aa r X i v : . [ h e p - t h ] F e b From Gauss Graphs to Giants
Robert de Mello Koch a,b, and Lwazi Nkumane b, a School of Physics and Telecommunication Engineering , South China Normal University, Guangzhou 510006, China b National Institute for Theoretical Physics,School of Physics and Mandelstam Institute for Theoretical Physics,University of the Witwatersrand, Wits, 2050,South Africa
ABSTRACT
We identify the operators in N = 4 super Yang-Mills theory that correspond to -BPSgiant gravitons in AdS × S . Our evidence for the identification comes from (1) countingthese operators and showing agreement with independent counts of the number of giantgraviton states, and (2) by demonstrating a correspondence between correlation functions ofthe super Yang-Mills operators and overlaps of the giant graviton wave functions. [email protected] [email protected] ontents The AdS/CFT correspondence[1] provides a beautiful realization of ’t Hooft’s proposal thatthe large N limit of Yang-Mills theories are equivalent to string theory[2]. Most studies of thecorrespondence have focused on the planar limit, which holds classical operator dimensionsfixed as we take N → ∞ . There are non-planar large N limits of the theory [3], which aredefined by considering operators with a bare dimension that is allowed to scale with N aswe take N → ∞ . These limits are relevant for the AdS/CFT correspondence. The limit onwhich we will focus in this study considers operators with a dimension that scales as N . Ourfocus is on operators relevant for the description of giant graviton branes[4, 5, 6].The worldvolume of the most general -BPS giant graviton can be described as theintersection of a holomorphic complex surface in C with the five sphere S of the AdS × S spacetime[7]. It is possible to quantize these giant graviton configurations and then to countthem[8]. Remarkably, this quantization leads to the Hilbert Space of N noninteracting Boseparticles in a 3d harmonic oscillator potential, a result conjectured in [9]. In [10] -BPSstates which carry three independent angular momenta on S were counted. This countingproblem can again be mapped to counting energy eigenstates of a system of N bosons in a3-dimensional harmonic oscillator. Both of these analysis [8, 10] make use of a world volumedescription of the branes. Finally, an index to count single trace BPS operators operators hasbeen constructed [11, 12]. The index has been computed both at weak coupling (using thegauge theory) and at strong coupling (as a sum over the spectrum of free massless particlesin AdS × S ) and the results again agree with [8, 10].Given the AdS/CFT correspondence, this counting should also arise in the dual N = 4super Yang-Mills theory, when the operators of a bare dimension of order N and vanishinganomalous dimension are considered. One of our goals in this study is to demonstrate this.A crucial ingredient in the study of operators with a bare dimension of order N , hasbeen the construction of bases of operators developed in [13, 14, 15, 16, 17, 18, 19, 20].These bases diagonalize the free field theory two point function to all order in 1 /N and mixweakly when the Yang-Mills coupling is switched on. Using these bases as a starting point,1he spectrum of anomalous dimensions for a class operators of bare dimension of order N has been computed in [21, 22, 23]. The operators are constructed using the three complexadjoint scalars Z , Y and X . We use n Z s, m Y s and p X s, fixing n ∼ N and m, p ≪ n . Thisimplies that we are focusing on small deformations of -BPS giant gravitons. The operatorsof a definite scaling dimension are labeled by a permutation σ ∈ S m × S p and a pair of Youngdiagrams R ⊢ n + m + p and r ⊢ n . The explicit form of these operators is O ~m,~pR,r ( σ ) = | H X × H Y |√ p ! m ! X j,k X s ⊢ m X t ⊢ p X ~µ ,~µ p d s d t Γ ( s,t ) jk ( σ ) × B ( s,t ) → HX × HY j~µ B ( s,t ) → HX × HY k~µ O R, ( r,s,t ) ~µ ~µ (1.1)The Young diagrams R and r both have q rows