OOscillating Multiple Giants
Ryo Suzuki
Shing-Tung Yau Center of Southeast University,15th Floor, Yifu Architecture Building, No.2 Sipailou,Xuanwu district, Nanjing, Jiangsu, 210096, China
Abstract
We propose a new example of the AdS/CFT correspondence between the system of multiplegiant gravitons in AdS × S and the operators with O ( N c ) dimensions in N = 4 super Yang-Mills.We first extend the mixing of huge operators on the Gauss graph basis in the su (2) sector to allloops of the ’t Hooft coupling, by demanding the commutation of perturbative Hamiltonians in aneffective U ( p ) theory, where p corresponds to the number of giant gravitons. The all-loop dispersionrelation 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 tonon-maximal giant gravitons. rsuzuki.mp at gmail.com a r X i v : . [ h e p - t h ] J a n ontents N = 4 SYM 5
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24A.2 Distant corners approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.3 Gauss graph basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B Explicit one-loop spectrum 29
B.1 Continuum case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B.2 Discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31B.3 Examples of the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
C Identities of hypergeometric functions 35D Details of strong coupling analysis 36
D.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36D.2 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36D.3 Classical solutions at j = 0 , E On non-abelian DBI 38 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, orby 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 theoryon AdS × S and N = 4 super Yang-Mills (SYM) theory. We depart from the planar regionof N = 4 SYM by studying huge operators whose dimensions are comparable to the rank of thegauge group N c . The operators with O ( N c ) dimensions correspond to the Lin-Lunin-Maldacena(LLM) geometry at strong coupling [3], while those with O ( N c ) dimensions correspond to the giantgraviton [4]. This correspondence continues to non-BPS operators in both of the O ( N c ) and O ( N c )cases. For the former case, an isomorphism between non-BPS states was conjectured between theLLM geometry and N = 4 SYM [5–9]. For the latter case, non-BPS states around the giant gravitonare 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 theoperator basis labeled by a Young diagram [10]. Similarly, a convenient way to describe non-BPSoperators with huge dimensions is the restricted Schur basis, labeled by a set of Young diagrams [11].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 theYoung diagram representing Z ’s has p long columns, this type of operators roughly corresponds toa system of p spherical giant gravitons in AdS × S . The Young diagram representing Y ’s describesa small fluctuation of the giant gravitons. The one-loop mixing in this setup is remarkably simple. First, the number of columns p doesnot change at large N c , because giant gravitons are semi-classical objects at strong coupling [12].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 [13, 14]. The lastobservation is called the non-planar integrability in the literature.We should emphasize that the mixing problem of huge operators is quite different from theplanar mixing problem, and the development of sophisticated techniques has been crucial. Themixing problem of Y ’s is solved by the Gauss graph basis, which counts the number of open stringsending on different giant graviton branes [15]. The same technique can be used to simplify themixing in the su (3) and su (2 |
3) sectors [16,17]. Explicit computation of the mixing matrix has beengiven up to two loops in g in [18]. There are also trial studies at higher loops [19, 20]. More Another promising approach to a non-planar large N c limit is the localization, which is valid at any N c and canextract non-BPS data [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 spoilthe color structure at the leading order of large N c . p giant gravitons can be described by an effective U ( p ) theory [21, 22]. The Hamiltonian of the effective U ( p ) theory has the symmetry U (1) p in thedistant corners approximation, namely when the differences of the length of the adjacent columnsare 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 [23]. The giant graviton is a classical solution of theD3-brane action moving in the AdS × S background. The spherical giant graviton wraps S insideS [4], and the AdS giant wraps S inside AdS [24, 25]. The quantum fluctuation modes aroundthe giant graviton have been studied in [26].The open strings ending on the giant graviton have been studied from two viewpoints. In thefirst viewpoint, we replace open strings with U (1) flux and study the D-brane. The classical motionof a D3-brane in such a background spacetime has been studied in the flat space [27], in the pp-wave [28] and in AdS × S [29]. The U (1) gauge fields typically become spiky, and they diverge atthe location where open strings end on the D-brane. In the second viewpoint, we study the openstring as a classical integrable system [30], or a boundary integrable system [31]. In both pointsof view, the brane-string system typically has divergent energy, which is canceled by a divergentangular momentum of open strings, just like the giant magnon [32].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 thispaper is to understand this theory more precisely by revisiting the analysis both in gauge and stringtheories.In Section 2 and 3, we study the perturbative Hamiltonians of the effective U ( p ) theory onthe 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 commutewith each other. We find that there are at most ( (cid:96) + 1) linearly-independent mutually-commutingoperators at (cid:96) -loops. In the continuum limit, these candidate operators reduce to the harmonicoscillators at one loop, which allows us to conjecture an all-loop ansatz,∆ − J = ˜ f ( λ ) N c m n ( σ ) , n ( σ ) ∈ Z ≥ , m = 1 , , . . . , (cid:108) N c − n Z (cid:109) (1.1)where we put p = 2 for simplicity, and ˜ f ( λ ) is an unknown function of the ’t Hooft coupling λ = N c g .Our ansatz (1 .
1) predicts two remarkable consequences. First, the anomalous dimensions remainnon-zero at the leading order of large N c , because m α can be an integer of O ( N c ). Second, theexcitations are gapless. Recall that the energy of an open string attached on the Z = 0 giantgravitons is gapped, because the open string stretching on S carries non-zero energy, equal to thelength times tension. This disagreement indicates that the previous analyses on the LLM geometry Here the energy is measured in the unit of string tension √ λ and not in the D3-brane tension N c /g s ∼ N c /λ .The string with a finite length produces a gap in the dispersion relation, even if g s (cid:28) λ (cid:28) N c .
4o not immediately apply to the system of multiple giant gravitons. What does the all-loop ansatzrepresent at strong coupling?In Section 4 we revisit a classical single D3-brane wrapping S inside AdS × S , and solve theequations of motion around the BPS spherical giant graviton. Following the steps similar to thestability analysis of [26], we found two types of classical solutions oscillating around the BPS giantgravitons. The first type is a point-like D-brane, and the second type is a fuzzy D-brane withnon-trivial KK modes on S . The energy of the latter solution is E − J = N c g s (cid:20) (cid:15) c k ( k + 1) − j )( k + 2) + O ( (cid:15) ) (cid:21) (1.2)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. N = 4 SYM
We collect known facts about the perturbative mixing of huge operators in N = 4 SYM. Ournotation and basis facts about the Gauss graph basis are summarized in Appendix A. We express the perturbative dilatation operator in the su (2) sector of N = 4 SYM by D ( g YM ) = ∞ (cid:88) (cid:96) =0 (cid:16) g YM π (cid:17) (cid:96) D (cid:96) (2.1)where [33] D = Tr Y ˇ Y + Tr Z ˇ Z (2.2) D = − Y, Z ][ ˇ
Y , ˇ Z ] : (2.3) D = − (cid:104) [ Y, Z ] , ˇ Z (cid:105)(cid:104) [ ˇ Y , ˇ Z ] , Z (cid:105) : − (cid:104) [ Y, Z ] , ˇ Y (cid:105)(cid:104) [ ˇ Y , ˇ Z ] , Y (cid:105) : − N c − D . (2.4)The fields (Φ , ˇΦ) satisfy the U ( N c ) Wick rulesTr ( A ˇΦ B Φ) = Tr ( A ) Tr ( B ) , Tr ( A ˇΦ) Tr ( B Φ) = Tr ( AB ) , Tr (1) = N c . (2.5)The explicit form of D (cid:96) has been known up to five loops [34, 35],We assume n Z = O ( N c ) , n Y = O (1). The operators in the Gauss graph basis are denoted by O R,r ( σ ) = O ( (cid:126)l ) = O ( l , l , . . . , l p ) (2.6)5s in (A . (cid:126)l specifies the Young diagram for Z . Let us write D (cid:96) acting on the Gaussgraph operators by D G(cid:96) . At the leading order of large N c , these dilatation operators are givenby [12, 15, 18, 36] D G = − p (cid:88) i,j =1 i (cid:54) = j n ij ( σ )∆ (1) ij D G = − p (cid:88) i,j =1 i (cid:54) = j n ij ( σ ) (cid:110) ( L − N c ) ∆ (1) ij + ∆ (2) ij (cid:111) (2.7)where ∆ (1) ij = ∆ + ij + ∆ ij + ∆ − ij ∆ (2) ij = (∆ + ij ) + ∆ ij ∆ + ij + ∆ + ij ∆ − ij + ∆ − ij ∆ + ij + ∆ ij ∆ − ij + (∆ − ij ) . (2.8)The difference operators ∆ ij , ∆ ∓ ij are defined by∆ ij O ( (cid:126)l ) = − (cid:16) h ( i, l i ) + h ( j, l j ) (cid:17) O ( (cid:126)l )∆ − ij O ( (cid:126)l ) = (cid:113) h ( i, l i ) h ( j, l j + 1) O ( . . . , l i − , . . . , l j + 1 , . . . )∆ + ij O ( (cid:126)l ) = (cid:113) h ( i, l i + 1) h ( j, l j ) O ( . . . , l i + 1 , . . . , l j − , . . . ) (2.9)where h ( i, l i ) is the box weight for spherical giants, h ( i, l i ) ≡ N c + i − l i , h ( i, l i ∓ − h ( i, l i ) = ± . (2.10)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 . Animportant feature of D G(cid:96) is that it consists of a sum over a pair of indices ( i, j ), and the thirdrow/column does not show up.We can simplify the difference operators in (2 .
