Anomalous dimensions from boson lattice models

Operators dual to strings attached to giant graviton branes in AdS$_5\times$S$^5$ can be described rather explicitly in the dual ${\cal N} = 4$ super Yang-Mills theory. They have a bare dimension of order $N$ so that for these operators the large $N$ limit and the planar limit are distinct: summing only the planar diagrams will not capture the large $N$ dynamics. Focusing on the one-loop $SU(3)$ sector of the theory, we consider operators that are a small deformation of a ${1\over 2}-$BPS multi-giant graviton state. The diagonalization of the dilatation operator at one loop has been carried out, but explicit formulas for the operators of a good scaling dimension are only known when certain terms which were argued to be small, are neglected. In this article we include the terms which were neglected. The diagonalization is achieved by a novel mapping which replaces the problem of diagonalizing the dilatation operator with a system of bosons hopping on a lattice. The giant gravitons define the sites of this lattice and the open strings stretching between distinct giant gravitons define the hopping terms of the Hamiltonian. Using the lattice boson model, we argue that the lowest energy giant graviton states are obtained by distributing the momenta carried by the $X$ and $Y$ fields evenly between the giants with the condition that any particular giant carries only $X$ or $Y$ momenta, but not both.


Introduction
Motivated by the AdS/CFT correspondence [1,2,3], there has been dramatic progress in computing the planar spectrum of anomalous dimensions in N = 4 super Yang-Mills theory. The planar spectrum is now known, in principle, to all orders in the 't Hooft coupling [4]. This has been possible thanks to the discovery of integrability [5,6] in the planar limit of the theory. This spectrum of anomalous dimensions reproduces classical string energies on the AdS 5 ×S 5 spacetime, in the dual string theory [7].
Much less is known about N = 4 super Yang-Mills theory outside the planar limit. There are many distinct large N but non-planar limits of the theory that could be considered and these correspond to a variety of fascinating physical problems. For example, the problem of considering new spacetime geometries (including black hole solutions) corresponds to considering operators with a bare dimension of order N 2 [8], while giant graviton branes [9,10,11] are dual to operators with a bare dimension of order N . The planar limit does not correctly capture the dynamics of these operators [12,13].
Although much less is known about these large N but non-planar limits, some progress has been made. Approaches based on group representation theory provide a powerful tool, essentially because they allow us to map the problem of the dynamics of the non-planar limit -summing the ribbon graphs contributing to correlation functions -into a purely algebraic problem in group theory. Typically, it can be phrased as the construction of a collection of projection operators and their properties. Once the algebraic problem is properly formulated, systematic approaches to it can be developed. As an example of this approach, bases of local gauge invariant operators have been given [14,15,16,17,18,19,20,21]. These bases provide a good starting point from which the anomalous dimensions can be studied. This is basically because they diagonalize the free field two point function and, at weak coupling, operator mixing is highly constrained [22,23,24,25,26]. The resulting operators have a complicated multi-trace structure, quite different to the single trace structure relevant for the planar limit and its mapping to an integrable spin chain. The spectrum of anomalous dimensions has been computed for operators that are small deformations of 1/2 BPS operators. Problems with 2 distinct characters have been solved: It is possible to simply treat all fields in the operator on the same footing, construct the basis and then diagonalize [27,28,29,30] or alternatively, one can build operators that realize a spacetime geometry or a giant graviton brane and use words constructed from the fields of the CFT to describe string excitations [22,31,32]. In the approach that treats all fields on the same footing, one simply defines the operators of the basis and considers the diagonalization of the dilatation operator with no physical input from the dual gravity description. When considering states dual to systems of giant gravitons, the Gauss Law of the dual giant world volume gauge theory emerges, so that in this approach we see open string and membranes are present in the CFT Hilbert space. When using words to describe string excitations, computations in the CFT reproduce the classical values of energies computed in string theory [31,32], the worldsheet S-matrix [33] and has lead to the discovery of integrable subsectors for closed string excitations of certain LLM backgrounds [32]. Clearly, this is a rich problem with hidden simplicity, so that further study of these limits are bound to be fruitful. The existence of this hidden simplicity is not unexpected: conventional lore of the large N limit identifies 1/N as the gravitational interaction, so that the N → ∞ limit, in which this interaction is turned off, should be a simple limit. One next step that can be contemplated, is to go beyond small perturbations of the 1/2 BPS sector. This problem is our main motivation in this study, and we will take a small step in this direction. We will study operators constructed from three complex adjoint scalars X, Y , Z of N = 4 super Yang-Mills theory. Operators that are a small perturbation of a 1/2 BPS operator are constructed using mainly Z fields. For these operators, interactions between the X, Y fields are subdominant to interactions between X, Z and between Y, Z fields and can hence be neglected. As we move further from the original 1/2 BPS operator, more and more X, Y fields are added. At some point the interactions between the X, Y fields can no longer be neglected. Dealing with these interactions is the focus of our study. We will argue that this is a well defined problem, that can be solved, often explicitly. This is accomplished by phrasing the new X, Y interactions as a lattice model, for essentially free bosons. Thus, we finally land up with a simple problem that is familiar and can be solved. This is the basic achievement of this paper.