for operators dual to a state of q giantgravitons. Each box in R is associated with one of the complex fields, so that we can talk ofa box as being a Z box, a Y box or an X box. r collects all of the Z boxes. The difference inthe row length of the q th row in R and q th row in r is equal to the number of X s (= p q ) and Y s (= m q ) in row q , so that R q − r q = m q + p q . The right most boxes are X boxes, the leftmost boxes Z boxes and the Y boxes are sandwiched in the middle. The q dimensional vector ~m collects the m i , while ~p collects the p i . The branching coefficients B ( s,t ) → HY × HX j~µ resolvethe operator that projects from ( s, t ), with s ⊢ m , t ⊢ p , an irreducible representationof S m × S p , to the trivial (identity) representation of the product group H Y × H X with H Y = S m × S m × · · · S m q and H X = S p × S p × · · · S p q , i.e.1 | H X × H Y | X γ ∈ H X × H Y Γ ( s,t ) ik ( γ ) = X ~µ B ( s,t ) → HX × HY i~µ B ( s,t ) → HX × HY k~µ (1.2)The operators O R, ( r,s,t ) ~µ ~µ are normalized versions of the restricted Schur polynomials [18] χ R, ( r,s,t ) ~µ ~µ ( Z, X, Y ) = 1 n ! m ! p ! X σ ∈ S n + m + p χ R, ( r,s,t ) ~µ ~µ ( σ )Tr( σZ ⊗ n Y ⊗ m X ⊗ p ) , (1.3)which themselves provide a basis for the gauge invariant operators of the theory. The re-stricted characters χ R, ( r,s,t ) ~µ ~µ ( σ ) are defined by tracing the matrix representing group el-ement σ in representation R over the subspace giving an irreducible representation ( r, s, t )of the subgroup S n × S m × S p . There is more than one choice for this subspace and themultiplicity labels ~µ ~µ resolve this ambiguity. The operators O R, ( r,s,t ) ~µ ~µ given by O R, ( r,s,t ) ~µ ~µ = s hooks r hooks s hooks t hooks R f R χ R, ( r,s,t ) ~µ ~µ (1.4)have unit two point function. Although the definition of the Gauss graph operators O R,r ( σ )is technically rather involved, they have a very natural and simple interpretation in termsof the dual giant graviton branes plus open string excitations. A Gauss graph operatorthat is labeled by a Young diagram R that has q rows corresponds to a system of q giant2ravitons. The Y and X fields describe the open string excitations of the giants. Eachsuch field corresponds to a directed edge, an open string, which can end on any two (notnecessarily distinct) of the q branes. The permutation σ ∈ S m × S p is a label which tellsus precisely how the m Y ’s and the p X ’s are draped between the q giant gravitons. Thepicture of directed edges stretched between q dots is highly suggestive of a brane plus openstring system, as reflected in our language. This interpretation is further supported by thefact that the only configurations that appear have the same number of strings starting orterminating on any given giant. This nicely implements the Gauss Law of the brane worldvolume theory implied by the fact that the giant graviton has a compact world volume. TheGauss graph operators which correspond to BPS states have all open strings described byloops that start at a given giant and loop back to the same giant, i.e. no open strings stretchbetween giants. In this case, we simply need to specify which brane the open string belongsto and this is most conveniently done by partially labeling Young diagram R : in each box weplace a z , an x or a y . Each row in the operator consists mainly of Z fields, correspondingto the fact that the unexcited giant graviton is dual to a half-BPS operator built only from Z s. The number of x and y boxes in a given row tell us how many X and Y strings attachto the corresponding giant.In the next section we will show the counting of these BPS states agrees with the countingof [8, 10]. Motivated by this observation, we explore the link between the N particle descrip-tion employing the 3d harmonic oscillator and the super Yang-Mills operators in section 3.Our results shed light on the attractive possibility of an N particle description of multi ma-trix models, suggesting that