Oscillating multiple giants

We propose a new example of the AdS/CFT correspondence between the system of multiple giant gravitons in AdS5× S5 and the operators with O(Nc) dimensions in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 super Yang-Mills. We first extend the mixing of huge operators on the Gauss graph basis in the su2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{su}(2) $$\end{document} sector to all loops of the ’t Hooft coupling, by demanding the commutation of perturbative Hamiltonians in an effective U(p) theory, where p corresponds to the number of giant gravitons. The all-loop dispersion relation remains gapless at any λ, which suggests that harmonic oscillators of the effective U(p) theory should correspond to the classical motion of the D3-brane that is continuously connected to non-maximal giant gravitons.


Introduction
Traditionally the AdS/CFT correspondence has been studied in the planar large N c limit [1]. Whether AdS/CFT holds true in a non-planar but still large N c limit is a challenging question. Such a nontrivial limit can be implemented by deforming the background theory or spacetime, or by introducing semiclassical objects carrying the dimensions or energy of order N c .
One of the most studied examples of AdS/CFT is the correspondence between string theory on AdS 5 × S 5 and N = 4 super Yang-Mills (SYM) theory. We depart from the planar region of N = 4 SYM by studying huge operators whose dimensions are comparable to the rank of the gauge group N c . 1 The operators with O(N 2 c ) dimensions correspond to the Lin-Lunin-Maldacena (LLM) geometry at strong coupling [3], while those with O(N c ) dimensions correspond to the giant graviton [4]. This correspondence continues to non-BPS operators in both of the O(N 2 c ) and O(N c ) cases. For the former case, an isomorphism between non-BPS states was conjectured between the LLM geometry and N = 4 SYM [5][6][7][8][9]. For the latter case, non-BPS states around the giant graviton are less well-understood. This is the main subject of this paper.
Let us first review recent progress on the weak coupling side. In AdS/CFT, the half-BPS operators with huge dimensions should be organized through the operator basis labeled by a Young diagram [10]. Similarly, a convenient way to describe non-BPS operators with huge dimensions is the restricted Schur basis, labeled by a set of Young diagrams [11,12]. The dilatation operator expressed in this basis mixes the Young diagrams with different shapes.
For simplicity, consider the operators in the su(2) sector, which consists of complex scalars Z and Y of N = 4 SYM. Suppose that a small number of Y 's are added to a large number of Z's. If the Young diagram representing Z's has p long columns, this type of operators roughly corresponds to a system of p spherical giant gravitons in AdS 5 × S 5 . The Young diagram representing Y 's describes a small fluctuation of the giant gravitons. 2 The one-loop mixing in this setup is remarkably simple. First, the number of columns p does not change at large N c , because giant gravitons are semi-classical objects at strong 1 Another promising approach to a non-planar large Nc limit is the localization, which is valid at any Nc and can extract non-BPS data [2]. 2 Here Y 's and Z's constitute a huge Young diagram whose shape wildly fluctuates due to the operator mixing. This situation is different from a single-trace operator coupled to det(Z), where the operator mixing does not spoil the color structure at the leading order of large Nc.

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coupling [13]. Second, the operator mixing splits into the mixing of Z's and the mixing of Y 's. Third, the one-loop spectrum eventually reduces to a set of p decoupled harmonic oscillators [14,15]. The last observation is called the non-planar integrability in the literature.
We should emphasize that the mixing problem of huge operators is quite different from the planar mixing problem, and the development of sophisticated techniques has been crucial. The mixing problem of Y 's is solved by the Gauss graph basis, which counts the number of open strings ending on different giant graviton branes [16]. The same technique can be used to simplify the mixing in the su(3) and su(2|3) sectors [17,18]. Explicit computation of the mixing matrix has been given up to two loops in g 2 YM in [19]. There are also trial studies at higher loops [20,21]. More generally, the mixing problem corresponding to p giant gravitons can be described by an effective U(p) theory [22,23]. The Hamiltonian of the effective U(p) theory has the symmetry U(1) p in the distant corners approximation, namely when the differences of the length of the adjacent columns are large.
Next, the key developments on the strong coupling side are summarized.
The D-brane motion is described by a low energy effective action which consists of Dirac-Born-Infeld (DBI) and Chern-Simons (CS) terms [24]. The giant graviton is a classical solution of the D3-brane action moving in the AdS 5 × S 5 background. The spherical giant graviton wraps S 3 inside S 5 [4], and the AdS giant wraps S 3 inside AdS 5 [25,26]. The quantum fluctuation modes around the giant graviton have been studied in [27].
The open strings ending on the giant graviton have been studied from two viewpoints. In the first viewpoint, we replace open strings with U(1) flux and study the D-brane. The classical motion of a D3-brane in such a background spacetime has been studied in the flat space [28], in the pp-wave [29] and in AdS 5 × S 5 [30]. The U(1) gauge fields typically become spiky, and they diverge at the location where open strings end on the D-brane. In the second viewpoint, we study the open string as a classical integrable system [31], or a boundary integrable system [32]. 3 In both points of view, the brane-string system typically has divergent energy, which is canceled by a divergent angular momentum of open strings, just like the giant magnon [36].
It is expected that the system of open strings with p giant gravitons corresponds to the effective U(p) theory, but there is still obscurity in this understanding as AdS/CFT. The purpose of this paper is to understand this theory more precisely by revisiting the analysis both in gauge and string theories.
In section 2 and 3, we study the perturbative Hamiltonians of the effective U(p) theory on the gauge theory side in detail. Possible forms of the effective Hamiltonian are constrained by the GL(p) algebra, and by demanding that the perturbative Hamiltonians at each loop order commute with each other. We find that there are at most ( + 1) linearly-independent mutually-commuting operators at -loops. In the continuum limit, these candidate operators reduce to the harmonic oscillators at one loop, which allows us