Our results show a fascinating structure that deserves to be discussed. The mapping to the lattice model associates a harmonic oscillator to both the X field and to the Y field. Earlier results [29] treating the leading term, performed the diagonalization by associating a harmonic oscillator to the Z field, so that in the end we seem to be seeing an equality in the description of the three scalar fields. An even-handed treatment of all three fields is a big step towards being able to treat operators constructed with equal numbers of X, Y and Z fields. This would most certainly go beyond the 1 2 -BPS sector, the main motivation for our study.
In the next section we review the action of the one loop dilatation operator D 2 . The action of D 2 in the SU (3) sector, in the Schur polynomial basis, has been evaluated previously [34] and we simply quote and use the result. We then move to the Gauss graph basis of [30], in which the terms in D 2 arising from Z, Y or Z, X interactions are diagonal. Again, this is a known result and we simply use it. The Gauss graph basis has a natural interpretation in terms of giant graviton branes and their open string excitations. We will often use this language of branes and strings. We then come to the central term of interest: the term in D 2 arising from X, Y interactions. Denote this term by D XY 2 . We will carefully evaluate this term, arriving at a rather simple formula, which is the starting point for section 3. The explicit expression for D XY 2 can easily be identified with a lattice model for a collection of bosons. The giant gravitons define the sites of this lattice, and the open string excitations determine the lattice Hamiltonian. Section 4 diagonalizes the dilatation operator for a number of giants plus open string configurations, arriving at detailed and explicit expressions both for the anomalous dimensions and for the operators of a definite scaling dimension. Our conclusions and some discussion are given in section 5.

Action of the One Loop Dilatation Operator
We combine the 6 hermitian adjoint scalars of N = 4 super Yang-Mills theory into three complex combinations, denoted X, Y, Z. The operators we consider are constructed using n Zs, m Y s and p Xs. Operators that are dual to giant graviton branes are constructed using n + m + p ∼ N fields. We will focus on operators that are small deformations of 1/2 BPS operators, achieved by choosing n m + p. We will fix m p ∼ 1 as N → ∞ and treat m n as a small parameter. The collection of operators constructed using X, Y, Z fields are often referred to as the SU (3) sector. This is not strictly speaking correct since these operators do mix with operators containing fermions. At one loop however, this is a closed sector.
Our starting point is the action of the one loop dilatation operator of the SU (3) sector on the restricted Schur polynomial basis. This has been evaluated in [34]. Further, the terms D Y Z 2 and D XZ 2 have been diagonalized. The operators of a definite scaling dimension O R,r (σ), called Gauss graph operators [28,30], are labeled by a pair of Young diagrams R n + m + p and r n as well as a permutation σ ∈ S m × S p . Although these labels arise when diagonalizing D Y Z 2 and D XZ 2 in the CFT, they have a natural interpretation in the dual gravitational description in terms of giant graviton branes plus open string excitations. A Young diagram R that has q rows corresponds to a system of q giant gravitons. The Y and X fields describe the open string excitations of these giants, so that there are m + p open strings in total. We can describe the state of the system using a graph, with nodes of the graph representing the branes (and hence rows of R) and directed edges of the graph describing the open string excitations (represented by X and Y fields in the CFT). Each directed edge ends on any two (not necessarily distinct) of the q branes. The only configurations that appear when D Y Z 2 and D XZ 2 are diagonalized have the same number of strings starting or terminating on any given giant, for the X and Y strings separately [30,34]. Thus the Gauss Law of the brane world volume theory implied by the fact that the giant graviton has a compact world volume [35] emerges rather naturally in the CFT description. Since every terminating edge endpoint can be associated to a unique emanating endpoint, we can give a nice description of how the open strings are connected to the giants by specifying how the terminating and emanating endpoints are associated. The permutation σ ∈ S m × S p describes how the m Y 's and the p X's are draped between the q giant gravitons by describing this association [30,34]. The explicit form of the Gauss graph operators is [30,34] O m, p R,r (σ) = Each box in R is associated with one of the complex fields. r is a label for the Z fields. The graph σ encodes important information. The number of Y (or X) strings terminating on