension of the operator; it includes L 0 z ] = Z = , z Y )(1 − z 1 − 2 l µ − | z O (1 + r ˙ α 2 · · · φ ( r (1 2 µ † α, − 14 =0 r Y ) ∞ µ we use ) X n − i − h µ 2 µ a n b +1 = L 1+ µ l = (1+ = Z  α β , z n 1+ a α † d † − 1+ Z – 18 – (1 + dZ · · · l n , 

) α . For 2 ( µ Y n N , z dσ z L 1 1 =1 =0)= l 1 X n b z µ, β | 0 µ, α The semiclassical sigma model action governing the low λ  (1 + Z ,y = d . dt 2 i d | = † dt =0 dZ L ∞ L ,x X n 1  H ββ , z Z , z =1+ x = ( Y † L z φ · · · · · ·  , V , = O αα 2 2 Tr S , z , z 1 1 background, these correlators can be computed by associati z z | 5 h i S  × limit the above sums become integrals leading to we use 5 dt b L Higgs fields and Cuntz oscillators with the
correction. For the loop we consider the full contribution c Z d dZ Y 2 YM = g in this action does not include the leading contribution to t S H this is also the case, except that the Cuntz oscillators are r , is (see appendix B) 14 -BPS and hence their one loop anomalous dimensions are non-z 1 2 square root of the weight divided by only the and for corresponding to a string localized on the inner edge of the a corresponding to a string localized at the outer edge of the a Cuntz oscillator Hamiltonian whoseThe spectrum computation of gives the thisB. relevant one In correlators the involving AdS thes In the large B Comparing this to theLLM fast geometry, we string read limit off of (recall the that Polyakov the actio outer radius of the a is dual to a coherent state. Indeed, it satisfies [ with the The Cuntz oscillator Hamiltonian whichN gives the one loop an and references therein. for strings localized at theoperator outer edge representation of is the valid, annulus. the In operator the the loop into suchresulting a state coherent by state with a different coherent s energy dynamics of this lattice model is given by JHEP11(2008)061 tor cks is called an  j b d sidered in the previous dZ  de can only add boxes an be used to grow the . entric rings. . Fields localized at an 2 n we probe the inner edge Tr , r e complete LLM answer for inner radius of the annulus ) 2 i is summed over all outward N j − 1 b R ider the Young diagram shown j √ 2 i N Z pointing corner. R √ j ). The corners identified in figure − X a Tr ( ( = + b j can be decomposed as X 2  ) r i + a Z µ ( N d − ) √ i dZ µ a Tr  N Z − √ – 19 – = 0) as felt by a string moving at the outer edge Tr Tr ( ,y N 1 2 i √ X =0)= ,x 1 i = x ,y X ( 2 ) φ = Z V ,x N 1  √ x ( Tr ( d φ dZ V corresponds, in the dual quantum gravity, to a sum of oscilla  ) Z ( N Tr √ N Tr 1 √ . Similarly, j b = 0). j This Young diagram is a generalization of the background con ,y ) is summed over all inward pointing corners and P 2 µ i + ,x √ † i 1 A corner of the Young diagram that we can erode by removing blo a x = ( i i φ section. The outer edge of each ring corresponds towhich an is inward the correct result for pointing corners. Eachat their of corner. the local fields on the right hand si 7 all correspond to inward pointing corners. P where of the annulus. ProbingR the inner edge gives (recall that the 3.2 An arbitrary number ofWe rings can generalize the results of the previous section to cons in figure 7. The geometry now corresponds tooutward a sequence pointing of corner; conc adiagram corner by of adding theoutward blocks (inward) Young pointing is diagram corner called that carry c an a subscript inward pointing corner The negative sign in thiswe do formula so comes with from counter theV excitations. fact Summing that these whe two gives th Figure 7: JHEP11(2008)061 1 2 ter (3.13) (3.14) B !+ configuration n ! has a vanishing i N † B b a d dZ E + n

) † i the Young diagram of a n † a can be described by a  Z . Is there a more direct ∼ ( i Z b , . is summed over all outward n nonrelativistic fermions [5, n, he bulk) operators? The d ) . i X i jk jk dZ a δ δ r insight, we expect that the N  ghly excited. A highly excited il il † i∼ Z  δ δ ( n