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to conjecture an all-loop ansatz, where we put p = 2 for simplicity, andf (λ) is an unknown function of the 't Hooft coupling λ = N c g 2 YM . Our ansatz (1.1) predicts two remarkable consequences. First, the anomalous dimensions remain non-zero at the leading order of large N c , because m can be an integer of O(N c ). Second, the excitations are gapless. Recall that the energy of an open string attached on the Z = 0 giant gravitons is gapped, because the open string stretching on S 5 carries non-zero energy, equal to the length times tension. 4 This disagreement indicates that the previous analyses on the LLM geometry do not immediately apply to the system of multiple giant gravitons. What does the all-loop ansatz represent at strong coupling?
In section 4 we revisit a classical single D3-brane wrapping S 3 inside AdS 5 × S 5 , and solve the equations of motion around the BPS spherical giant graviton. Following the steps similar to the stability analysis of [27], we found two types of classical solutions oscillating around the BPS giant gravitons. The first type is a point-like D-brane, and the second type is a fuzzy D-brane with non-trivial KK modes on S 3 . The energy of the latter solution is where c k is a numerical constant that remains finite as k → ∞. We argue that the latter solution (1.2) is a good candidate for the string theory state corresponding to the all-loop harmonic oscillator (1.1) at strong coupling. Our reasoning will be presented in section 5. This paper is supplemented by Mathematica files used for the computations in sections 3, 4.

Mixing of huge operators in N = 4 SYM
We collect known facts about the perturbative mixing of huge operators in N = 4 SYM. Our notation and basis facts about the Gauss graph basis are summarized in appendix A.

Perturbative dilatation operator
We express the perturbative dilatation operator in the su(2) sector of N = 4 SYM by where [37] D 0 = Tr YY + Tr ZŽ (2.2) Here the energy is measured in the unit of string tension √ λ and not in the D3-brane tension Nc/gs ∼ N 2 c /λ. The string with a finite length produces a gap in the dispersion relation, even if gs 1 or λ Nc.

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The fields (Φ,Φ) satisfy the U (N c ) Wick rules Tr (AΦBΦ) = Tr (A) Tr (B), Tr (AΦ) Tr (BΦ) = Tr (AB), The explicit form of D has been known up to five loops [38,39], (1). The operators in the Gauss graph basis are denoted by as in (A.20), where l specifies the Young diagram for Z. Let us write D acting on the Gauss graph operators by D G . At the leading order of large N c , these dilatation operators are given by [13,16,19,40] where h(i, l i ) is the box weight for spherical giants, We find it convenient to keep O(1) terms in (2.9) although h(i, l i ) ∼ h(i, l i ∓ 1) at large N c . An important feature of D G is that it consists of a sum over a pair of indices (i, j), and the third row/column does not show up. We can simplify the difference operators in (2.9) by introducing (2.11) These operators satisfy the relations 5 We slightly modified the result of [19] which computed only the first term of D2 in (2.4). We added the third term of D2 and symmetrized ∆ (2) ij with respect to i ↔ j, owing to nij(σ) = nji(σ). Note that we sum over i = j and not i < j.

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which follow from (2.14) We can rewrite the difference operators in (2.8) as If we introduce The dilatation operator D G written in terms of {d + i , d − i } can be regarded as the Hamiltonian of an effective U(p) theory. This is because (2.20) We assume that n ij (σ) are general non-negative integers. Then the perturbative Hamiltonians in (2.17), (2.18) are invariant only under U(1) p .

Commutation relations
The one-and two-loop dilatations at p = 2 in (2.17), (2.18) satisfy the relation and the perturbative dilatation operators on the Gauss graph basis commute. This is a surprising feature because the planar dilatation operators (on the single-trace operators) at one-and two-loop do not commute [37]. 7 6 Roughly speaking, Eij is a p × p matrix whose entries are zero except at the i-th row, j-th column.
The Hermitian combinations generate U(p). 7 In general, the operator C(g) = A + gB has the eigenvectors independent of g when [A, B] = 0.
Conversely, if [A, B] = 0, the matrix elements of A and B in the eigenbasis of C(g) have off-diagonal elements depending on g.