there maybe an extension of the famous free fermion/eigenvaluedescription of single matrix models [24]. Finally, we refer the reader to [25] and [26] forfurther related background dealing with BPS giant gravitons and to [27, 28, 29] for furtherbackground relevant for the counting and construction of and -BPS operators for theregime where operator dimensions are less than N . As discussed in the introduction, our description of -BPS operators is in terms of a Youngdiagram R with partially labeled boxes. When the boxes corresponding to Y and X fieldsare removed from the rows of R , we are left with the valid Young diagram r . An example ofa valid -BPS operator is z z z z z z z z y y xz z z z z z z y xz z z y y (2.1)The boxes with label z belong to the Young diagram r and the boxes with label y or x arethe ones that are removed from the Young diagram R to obtain r . The operator labeled bythe Young diagram shown in (2.1) corresponds to a system of 3 giant gravitons, with 2 Y X string attached to the first giant, a Y and an X string attached to thesecond giant and 2 Y strings attached to the third giant. This description in terms of Gaussgraph operators is valid in the case where n the total number of boxes of the Young diagram r and m + p the total number of the boxes that are added to the Young diagram r to form R , are both large and of order N ≫
1. In addition, m + p ≪ n and the number of rows ofthe Young diagram R is of order 1 = N . Finally, the length of any row of R is of order N ,as is the difference between the length of any two consecutive rows.Let us first start by fixing our notation. We will denote by R i the number of boxes in the i th row of R , and we will denote by m i and p i the number of Y and X boxes to be removedfrom the i th row of R to obtain r . Furthermore, q will stand for the number of rows of R , n will stand for the total number of boxes of r , m = P qi =1 m i and p = P qi =1 p i . Hence, thetotal number of boxes of R is then n + m + p . If we denote by r i the number of boxes in the i th row of r , then we have r i = R i − m i − p i In our conventions, we start the numbering of rows from top to bottom. As already mentionedabove, this description of -BPS states is proved to work[22] in the cases that R i ∼ N R i +1 − R i ≫ m + p ∼ N q ∼ N (2.2)We call this the displaced corners approximation because the neighboring corners of R areseparated by a huge number of columns. Outside this regime, things are more complicatedand it is not even known if partially labeled Young diagrams can be used to describe these -BPS states. The number of -BPS operators is the same as the number of possible pairs( R ; r ) counted with multiplicity equal to the number of ways of assigning a valid vector ~m = ( m , m , . . . , m q ). Note that once the pair ( R ; r ) and the vector ~m are given, the vector ~p = ( p , p , . . . , p q ) is determined. The first step towards counting the number of Gaussgraph operators entails writing a generating function for the number of pairs ( R ; r ). Ourstarting point is the observation that the Young diagrams are in one to one correspondencewith partitions of integers. The generating function of the latter is given by Z = ∞ Y n =1 − q n = ∞ X k =0 D k q k (2.3)where D k is the number of possible ways to partition an integer k . This counting is toocoarse for us to reach our goals: we need to track the number of parts in the partition whichcorresponds to the number of rows in the Young diagram. Indeed, we must encode theinformation about q the number of rows of R , as well as the information about the differentpossible m i ’s and p i ’s in such a partition, to ensure that we are counting states in the regimein which the Gauss graph operators provide a trustworthy description. Both modificationsare easy to take into account in our case of interest where m i + p i + r i ≪ m i +1 + p i +1 + r i +1 for all values of i = 1 , , . . . , q . The number of ways to partition an integer k