9) by introducing d − i O ( (cid:126)l ) = (cid:112) h ( i, l i ) O ( . . . , l i − , . . . ) d + i O ( (cid:126)l ) = (cid:112) h ( i, l i + 1) O ( . . . , l i + 1 , . . . )ˆ h i O ( (cid:126)l ) = h ( i, l i ) O ( (cid:126)l ) . (2.11)These operators satisfy the relations d + i d − i = ˆ h i , (cid:2) d + i , d − j (cid:3) = δ ij (2.12)which follow from (cid:2) d + i , d − i (cid:3) O ( (cid:126)l ) = (cid:110) h ( i, l i ) − h ( i, l i + 1) (cid:111) O ( (cid:126)l ) = O ( (cid:126)l ) . (2.13)Note that p (cid:88) i =1 ˆ h ( i, l i ) O ( (cid:126)l ) = n Z O ( (cid:126)l ) . (2.14) We slightly modified the result of [18] which computed only the first term of D in (2 . D and symmetrized ∆ (2) ij with respect to i ↔ j , owing to n ij ( σ ) = n ji ( σ ). Note that we sum over i (cid:54) = j andnot i < j .
6e can rewrite the difference operators in (2 .
8) as∆ (1) ij = − (cid:0) d + i − d + j (cid:1) (cid:0) d − i − d − j (cid:1) (2.15)∆ (2) ij = − (cid:0) d + i − d + j (cid:1) (cid:16) d + i d − j + d + j d − i (cid:17) (cid:0) d − i − d − j (cid:1) . (2.16)If we introduce H = p (cid:88) i (cid:54) = j n ij ( σ ) H ,ij ≡ p (cid:88) i (cid:54) = j n ij ( σ ) (cid:0) d + i − d + j (cid:1) (cid:0) d − i − d − j (cid:1) (2.17) H = p (cid:88) i (cid:54) = j n ij ( σ ) H ,ij ≡ p (cid:88) i (cid:54) = j n ij ( σ ) (cid:0) d + i − d + j (cid:1) (cid:16) d + i d − j + d + j d − i (cid:17) (cid:0) d − i − d − j (cid:1) (2.18)we obtain D G = −H , D G = − ( L − N c ) H − H . (2.19)The dilatation operator D G(cid:96) written in terms of { d + i , d − i } can be regarded as the Hamiltonian of aneffective U ( p ) theory. This is because E ij ≡ d + i d − j satisfies the GL ( p ) commutation relations, [ E ij , E kl ] = δ jk E il − δ il E kj . (2.20)We assume that n ij ( σ ) are general non-negative integers. Then the perturbative Hamiltonians in(2 . .
18) are invariant only under U (1) p . Since the dilatation operator D ( g YM ) in (2 .
1) is diagonalizable for any g YM , the perturbative coef-ficients commute at any N c , e.g. [ D , D ] = 0. Surprisingly, this is not true for the Hamiltoniansof an effective U ( p ) theory. They do not commute for general { n ij ( σ ) } , (cid:2) D G , D G (cid:3) (cid:54) = 0 . (2.21)We will see in Section 3 that this is a generic feature of effective U ( p ) theory Hamiltonians undersome ansatz, and not due to potentially missing terms in D G .Let us take a closer look at the situation. The condition (cid:2) D G , D G (cid:3) = 0 reduces to [ H , H ] = 0.If we impose this condition for any { n ij ( σ ) } , we get0 = (cid:88) ij,i (cid:48) j (cid:48) n ij ( σ ) n i (cid:48) j (cid:48) ( σ ) [ H ,ij , H ,i (cid:48) j (cid:48) ] (2.22)= (cid:88) ij n ij ( σ ) [ H ,ij , H ,ij ] + (cid:88) ijk n ij ( σ ) n ik ( σ ) ([ H ,ij , H ,ik ] + [ H ,ik , H ,ij ]) (2.23)giving us 0 = [ H ,ij , H ,ij ] = [ H ,ij , H ,ik ] + [ H ,ik , H ,ij ] . (2.24) Roughly speaking, E ij is a p × p matrix whose entries are zero except at the i -th row, j -th column. The Hermitiancombinations generate U ( p ).
7y explicit computation, one finds[ H ,ij , H ,ij ] = 0 but [ H ,ij , H ,ik ] + [ H ,ik , H ,ij ] (cid:54) = 0 . (2.25)This shows that D G , D G in (2 .
7) can be regarded as the coefficients of the weak coupling expansion D G ( g YM ) = ∞ (cid:88) (cid:96) =0 (cid:16) g YM π (cid:17) (cid:96) D G(cid:96) (2.26)only if p = 2, when Young diagrams have two columns.This puzzle can be solved in the following way. In the displaced corners approximation, wetruncate the Hilbert space to a fixed number of columns, then take the large N c limit. We obtainedthe Hamiltonians D G(cid:96) after the Hilbert space truncation, but without taking the limit. The aboveinconsistency shows that taking the large N c limit is a necessary step in the displaced cornersapproximation. In fact, even if we pick up an operator in the distant region l (cid:29) l , the difference( l − l ) keep decreasing due to the operator mixing, until it hits the Young diagram constraints l ≥ l . The original one-loop mixing matrix no longer takes the simple form (2 .
7) when two columnshave comparable lengths. We expect that these boundary effects on the anomalous dimensions arenegligible at the leading order of large N c . We take the continuum limit following [13, 14, 18].We begin with the ansatz for the dilatation eigenstates, O f ( σ ) = (cid:88) l ,l ,...,l p (cid:48) f ( l , l , . . . , l p ) O R,r ( σ ) , N c ≥ l ≥ l ≥ · · · ≥ l p ≥ , p (cid:88) i =1 l i = n Z (2.27)where we specify the column lengths of r by ( l , l , . . . , l p ), and Σ (cid:48) means the sum over { l i } underthe constraints shown in (2 . .
11) on O f can be written as d − i O f ( σ ) (cid:39) (cid:88) l ,l ,...,l p (cid:48) (cid:112) h ( i, l i ) f ( . . . , l i + 1 , . . . ) O R,r ( σ ) d + i O f ( σ ) (cid:39) (cid:88) l ,l ,...,l p (cid:48) (cid:112) h ( i, l i + 1) f ( . . . , l i − , . . . ) O R,r ( σ ) (2.28)where (cid:39) means that we neglect potential contributions from the boundary of the summation range.Consider the following large N c limit n Z ∼ O ( N c ) , l ∼ O ( N c ) , l i ∼ O ( (cid:112) N c ) ( i = 2 , , . . . , p ) (2.29)which is similar to the limit discussed in [12]. We prefer the square-root scaling l i ∼ O ( √ N c ) tothe 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 sphericalgiant gravitons. The constraint (cid:80) i l i = n Z becomes somewhat trivial because n Z ∼ l .8e introduce the rescaled variables and functions y i = l i √ αN c , (cid:114) N c α ≥ y ≥ y ≥ · · · ≥ y p ≥ F ( y , y , . . . , y p ) ≡ f (cid:18) l √ αN c , l √ αN c , . . . , l p √ αN c (cid:19) . (2.31)We keep y to simplify our notation, even though y = O ( √ N c ) (cid:29)
1. It follows that h ( i, l i ) = N c + i − y i (cid:112) αN c f ( . . . , l i ± , . . . ) = F (cid:18) . . . , y i ± √ αN c , . . . (cid:19) . (2.32)In the continuum limit, the difference operators H ,ij , H ,ij in (2 . .
18) become H ,ij → D ij , H ,ij → N c D ij , D ij ≡ α y ij − α ∂ ∂y ij (2.33)where y ij = y i − y j . This suggests that the one-loop and two-loop dilatations commute in this limit, (cid:2) D G , D G (cid:3) → . (2.34)The spectrum of D G is discussed in detail in Appendix B. We conjecture that perturbative dilatation operators at all loops in the continuum limit (2 .