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Let us take a closer look at the situation for p > 2. The condition D G 1 , D G 2 = 0 reduces to [H 1 , H 2 ] = 0. If we impose this condition for any {n ij (σ)}, we get and hence D G 1 , D G 2 = 0 for p > 2. We will see in section 3 that this is a generic feature of effective U(p) theory Hamiltonians under some ansatz, and not due to potentially missing terms in D G 2 . This trouble can be solved in the following way. In the displaced corners approximation, we truncate the Hilbert space to a fixed number of columns, then take the large N c limit. We obtained the Hamiltonians D G after the Hilbert space truncation, but without taking the limit. In fact, in the continuum limit we find showing that taking the large N c limit is a necessary step. Note that there is a caveat in the displaced corners approximation. Even if we pick up an operator in the distant region l 1 l 2 , the difference (l 1 −l 2 ) keeps decreasing due to the operator mixing, until it hits the Young diagram constraints l 1 ≥ l 2 . The original one-loop mixing matrix no longer takes the simple form (2.7) when two columns have comparable lengths. We expect that these boundary effects on the anomalous dimensions are negligible at the leading order of large N c .
We begin with the ansatz for the dilatation eigenstates, (2.27) where we specify the column lengths of r 1 by (l 1 , l 2 , . . . , l p ), and Σ means the sum over {l i } under the constraints shown in (2.27). The action of the operators (2.11) on O f can be written as

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where means that we neglect potential contributions from the boundary of the summation range. Consider the following large N c limit which is similar to the limit discussed in [13]. We prefer the square-root scaling l i ∼ O( √ N c ) to the linear scaling l i ∼ O(N c ), because the difference equations are rather trivial in the latter limit. Physically, the system (2.29) consists of one nearly maximal, and (p − 1) far-from maximal spherical giant gravitons. The constraint i l i = n Z becomes somewhat trivial because n Z ∼ l 1 .
We introduce the rescaled variables and functions We keep y 1 to simplify our notation, even though In the continuum limit, the difference operators H 1,ij , H 2,ij in (2.17), (2.18) become where y ij = y i − y j . This suggests that the one-loop and two-loop dilatations commute in this limit, The spectrum of D G 1 is discussed in detail in appendix B.

All-loop ansatz
We conjecture that perturbative dilatation operators at all loops in the continuum limit (2.29) takes the form (3.1) A related argument was given in [20], where they showed that the mixing of Y 's at higher loops takes the same form as the one-loop mixing.

Constraints on higher-loop dilatations
We expand the dilatation operator on the Gauss graph basis at weak coupling as and make the following ansatz for D G in the leading order of large N c , where {x k } are numerical constants of O(1). The first equation (3.3) means that the -loop dilatation contains lower-loop difference operators multiplied by powers of N c . The second equation means that H depends only on a pair of column labels (i, j) coupled to n ij (σ). We impose this condition because n ij (σ) should count the number of open string modes stretching between the i-th and j-th giant graviton brane. We further assume that where {x ,m } are numerical constants of O(1), and P ,m is a polynomial of the difference operators. The form of P ,m originates from the perturbative dilatation operators of N = 4 SYM discussed in section 2.1. It is known that there is a correspondence between the terms of ∆ (2) ij in (2.8) and those of D 2 , according to the two-loop computation [19]. 8 Since the -loop dilatation operator D should remove at most fields and add fields, we arrive at the ansatz of P ,m in (3.4). 9 Let us revisit the commutation relations in section 2.2. Now we impose by generalizing (2.26). This is a crucial assumption in our discussion, and should be justified in future. The equation (3.5) implies that the mixing problem can be "one-loop exact" in the sense that the eigenvectors remain unchanged at higher loop orders. 10 The first equation of (2.24) is generalized to It is straightforward to enumerate all possible solutions of (3.6), or equivalently the general form of P ,m , with the help of Mathematica. 8 For example, (d † i ) 2 d 2 j removes two boxes from the j-th column and add two boxes to the i-th column. This term comes from (∆ + ij ) 2 which roughly corresponds to Tr (ZZWŽŽW ). 9 We can also explain the powers of Nc in (3.3) from the fact that D removes fields and adds fields. 10 The author thanks an anonymous referee of JHEP for correcting mistakes in the previous version and emphasizing this point.

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It turns out that at -loop, there are ( +1) independent solutions of the equation (3.6). At one-loop, there are two solutions where H 1,ij is given in (2.17) and We also define At two-loop, there are three solutions, where H 2,ij is given in (2.18) and At higher loops, we find that all solutions at -loop can be written in the form We checked that no more solutions exist up to four-loop. The two-parameter family of difference operators (3.16) mutually commute, which follows from (3.10) and (3.11). Our ansatz for the -loop dilatation in (3.3) becomes The second equation of (2.24) generalized to higher loops reads Some of Q ab,ij in (3.16) up to two loops satisfy these conditions. At three-loops, no linear combinations of (Q 3,0 , Q 2,1 , Q 1,2 , Q 0,3 ) satisfy (3.19) against the two-loop dilatation. Thus, we should trust the discrete form of our all-loop ansatz (3.3) only at p = 2.

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This conclusion is not surprising. The effective Hamiltonian H ,ij is a linear combination of Q ab,ij in (3.16), which are polynomials of J ij . However, J 12 and J 23 as the su(p) generators (2.20) do not commute: which makes it hard to solve (3.19). See also the discussion in section 2.2. We can determine the numerical coefficients in (3.3), (3.18) up to two loops. By comparing Q ab with the perturbative results (2.19), we find implying that most coefficients vanish in N = 4 SYM. This result is also consistent with our assumption in (3.3) that {x m } are numerical constants of O(1).

Continuum limit at higher loops
By combining (3.3) and (3.18), we obtain a conjecture of the -loop dilatation operator, The terms with k < are part of the dilatation at lower loop orders, combined with powers of N c . Let us take the continuum limit (2.29). The commuting operators Q ab in (3.16) scale as 11 Q −m,m ∼ N m c (3.23) and in particular where D ij is given in (2.33). From this observation, we can refine (3.23) as where : D ij : means that the derivative (∂/∂y ij ) should not hit y ij in the subsequent D ij 's. 11 Before the continuum limit, Q ab scales as N a+b c .