i | j  b a ) ZZ N N c c a 2 1 Tr † Tr a = = D * however we find − , , + 2 B B i † j ≡ ≡ where again † a b is large. E N X N + c dt c j dZ  kl + a aa kl n ) † ( † j i | ) ) †   a dt ]= th outward pointing corner. Finally, arbitrary ]= th inward pointing corner and B dZ n B j Z i † i Z † j j P h Z f † f b ( ( 2 1 – 20 – ( d , b + , a B √ B √ = ij  i j † i dZ ) χ χ b b a i i [ [ i a   n  into local operators has been defined by stating what ji Tr | is large if n Z 2 # ij
P is governed by the singlet sector dynamics of the action ) ( ) dt 2 n i †  D d † 3 a are related to the positions of the fermions. Now, recall dZ a j ! X S b Z Z i † ( d b Z + × n d = dZ ) a dZ i R (  S | a n Z * h " " ( is summed over all inward pointing corners. It is a simple mat  ) ) and Tr B Z j Z limit, we expect that the eigenvalues of ( f Tr B ) is a “picture” of the eigenvalues of the classical large √ χ N B Z ( f B ij B √  χ n   i is the weight of the boxes at the is the weight of the boxes at the b d i j a b dZ c c In the large  * expectation value for any energy eigenstate; for a few facts from the usual harmonic oscillator: the position are easily computed using the modified ribbon graphs classical configuration. Armedeigenvalues with will our be harmonic largefermion if oscillato corresponds the corresponding to fermion athe is long background hi row in the Young diagram. Thus, 6, 21, 22]. The eigenvalues of 3.3 Eigenvalues The split of both Tr ( corresponds to the sum of oscillator action these operators have on the background Young diagram definition which would allow us to construct these local (in t The eigenvalue dynamics of this model is the dynamics of where BPS sector of the theory on so that the expectation value of pointing corners and to check that where mixed correlators, for example of the form JHEP11(2008)061 1 4 . (4.1) ) values m + m n + ( into local n orrespond σ i i Z X · · · . ) +1) with n Z ng diagram and the +1 ( ( b , n σ i ) i separated eigenvalues, Z Z X ( ) h these geometries. The + ave eigenvalues clumped sider a matrix χ n example, the background ( a n σ ry, to operators built using i . i ) Z ely the split of ls of [9] χ )=  Z Z = ( 2 Z = ( λ · · · on rule. Our local operators will b Z 2 λ χ (1) ) ) 0 0 0 χ 1 σ i i Z Z  , . ( ( Z 2  )+ = )) ) λ 2 b ) Z . These backgrounds will include the σ 0 Z ( λ ( ( Z Z R ( 1 (Γ 0 χ λ ) χ and , χ , χ χ , Z m – 21 –  ) ) ,r  )= Z Z n X = = 0 ( ( r ( Z ( Z F . 1 Tr 2 0 0 χ very rapidly approaches 1, confirming our proposal. λ ) λ χ χ m  + Z F . Then our proposal says n ( = ≫ 2 = S X )= 1 λ 1 ∈ a λ σ Z λ ) Z ( χ ! ≫ a Z m 1 ( χ 1 ! ) λ n can thus be split into four submatrices which each have eigen Z increases, ( χ Z y )= . In this section we would like to consider geometries which c χ Z . As Z, X 2 1 ( λ λ ) m = ,r n y r ( in the sense that there is an eigenvalue for each row in the You R, It is easy to gather numerical support for this proposal. Con In figure 8 we have plotted Thus, locality on the Young diagram maps, at least for widely Z χ We thus have the predictions to operators built using two matrices operators. into locality on the eigenvalues. 