29) takesthe form D G = D + f c ( λ ) p (cid:88) i,j =1 i (cid:54) = j n ij ( σ ) D ij , λ ≡ N c g . (3.1)A related argument was given in [19], where they showed that the mixing of Y ’s at higher loopstakes the same form as the one-loop mixing. We make the following ansatz for D G(cid:96) in the leading order of large N c , D G(cid:96) = (cid:96) (cid:88) k =1 N (cid:96) − kc x k H k , H (cid:96) = p (cid:88) i (cid:54) = j n ij ( σ ) H (cid:96),ij (3.2)where { x k } are numerical constants of O (1). The first equation (3 .
2) means that the (cid:96) -loop dilatationcontains lower-loop difference operators multiplied by powers of N c . The second equation meansthat H (cid:96) depends only on a pair of column labels ( i, j ) coupled to n ij ( σ ). We impose this conditionbecause n ij ( σ ) should count the number of open string modes stretching between the i -th and j -thgiant graviton brane. 9e further assume that H (cid:96),ij ≡ (cid:88) m ˜ x (cid:96)m P (cid:96),m ( d † i , d † j , d i , d j ) , P (cid:96),m contains (cid:96) d † ’s followed by (cid:96) d ’s (3.3)where { ˜ x (cid:96),m } are numerical constants of O (1), and P (cid:96),m is a polynomial of the difference operators.The form of P (cid:96),m originates from the perturbative dilatation operators of N = 4 SYM discussed inSection 2.1. It is known that there is a correspondence between the terms of ∆ (2) ij in (2 .
8) and thoseof D , according to the two-loop computation [18]. Since the (cid:96) -loop dilatation operator D (cid:96) shouldremove at most (cid:96) fields and add (cid:96) fields, we arrive at the ansatz of P (cid:96),m in (3 . Let us revisit the consistency conditions in Section 2.2. Since all perturbative Hamiltonians aresimultaneously diagonalizable for any n ij ( σ ), the first equation of (2 .
24) is generalized to[ H (cid:96),ij , H (cid:96) (cid:48) ,ij ] = 0 ( ∀ (cid:96), (cid:96) (cid:48) ) . (3.4)It is straightforward to enumerate all possible solutions of (3 . P (cid:96),m , with the help of Mathematica .It turns out that at (cid:96) -loop, there are ( (cid:96) + 1) independent solutions of the equation (3 . P ij, = d † i d j + d † j d i ≡ J ij (3.5) P ij, = d † ij d ij = H ,ij (3.6)where H ,ij is given in (2 .
17) and d † ij = d † i − d † j , d ij = d i − d j . (3.7)We also define I ij ≡ d † i d i + d † j d j = d † ij d ij − J ij (3.8)which satisfies [ I ij , J ij ] = 0 . (3.9)At two-loop, there are three solutions, P ij, = ( d † i ) ( d j ) + 2 d † i d † j d i d j + ( d † j ) ( d i ) = J (2) ij (3.10) P ij, = d † ij J ij d ij = H ,ij (3.11) P ij, = ( d † ij ) ( d ij ) = : H ,ij : (3.12)where H ,ij is given in (2 .
18) and J ( n ) ij ≡ : ( J ij ) n : = n (cid:88) m =0 (cid:18) nm (cid:19) ( d † i ) m ( d † j ) n − m ( d i ) n − m ( d j ) m . (3.13) For example, ( d † i ) d j removes two boxes from the j -th column and add two boxes to the i -th column. This termcomes from (∆ + ij ) which roughly corresponds to Tr ( ZZW ˇ Z ˇ Z ˇ W ). We can also explain the powers of N c in (3 .
2) from the fact that D (cid:96) removes (cid:96) fields and adds (cid:96) fields.
10t higher loops, we find that all solutions at (cid:96) -loop can be written in the form Q ab = Q ab,ij ≡ ( d † ij ) a J ( b ) ij ( d ij ) a , ( a = (cid:96) − b = 0 , , . . . , (cid:96) ) . (3.14)We checked that no more solutions exist up to four-loop. The two-parameter family of differenceoperators (3 .
14) mutually commute,[ Q ab , Q a (cid:48) b (cid:48) ] = 0 ( ∀ a, b, a (cid:48) , b (cid:48) ) (3.15)which follows from (3 .
8) and (3 . (cid:96) -loop dilatation in (3 .
2) becomes H (cid:96),ij = (cid:96) (cid:88) m =0 ˜ x (cid:96)m Q (cid:96) − m,m . (3.16)The second equation of (2 .
24) generalized to higher loops reads[ H (cid:96),ij , H (cid:96) (cid:48) ,ik ] + [ H (cid:96),ik , H (cid:96) (cid:48) ,ij ] = 0 ( ∀ (cid:96), (cid:96) (cid:48) ) . (3.17)Some of Q ab,ij in (3 .
14) up to two loops satisfy these conditions. At three-loops, no linear combina-tion of ( Q , , Q , , Q , , Q , ) satisfy (3 .
17) against the two-loop dilatation. Thus, we should trustthe discrete form of our all-loop ansatz (3 .
2) only at p = 2.This conclusion is not surprising. The effective Hamiltonian H (cid:96),ij is a linear combination of Q ab,ij in (3 . J ij . However, J and J as the su ( p ) generators (2 . J , J ] ∼ , = − (cid:54) = 0 (3.18)which makes it hard to solve (3 . . .
16) up to two loops. By comparing Q ab with the perturbative results (2 . D G = − (cid:88) i (cid:54) = j n ij ( σ ) { · Q , + 1 · Q , } D G = − (cid:88) i (cid:54) = j n ij ( σ ) { · Q , + ( L − N c ) Q , + 0 · Q , + 1 · Q , + 0 · Q , } (3.19)implying that most coefficients vanish in N = 4 SYM. This result is also consistent with ourassumption in (3 .
2) that { ˜ x (cid:96)m } are numerical constants of O (1). By combining (3 .
2) and (3 . (cid:96) -loop dilatation operator, D G(cid:96) = p (cid:88) i (cid:54) = j n ij ( σ ) (cid:96) (cid:88) k =1 k (cid:88) m =0 N (cid:96) − kc C k,m Q k − m,m , C k,m = x k ˜ x km = O ( N c ) . (3.20)11he terms with k < (cid:96) are part of the dilatation at lower loop orders, combined with powers of N c .Let us take the continuum limit (2 . Q ab in (3 .
14) scale as Q (cid:96) − m,m ∼ N mc (3.21)and in particular Q ,m = (2 N c ) m + O ( N m − / c ) Q ,m = (2 N c ) m D ij + O ( N m − / c ) Q ,m = (2 N c ) m (cid:18) α y ij − y ij ∂ ∂y ij + 1 α ∂ ∂y ij (cid:19) + O ( N m − / c ) Q ,m = (2 N c ) m (cid:18) α y ij − y ij ∂ ∂y ij + 3 y ij ∂ ∂y ij − α ∂ ∂y ij (cid:19) + O ( N m − / c ) (3.22)where D ij is given in (2 . .
21) as Q (cid:96),m = (2 N c ) m : D (cid:96)ij : + O ( N m − / c ) (3.23)where : D (cid:96)ij : means that the derivative ( ∂/∂y ij ) should not hit y ij in the subsequent D ij ’s.Then, our conjectured dilatation operator (3 .
20) becomes D G(cid:96) = p (cid:88) i (cid:54) = j n ij ( σ ) (cid:96) (cid:88) k =1 k (cid:88) m =0 N (cid:96) − k + mc (2 m C k,m ) : D k − mij : . (3.24)We find that the terms m = k are leading at large N c . However, perturbative data (3 .
19) showsthat C k,k = 0. The terms m = k − D G . The terms m ≤ k − C k,m = O ( N c ).Given (3 .
24) we can formally sum up the perturbation series, D G = D + ∞ (cid:88) (cid:96) =1 g (cid:96) YM D G(cid:96) → D + N − c f c ( λ ) (cid:88) i (cid:54) = j n ij ( σ ) D ij (3.25)where λ = N c g is the ’t Hooft coupling and f c ( λ ) = ∞ (cid:88) (cid:96) =1 λ (cid:96) (cid:96) (cid:88) k =1 k − C k,k − . (3.26)This is the result quoted in (3 . D ≡ (cid:80) n ij ( σ ) D ij hasthe eigenvalues (B . D G − D = N − c f c ( λ ) (cid:40) n Y + 2 p − (cid:88) a =1 (2 m a + 1) ˜ λ a ( { n ij } ) (cid:41) (3.27) Before the continuum limit, Q ab scales as N a + bc . λ a depends on n ij ( σ ) and has the same order as n Y . We will see in Appendix B.2 thatthe non-negative integers m a should be bounded from above, and at most O ( N c ). Neglecting O (1)quantities, the equation (3 .