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Then, our conjectured dilatation operator (3.22) becomes We find that the terms m = k are leading at large N c . However, the perturbative data (3.21) shows that C k,k = 0. The terms m = k − 1 gives the first non-vanishing term, which is proportional to the one-loop result D G 1 . The terms m ≤ k − 2 are negligible as long as C k,m = O(N 0 c ). Given (3.26) we can formally sum up the perturbation series, where λ = N c g 2 YM is the 't Hooft coupling and This is the result quoted in (3.1). According to appendix B.1, the operator D ≡ n ij (σ) D ij has the eigenvalues (B.14). Thus where λ a depends on n ij (σ) and has the same order as n Y . We will see in appendix B.2 that the non-negative integers m a should be bounded from above, and at most O(N c ).
Neglecting O(1) quantities, the equation (3.29) becomes where we used (2.1) and (B.37). The factor m a /N c in (3.30) has the spacing of order 1/N c , which becomes continuous at large N c . If f c (λ) remains non-zero at λ 1, then the above ansatz should describe (semi)classical motion of the system with D-branes and strings. Importantly, this excitation spectrum should be gapless.

Strong coupling
We want to reproduce the dilatation spectrum (3.30) at strong coupling. Since the energy of excited states is continuously connected to the BPS state, we take a classical D3-brane action and study the solution around the BPS giant graviton.

D3-brane action
The action for a single D3-brane is given by which includes the spherical giant graviton. There is no B ab and F ab in the background, and the dilaton is constant, e ϕ = g s . The constant T 3 is given by [4] where R is the radius of AdS 5 × S 5 and Ω 3 = 2π 2 is the volume of S 3 with the unit radius.
Our notation for the AdS 5 × S 5 geometry is explained in appendix D.1.
The induced metric is written as We choose the static gauge and assume the ansatz The D3-brane action effectively becomes two-dimensional, 12 The conserved charges can be computed in the standard way, JHEP04(2021)293

Classical solutions
We study the effective two-dimensional action (4.7) around the spherical giant graviton solution as follows. We assume the static gauge, introduce the deformation parameter , and solve the equations of motion (EoM) as a formal series of small . The linearized EoM are given by a set of homogeneous partial differential equations, whose coefficients may depend on η. We remove the η dependence by the separation of variables for the deformed degrees of freedom. This procedure looks similar to the analysis of one-loop fluctuation [27,29]. Generally, however, not all off-shell fluctuations become the deformed solutions of the classical equations of motion.
Other deformations of the spherical giant graviton solution might be possible if the ansatz and the gauge choice are generalized. 13 One can try to study the deformation in the AdS directions, and to search for the solutions with non-zero U(1) field strength or other components of SO(6) angular momenta.

Ground state
The ansatz for the ground (or BPS) state of a spherical giant graviton is [4] The energy as a function of ρ has a local minimum at ρ = N c /(g s J), and the minimum value is E = J. (4.10)

Excited states
We are interested in the non-BPS states which are continuously connected to the BPS state. Let us generalize the ansatz by expanding around the ground state solution as We consider the EoM for three cases, j = 0, 0 < j < 1 and j = 1. No non-trivial solutions are found for the cases j = 0, 1, as discussed in appendix D.3. When 0 < j < 1, the EoM for φ and ρ take the form which can be solved by separation of variables. The r.h.s. of (4.12), (4.13) are identical to the Laplacian on S 3 , whose normalizable solutions are given by the spherical harmonics (D.6). Since our φ 1 and ρ 1 are independent of θ 1 , θ 2 , we set (4.14) 13 The gauge choice may change the CS term.

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Then the general solution of the equations (4.12), (4.13) is given bỹ where c (i) (i = 1, 2, 3, 4) are integration constants. Let us compute the corrections to the conserved charges from (4.14), It follows that Only the k = 0 term remains non-zero after the integration over S 3 owing to the orthogonality (D.9). We remove the first-order correction to the conserved charges by setting The difference (E − J) is non-zero at the second order in the expansion, where denotes the equality after the integration over S 3 coming from (D.10).
Here we encounter apparent inconsistency. The corrections to the conserved charges (4.20) may depend on t even after the integration over S 3 . This is partly because our ansatz (4.11) solves the EoM only at O( ) whereas the corrections are O( 2 ). This explanation is not entirely correct because the solutions at O( 2 ) do not seem to change E − J at O( 2 ). Fortunately we can remove the t dependence either by adjusting the constants {c i }, or by setting k = 0.

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which has the dispersion relation . (4.22) Here k should be a positive even integer as in appendix D.2, and c k should remain finite as k 1 in order to keep (4.21) finite. The general k = 0 solution is In (4.23), the parameter c (3) is redundant because it just shifts the origin of φ. Also, the terms proportional to c (2) coincide with the k = 0 case of the previous solution (4.21).
The profile of those with KK modes is depicted in figure 1, and the profile of the oscillating giant gravitons without KK modes is in figure 2. The former solutions expand and shrink over the area of O( 2 ). The latter solutions are point-like and oscillate around the BPS configuration.