4. Beyond LLM The LLM geometries correspond,a in single the matrix super Yang-Mills theo The product for follows as a simple applicationsatisfy of the Littlewood-Richards Consider the case that which should be true when BPS geometries of [23];operators we we are have not in yet mind are able the to restricted make Schur contact polynomia wit size of the eigenvaluecorresponding is to related the to Youngaround the diagram four length values. pictured of in the figure row. 7 will For h of roughly the same size. We suggest that this split is precis against JHEP11(2008)061 m M and n M s. . ) Z is a Young diagram X . ( ) n m r ompute the correlators r tion 3. We will not be s and χ (1). The backgrounds we ) m of the product rule for . Z, X te using the results of [9]. X tttlewood-Richardson rule. hat ( 2 O ed. Indeed, all traces that Z ( B d operator) /λ n are both rectangles that have χ 1 r estricted traces; we do not yet λ χ m ≡ r = m ) r boxes we can write [24] y m X r ( f and m columns. We take both N r m r n + p m r χ (hooks) ) against M n n are numbers of r Z r F ( m f m n r N r (hooks) p M – 22 – χ ≡ (hooks) )= m A plot of µ )= Z, X ( ) and columns and m Z, X n s. It would be much more interesting to consider backgrounds ( ,r n ) N n M Z r m Figure 8: ( M , ,r ≡ n m r r s or ( n , + has µ X m ˆ χ r n + r χ ) so that N columns and each column contains exactly ( O M rows and that consider are perhaps theappear contain most only trivial that could be consider with The two point functionThey of also these satisfy operators aneeded product is to rule trivial probe as to therestricted shown compu geometry Schur in polynomials with [24]. involves gravitons.have the Thus efficient computation we The methods of can precise for r c for this. However, in the special case t Clearly, in thisThus, case we the can required studyso product these general: rule backgrounds in is using what just the follows the methods we will of Li assume sec that After normalizing this becomes (the hat denotes a normalize which are dual to operators that have traces over a product of to be N JHEP11(2008)061 . Z (4.2) is read from ular, we have c ) Z eling of the operators hich keeps track of this efine operators dual to e the case. This result , , ) ) f terms which each have a . , chur polynomials we have ) ) ual to localized excitations BPS geometries by cleverly ) whilst the gravitons built wo properties that are ulti- r of fields in each trace vary sing the Schur polynomials, geometry dual to operators Z 4 1 Z, X ated by Tr ( al is constructed. ( Z, X ( l ons) provide a convenient way n ( Z, X ey satisfy a simple product rule their two point function exactly Z, X B r ( times a Schur polynomial in ( B . This result is easy to interpret: χ χ B χ local B p m χ p X χ r  p p  ) ) fields. Direct confrontation of these ) ! ! 2 X N
Z
( N -point correlators. N ( ( √ d d n √ dZ dX χ N N χ O  √ √ ). This is a consequence of the fact that our  ! ! Tr Tr X p p ( – 23 – is read from

√ √ m 2 ! ! c r 2 p p χ p/ ) p/ ) √ ) √ 1 1 1 X 2 1 2 m n ) create these gravitons and the state must again be µ µ p/ n m p/ Z µ µ that appeared above; this suggests that these gravitons = = (1 + (1 + p p s feel only the background of − − c/N = = Z O O p p ) and Tr ( O O X graviton states, as are p -charge emerges from the super Yang-Mills theory. In partic R s feel only the background of X It is straight forward to verify that The Schur polynomials are a nontrivial linear combination o We have demonstrated some features of how the ; for the gravitons created by Tr ( n r propagate on some classical geometry. For the gravitons cre using normalized by the factor Thus, even in thislocal more gravitons. general two Tr matrix ( background we can d the gravitons built using combining LLM geometries. 