the ith node which equals the number of Y (or X) strings emanating from the ith node is denoted by m i (or p i ). m i (or p i ) also counts the number of boxes in the ith row of R that correspond to Y (or X) fields. We will often assemble m i and p i into the vectors m and p. The number of Y (or X) strings stretching between nodes i and k is denoted m ik (or p ik ), while the number of strings stretching from node i to node k are denoted m i→k (or p i→k ). A Young diagram with k boxes a k labels an irreducible representation of S k with dimension d a . The branching coefficients B resolve the operator that projects from (s, t), with s m, t p, an irreducible representation of S m ×S p , to the trivial (identity) representation of the product group jk (σ) is a matrix (with row and column indices jk) representing σ ∈ S m × S p in irreducible representation (s, t). The operators O R,(r,s,t) µ 1 µ 2 are normalized versions of the restricted Schur polynomials [18] which themselves provide a basis for the gauge invariant operators of the theory. The restricted characters χ R,(t,s,r) µ 1 µ 2 (σ) are defined by tracing the matrix representing group element σ in representation R over the subspace giving an irreducible representation (r, s, t) of the S n × S m × S p subgroup. There is more than one choice for this subspace and the multiplicity labels µ 1 µ 2 resolve this ambiguity, for the row and column index of the trace. The operators O R,(t,s,r) µ 1 µ 2 given by have unit two point function. hooks r stands for the product of hook lengths of Young diagram r and f R stands for the product of the factors of Young diagram R. The action of the dilatation operator on the Gauss graph operators is [28,30,34] [29]. We will now spell out the action of the operators ∆ + ij , ∆ 0 ij and ∆ − ij . Denote the row lengths of r by l r i . The Young diagram r + ij is obtained by deleting a box from row j and adding it to row i. The Young diagram r − ij is obtained by deleting a box from row i and adding it to row j. In terms of these Young diagrams we have are not yet diagonal: they still mix operators with different R, r labels. This last diagonalization however, is rather simple: it maps into diagonalizing a collection of decoupled oscillators as demonstrated in [29]. We will call these Z oscillators, since they are associated to the r label which organizes the Z fields. It is clear that D XY 2 does not act on the r label so that in the end, the contribution from D XY 2 simply shifts the ground state eigenvalue of the Z oscillators.
We will now focus on the term D XY 2 . Recall that our operators are built with many more Z fields, than X or Y fields (n p + m). Since this term contains no derivatives with respect to Z it is subleading (of order m n ) when compared to D Y Z 2 and D XZ 2 . Diagonalizing this operator is the main goal of this article, so it is useful to sketch the derivation of the matrix elements of D XY 2 in the Gauss graph basis. We will simply quote existing results that we need, giving complete details only for the final stages of the evaluation, which are novel. The reader will find useful background material in [34]. The action of this term on the restricted Schur polynomial basis was computed in [34]. The result is Young diagram R is obtained from Young diagram R by dropping a single box, with c RR denoting the factor of this box. I T R , I R T , P 1 and P 2 are intertwining maps. I T R maps from the carrier space of R to the carrier space of T . It is only non-vanishing if T and R are equal as Young diagrams implying that operators labeled by R and T can only mix if they differ by the placement of a single box. The operators P 1 and P 2 are the intertwining maps used in the construction of the restricted Schur polynomials. It is challenging to evaluate the above expression explicitly, basically because it is difficult to construct P 1 and P 2 . However, the above expression has not yet employed the simplifications of large N . To do this, following [28] we will use the displaced corners approximation. After applying the approximation we obtain[34] The trace in this expression is over the tensor product V ⊗n+m p where V p is the fundamental representation of U (p). The intertwining maps used to define the restricted Schur polynomials (P 1 and P 2 above) factor into an action on the boxes associated to the Z fields, an action on the boxes associated to the Y fields and an action on the boxes associated to the X fields. The intertwining maps 1 P ( p, m) tα 1 β 1 ;sα 2 β 2 and P ( p , m ) wµ 1 ν 1 ;vµ 2 ν 2 are the actions of the intertwining maps on the X and Y fields only. This happens because the trace over the Z field indices, which is simple as the dilatation operator D XY 2 does not act on the Z fields, has been performed. Young diagram R i is obtained from R by dropping a single box from row i and T k from T by dropping a single box from row k.