is given by the4umber of solutions to the equation k = X i χ i n i n ≥ n ≥ · · · > χ i ≥ χ i n i in the above equation is associated to the term ( q n i ) χ i in theexpansion of Z . This term appears in the expansion for the term (1 − q n i ) − . Clearly then,to keep track of contributions from different rows χ i we just need to multiply q n by an extraparameter χ and track the power of χ . So, we consider the following modification of thepartition function Z Z = ∞ Y i =1 − χq n = ∞ X k,d =0 D k ; d χ d q k (2.5)where D k ; d counts the number of Young diagrams with k boxes and d rows. Next considerthe information associated to the m i ’s and p i ’s. There is a potential complication becausewe want both R and r to be Young diagrams. However, in the displaced corners limit, wecan ensure that this is not an issue. Indeed, by taking m, p ≪ | r i +1 − r i | for all i , we ensurethat we can never pile enough Y and X boxes onto a row to make it longer than the rowabove it. Thus, we may treat the m i ’s and p i ’s as independent, except for the requirementthat P qi =1 m i = m and P qi =1 p i = p . In terms of the partition function Z , this is equivalentto associating to each term q χ i n i , a term p b i m i r c i p i , where b i = χ i and c i = χ i in general.The latter condition is equivalent to associating the term p l r m , with l, m = 0 , , . . . for eachterm q n in the product form of Z in equation (2.5). Thus, we finally obtain the generatingfunction Z = ∞ Y l =0 ∞ Y m =0 ∞ Y n =0 − χp l r m q n = X d,m,p,n D m,p,n ; d χ d p m r p q n (2.6)where D m,p,n ; d counts the number of diagrams R with ( n + m + p ) boxes and d rows, that isthe result of adding m + p boxes that are randomly distributed over the d rows of the Youngdiagram r with n boxes. Our construction of the Gauss graph operators only holds whenthe displaced corners approximation holds. Thus, we trust D m,p,n ; d to count the numberof Gauss graph operators for a system of d ∼ N giant gravitons when n, m, p ∼ N and n ≫ m + p . This is the main result of this section.We want to compare this to the counting of -BPS giant gravitons. As we discussed inthe introduction, this counting problem can be mapped to counting energy eigenstates of asystem of N bosons in a 3-dimensional harmonic oscillator. The grand canonical partitionfunction for bosons in a 3-dimensional simple harmonic oscillator is given by Z ( ζ , q , q , q ) = ∞ Y n =0 ∞ Y n =0 ∞ Y n =0 − ζ q n q n q n (2.7)with the fugacity ζ being dual to particle number[10]. Notice that (2.6) exactly matches thegrand canonical partition function (2.7) for bosons in a harmonic oscillator potential with χ q , q and q . These map into the three types of boxes ( X , Y or Z boxes) appearing in R , counted by p , q and r . Thus, long rows in R map to highlyexcited particles. This proves our first claim: the counting of the Gauss graph operatorsmatches the counting of BPS giant gravitons.It is straightforward to consider the restriction to the -BPS giant gravitons. Theseoperators are constructed using only Z and Y fields. Arguing as above and counting partiallylabeled Young diagrams with boxes labeled z or y , in the displaced corners approximation,we obtain the generating function Z = ∞ Y l =0 ∞ Y n =0 − χp l q n = X d,m,n D m,p,n ; d χ d p m q n (2.8)This counting can be compared to the counting of -BPS giant gravitons. This countingproblem can be mapped to counting energy eigenstates of a system of N bosons in a 2-dimensional harmonic oscillator. The counting (2.8) does indeed match the grand canonicalpartition function for bosons in a 2-dimensional simple harmonic oscillator, which is givenby Z ( ζ , q , q , q ) = ∞ Y n =0 ∞ Y n =0 − ζ q n q n (2.9)Thus, restricting the counting we demonstrates that the counting of the -BPS Gauss graphoperators matches the counting of BPS giant gravitons, as it should.