5. Discussion The Schur polynomials (andto their organize multimatrix the generalizati interactionsbeen of able a to matrix compute model correlators - that by involve using the S background takes the form of a Schur polynomial in For more general backgroundssuggests we that it do may not be possible expect to that construct some this nontrivia will b with a large are normalized correlators using standard Feynman diagrams is daunting; u these computations are simple.mately The responsible for Schur this polynomials simplicity: haveand (i) t further one they can diagonalize compute thewhich two allows point a function; straight (ii) forward th computation of different trace structure - thefrom number term of to traces term. andstructure The the Young numbe i.e. diagram it is summarizes a exactly book how keeping the device Schur w polynomi described how to definein super the Yang-Mills operators bulk that of are the d dual quantum gravity. The Young diagram lab JHEP11(2008)061 ions that we =1 at a fixed ˙ φ s, Michael Stephanou gy with respect to the ped, so that there are nto an LLM boundary . Of course, as we have fy the background. We 2 that they also tell us the o , we can ensure that the y constructing excitations r of the theory. In this way, ct Our work might provide a into a particular eigenvalue rated values. This matches e the excitation is localized. gram. Localized excitations dom that we have discussed s. Indeed, the multimatrix tation at a specific radius in c way in which the indices of ed to write the product rule at add or remove boxes at a heory. ormalized with a factor equal LM boundary condition that satisfy a generalization of the = ng group theory techniques. red ed at a specific radius and by espond to a specific eigenvalue ture of the eigenvalue dynamics r the various sectors of the field m. See also [7, 8] for alternative τ is the weight of the box corresponding c limit of a matrix model is in terms of a N rings will correspond to a distribution in which = 0 LLM plane and orbit with – 24 – m y where c N “clumps” with eigenvalues in a particular clump having m minima. The locality that we have seen emerge from the field m is the weight of the box which was added/removed. The excitat c . This normalization factor o r where A standard way to describe the large The Hamiltonian obtained from the field theory measures ener Finally, note that the reorganization of the degrees of free c N condition - it correspondsis to a a set set of of concentricin concentric rings. spacetime rings are can Any also L be localizedthe translated on LLM into the plane a Young has diagram. Young an dia of energy An equal a exci to specific the energy radiusmeasuring squared; we the thus have energy b constructed of the excitationsWe excitation have localiz we defined can local determine creationspecific wher location and on annihilation the operators Young diagram.to th These operators are n has played a crucial role. A Young diagram can be translated i Hamiltonian measures energy withwe respect are to lead the rather true naturallytheory. vacuum to rescaling The time differently LLMvarious fo geometries “local” have times a asnatural time way well coordinate to as understand that the these is time global war rescalings time in coordinate. the field t to the localized excitation, is also equal to the radius squa density of eigenvalues. A particular background would corr density. In thishave article seen however, we that have thisdistribution. used Young diagram A a can Young diagram be Young describing translated diagram to speci reviewed, the weights of thethe Young diagram matrices encode making the up specifi radii the of operator the were rings symmetrized; of we the see LLM boundary condition. background. By introducing a rescaled time coordinate radius have studied are confined to move on the the eigenvalues are clustered into similar magnitudes. This suggests that the free fermion pic has a potential with theory emerges when the clumps of eigenvalues have well sepa nicely with other recent results [25]. in this article hasoperators given a in natural [9] have extensionLittlewood-Richardson a diagonal to rule two [24]. multimatrix point function model Moresatisfied and by work the is restricted however Schur need multimatrix polynomials