Excited states at higher orders
We can take a linear combination of the solutions at O( ) and proceed to higher orders, (4.25) When ∂ η ρ = ∂ η φ = 0, or equivalently if c k = 0 for k > 0, we could solve EoM at higher orders of . In this case, φ(t) is fixed by the angular momentum (4.8), We can solve the EoM for ρ(t) as  The parameters (c, d) are arbitrary constants of O(1) which may depend on . The energy of an oscillating D3 brane is We should renormalize the frequencies in order to keep ρ(t) finite at large t.
When the first-order solution has KK modes, namely if c k = 0, we do not find higherorder classical solutions. One possible interpretation is that the solutions with non-trivial KK modes on S 3 are not purely classical, and hence they cannot produce E − J = O(N c ) > 0. If we think of λ/N c ∼ g s , then the higher-order corrections are mixed up with g s corrections.

Comments on AdS/CFT
We look for the strong coupling counterpart of the all-loop ansatz (3.30) where α labels the eigenvalues of the (p−1) coupled oscillators, and n ij (σ) is a non-negative integer satisfying The second equation is the Gauss law constraints (A.13), which suggests that n ij (σ) is the number of open strings stretching between the i-th and j-th branes. At p = 2, the ansatz (5.1) becomes

Comparison with the oscillating D3-brane
In section 4, we found that there are two types of classical D-brane motion around the BPS configuration, whose energies are given by (5.4) The first solution can be easily extended to higher orders of , whereas the second solution cannot be extended to the next order by means of the simple separation of variables.
We argue that the oscillating D-brane should correspond to the harmonic oscillator of the effective U(p) theory. More explicitly, we relate the energy of oscillating D-brane (5.4) at large k and the all-loop ansatz (5.3) at p = 2 and large m, Secondly, let us recall the AdS/CFT correspondence for the half-BPS states. At weak coupling, the Young diagrams with different shapes start mixing at one-loop. The column length of a Young diagram can be interpreted as the radial direction of droplet patterns in the LLM plane [3]. This interpretation shows that the D-brane itself should oscillate.
Thirdly, we cannot deform the j = 1 solution, i.e. the maximal giant graviton. This corresponds to the fact that one cannot attach a box representing Y to the Gauss graph operator O R,r (σ) if r has the column of length equal to N c in (A.9).
We make some comments on the parameter n i→j (σ) in (5.2). On the weak coupling side, the parameter n i→j (σ) is interpreted as the number of open strings from the i-th brane to the j-th brane. On the strong coupling side, it is not clear whether we can introduce an open string as a probe, because non-maximal giants start oscillating by perturbing with infinitesimal energy; see figure 3. Moreover, the length of the probe string must be negligibly small, in order to maintain the gapless property of the dispersion relation. We will discuss a related issue in section 5.2.
The parameter n i→j (σ) also counts the number of Y -fields, which should correspond to (part of) the angular momentum in S 3 denoted by J Y = n Y . At strong coupling, the D-branes wrapping S 3 inside S 5 have the zero angular momentum in θ 1 due to the static gauge (4.5). This conclusion is reasonable because we do not see J Y ∼ O(1) at strong coupling, in agreement with the assumption n Y n Z ∼ O(N c ) in the all-loop ansatz. If we still want to explain J Y , we may also add a point particle rotating S 3 carrying J Y . This particle does not interact with D-branes at the leading order of large N c . 14 When p > 2 as in (5.1), we should find p D3-branes oscillating individually, corresponding to the U(1) p symmetry of the effective U(p) theory. The symmetry can be enhanced to non-abelian, e.g. U(2) × U(1) p−2 , if some D-branes stay on top of each other at strong coupling, or if we give up the distant corners approximation at weak coupling. This point will be discussed in appendix E.

On reflecting magnons
At strong coupling, there is another known situation of open strings ending on a giant graviton, called the reflecting magnons in the literature [32].
where M is the number of bulk magnons. Note that the boundary terms can be interpreted as extra magnons with p = π. At strong coupling λ 1, the magnon energy (5.7) is equal to the sum of the open string length multiplied by the string tension. This dispersion relation cannot be gapless, because an open string should connect the equator and the north pole of S 5 .
At weak coupling, the system (5.6) is expected to be dual to a long operator attached to the determinant of Z's, j Nc i Nc (5.8) where χ L , χ R represent the boundary magnons and ψ 1 , ψ 2 , . . . represent the bulk magnon.
Consider the expansion of the determinant-like operator (5.8) in the Gauss graph basis. It is known that the determinant of Z corresponds to r = N c , a single column of length N c , and a single-trace operator is a linear combination of single hook Young diagrams [43]. 15 Thus we expect that the determinant-like operator We can interpret the first term as (5.7) and the second term as (5.5) if J string (the length of a single-hook) is large.
Below are a few remarks on the su(2|2) symmetry. We expect that the giant graviton possesses the residual superconformal symmetry psu(2|2) 2 , based on the κ-symmetric formulation of the D3-brane action on AdS 5 × S 5 [44]. However, we do not find any reasons that this symmetry should be promoted to the centrally-extended su(2|2). In other words, it is likely that the oscillating D-brane solutions are intrinsically non-BPS, and not centrally-extended BPS.
In [45], they constructed the su(2|2) generators of the effective U(p) theory. They proposed the central extension by and interpreted ∆ ± ij as the centers