27) becomes D G → D + ˜ f ( λ ) p − (cid:88) a =1 m a N c ˜ λ a ( { n ij } ) (3.28)with ˜ f ( λ ) = 4 f c ( λ ). When p = 2, we obtain D G = L + ˜ f ( λ ) mN c n ( σ ) , ˜ f ( λ ) = λ π + O ( λ ) n ( σ ) ∈ Z ≥ , m = 1 , , . . . , (cid:108) N c − n Z (cid:109) (3.29)where we used (2 .
1) and (B . m a /N c in (3 .
28) has the spacing of order 1 /N c , which becomes continuous at large N c . If f c ( λ ) remains non-zero at λ (cid:29)
1, then the above ansatz should describe (semi)classicalmotion of the system with D-branes and strings. Importantly, this excitation spectrum should begapless.
We want to reproduce the dilatation spectrum (3 .
28) at strong coupling. Since the energy of excitedstates is continuously connected to the BPS state, we take a classical D3-brane action and studythe solution around the BPS giant graviton.
The action for a single D3-brane is given by S = S DBI + S CS = − T (cid:90) Σ d ξ e − ϕ (cid:112) − det ( G ab + B ab + 2 πα (cid:48) F ab ) + T (cid:90) Σ C (4) (4.1)where Σ is the worldvolume, G ab = g µν ∂ a X µ ∂ b X ν is the induced metric. We consider a D3-branewrapping S inside AdS × S , D3 : Σ → R × S ⊂ AdS × S (4.2)which includes the spherical giant graviton. There is no B ab and F ab in the background, and thedilaton is constant, e ϕ = g s . The constant T is given by [4] T = 2 πg s (4 π α (cid:48) ) = N c R Ω (4.3)where R is the radius of AdS × S and Ω = 2 π is the volume of S with the unit radius. Ournotation for the AdS × S geometry is explained in Appendix D.1.13he induced metric is written as G ab = R (cid:110) − ∂ a t ∂ b t + ∂ a ρ ∂ b ρ ρ − ρ + ( ρ − ∂ a φ ∂ b φρ + ∂ a η ∂ b η + cos η ∂ a θ ∂ b θ + sin η ∂ a θ ∂ b θ ρ (cid:111) . (4.4)We choose the static gauge t = ξ , θ = ξ , θ = ξ , η = ξ (4.5)and assume the ansatz ρ = ρ ( t, η ) , φ = φ ( t, η ) . (4.6)The D3-brane action effectively becomes two-dimensional, S ≡ (cid:90) R × S d ξ L = N c g s Ω (cid:90) d ξ (cid:32) −√− det G + δ a sin η cos η ( ∂ a φ ) ρ (cid:33) . (4.7)The conserved charges can be computed in the standard way, J = (cid:90) π/ dη j ( t, η ) = (cid:90) dη δSδ∂ φE = (cid:90) π/ dη h ( t, η ) = (cid:90) dη (cid:40) (cid:88) X = ρ,φ ∂ a X δSδ∂ a X − L (cid:41) . (4.8) We study the effective two-dimensional action (4 .
7) around the spherical giant graviton solutionas follows. We assume the static gauge, introduce the deformation parameter (cid:15) , and solve theequations of motion (EoM) as a formal series of small (cid:15) . The linearized EoM are given by a set ofhomogeneous 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 [26, 28]. 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 andthe gauge choice are generalized. One can try to study the deformation in the AdS directions, andto search for the solutions with non-zero U (1) field strength or other components of SO (6) angularmomenta. The ansatz for the ground (or BPS) state of a spherical giant graviton is [4] ρ = constant , φ = t. (4.9) We chose the CS coupling so that the ground state satisfies E = J including the sign, which can be flipped by φ → − φ . The physical brane tension is proportional to N c /g s . The gauge choice may change the CS term. ρ has a local minimum at ρ = N c / ( g s J ), and the minimum value is E = J. (4.10) We are interested in the non-BPS states which are continuously connected to the BPS state. Letus generalize the ansatz by expanding around the ground state solution as ρ = 1 j + (cid:15) ρ ( t, η ) , φ = t + (cid:15) φ ( t, η ) , j ≡ g s JN c . (4.11)We consider the EoM for three cases, j = 0, 0 < j < j = 1. No non-trivial solutions arefound 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 j ∂ t ρ j − ∂ t φ = ∂ η φ + 2 cot(2 η ) ∂ η φ (4.12) − j − ∂ t φ j + ∂ t ρ = ∂ η ρ + 2 cot(2 η ) ∂ η ρ (4.13)which can be solved by separation of variables. The RHS of (4 . .
13) are identical to theLaplacian on S , whose normalizable solutions are given by the spherical harmonics (D . φ and ρ are independent of θ , θ , we set ρ ( t, η ) = ˜ ρ ( t ) Φ k, , ( η ) , φ ( t, η ) = ˜ φ ( t ) Φ k, , ( η ) . (4.14)Then the general solution of the equations (4 . .
13) is given by˜ ρ ( t ) = 1 j (cid:34) c (1) (cid:16) ( k + 2) cos( kt ) + k cos(( k + 2) t ) (cid:17) + c (2) cos t sin(( k + 1) t )+ c (3) (cid:16) k sin(( k + 2) t ) − ( k + 2) sin( kt ) (cid:17) + c (4) sin t sin(( k + 1) t ) (cid:35) (4.15)˜ φ ( t ) = 12( j − (cid:34) c (1) (cid:16) ( k + 2) sin( kt ) − k sin(( k + 2) t ) (cid:17) − c (2) sin t sin(( k + 1) t )+ c (3) (cid:16) ( k + 2) cos( kt ) + k cos(( k + 2) t ) (cid:17) + c (4) cos t sin(( k + 1) t ) (cid:35) (4.16)where c ( i ) ( i = 1 , , ,
4) are integration constants.Let us compute the corrections to the conserved charges from (4 . E = (cid:88) n =0 (cid:15) n E ( n ) = (cid:88) n =0 (cid:15) n (cid:90) π/ dη h ( n ) , J = (cid:88) n =0 (cid:15) n J ( n ) = (cid:88) n =0 (cid:15) n (cid:90) π/ dη j n ) . (4.17)It follows that h (1) = j = N c g s Ω sin η cos η Φ k, , ( η ) (cid:110) (1 − j ) ∂ t ˜ φ − j ˜ ρ ( t ) (cid:111) . (4.18)15nly the k = 0 term remains non-zero after the integration over S owing to the orthogonality(D . E (1) = J (1) ∝ (4 c (1) + c (4) ) = 0 . (4.19)The difference ( E − J ) is non-zero at the second order in the (cid:15) expansion, h (2) − j = N c g s Ω sin 2 η − j ) (cid:110) Φ k, , ( η ) (cid:16) j ( ∂ t ˜ ρ ) + 4( j − ( ∂ t ˜ φ ) (cid:17) − ( ∂ η Φ k, , ) (cid:16) j ˜ ρ + 4( j − ˜ φ (cid:17)(cid:111) (4.20) (cid:39) N c g s Ω sin 2 η Φ k, , ( η ) − j ) (cid:110)(cid:16) j ( ∂ t ˜ ρ ) + 4( j − ( ∂ t ˜ φ ) (cid:17) − k ( k + 2) (cid:16) j ˜ ρ + 4( j − ˜ φ (cid:17)(cid:111) where (cid:39) denotes the equality after the integration over S coming from (D . .
20) maydepend on t even after the integration over S . This is partly because our ansatz (4 .
11) solvesthe EoM only at O ( (cid:15) ) whereas the corrections are O ( (cid:15) ). This explanation is not entirely correctbecause the solutions at O ( (cid:15) ) do not seem to change E − J at O ( (cid:15) ). Fortunately we can removethe t dependence either by adjusting the constants { c i } , or by setting k = 0.The general k > ρ ( t, η ) = c k ( k + 1) j ( k + 2) sin(( k + 2) t ) Φ k, , ( η ) φ ( t, η ) = c k ( k + 1)2( j − k + 2) cos(( k + 2) t ) Φ k, , ( η ) (4.21)which has the dispersion relation E − J = N c g s (cid:15) c k ( k + 1) − j )( k + 2) . (4.22)Here k should be a positive even integer as in Appendix D.2, and c k should remain finite as k (cid:29) .