operators in a that useful can for also be analyzed by exploiti Acknowledgments I would like to thank Norman Ives, Jeff Murugan, Joao Rodrigue JHEP11(2008)061 2 B ,x 1 (A.2) (A.1) x , . we have 2 3 , ˜   Ω 1 d   G 1 − − 2 l − ,...,E ye . R 2 2 l 2 G, ,  + r R ¯ z 4 2 2 3 z. work is also supported by σ r y = 1 Ω 2 l findings and conclusions or − 4 ∂ be a general background moving in an LLM geometry tanh d l z∂ R 2 e Department of Science and ij 2 l , G − l σ ) 2 1 k is based upon research sup- ǫ or for the complex coordinates ) authors and therefore the NRF 2 l ∂ R + 2 R L ye ) 2 R = 2 n a 2 l λ − L y )= r, φ + R 2 i )+ . y + V − i Y Z 2 SO(4) isometry group. The metric is j + ˙ y 2 z ∂ + 2 r dx × = 0 · · · ¯ i V y + 2 − r i 2 2 z Y dx j + r y y 2 ( V 2 n ∂ − a i + = 0 plane. We will often trade the r y ˙ ¯ ∂ SO(4) z ( q 2 ( y Y Z V y∂ ×   q edges with radii 1 dy – 25 – i 2 ( n a R + BPS solutions of type IIB string theory that are 2    E +1 z h l l i 1 2 − − Y Z ∂ dσ G, z + i E E 2 on the 1 ∂ 2 2 0 ) 1) 1) z, y i 2 1 Z j − − cosh ∂ = Tr ( ( ( ± dx y i dτ ij ǫ V O E E = =1 =1 l l Z X X + = 2 = . z L 2 i = 2 . They have an dt 5 V − ix ( = ) = z y S 2 h S − ,y − × = 0 plane for an angle and a radius y∂ 2 5 h 1 y x − ,x 1 = is obtained by solving = x ( z z 2 φ , ¯ V 2 ds ix + 1 The fast string limit [26] of the Polyakov action for a string For a set of rings having a total of x = asymptotically AdS ported by the SouthTechnology African and Research National Chairs Research Initiative Foundation. ofrecommendations expressed th Any in opinion, this material areand those DST of the do notNRF accept grant any number liability Gun with 2047219. regard thereto. This A. A review of the LLMThe geometries LLM geometries [2] are regular and Alex Welte for enjoyable, helpful discussions. This wor B. Correlators In this section wewith will a compute closed the string. correlators In needed terms of to the pro loop given by z Regularity requires that is given by [11, 12] coordinates of the where The function JHEP11(2008)061 ed (B.1) . i n ) up to addition of µ c iltonian quoted in (1 + . is by introducing Cuntz +1 comes from the following e correlators are easy to i + n N B . N † B f ) | eded when probing the back- χ . 0 L √ . reproduced if we take =1 i Y ) the Cuntz oscillator algebra. In , ih + i jk 0 δ n jk = B ) δ il † B ward to see that all such correlators † f a δ il + − | χ ) δ . Z √ ( µ B † i c † B jk (1 f N n δ a χ c √ il N Z = OO δ ) i = B E n = B ) f Tr ( = (1+ † kl † α a χ ) E † √ E L Z =1 † kl – 26 – i Y ( loc † kl * i Z ) n Z a B a ( ij B ≡ f Z Z χ ij Z ( B √ ) is the annulus. With the result (B.1) in hand we find , α D ij E Tr ( * c ) † loc N B a Z ≡ L =1 Z ( i = Y ( OO D B † D D B E B (1) number. This clearly reproduces the correlators comput f † χ αα O √ ) * OO D N = s B Y E † are all distinct. The leading contribution at large OO i n D acts locally on the Young diagram, adding boxes with weight loc Z Look back at the correlation functions (2.1) and (2.5). Thes A very convenient way to account for the planar contractions Focus again on the case that contraction of the Assume the reproduce using the two point function Clearly it is nowground straight with forward a to closed compute string correlators state. ne oscillators [28]. The matrixterms two point of function Cuntz determines oscillators (B.1) becomes where we wish to compute By considering more general correlators, itare is straight reproduced for by the rule Now consider (3.1) and (3.9). These correlators are clearly an irrelevant (at large using Schur technology. The analysis ofsection [27] 3.1. can now be applied to obtain the Cuntz Ham JHEP11(2008)061 ]; 59 ]; ]. 03 ]; JHEP Adv. B 746 , , JHEP , AIP Conf. Proc. ]. 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