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which correspond to the second term of (5.10). However, the proposal (5.12) is inconsistent because l.h.s. of (5.11) is symmetric under (a, α, i) ↔ (b, β, j) whereas the r.h.s. is antisymmetric. 16 An alternative approach to the question of the central extension is as follows. Here we gave several expressions of the D-brane energy; the conjectured all-loop formula (5.1), ∆(Oscillating giants) in (5.10), and the boundary term in the spin chain energy (5.7). Only the last expression takes the square-root form inherited from the centrally extended su(2|2) symmetry. 17 For the other cases we do not find square-roots, and one possible reason is the wrapping corrections to the Z = 0 brane [46]. Roughly speaking, the oscillating giants without open strings would correspond to the J string → 0 limit of the integrable spin chain. In other words, it may be possible to refine the proposal (5.12) by studying the states like (5.9) in the limit of J string 1. 18

Summary
In this paper, we studied a non-planar large N c limit of N = 4 SYM as a new example of the AdS/CFT correspondence. First, we reviewed the Hamiltonian of an effective U(p) theory coming from the perturbative dilatation operator acting on the Gauss graph basis. When p = 2, this model is related to the finite harmonic oscillator. Second, we proposed an all-loop ansatz based on the effective U(p) theory. We found mutually commuting charges generated by the difference operators. By taking the continuum limit, we argue that higher loop terms should be proportional to the one-loop result. In our all-loop ansatz, the harmonic oscillators remain non-vanishing in the large N c limit, giving a gapless dispersion relation. In particular, it indicates that non-BPS excited giant gravitons should be continuously connected to the BPS giant graviton at strong coupling.
We investigated the classical D3-brane action on AdS 5 × S 5 and found that a nonmaximal spherical giant graviton can be excited in a gapless way. We argued that this new oscillating brane with KK modes on S 3 is a good candidate for the AdS/CFT dictionary which corresponds to the harmonic oscillator in the effective U(p) theory.
Possible future directions are sketched as follows.
One direction is to investigate this correspondence further. At weak coupling, the mixing matrix on the Gauss graph basis should be evaluated in a more general setup. This includes higher loop effects, a larger set of operators including the sl(2) sector [47], and the corrections from higher orders in n Y /n Z [48]. At strong coupling, the dynamics of D3-brane on AdS 5 × S 5 should be studied in a comprehensive way. This includes to resum the series in the k = 0 solution (4.27), and to investigate the non-abelian DBI action [49]. The non-abelian analysis of the pp-wave background [29], the F1-D3 system [50] and the D1-D3 system [51] may be helpful. A closely related method is the matrix regularization of the worldvolume theory in the pp-wave background [52][53][54], which should capture part of the energy spectrum at strong coupling.

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It is interesting to generalize the computation of the three-point function between two giants and one graviton, into the correlators with oscillating giants [55][56][57], which serves as a non-trivial check of our proposal.
Another direction is to determine the general spectrum of the Hamiltonian of the effective U(p) theory. This includes solving the finite oscillator for p > 2, finding consistent wave-functions for AdS giants, and understanding the structure of 1/N c corrections.
The role of the superconformal symmetry needs to be examined. In particular, the N = 4 SYM theory can be deformed while keeping su(2|2) 2 [58]. It is worth investigating the corresponding deformation at strong coupling and finding the relation to the system of giant gravitons in AdS 5 × S 5 .
A challenging question is whether the "non-planar integrability" can be found at strong coupling. One starting point is the κ-symmetric D3-brane action in AdS 5 × S 5 [44]. Then, through the reduction to two-dimensions (4.7), we may be able to find a classical integrable system. The two-dimensional reduction may not be necessary if one can construct an integrable system in 4d along the line of [59][60][61].

RS thanks the organizers of the workshops YITP-W-20-03 on Strings and Fields 2020, and
Online 2020 NTU-Kyoto high energy physics workshop for stimulating discussions. He is grateful to Robert de Mello Koch, Arkady Tseytlin and Keisuke Okamura for comments on the manuscript. This research is supported by NSFC grant no. 12050410255.

A Review of the Gauss graph basis
We briefly review the construction of the Gauss graph basis, and how it simplifies the action of the perturbative one-loop dilatation operators of N = 4 SYM.

A.1 Notation
Let S L be a permutation group of degree L. Its irreducible representations are labeled by a partition (Young diagram) R L, whose dimensions are denoted by d R . D R IJ (σ) denotes the matrix representation of σ in the irreducible representation R with the component (I, J), where I, J = 1, 2, . . . , d R .
Consider the restriction S L ↓ (S m ⊗ S n ) with m + n = L. We denote the irreducible decomposition by where ν is a multiplicity label and g(r, s; R) is the Littlewood-Richardson coefficient. The branching coefficients are defined by the overlap between the components See [9] for the properties of these quantities.