21) finite. The general k = 0 solution is ρ ( t, η ) = c (2) sin(2 t ) − c (4) cos(2 t )2 √ πj , φ ( t, η ) = (4 c (3) − c (2) ) + c (2) cos(2 t ) + c (4) sin(2 t )4 √ π ( j −
1) (4.23)with the dispersion relation E − J = N c g s (cid:15) (cid:16) c + c (cid:17) π (1 − j ) . (4.24)In (4 . c (3) is redundant because it just shifts the origin of φ . Also, the termsproportional to c (2) coincide with the k = 0 case of the previous solution (4 . j, k ) = (0 . ,
2) for top left, ( j, k ) = (0 , ,
4) for top right, ( j, k ) = (0 . , j, k ) = (0 , ,
6) for bottom right. Other parameters are R = 1 , (cid:15) = 0 . , c k = 1.Figure 2: Giant gravitons at the vacuum (blue) and excited (orange) states. We put j = 0 . j = 0 . R = 1 , (cid:15) = 0 . , c = 1 , d = 0.The profile of those with KK modes is depicted in Figure 1, and the profile of the oscillating17iant gravitons without KK modes is in Figure 2. The former solutions expand and shrink over thearea of O ( (cid:15) ). The latter solutions are point-like and oscillate around the BPS configuration. We can take a linear combination of the solutions at O ( (cid:15) ) and proceed to higher orders, ρ = 1 j + (cid:15) (cid:40) C cos(2 t )2 √ π j − (cid:88) k c k (1 + k ) sin(( k + 2) t ) Φ k, , ( η ) j ( k + 2) (cid:41) + (cid:88) n =2 (cid:15) n ρ n ( t, η ) φ = t + (cid:15) (cid:40) C sin(2 t )4 √ π (1 − j ) − (cid:88) k c k (1 + k ) cos(( k + 2) t ) Φ k, , ( η )2(1 − j )( k + 2) (cid:41) + (cid:88) n =2 (cid:15) n φ n ( t, η ) . (4.25)When ∂ η ρ = ∂ η φ = 0, or equivalently if c k = 0 for k >
0, we could solve EoM at higher ordersof (cid:15) . In this case, φ ( t ) is fixed by the angular momentum (4 . j = 1 ρ + ( ρ − R ˙ φρ / (cid:32) κ − R ˙ ρ ρ − ρ − ( ρ − R ˙ φ ρ (cid:33) − / . (4.26)We can solve the EoM for ρ ( t ) as ρ = 1 j + (cid:15) { c cos(2 t ) + d sin(2 t ) } + (cid:15) j (2 j − c + d )2( j − − (cid:15) j ( c + d )4( j − { ( d + 4 c t ) sin(2 t ) + ( c − d t ) cos(2 t ) }− (cid:15) j ( c + d )8( j − (cid:8) (2 j ( j (4 j −
9) + 6) − c + d ) + 2( j − (( d − c ) cos(4 t ) − cd sin(4 t )) (cid:9) + O ( (cid:15) ) . (4.27)The parameters ( c , d ) are arbitrary constants of O (1) which may depend on (cid:15) . The energy of anoscillating D3 brane is E = N c g s (cid:26) j + (cid:15) ( c + d ) j − j ) − (cid:15) ( c + d ) j (7( j − j + 15)8(1 − j ) + O ( (cid:15) ) (cid:27) . (4.28)The solution (4 .
27) contains secular terms like(polynomial of t ) × cos(2 mt ) , (polynomial of t ) × sin(2 mt ) , ( m ∈ Z ) . (4.29)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 (cid:54) = 0, we do not find higher-orderclassical solutions. One possible interpretation is that the solutions with non-trivial KK modes onS are not purely classical, and hence they cannot produce E − J = O ( N c ) >
0. If we think of (cid:15) (cid:38) λ/N c ∼ g s , then the higher-order corrections are mixed up with g s corrections.18 Comments on AdS/CFT
We look for the strong coupling counterpart of the all-loop ansatz (3 . − J = ˜ f ( λ ) N c p − (cid:88) α =1 m α λ α ( { n ij } ) (5.1)where α labels the eigenvalues of the ( p −
1) coupled oscillators, and n ij ( σ ) is a non-negative integersatisfying n ij ( σ ) = n i → j ( σ ) + n j → i ( σ ) , p (cid:88) j =1 n i → j = p (cid:88) j =1 n j → i , n Y = p (cid:88) i =1 p (cid:88) j =1 n i → j ( σ ) . (5.2)The second equation is the Gauss law constraints (A . n ij ( σ ) is the numberof open strings stretching between the i -th and j -th branes.At p = 2, the ansatz (5 .
1) becomes∆ − J = ˜ f ( λ ) N c m n , n ( σ ) ∈ Z ≥ , m = 1 , , . . . , (cid:108) N c − n Z (cid:109) . (5.3) In Section 4, we found that there are two types of classical D-brane motion around the BPS con-figuration, whose energies are given by E − J = N c g s (cid:20) (cid:15) ( c + d ) j − j ) − (cid:15) ( c + d ) j (7( j − j + 15)8(1 − j ) + O ( (cid:15) ) (cid:21) ( k = 0) N c g s (cid:20) (cid:15) c k ( k + 1) − j )( k + 2) + O ( (cid:15) ) (cid:21) ( k ≥ . (5.4)The first solution can be easily extended to higher orders of (cid:15) , whereas the second solution cannotbe 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 effec-tive U ( p ) theory. More explicitly, we relate the energy of oscillating D-brane (5 .
4) at large k andthe all-loop ansatz (5 .
3) at p = 2 and large m , E − J (cid:39) N c (cid:15) λ π c k − j ) k ↔ ∆ − J = ˜ f ( λ ) N c n ( σ ) m . (5.5)where we used λ ≡ R /α (cid:48) = 4 πg s N c . We regard (cid:15) as the quantity of O (1 /N c ), which is an effect ofthe fundamental strings moving around the D3-brane. Then k should be less than O ( (cid:15) − / ) to keepthe corrections to ( E − J ) small. This bound corresponds to the fact that the mode number m isbounded from above at O ( N c ).Let us present several lines of reasoning behind this identification.19 ⇒ Figure 3: (Left) Open string stretching between two D3-branes as a probe. (Right) D3-branes startexpanding and shrinking.Firstly, both dispersion relations are gapless, and one can excite the BPS state by supplying anarbitrarily small amount of energy.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 Youngdiagram can be interpreted as the radial direction of droplet patterns in the LLM plane [3]. Thisinterpretation shows that the D-brane itself should oscillate.Thirdly, we cannot deform the j = 1 solution, i.e. the maximal giant graviton. This correspondsto 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 . n i → j ( σ ) in (5 . n i → j ( σ ) is interpreted as the number of open stringsfrom the i -th brane to the j -th brane. On the strong coupling side, it is not clear whether we canintroduce an open string as a probe, because non-maximal giants start oscillating by perturbingwith infinitesimal energy; see Figure 3. Moreover, the length of the probe string must be negligiblysmall, in order to maintain the gapless property of the dispersion relation. We will discuss a relatedissue 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 denoted by J Y = n Y . At strong coupling, the D-branes wrapping S inside S have the zero angular momentum in θ due to the static gauge (4 . J Y ∼ O (1) at strong coupling, in agreement with the assumption n Y (cid:28) n Z ∼ O ( N c ) in the all-loop ansatz. If we still want to explain J Y , we may also add a pointparticle rotating S carrying J Y . This particle does not interact with D-branes at the leading orderof large N c . When p > . p D3-branes oscillating individually, corresponding tothe 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 − , if some D-branes stay on top of each other at strong coupling, or if we give upthe distant corners approximation at weak coupling. This point will be discussed in Appendix E. If we think of the point particle as a closed string, this may also correspond to the term n ii ( σ ) in the effective U ( p ) theory Hamiltonian, which shows up in the subleading order of large N c [21, 37]. .2 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 [31]. An example can be depicted as (cid:32)
Spinning open strings ending onthe Z = 0 maximal giant graviton (cid:33) = (5.6)where the red dashed lines represent boundary magnons, and the blue thick lines represent bulkmagnons. This figure is the same as the top view of Figure 2 if the angular momentum J φ takesthe maximal value at r = 0. The energy of open strings in (5 .
6) having a large angular momentum J string is given by the energy of an integrable open spin chain with su (2 |
2) symmetry [38] as E − J string = (cid:88) α = L,R (cid:114) Q α + λπ + M (cid:88) i =1 (cid:114) λπ sin p i , ( N c (cid:29) J string (cid:29)
1) (5.7)where M is the number of bulk magnons. Note that the boundary terms can be interpreted as extramagnons with p = π .At strong coupling λ (cid:29)
1, the magnon energy (5 .
7) is equal to the sum of the open string lengthmultiplied by the string tension. This dispersion relation cannot be gapless, because an open stringshould connect the equator and the north pole of S .At weak coupling, the system (5 .