The fact that the number of Gauss graph operators matches the number of energy eigenstatesstates of a system of bosons in a 3-dimensional harmonic oscillator potential, motivates usto look for a correspondence between the two. To start we will consider operators O ~m,~pR,r ( σ )labeled by Young diagrams that have a single row. In this case we don’t need to encode acomplicated shape for R , so we will simply list the number of Z s, Y s and X s in the operatoras O n,m,p . Since this row has O ( N ) boxes, we have a system of N bosons and one of them ishighly excited. The idea is that since we have one highly excited particle, we can use a singleparticle description and overlaps of the single particle wave functions will match correlationfunctions of Gauss graph operators in the CFT. We focus on R ’s with a single row becausethe computations are so simple to carry out in this case that we can compute many quantitiesexactly. There is a simple formula for the Gauss graph operators we consider, in terms ofthe Schur polynomials O n,m,p ( Z, Y, X ) = N Tr (cid:18) Y ddZ (cid:19) m Tr (cid:18) X ddZ (cid:19) p χ ( n + m + p ) ( Z ) (3.1)6here N = s n !( N − m ! p !( n + m + p )!( N + n + m + p − k ) to denote a Young diagram that has a single row of k boxes.There are a number of natural operators that act on the Gauss graphs. For example, wehave Tr( Y ddZ ) k Tr(
X ddZ ) k O n,m,p ( Z, Y, X ) ∝ O n − k − k ,m + k ,p + k ( Z, Y, X ) (3.3)Thus, a natural correlator to consider is given by h O † n − k − k ,m + k ,p + k Tr (cid:18) Y ddZ (cid:19) k Tr (cid:18) X ddZ (cid:19) k O n,m,p i = s ( m + k )!( p + k )! n ! m ! p !( n − k − k )! (3.4)To describe a single particle in a 3d harmonic oscillator, we need three sets of creation andannihilation operators (cid:2) a z , a † z (cid:3) = (cid:2) a y , a † y (cid:3) = (cid:2) a x , a † x (cid:3) = 1 (3.5)Using the above oscillators we can create a state with an arbitrary number of z quanta, y quanta or x quanta. We suggest that the correspondence between Gauss graph operatorsand particle states is as follows O n,m,p ↔ | O n,m,p i = 1 √ n ! m ! p ! ( a † x ) p ( a † y ) m ( a † z ) n | i (3.6)The correspondence identifies the number of z , y or x quanta in the particle state with thenumber of Z s, Y s or X s in the Gauss graph operator. There is a natural extension to includeoperators, suggested by this identification. For exampleTr( Y ddZ ) k Tr(
X ddZ ) k ↔ ( a † y ) k ( a † x ) k ( a z ) k + k (3.7)As a test of the proposed correspondence, note that h O n − k − k ,m + k ,p + k | ( a † y ) k ( a † x ) k ( a z ) k + k | O n,m,p i = h | ( a z ) n ( a y ) m + k ( a x ) p + k ( a † z ) n ( a † y ) m + k ( a † x ) p + k | i p n ! m ! p !( n − k − k )!( m + k )!( p + k )!= s n !( m + k )!( p + k )!( n − k − k )! m ! p ! (3.8)7hich is in complete agreement with (3.4). Very similar computations comparing, for exam-ple h O † n − k ,m − k ,p − k (cid:18) Tr ddZ (cid:19) k (cid:18) Tr ddY (cid:19) k (cid:18) Tr ddX (cid:19) k O n,m,p i (3.9)and h O n − k ,m − k ,p − k | ( a z ) k ( a y ) k ( a x ) k | O n,m,p i (3.10)show that we should identify a x ↔ r m + n + pN + m + n + p Tr (cid:18) ddX (cid:19) a y ↔ r m + n + pN + m + n + p Tr (cid:18) ddY (cid:19) a z ↔ r m + n + pN + m + n + p Tr (cid:18) ddZ (cid:19) (3.11)These computations make use of the reduction rule of [30, 31].We now want to argue that the identifications we have developed above have a natu-ral extension which identifies Gauss graph operators with q rows with a q particle system.Towards this end, we first point out a dramatic simplification in the formula for the Gaussgraph operators, arising when we specialize to BPS operators. As discussed in the intro-duction, in this case we set the permutation σ appearing in (1.1) to the identity. Using theorthogonality of the branching coefficients we then find X j,k Γ ( s,t ) jk ( ) B ( s,t ) → HX × HY j~µ B ( s,t ) → HX × HY k~µ = X j,k δ jk B ( s,t ) → HX × HY j~µ B ( s,t ) → HX × HY k~µ = δ ~µ ~µ (3.12)This leads to the following formula (the operators below are normalized to have a unit