The result (2.13) gives the D XY 2 term in the dilatation operator, as a matrix that must be diagonalized. As we will see, all three terms in D 2 are simultaneously diagonalizable at large N so that it is convenient to employ the Gauss graph basis which already diagonalizes both D ZY 2 and D ZX 2 . The problem of diagonalizing D XY 2 then amounts to a diagonalization on degenerate subspaces of D ZY 2 and D ZX 2 . Thus, the original diagonalization of an enormous matrix is replaced by diagonalizing a number of smaller matrices -a significant simplification. Applying the results of [34], we find that, after the change in basis Here the Gauss graph operatorsÔ m, p R,r (σ 1 ) are normalized to have a unit two point function. They are related to the operators introduced in (2.3) as follows Introduce the vectors (v (i) ) a = δ ia which form a basis for V p . The vector | p, m is defined as follows We will now explain how the sums over ψ 1 and ψ 2 in (2.15) can be evaluated. This discussion is novel and is one of the new contributions of this paper. Consider the term The dependence on the permutations σ 1 , σ 2 can be simplified with the following change of variables: replace ψ 2 withψ 2 wherẽ After relabelingψ 2 → ψ 2 and taking the transpose of the first factor which is a real number, we find Indicate this explicitly as follows To simplify this expression further, note that E ii | p, m is only non-zero if vector v (i) occupies slot one in the vector | p . In this case E (1) ii | p, m = | p, m . Since ψ 1 and ψ 2 shuffle the vectors in | p, m into all possible locations, E (1) ii will in the end count how many times the vector v (i) appears in | p, m . This is given by p i introduced above. A similar argument applies to E (p+1) | p, m , Thus, we obtain Now, perform the following change of summation variables ψ 1 →ψ 1 with The summand is now independent of ψ 2 so that after summing over ψ 2 and relabelingψ 1 → ψ 1 we find Summing over ψ 1 now gives We also need to consider the term shows that T 4 = T 1 and hence The next sum we consider is the sum becomes Change variables ψ 2 → ρ with ρ = ψ −1 1 ψ 2 and relabel ρ → ψ 2 to find We will useb to denote the q dimensional vector that has all entries zero except the bth entry which is 1. For a non-zero contribution the first factor requires For case 1 Consider the sum over ψ 1 . Due to the factor E ψ −1 1 (p+1) ki we get a non-zero contribution from the slots p + 1, p + 2, · · · , p + m (a Y string) if a string starts from node k and ends at node i. Thus, the sum over ψ 1 gives For case 2 Consider the sum over ψ 1 . We get a non-zero contribution for each Y string starting from node k which ends at node i. After summing over ψ 1 we have Armed with these sums, we now obtain a rather explicit expression for the matrix elements of D XY 2 in the Gauss graph basis This is the key result of this section and one of the key results of this paper.
We will now describe how the above matrix can be diagonalized.

Boson Lattice
Our goal in this section is to diagonalize (2.28). This is achieved by interpreting (2.28) as the matrix elements of a Hamiltonian for bosons on a lattice. Towards this end, first note that the matrix elements M m, p R,r,σ 1 T,t,σ 2 are only non-zero if we can choose coset representatives such that σ 1 and σ 2 describe the same element of S m × S p . This implies that the brane-string systems described by σ 1 and σ 2 differ only in the number of strings with both ends attached to the same brane, but not in the number of string stretching between distinct branes. This already implies that the contribution D XY The number of strings stretching between the branes m ik (for Y strings) and p ki (for X strings) are the same for both systems so that It is the number of closed loops (m ii for Y loops and p ii for X loops) that can differ between the operators that mix. Finally, we have introduced the notation From the structure of the operator mixing problem, we would expect that M m, p R,r,σ 1 T,t,σ 2 = M m, p T,t,σ 2 R,r,σ 1 . This is indeed the case, as a consequence of the easily checked identity which holds for any i, k.
The lattice model consists of two distinct species of bosons, one for X and one for Y , hopping on a lattice, with a site for every brane, or equivalently, a site for every row in the Young diagram R labeling the Gauss graph operatorÔ m, p R,r (σ). The bosons are described by the following commuting sets of operators Using these boson oscillators, we have The vacuum of the Fock space |0 obeys The Hamiltonian of the lattice model is given by Notice that this Hamiltonian is quadratic in each type of oscillator. It has a nontrivial repulsive interaction given by the i a † i a i b † i b i term, which makes it energetically unfavorable for a and b type particles to sit on the same site. Also, the full Fock space is a tensor product between the Fock space for the a oscillator and the Fock space for the b oscillator. We will use the occupation number representation to describe the boson states. To complete the mapping to the lattice model, we need to explain the correspondence between Gauss graph operators and states of the boson lattice. This map is given by reading the boson occupation numbers for each site from the number of closed strings with both ends attached to the node corresponding to that site. In the next subsection we consider an example which nicely illustrates this map.