6) is expected to be dual to a long operator attached to thedeterminant of Z ’s, O det = N c (cid:88) i ,i ,...,i Nc ,j ,j ,...,j Nc =1 (cid:15) i i ...i Nc j j ...j Nc Z j i Z j i . . . Z j Nc − i Nc − ( χ L . . . ZZ . . . ψ . . . ψ . . . ZZ . . . χ R ) j Nc i Nc (5.8)where χ L , χ R represent the boundary magnons and ψ , ψ , . . . represent the bulk magnon.Consider the expansion of the determinant-like operator (5 .
8) in the Gauss graph basis. It isknown that the determinant of Z corresponds to r = (cid:74) N c (cid:75) , a single column of length N c , and asingle-trace operator is a linear combination of single hook Young diagrams [39]. Thus we expectthat the determinant-like operator (5 .
8) should be expanded by O R,r ( σ ), where both R and r consist of a single hook attached to the column of length O ( N c ). We can generalize this systemby introducing multiple giant gravitons. The Young diagrams ( R, r ) for Gauss graph basis (A . r = l ... l p , R = ... p (5.9)Recall that in the distant corners approximation, we can neglect the mixing of Y fields in thedifferent columns. Thus, the mixing matrix of the system (5 .
9) at large N c should factorize betweenthe single-trace part and the effective U ( p ) theory part,∆ − n Z = ∆(Reflecting magnons) + ∆(Oscillating giants) . (5.10)The first term represents (5 .
7) and the second term represents (5 . su (2 |
2) symmetry.We expect that the giant graviton possesses the residual superconformal symmetry psu (2 | ,based on the κ -symmetric formulation of the D3-brane action on AdS × S [40]. However, we donot find any reasons that this symmetry should be promoted to the centrally-extended su (2 | ± ij are interpreted as the central charges of the centrally-extended su (2 |
2) algebra, P ij = α ( d + i − d + j ) , K ij = β ( d − i − d − j ) , ∆ − J ? = 12 (cid:88) ij (cid:112) P ij K ij . (5.11)One evidence is that the one-loop anomalous dimensions in the su (2 |
3) sector still take the sameform as (2 . n ij by a sum over four types of excitations [22] D G = − p (cid:88) i,j =1 i (cid:54) = j n ij ( σ )∆ (1) ij , n ij = (cid:88) α =1 n ( α ) ij (5.12)22t the leading order of large N c . This proposal has the following difficulties. First, it producesonly the m = 0 terms in (3 . su (2 | m = 0 in (B .
36) has a parity-evenwave-function. In this paper, we studied a non-planar large N c limit of N = 4 SYM as a new example of theAdS/CFT correspondence. First, we reviewed the Hamiltonian of an effective U ( p ) theory comingfrom the perturbative dilatation operator acting on the Gauss graph basis. When p = 2, this modelis related to the finite harmonic oscillator. Second, we proposed an all-loop ansatz based on theeffective 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 theone-loop result.In our all-loop ansatz, the harmonic oscillators remain non-vanishing in the large N c limit, givinga gapless dispersion relation. In particular, it indicates that non-BPS excited giant gravitons shouldbe continuously connected to the BPS giant graviton at strong coupling.We investigated the classical D3-brane action on AdS × S and found that a non-maximalspherical giant graviton can be excited in a gapless way. We argued that this new oscillating branewith KK modes on S is a good candidate for the AdS/CFT dictionary which corresponds to theharmonic 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 matrixon the Gauss graph basis should be evaluated in a more general setup. This includes higher loopeffects, a larger set of operators including the sl (2) sector [42], and the corrections from higherorders in n Y /n Z [43]. At strong coupling, the dynamics of D3-brane on AdS × S should be studiedin a comprehensive way. This includes to resum the (cid:15) series in the k = 0 solution (4 . U ( p )theory. This includes solving the finite oscillator for p >
2, finding consistent wave-functions for This is a problem of the spin-chain interpretation and not of N = 4 SYM. We can throw away the unphysicalpart from the BPS states in N = 4 SYM, because any superpositions of BPS states are again BPS. /N c corrections.The role of the superconformal symmetry needs to be examined. In particular, the N = 4 SYMtheory can be deformed while keeping su (2 | [53]. It is worth investigating the correspondingdeformation at strong coupling and finding the relation to the system of giant gravitons in AdS × S .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 × S [40]. Then, through thereduction to two-dimensions (4 . Acknowledgments
RS thanks the organizers of the workshops YITP-W-20-03 on
Strings and Fields 2020 , and
Online2020 NTU-Kyoto high energy physics workshop for stimulating discussions. He is grateful to Robertde Mello Koch, Arkady Tseytlin and Keisuke Okamura for comments on the manuscript. Thisresearch 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 theperturbative 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 (cid:96) L , whose dimensions are denoted by d R . D RIJ ( σ ) denotes the matrix represen-tation of σ in the irreducible representation R with the component ( I, J ), where
I, J = 1 , , . . . , d R .Consider the restriction S L ↓ ( S m ⊗ S n ) with m + n = L . We denote the irreducible decompositionby R = (cid:77) r (cid:96) ms (cid:96) n g ( r,s ; R ) (cid:77) ν =1 ( r ⊗ s ) ν (A.1)where ν is a multiplicity label and g ( r, s ; R ) is the Littlewood-Richardson coefficient. The branchingcoefficients are defined by the overlap between the components B R → ( r,s ) ,νI → ( i,j ) = (cid:28) RI (cid:12)(cid:12)(cid:12) r si j ν (cid:29) , ( B T ) R → ( r,s ) ,νI → ( i,j ) = (cid:28) r si j ν (cid:12)(cid:12)(cid:12) RI (cid:29) . (A.2)See [9] for the properties of these quantities.We denote partitions of an integer, or Young diagrams, in two ways. The symbol y = (cid:74) l , l , . . . , l p (cid:75) means that the i -th column of the Young diagram y has the length l i . The symbol y = [ m , m , . . . , m q ]24eans that the j -th row of y has the length m j . It follows that y (cid:96) L = p (cid:88) i =1 l i = q (cid:88) j =1 m j , l ≥ l ≥ · · · ≥ l p , m ≥ m ≥ · · · ≥ m q . (A.3) A.2 Distant corners approximation
We introduce the collective index( Z ⊗ n ) (cid:126)ı(cid:126) = Z i j Z i j . . . Z i n j n , ( U α ) (cid:126)(cid:126)k = δ j k α (1) δ j k α (2) . . . δ j n k α ( n ) ( α ∈ S n ) (A.4)with i p , j p = 1 , , . . . , N c . The matrix U α satisfies the composition rules( U α ) (cid:126)ı(cid:126) ( U β ) (cid:126)k(cid:126)ı = ( U αβ ) (cid:126)k(cid:126) , ( U α ) (cid:126)ı(cid:126) ( U β ) (cid:126)(cid:126)k = ( U βα ) (cid:126)ı(cid:126)k tr n ( U α ) = N C ( α ) c . (A.5)We denote multi-trace operators in the su (2) sector bytr L (cid:0) U α · Y ⊗ n Y Z ⊗ n Z (cid:1) = N c (cid:88) i ,i ,...,i L =1 Y i i α (1) Y i i α (2) . . . Y i nY i α ( nY ) Z i nY +1 i α ( nY +1) Z i nY +2 i α ( nY +2) . . . Z i L i α ( L ) (A.6)with L = n Y + n Z and α ∈ S L . We define the restricted Schur basis of operators by O R, ( r,s ) ,ν + ,ν − = 1 n Y ! n Z ! (cid:88) α ∈ S L χ R, ( r,s ) ,ν + ,ν − ( α ) tr L (cid:0) U α · Y ⊗ n Y Z ⊗ n Z (cid:1) (A.7) χ R, ( r,s ) ,ν + ,ν − ( α ) = d R (cid:88) I,J =1 d r (cid:88) i =1 d s (cid:88) j =1 B R → ( r,s ) ν + I → ( i,j ) ( B T ) R → ( r,s ) ,ν − J → ( i,j ) D RIJ ( α ) (A.8)coming from the restriction S L ↓ ( S n Y ⊗ S n Z ).It is expected that the half-BPS operators dual to p spherical giant gravitons consist of p longcolumns, with n Z = O ( N c ) with N c (cid:29)
1. Non-BPS operators can be constructed by attaching Y fields. We write r = (cid:74) l , l , . . . , l p (cid:75) where l i is the length of the i -th column. In the distant cornersapproximation, we assume that l i − l i − (cid:29)
1, so that the corners of r are well separated. Therefore,25e typically work with Young diagrams r = l l ... l p , R = ... p (A.9)where we construct R by adding s (cid:96) n Y (gray boxes) to r (cid:96) n Z (white boxes). A.3 Gauss graph basis
We introduce the Gauss graph basis following [15].