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We denote partitions of an integer, or Young diagrams, in two ways. The symbol y = l 1 , l 2 , . . . , l p means that the i-th column of the Young diagram y has the length l i . The symbol y = [m 1 , m 2 , . . . , m q ] means that the j-th row of y has the length m j . It follows that

A.2 Distant corners approximation
We introduce the collective index with i p , j p = 1, 2, . . . , N c . The matrix U α satisfies the composition rules We denote multi-trace operators in the su(2) sector by (A.6) with L = n Y + n Z and α ∈ S L . We define the restricted Schur basis of operators by It is expected that the half-BPS operators dual to p spherical giant gravitons consist of p long columns, with n Z = O(N c ) with N c 1. Non-BPS operators can be constructed by attaching Y fields. We write r = l 1 , l 2 , . . . , l p where l i is the length of the i-th column. In the distant corners approximation, we assume that l i − l i− 1 1, so that the corners of where we construct R by adding s n Y (gray boxes) to r n Z (white boxes).

A.3 Gauss graph basis
We introduce the Gauss graph basis following [16].

A.3.1 Skew Young diagrams
We can specify the representation of Y fields in two ways, s or R/r. The states of s are labeled by the standard Young tableaux, and those of R/r are by the skew Young tableaux. In the restricted Schur polynomial (A.8), we may keep track of which box of s goes to which box of R/r, before summing over the indices (I, J, i, j).
In the distant corners approximation, R/r consists of p columns well separated from each other. This indicates that only the column position, 1, 2, . . . , p, should be important in finding the eigenstates of the perturbative dilatation operator of N = 4 SYM.

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Consider an example of p = 3, with s = 4, 2, 1 and R/r = 3, 2, 2 . We parameterize a state of s and R/r using only the column labels, as In other words, we project the standard Young diagrams of shape s n Y onto the trivial (totally symmetric) representation of The group H is an extra symmetry that emerges in the distant corners approximation [23]. We should refine the label of the restricted Schur operator O R,(r,s),ν + ,ν − by adding s = (s 1 , s 2 , . . . , s p ) which specifies how s n Y shows up in the skew Young diagram R/r.

A.3.2 Adjacency matrix
We can define the adjacency matrix n i→j by counting how many i's appear in the j-th column of the skew tableau R/r. In the above example (A.10) we find The adjacency matrix satisfies a conservation law. When a box with the label i goes to the j-th column (j = i), then there must be a box k which comes to the i-th column. This implies the relation called the Gauss law constraints, Intuitively, we may regard the diagonal elements n i→i as the number of closed strings on the i-th brane, and the off-diagonal elements n i→j as the number of open strings between the i-th and j-th brane.
Here we defined the adjacency matrix from a skew Young diagram. The skew Young diagram is in one-to-one correspondence with the Gelfand-Testlin basis, as discussed in [15].

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where⊗ represents the symmetrized tensor product and ⊗ is the usual tensor product. The symmetrization is equivalent to a sum over the states in V ⊗n Y p where V p = {1, 2, . . . , p}. We can permute the state | s by applying σ ∈ S n Y to each summand as This result consists of p tensor product of symmetrized components. It makes sense to count the number of i's in the j-th symmetrized component, and call it n i→j .
Owing to the symmetrization, the action of the permutation σ ∈ S n Y reduces to the action of an element in the double cosetσ ∈ H\S n Y /H, We can compute n i→j graphically as which corresponds to (s 1 , s 2 , . . . , s p ) = (3, 2, . . . , 2).

A.3.4 Operator mixing in the Gauss graph basis
We define operators in the Gauss graph basis by 19 We consider the case n Z = O(N c ) and n Y = O(1), which should correspond to excited multiple giant gravitons. The one-loop dilatation acting on the restricted Schur polynomial factorizes into the mixing of Y 's and the mixing of Z's at the leading order of large N c . 20 As shown in [16], the mixing of Y 's can be solved by taking the Gauss graph basis. The eigenvalues are labeled by the symmetrized adjacency matrix, Our definition looks slightly different from [16] because our restricted Schur basis (A.7) is not normalized. 20 The factorization property is violated at the subleading order of nY /nZ [48].

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The mixing of Z's changes the shape of r = l 1 , l 2 , . . . , l p . We use the simplified notation in place of (A.18).

B Explicit one-loop spectrum
We studying the spectrum of D G 1 = −H 1 following [14,15] with minor improvement. We consider both continuum and discrete cases. The dilatation operator reduces to a set of harmonic oscillators with boundary conditions in the continuum limit (2.29). In order to determine the dilatation spectrum before taking the limit, we solve the discrete case. The discrete case has been solved for p = 2 by using the finite oscillator [62,63]. The spectra of the two cases agree, implying that the one-loop dimensions do not depend on the details of the continuum limit.

B.1 Continuum case
The spectrum of one-loop dilatation in the continuum limit for general p has been studied in [15]. In this limit, we find D G We solve this equation in the region of {y i } given in (2.30). We rewrite the differential operator D ij in (2.33) as where y ij = y i − y j . The new differential operators {A ± (y)} satisfy The operator D ij has the symmetry Thus, Λ ± represent the zero modes of D. Roughly speaking, Λ ± correspond to the addition or removal of a box from each of the p columns, suggesting that the spectrum of H 1 depends only on the difference of column lengths. From (B.2) one finds that the eigenvalues of D ij are written as (m + 1 2 ). The mode number m should be chosen so that the variables y ij satisfy the Young diagram constraints (2.30). These constraints are expressed in terms of the variables (y 12 , . . . , y p−1,p , y p ) as y 12 ≥ 0, y 23 ≥ 0, . . . , y p−1,p ≥ 0, y p ≥ 0. (B.5)

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Consider the operator 21 This operator can be written as the following quadratic form where By diagonalizing M , we obtain the non-zero eigenvalues ofD . The physical states should contain an odd number of creation operators for each a. Thus, the spectrum ofD is The eigenvalues of the original equation (B.1) are (B.14) 21 We used i =j nij(σ) = 2nY .