twopoint function; they differ from the operators in (1.1) that are not normalized, by a factorof p | H X × H Y | ) O ~m,~pR,r ( X, Y, Z ) = 1 n ! m ! p ! s | H X × H Y | hooks r hooks R f R X σ ∈ S n + m + p Tr ( P R,r Γ R ( σ )) Tr( σX ⊗ p Y ⊗ m Z ⊗ n )(3.13) P R,r is a projector on the carrier space of R . It projects to the subspace of Young-Yammonouchistates that have 1 , , ..., m + p distributed in the boxes that belong to R but not r and m + p + 1 , ..., m + p + n distributed in the boxes that belong to R and r . Using this formula,it is straight forward to prove thatTr (cid:18) ddX (cid:19) O ~m,~pR,r ( X, Y, Z ) = q X i =1 s p i c RR (1) i n i + m i + p i O ~m,~p (1) i R (1) i ,r ( X, Y, Z ) (3.14)8r (cid:18) ddY (cid:19) O ~m,~pR,r ( X, Y, Z ) = q X i =1 s m i c RR (1) i n i + m i + p i O ~m (1) i ,~pR (1) i ,r ( X, Y, Z ) (3.15)Tr (cid:18) ddZ (cid:19) O ~m,~pR,r ( X, Y, Z ) = q X i =1 s n i c RR (1) i n i + m i + p i O ~m,~pR (1) i ,r (1) i ( X, Y, Z ) (3.16)The first formula above is exact. The last two hold only in the large N limit. We haveintroduced some new notation: the Young diagram R ( n ) i is obtained from R by dropping n boxes from row i of R . Further, ~p ( n ) i is obtained from vector ~p by replacing p i → p i − n andsimilarly for ~m ( n ) i . Finally, c RR (1) i is the factor of the box that belongs to R but not to R (1) i .Recall that a box in row i and column j has factor N − i + j . For the proof of these formulas,we use the notation N = 1 n ! m ! p ! s | H X × H Y | hooks r hooks R f R andTr( σ · X ⊗ p ⊗ Y ⊗ m ⊗ Z ⊗ n ) = X i i σ (1) · · · X i p i σ ( p ) Y i p +1 i σ ( p +1) · · · Y i p + m i σ ( p + m ) Z i p + m +1 i σ ( p + m +1) · · · Z i p + m + n i σ ( p + m + n ) We will now prove (3.14). A simple computation shows dO ~m,~pR,r dX ii = p N X σ ∈ S n + m + p Tr( P R,r Γ ( R ) ( σ ) )Tr( σ · X ⊗ p − ⊗ Y ⊗ m ⊗ Z ⊗ n )= p N X σ ∈ S n + m + p − n + m X i =1 Tr( P R,r Γ ( R ) ( σ ( i, σ ( i, · X ⊗ p − ⊗ Y ⊗ m ⊗ Z ⊗ n )= p N X σ ∈ S n + m + p − Tr( P R,r Γ ( R ) ( σ )[ N + n + m X i =2 ( i, σ · X ⊗ p − ⊗ Y ⊗ m ⊗ Z ⊗ n )Since we are summing over elements of the subgroup S n + m + p − ⊂ S n + m + p we can decomposethe trace over the irreducible representation of S n + m + p as a sum of traces over irreduciblerepresentation R (1) i of the subgroup S n + m + p − . Now use the fact that N + P n + mi =2 ( i,
1) gives c RR (1) i = the factor of the box dropped from R when acting on any state in the carrier spaceof R that also belongs to the R (1) i subspace. We find dO ~m,~pR,r dX jj = q X i =1 f ( i ) N c RR (1) i O ~m,~p (1) i R (1) i ,r (3.17)where the factor f ( i ) N = s p i ( n i + m i + p i ) c RR (1) i (3.18)9ccounts for the change in the normalization factor N of the operator. This is an exactformula - it does not depend on large N or on the displaced corners approximation. Nextconsider the proof of (3.15) and (3.16). Consider dO ~m,~pR,r dY ii = m N X σ ∈ S n + m + p Tr( P R,r Γ ( R ) ( σ ) )Tr( σ · X ⊗ p ⊗ ⊗ Y ⊗ m − ⊗ Z ⊗ n )= m N X σ ∈ S n + m + p − n + m X i =1 Tr( P R,r Γ ( R ) ( σ )Tr(( p + 1 , σ ( p + 1 , · ⊗ X ⊗ p ⊗ Y ⊗ m − ⊗ Z ⊗ n )= m N X σ ∈ S n + m + p − Tr( P R,r Γ ( R ) ((1 , p + 1) σ )(1 , p + 1))Tr( σ · ⊗ X ⊗ p ⊗ Y ⊗ m − ⊗ Z ⊗ n )The new feature in the above derivation is the presence of the (1 , p + 1) ∈ S n + m + p factorsneeded to swap the removed Y box to the end of the row so that it can be removed, usingthe same manipulations as above. The evaluation of the action of these factors is mosteasily performed using Young’s orthogonal representation, which gives a rule for the actionof adjacent permutations (i.e. permutations of the form ( i, i + 1)) on Young-Yamanouchi(hereafter abbreviated YY) states. Let | Y i denote a YY state, and let | Y ( i ↔ i + 1) i denotethe YY state obtained by swapping boxes i and i + 1. A box in row a and column b hascontent given by b − a . Denote the content of the box in | Y i filled with j by c j . The rule is( i, i + 1) | Y i = 1 c i − c