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 bythe standard Young tableaux, and those of R/r are by the skew Young tableaux. In the restrictedSchur polynomial (A . s goes to which box of R/r , beforesumming 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 , , . . . , p , should be important in finding the eigen-states of the perturbative dilatation operator of N = 4 SYM.Consider an example of p = 3, with s = (cid:74) , , (cid:75) and R/r = (cid:74) , , (cid:75) . We parameterize a state26f s and R/r using only the column labels, as s = 1 2 31 211 , R/r = 1212113 (A.10)In other words, we project the standard Young diagrams of shape s (cid:96) n Y onto the trivial (totallysymmetric) representation of H = S s ⊗ S s ⊗ . . . S s p ⊂ S n Y , s = (cid:74) s , s , . . . , s p (cid:75) . (A.11)The group H is an extra symmetry that emerges in the distant corners approximation [22]. Weshould refine the label of the restricted Schur operator O R, ( r,s ) ,ν + ,ν − by adding (cid:126)s = ( s , s , . . . , s p )which specifies how s (cid:96) 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 ofthe skew tableau R/r . In the above example (A .
10) we find { n i → j } = . (A.12)The adjacency matrix satisfies a conservation law. When a box with the label i goes to the j -thcolumn ( j (cid:54) = i ), then there must be a box k which comes to the i -th column. This implies therelation called the Gauss law constraints, p (cid:88) j =1 n i → j = p (cid:88) j =1 n j → i . (A.13)Intuitively, we may regard the diagonal elements n i → i as the number of closed strings on the i -thbrane, and the off-diagonal elements n i → j as the number of open strings between the i -th and j -thbrane.Here we defined the adjacency matrix from a skew Young diagram. The skew Young diagram isin one-to-one correspondence with the Gelfand-Testlin basis, as discussed in [14]. A.3.3 Permutation and the double coset
We can determine the adjacency matrix { n i → j } from a permutation element σ ∈ S n Y as follows.Given s = (cid:74) s , s , . . . , s p (cid:75) , we introduce a state | (cid:126)s (cid:105) ≡ | ⊗ . . . ¯ ⊗ (cid:105) ⊗ | ⊗ . . . ¯ ⊗ (cid:105) ⊗ · · · ⊗ | p ¯ ⊗ . . . ¯ ⊗ p (cid:105) = (cid:12)(cid:12) ¯ ⊗ s ⊗ ¯ ⊗ s ⊗ · · · ⊗ p ¯ ⊗ s p (cid:11) (A.14)27here ¯ ⊗ represents the symmetrized tensor product and ⊗ is the usual tensor product. The sym-metrization is equivalent to a sum over the states in V ⊗ n Y p where V p = { , , . . . , p } . We can permutethe state | (cid:126)s (cid:105) by applying σ ∈ S n Y to each summand as | σ , (cid:126)s (cid:105) = | σ (1) ⊗ σ (2) ⊗ · · · ⊗ σ ( n Y ) (cid:105) symm (A.15) ≡ | σ (1) ¯ ⊗ . . . ¯ ⊗ σ ( s ) (cid:105) ⊗ | σ ( s + 1) ¯ ⊗ . . . ¯ ⊗ σ ( s + s ) (cid:105) ⊗ · · · ⊗ | σ ( n Y − s p + 1) ¯ ⊗ . . . ¯ ⊗ σ ( s Y ) (cid:105) . This result consists of p tensor product of symmetrized components. It makes sense to count thenumber 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 ofan element in the double coset ¯ σ ∈ H \ S n Y /H ,¯ σ = 1 | H | (cid:88) γ ,γ ∈ H γ σγ , σ ∈ S n Y . (A.16)We can compute n i → j graphically as n Y − n Y σ . . .. . . γ (1)2 γ (2)2 γ ( p )2 γ (1)1 γ (2)1 γ ( p )1 n i → j ( σ ) ∼ | H | X γ ,γ ∈ H (A.17)which corresponds to ( s , s , . . . , s p ) = (3 , , . . . , A.3.4 Operator mixing in the Gauss graph basis
We define operators in the Gauss graph basis by O R,r ( σ ) = | H | (cid:112) n Y ! (cid:88) j,k (cid:88) s (cid:96) n Y (cid:88) ν − ,ν + D sjk ( σ ) B s → H ,ν − j ( B T ) s → H ,ν + k O R, ( r,s ) ,ν + ,ν − . (A.18)We consider the case n Z = O ( N c ) and n Y = O (1), which should correspond to excited multiplegiant gravitons. The one-loop dilatation acting on the restricted Schur polynomial factorizes intothe mixing of Y ’s and the mixing of Z ’s at the leading order of large N c . As shown in [15], themixing of Y ’s can be solved by taking the Gauss graph basis. The eigenvalues are labeled by thesymmetrized adjacency matrix, n ij ( σ ) ≡ n i → j ( σ ) + n j → i ( σ ) , n ij ( σ ) = n ji ( σ ) . (A.19) Our definition looks slightly different from [15] because our restricted Schur basis (A .
7) is not normalized. The factorization property is violated at the subleading order of n Y /n Z [43]. Z ’s changes the shape of r = (cid:74) l , l , . . . , l p (cid:75) . We use the simplified notation O ( (cid:126)l ) = O ( l , l , . . . , l p ) , p (cid:88) i =1 l i = n Z (A.20)in place of (A . B Explicit one-loop spectrum
We studying the spectrum of D G = −H following [13, 14] with minor improvement.We consider both continuum and discrete cases. The dilatation operator reduces to a set ofharmonic oscillators with boundary conditions in the continuum limit (2 . p = 2 by using the finite oscillator [57, 58]. 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 [14].In this limit, we find D G = −H → −D where D F ( (cid:126)y ) = E ( { n ij } ) F ( (cid:126)y ) , D ≡ p (cid:88) i (cid:54) = j n ij ( σ ) D ij . (B.1)We solve this equation in the region of { y i } given in (2 . D ij in (2 .
33) as D ij F = (cid:18) A + ( y ij ) A − ( y ij ) + 12 (cid:19) F, A ± ( y ) = 1 √ α (cid:26) αy ± ∂∂y (cid:27) (B.2)where y ij = y i − y j . The new differential operators { A ± ( y ) } satisfy[ A + ( y ij ) , A − ( y kl )] = δ ik + δ jl − δ il − δ jk . (B.3)The operator D ij has the symmetry[ D ij , Λ ± ] = 0 , Λ + ≡ p (cid:88) i =1 y i , Λ − ≡ p (cid:88) i =1 ∂∂y j . (B.4)Thus, Λ ± represent the zero modes of D . Roughly speaking, Λ ± correspond to the addition orremoval of a box from each of the p columns, suggesting that the spectrum of H depends only onthe difference of column lengths.From (B .