Finally, lets make an important observation regarding (3.9). Although the eigenvalues of this Hamiltonian are subleading contributions to the anomalous dimension, there is an important situation in which this correction is highly significant: for BPS states the leading contribution to the anomalous dimension vanishes and this subleading correction is important. The BPS operators are labeled by Gauss graphs that have p ik = m ik = 0 whenever i = k, i.e. there are no strings stretching between branes. In this case, it is clear that (3.9) vanishes so that the BPS operators remain BPS.

Example
In this section we will consider an example for which R has q = 3 rows and p = m = 3. In this problem, 10 operators mix. The Gauss graph labels for the operators that mix are displayed in Figure 1. Figure 1: Each Gauss graph label is composed of two graphs, the first for the X strings and the second for the Y strings. Each graph has 3 nodes (because q = 3). There are no b type particles because there are no closed X strings. There are 3 a type particles because there are three closed Y strings. All operators share the same r label.
For the Gauss graph operators shown, we have the following correspondence with boson lattice states It is now rather straight forwards to compute matrix elements of the lattice Hamiltonian. For example It is instructive to compare this to the answer coming from (2.28). To move from state 2 to state 1, a string must detach from node 2 and reattach to node 3. Thus, we should plug i = 2 and k = 1 into (2.28). The Gauss graph σ 1 corresponds to |1 while σ 2 corresponds to |2 . In addition R 2 = T 1 and from the Gauss graphs we read off p 32 = 1 and m 22 (σ 1 ) = 1. It is now simple to see that (2.28) is in complete agreement with the above matrix element. Finally, the state corresponding to the Gauss graph in Figure 2 is

Diagonalization
In this section we will consider a class of examples that can be diagonalized explicitly. Our main motivation is to show that working with the lattice is simple, so the mapping we have found is useful.

Exact Eigenstates
For these examples take with A and B two positive integers. For examples of Gauss graphs that obey this condition, see Figure 3. There are two cases we will consider: we will fix the number of a particles to zero and leave the number of b particles arbitrary, or, fix the number of b particles to zero and leave the number of a particles arbitrary. We will also specialize to labels R that have the difference between any two row lengths In this case our lattice Hamiltonian simplifies to This Hamiltonian is easily diagonalized by going to Fourier space. Indeed, in terms of the new oscillators the Hamiltonian becomes (we have set the number of a particles to zero) Eigenstates of the lattice Hamiltonian are given by arbitrary momentum space excitations where the occupation numbers α n , β n are arbitrary. This state can be translated back into the Gauss graph language to give operators of a definite scaling dimension.

General Properties of Low Energy Eigenstates
In this section we will sketch the features of generic low energy states of the lattice Hamiltonian. We begin by relaxing the constraint that only one species is hopping. In the end we will also make comments valid for the general Gauss graph configuration. The Hamiltonian becomes The constant E 0 = 2ABq is not important for the dynamics but must be included to obtain the correct anomalous dimensions. To start, consider H a which is a kinetic term for the a particles. The first term in the Hamiltonian implies that it costs energy to have an a particle occupying a site, while the second and third terms tell us this energy can be lowered by hopping between sites i and i + 1. Consequently, to minimize H a , the a particles will spread out as much as is possible. This is in perfect accord with the results of the last section. The lowest energy single particle state is the zero momentum state, which occupies each site with the same probability: the particle spreads out as much as is possible. Very similar reasoning for H b implies that the b particles will also spread out as much as is possible. Finally, the term H ab is a repulsive interaction, telling us that it costs energy to have as and bs occupying the same site. So there is a competition going on: The terms H a and H b want to spread the as and bs uniformly on the lattice which would certainly distribute as and bs to the same site. The term H ab wants to ensure that any particular site will have only as or bs but not both. Who wins?