i +1 | Y i + s − c i − c i +1 ) | Y ( i ↔ i + 1) i (3.19)This rule simplifies dramatically in the displaced corners limit, at large N . If the two boxesbelong to the same row we find ( i, i + 1) | Y i = | Y i and if not ( i, i + 1) | Y i = | Y ( i ↔ i + 1) i . This is all that is needed to complete the proof of (3.15) and (3.16) and it proceeds exactlyas for the first rule proved above. Note that because we used simplifications of the large N limit, (3.15) and (3.16) are not exact statements but hold only at large N . The threestatements derived above admit some natural generalizations. For example, we can considertracing over a product of derivatives to obtainTr (cid:18) d k dX k (cid:19) O ~m~pR,r ( X, Y, Z ) = q X i =1 (cid:18) c RR (1) i n i + m i + p i (cid:19) k k − Y a =0 √ p i − a O ~p ( k ) i ~mR ( k ) i ,r ( X, Y, Z ) (3.20)There are obvious generalization when we have a product of Y or Z derivatives. We couldalso allow more than one type of derivative in a given trace, for example (in what follows k = k + k )Tr (cid:18) d k dX k d k dY k (cid:19) O ~m~pR,r ( X, Y, Z )= q X i =1 (cid:18) c RR (1) i n i + m i + p i (cid:19) k k − Y a =0 √ p i − a k − Y b =0 p m i − b O ~m ( k i ~p ( k i R ( k ) i ,r ( X, Y, Z ) (3.21)10y using these formulas for each trace successively, we can also easily evaluate expressionsof this form Tr (cid:18) d k dX k d k dY k (cid:19) · · · Tr (cid:18) d k dZ k (cid:19) O ~m~pR,r ( X, Y, Z ) (3.22)To compare to a multi particle system of q noninteracting particles, again in a 3-dimensionalharmonic oscillator potential, we need to introduce q copies of the oscillators ( I, J = 1 , ..., q ) (cid:2) a ( I ) z , a ( J ) † z (cid:3) = (cid:2) a ( I ) y , a ( J ) † y (cid:3) = (cid:2) a ( I ) x , a ( J ) † x (cid:3) = δ IJ (3.23)one copy for each particle. Each Gauss graph operator O ~m~pR,r is specified by giving the num-ber of Z boxes ( r i ), Y boxes ( m i ) and X boxes ( p i ) in the i th row for i = 1 , ..., q . Thecorresponding multi particle state is | O ~m~pR,r i = q Y I =1 ( a ( I ) † z ) r I √ r I ! ( a ( I ) † y ) m I √ m I ! ( a ( I ) † x ) p I √ p I ! | i (3.24)Using these formulas we can compare (for example) the matrix elements h O ~m~pR ( k ) q ,r ( k ) q | ( a ( I ) z ) k | O ~m~pR,r i (3.25)to the correlation functions h O ~m~p † R ( k ) q ,r ( k ) q Tr (cid:18) d k dZ k (cid:19) O ~m~pR,r i (3.26)to learn that we should identifyTr (cid:18) d k dZ k (cid:19) ↔ q X I =1 s N + m I + n I + p I m I + n I + p I ! k ( a ( I ) z ) k (3.27)In the above formula n I is the number of Z boxes in row I , m I the number of Y boxes and p I the number of X boxes. The general rule is ( k = k + k + k )Tr (cid:18) d k dX k d k dY k d k dZ k (cid:19) ↔ q X I =1 s N + m I + n I + p I m I + n I + p I ! k ( a ( I ) x ) k ( a ( I ) y ) k ( a ( I ) z ) k (3.28)It is easy to check that the ordering of operators inside the trace on the left hand side abovedoes not matter, when acting on the operators we consider, at large N . Multi trace formulasuse the above identification for each trace separately. For exampleTr (cid:18) d k dX k d k dY k d k dZ k (cid:19) Tr (cid:18) d k dX k (cid:19) q X I =1 s N + m I + n I + p I m I + n I + p I ! k ( a ( I ) x ) k ( a ( I ) y ) k ( a ( I ) z ) k × q X J =1 s N + m J + n J + p J m J + n J + p J ! k ( a ( I ) x ) k (3.29)By comparing overlaps between states with polynomials of creation and annihilation opera-tors sandwiched in between and correlators of Gauss graph operators with traces of polyno-mials of the matrices and derivatives with respect to the matrices acting on the Gauss graphoperators as in the examples we studied above, we can build any entry in the dictionarybetween the q particle system and Gauss graph operators with q rows. The description of giant gravitons, constructed using a world volume analysis, allows oneto count the set of all -BPS giant gravitons. This counting matches N bosons in a 3-dimensional harmonic oscillator. It is also possible