B.2 Discrete case
The main difficulty in computing the discrete spectrum lies in how to impose the Young diagram constraints. The Fock space created by the oscillator representation of H 1 in (2.17) is not useful, because it does not immediately solve the constraints. Instead, we directly look for the wave functions. Some functional identities are summarized in appendix C. For simplicity, we consider the case of p = 2. We take the linear combination of O(l 1 , l 2 ) in (2.9) as The operator H 1,12 in (2.17) acts on O f as which gives the following discrete eigenvalue equation This equation determines the eigenvalue of D G 1 = −H 1 at p = 2 as where we used n 12 = n 21 . We solve (B.17) by relating it to a finite oscillator [62,63]. Let us take a basis of states in the irreducible representations of su(2), and a rotated basis In view of the effective U(p) theory, the su(2) generators can be interpreted as The generator (2J 1 ) is identical to J 12 in (3.7). The rotation matrix is given by The su(2) generators acts on the states |j, j 1 1 in the standard way,

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We define and assume n Z ≤ 2N c , which is trivial at p = 2. Let us assign where the new variables run the ranges Note that The equations (B.23) become The last two lines agree with the off-diagonal terms in (B.17) with the help of (2.10).
We can immediately solve (B.17) by relating the wave function to the rotation matrix (B.22). If we define this function satisfies the recursion relation which is equivalent to the discrete eigenvalue equation (B.17) with E = 2m. We should impose the Young diagram constraints at p = 2, namely Generally, the function F m (l 1 , l 2 ) does not vanish even if l 2 < 0, and it slowly decreases to zero when l 2 is large and negative. This function has a special zero at 22 meaning that we can solve the Young diagram constraints if all of the following three conditions are satisfied, • The mode number m is even, • n Z = l 1 + l 2 is odd, The last condition may circumvented by using the translation symmetry of the recursion relation 23 although this operation changes the value of N c . See figure 4 for the behavior of F m (l 1 , l 2 ). One finds that F m (l 1 , l 2 ) for l 1 , l 2 , m ∈ Z ≥0 satisfies In summary, at p = 2 we find where m runs over the range (B.36). The result (B.37) reproduces [19], and is valid for any scaling of (l 1 , l 2 ) with respect to N c . In particular, the eigenvalues are O(1) ∼ O(λ/N c ) as long as m ∼ O(1).

B.3 Examples of the eigenvalues
If we introduce a reference point 0 so that all edges pass through that point, we can write where n E is the total number of edges. Then N ij simplifies a bit, Consider some special cases. The first case is The residual symmetry is SO(p − 1). The second case is

C Identities of hypergeometric functions
A special case of Gauss hypergeometric function 2 F 1 is called the Kravchuk (or Krawtchouk) polynomial used in [62], which satisfies the recursion relation As a corollary, where (x) a = Γ(x + a)/Γ(x). We also have Now we can rewrite the rotation matrix (B.22) as and (B.29) as Note that the original expression in terms of 2 F 1 (a, b; c; 2) is more suitable for numerical evaluation. By combining the above identities, we find which vanishes at m = 0, 1, 2, . . . due to Γ(1 − m ) = ∞.
(D. 16) The non-trivial solution of the first equation is for a constant c. This function diverges around η = 0, π/2, and is inconsistent with r ∈ [0, 1]. Again, we find no non-trivial solution.

E On non-abelian DBI
The symmetry of the effective U(p) theory becomes non-abelian if some of the column lengths become equal, l i = l i+1 for some i in (A.9). Correspondingly, some of the p giant graviton branes coincide at strong coupling. Let us make a short digression about nonabelian DBI action to examine this situation. 25 The DBI action is a low energy effective action of closed and open string massless modes on the brane. As a worldvolume theory, the DBI action without U(1) flux is also an example of 4d conformal theory. In AdS 5 × S 5 , the conformal symmetry of the target spacetime is nonlinearly realized [66,67]. The DBI action can be made supersymmetric in the sense of κ symmetry [44,68,69] and of the worldvolume symmetry [70,71]. The fundamental strings can be coupled to multiple coincident D-branes by introducing nonabelian flux F µν [72]. The addition of the CS term to the non-abelian DBI induces dielectric effects [49].
One way to define non-abelian DBI is to expand the DBI action in a formal series of F , Then we promote F to a non-abelian field, and take the trace. This procedure suffers from the ordering ambiguity, which should be fixed by the consistency with the open string amplitude [73][74][75][76].
In section 4, we want to find classical solutions continuously connected to the spherical giant graviton. From the above prescription for the non-abelian DBI in (E.1), we do not see any significant difference between U(1) p and U(p) theories at the order of α 2 .

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The induced metric G ab should not have the U(p) structure for the following reasons. First, the induced metric comes from closed string massless modes, which do not see nonabelian symmetry. Second, the attempts to make G ab a matrix-valued field suffer from various difficulties [77,78]. Third, in our deformation problem, we are only interested in the corrections of O(α 2 ), and the commutator terms do not show up at this order. Then, all matrices are simultaneously diagonalizable, unless they couple to other objects.
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