Consider a thermodynamic like limit where we consider a very large number of both species of particles, n a and n b . In the end, the low energy state will be a "demixed" state with no sites holding both as and bs. To see this, note that H a grows like n a and H b like n b . This is much smaller than the growth of the term H ab which grows like n a n b , so the repulsive interaction wins. This conclusion is nicely borne out by numerical results for the two component Bose-Hubbard model [36,37]. The ground state phase diagram of the Hamiltonian of [36], shows four distinct phases: double super fluid phase, supercounterflow phase, demixed Mott insulator phase and a demixed superfluid phase. Comparing our Hamiltonian to that of [36], we are always in the demixed superfluid phase: the a and b particles do not mix, but are free to move in their respective domains.
For the generic Gauss graph, with any choices for the values of m ik and p ik , it is clear that H a and H b will still cause the a and b particles to spread out as much as possible. The term H ab will again dominate when we have large numbers of as and bs so we again expect a demixed gas. We can translate this structure of the generic state back into the language of the giant graviton description. Up to now we have considered dual giant gravitons which correspond to operators labeled by Young diagrams with long rows. Recall that dual giant gravitons wrap an S 3 ⊂AdS 5 . In this context, l R 1 is the momentum of each giant and N + l R 1 is the radius on the LLM plane at which the giant orbits. The Hamiltonian for giant gravitons, which wrap S 3 ⊂S 5 is given by These operators are labeled by Young diagrams with long columns. The giants orbit on the LLM plane with a radius of N − l R 1 . The X and Y fields are each charged under different U (1)s of the R-symmetry group. The Rsymmetry of the CFT translates into angular momentum of the dual string theory, so that attaching the particles to a given giant corresponds to giving the giant angular momentum. The lowest energy giant graviton states are obtained by distributing the momenta carried by the X and Y fields evenly between the giants with the condition that any particular giant carries only X or Y momenta, but not both. These conclusions hold for the generic state where there are enough p ik and m ik non-zero, allowing the Xs and Y s to hop between any two giants, possibly by a complicated path. Thus in the end we see that the mapping to the boson lattice model has allowed a rather detailed understanding of the operator mixing problem.

Conclusions
In this article we have studied the action of the one loop dilatation operator D 2 on Gauss graph operators O m, p R,r (σ) which belong to the SU (3) sector. The term we have studied, D XY 2 , is diagonal in the r label, mixing operators labeled by distinct graphs. It makes a subleading contribution as compared to D XZ on r leads to a collection of decoupled harmonic oscillators, which we refer to as the Z oscillators, since the r label is associated with Z. The spectrum of the Z oscillators gives the leading contribution to the anomalous dimensions. The new contribution that we have studied in this paper can also be mapped to a collection of oscillators, describing a lattice boson model. This is done by introducing two sets of oscillators, the X and Y oscillators associated to the X and Y fields. Diagonalizing the X and Y oscillators breaks degeneracies among different copies of Z oscillators and leads to a constant addition to their ground state energy. This is then a constant shift of the anomalous dimension. Although this shift is subleading (it is of order m n ), it could potentially show that certain states are not in fact BPS. This was investigated in detail and it turns out that states that are BPS (their leading order anomalous dimension vanishes) at leading order, remain BPS when the subleading correction is computed (it too vanishes).
The mapping that we have found to a lattice boson model has achieved an enormous simplification of the operator mixing problem and we have managed to understand it in some detail. Indeed, using the lattice boson model, we have argued that the lowest energy giant graviton states are obtained by distributing the momenta carried by the X and Y fields evenly between the giants with the condition that any particular giant carries only X or Y momenta, but not both. Since states with two charges are typically 1/4-BPS while states with 3 charges are typically 1/8-BPS, it maybe that the solution is locally trying to maximize susy. It would be interesting to arrive at the same picture, employing the dual string theory description.
Perhaps the most interesting consequence of our results is that they suggest ways in which one can go beyond the 1/2 BPS sector. Indeed, all three types of fields considered have been mapped to oscillators, so perhaps there is a more general description of this sector that treats all three types of oscillators on the same footing. This would relax the constraint n p + m which allows for operators that are far from the 1/2 BPS limit. Deriving this picture is a fascinating open problem, since it will require that we go beyond the displaced corners approximation, or alternatively, that we generalize it.
As a final comment, recall that Mikhailov [38] has constructed an infinite family of 1/8 BPS giant graviton branes in AdS 5 ×S 5 . Quantizing the space of Mikhailov's solutions leads to N non-interacting bosons in a harmonic oscillator [39,40,41]. It is tempting to speculate that it is precisely these oscillators that we are uncovering in our study; for evidence in harmony with this suggestion see [42]. It would be interesting to make this speculation precise.