to define an index to count single traceBPS operators, and it can be computed both at weak coupling (using the gauge theory)and at strong coupling (as a sum over the spectrum of free massless particles in AdS × S ).The results of these different computations are in complete accord. One can compute thespectrum of anomalous dimensions, for operators with a bare dimension of order N , inthe N = 4 super Yang-Mills theory[21, 22, 23]. In this study we have demonstrated thatexactly the same counting (i.e. N bosons in a 3-dimensional harmonic oscillator) results fromcounting operators of vanishing anomalous dimension in this spectrum. Motivated by thisagreement, we have looked for a relation between multi particle wave functions and Gaussgraph operators. Our basic result is that a map between particle wave functions for particlesin a 3-dimensional harmonic oscillator and Gauss graph operators is easily constructed bycomparing overlaps of wave functions of the particle system with correlators of Gauss graphoperators. The correlator computations have made use of significant simplifications thatarise for the BPS Gauss graph operators. The number of particles match the number ofrows in the Young diagram labeling the Gauss graph operator. In our opinion, these resultsprovide concrete evidence that the Gauss graph operators are indeed the operators dual tothe -BPS giant gravitons. To interpret the link between the particle system and the Gaussgraph operators, recall the link between giant gravitons and an eigenvalue description of themulti matrix dynamics, which has been pursued in [32, 33]. Thus, the fact that the matrixmodel computations appear to be related to the dynamics of non-interacting particles giveshints as to how matrix model dynamics may simplify, along the line of the proposals of[34, 35, 36, 37].Any computation of overlaps performed with our wave functions can be mapped into acomputation of Gauss graph correlators. However, the wave function picture does clarify12he structure of the -BPS operators in ways that are not manifest in the Gauss graphdescription. For example, our wave functions make it clear that a Hilbert space for N -BPSsector has a high degree of supersymmetry and so is relatively simple and has often servedas a bridge connecting the gauge theory and supergravity regimes. In the CFT this sectorcan be consistently decoupled resulting in a system that admits a description in terms of freefermions moving in a harmonic oscillator potential. This is well understood from the gaugetheory point of view where the Lagrangian of the decoupled theory is that of a complexmatrix whose eigenvalues obey Fermi-Dirac statistics, with the statistics induced from theintegration measure. On the gravity side the symplectic form of Type IIB SUGRA encodesthe commutation relations that must be imposed to quantize the system. Restricting thissymplectic form to the LLM family of solutions defines a symplectic structure that fixes aquantization and ultimately reproduces the free fermion Hilbert space[38]. In the same waythat free fermion quantum mechanics is equivalent to the singlet sector of a single matrixquantum mechanics, the wave functions we have written down are equivalent to the BPSGauss graph operators. It would be interesting to recover the Hilbert space of our wavefunctions by quantizing using the symplectic form of Type IIB SUGRA, after restricting tothe -BPS family of solutions. Acknowledgements:
RdMK would like to thank Ilies Messamah and Sanjaye Ramgoolamfor many useful discussions on the topic of this study. This work is based upon researchsupported by the South African Research Chairs Initiative of the Department of Science andTechnology and National Research Foundation. Any opinion, findings and conclusions orrecommendations expressed in this material are those of the authors and therefore the NRFand DST do not accept any liability with regard thereto.
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