2) one finds that the eigenvalues of D ij are written as ( m + ). The mode number m should be chosen so that the variables y ij satisfy the Young diagram constraints (2 . y , . . . , y p − ,p , y p ) as y ≥ , y ≥ , . . . , y p − ,p ≥ , y p ≥ . (B.5)29onsider the operator ˜ D ≡ D − n Y = 2 p (cid:88) i 1) matrix which depends on { n i,j } with i < j . By diagonalizing M , we obtain the non-zero eigenvalues of ˜ D .˜ λ u δ uv = (cid:0) S M S T (cid:1) uv , ˜ D = 2 p − (cid:88) a =1 ˜ λ a A + (˜ z a ) A − (˜ z a ) . (B.9)The new coordinates { ˜ z a } are written as y a,a +1 = p (cid:88) b =1 ( S T ) ab ˜ z a . (B.10)We define the vacuum of H by requiring A − (˜ z a ) | (cid:105) = 0 ( ∀ a ) (B.11)and define excited states by applying the creation operators A + (˜ z a ). We solve the Young diagramconstraints (B . 5) as follows. The boundary of the constraints lies on y a,a +1 = 0, or equivalently˜ z a = 0. The wave function ψ ( { ˜ z a } ) should vanish at ˜ z a = 0 for all a , because it is ill-defined in theregion y a,a +1 < 0. Recall that the creation operators are parity odd, A + ( − ˜ z a ) = − A + (˜ z a ) . (B.12)The physical states should contain an odd number of creation operators for each a . Thus, thespectrum of ˜ D is ˜ D ψ (cid:126)m ( { ˜ z a } ) = 2 p − (cid:88) a =1 (2 m a + 1) ˜ λ a ψ (cid:126)m ( { ˜ z a } ) , m a ∈ Z ≥ . (B.13)The eigenvalues of the original equation (B . 1) are E ( { n ij } ) = n Y + 2 p − (cid:88) a =1 (2 m a + 1) ˜ λ a ( { n ij } ) . (B.14) We used (cid:80) i (cid:54) = j n ij ( σ ) = 2 n Y . .2 Discrete case The main difficulty in computing the discrete spectrum lies in how to impose the Young diagramconstraints. The Fock space created by the oscillator representation of H in (2 . 17) is not useful,because it does not immediately solve the constraints. Instead, we directly look for the wavefunctions. 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 , l ) in (2 . O f = (cid:100) ( l − l ) / (cid:101) (cid:88) x = − l f ( x ) O ( l − x, l + x ) . (B.15)The operator H , in (2 . 17) acts on O f as H , O f = (cid:16) h (1 , l ) + h (2 , l ) (cid:17) O f − (cid:112) h (1 , l ) h (2 , l + 1) (cid:88) x f ( x ) O ( l − x − , l + x + 1) − (cid:112) h (1 , l + 1) h (2 , l ) (cid:88) x f ( x ) O ( l − x + 1 , l + x − 1) (B.16)which gives the following discrete eigenvalue equation H , = h (1 , l ) + h (2 , l ) − (cid:112) h (1 , l ) h (2 , l + 1) e − ∂ x − (cid:112) h (1 , l + 1) h (2 , l ) e + ∂ x H , f ( x ) = E f ( x ) . (B.17)This equation determines the eigenvalue of D G = −H at p = 2 as D G = − n E (B.18)where we used n = n .We solve (B . 17) by relating it to a finite oscillator [57, 58]. Let us take a basis of states in theirreducible representations of su (2), J | j, j (cid:105) = j | j, j (cid:105) , ( j = − j, − j + 1 , . . . , j ) (B.19)and a rotated basis J | j, j (cid:105) = j | j, j (cid:105) , ( j = − j, − j + 1 , . . . , j ) . (B.20)In view of the effective U ( p ) theory, the su (2) generators can be interpreted as J = 12 (cid:0) d +1 d − + d +2 d − (cid:1) , J = i (cid:0) d +1 d − − d +2 d − (cid:1) , J = 12 (cid:0) d +1 d − − d +2 d − (cid:1) . (B.21)The generator (2 J ) is identical to J in (3 . (cid:104) j, j (cid:12)(cid:12) j, j (cid:105) = ( − j + j j (cid:115)(cid:18) jj + j (cid:19)(cid:18) jj + j (cid:19) F ( − j − j , − j − j ; − j ; 2) . (B.22)The su (2) generators acts on the states | j, j (cid:105) in the standard way, J | j, j (cid:105) = j | j, j (cid:105) , J ± | j, j (cid:105) = (cid:112) ( j ∓ j )( j ± j + 1) | j, j ± (cid:105) . (B.23)31e define n Z ≡ l + l , l = l − l (B.24)and assume n Z ≤ N c , which is trivial at p = 2. Let us assign j = N c − n Z − , j = − l − , j + j = m − n Z − N c − ≤ l ≤ N c − n Z , ≤ m ≤ N c − n Z + 2 . (B.26)Note that l = n Z − N c − ⇔ l = N c + 1 , l = 2 N c − n Z ⇔ l = N c . (B.27)The equations (B . 23) become J (cid:12)(cid:12)(cid:12) N c − n Z − , − l − (cid:69) = − l − (cid:12)(cid:12)(cid:12) N c − n Z − , − l − (cid:69) J + (cid:12)(cid:12)(cid:12) N c − n Z − , − l − (cid:69) = (cid:112) ( N c − l + 1)( N c − l + 1) (cid:12)(cid:12)(cid:12) N c − n Z − , − l + 12 (cid:69) J − (cid:12)(cid:12)(cid:12) N c − n Z − , − l − (cid:69) = (cid:112) ( N c − l )( N c − l + 2) (cid:12)(cid:12)(cid:12) N c − n Z − , − l − (cid:69) . (B.28)The last two lines agree with the off-diagonal terms in (B . 17) with the help of (2 . . 17) by relating the wave function to the rotation matrix (B . F m ( l , l ) = ( − m − − N c + ( n Z − (cid:115)(cid:18) N c − n Z + 1 m − (cid:19)(cid:18) N c − n Z + 1 N c − l (cid:19) × F ( − m + 1 , − N c + l ; − N c + n Z − 1; 2) . (B.29)this function satisfies the recursion relation (cid:112) h (1 , l ) h (2 , l + 1) F m ( l − , l + 1) + (cid:112) h (1 , l + 1) h (2 , l ) F m ( l + 1 , l − − (cid:16) h (1 , l ) + h (2 , l ) (cid:17) F m ( l , l ) = − m F m ( l , l ) (B.30)which is equivalent to the discrete eigenvalue equation (B . 17) with E = 2 m .We should impose the Young diagram constraints at p = 2, namely N c ≥ l ≥ l ≥ ( N c − l ) ( n Z ≥ N c ) l ≥ l ≥ n Z ≤ N c ) . (B.31)Note that the operator mixing does not change the value of n Z = l + l . It turns out that oursolution (B . 29) can solve these constraints only in limited cases.32 l - - F m ( l , l ) Nc = = = = l - F m ( l , l ) Nc = = = = m - - F m ( l , l ) Nc = = = - x - - F m ( l + x , l - x ) m = = = Figure 4: Plotting the function F m ( l , l ) against l , l (above) and m, x (below).Generally, the function F m ( l , l ) does not vanish even if l < 0, and it slowly decreases to zerowhen l is large and negative. This function has a special zero at F m (cid:48) ( l, l + 1) ( l, m (cid:48) ∈ Z ≥ ) (B.32)meaning that we can solve the Young diagram constraints if all of the following three conditions aresatisfied, • The mode number m is even, • n Z = l + l is odd, • n Z ≥ N c .The last condition may circumvented by using the translation symmetry of the recursion relation ( l , l , N c ) → ( l + x , l + x , N c + x ) (B.33)although this operation changes the value of N c . See Figure 4 for the behavior of F m ( l , l ).One finds that F m ( l , l ) for l , l , m ∈ Z ≥ satisfies0 = F m ( l , l ) ( l > N c ) (B.34)0 = F m ( l , l ) ( m < m > N c − n Z + 2) (B.35)as well as (B . m in the eigenvalue equation (B . 30) shouldbe chosen from m = 2 m (cid:48) , m (cid:48) = 1 , , . . . , (cid:108) N c − n Z (cid:109) . (B.36) There are other loci of zeroes, such as F m ( l , − N c ) = 0 if m ≤ N c and l ≥ N c − m + 2, which is notmeaningful. Practically there is no lower bound for l . This is different from the symmetry (cid:104) H , (cid:80) pi =1 d † i (cid:105) = 0. 33e can derive the identities (B . . 34) and (B . 35) from the hypergeometric identities in Ap-pendix C.In summary, at p = 2 we find D G O F m (cid:48) = 8 m (cid:48) n O F m (cid:48) , O F m (cid:48) = (cid:100) ( l − l ) / (cid:101) (cid:88) x = − l F m (cid:48) ( l − x, l + x ) O ( l , l ) (B.37)where m (cid:48) runs over the range (B . . 37) reproduces [18], and is valid for any scaling of( l , l ) 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 n ij = n i + n j ≡ ˜ n i + ˜ n j , ˜ n E ≡ p (cid:88) k =1 ˜ n k (B.38)where n E is the total number of edges. Then N ij simplifies a bit, N ij = ˜ n E δ ij + ˜ N ij , ˜ N ij ≡ ( p − 2) ˜ n i ( i = j ) − (˜ n i + ˜ n j ) ( i (cid:54) = j ) . (B.39)Consider some special cases. The first case is n i = m, (B.40)then λ a = pm ( a = 1 , , . . . p − a = p ) . (B.41)The residual symmetry is SO ( p − n (cid:29) , ˜ n i ∼ O (1) for ( i (cid:54) = 1) . (B.42)Then N ij ∼ ( p − n ( i = j = 1)˜ n ( j = j ≥ − ˜ n ( i = 1 , j (cid:54) = 1 or i (cid:54) = 1 , j = 1) (B.43)whose eigenvalues are p ˜ n , ˜ n , ˜ n , . . . , ˜ n (cid:124) (cid:123)(cid:122) (cid:125) p − , O (1) corrections. The residual symmetry is SO ( p − Identities of hypergeometric functions A special case of Gauss hypergeometric function F is called the Kravchuk (or Krawtchouk) poly-nomial used in [57], K n ( x ; p, q ) ≡ F (cid:16) − n, − x ; − q ; 1 /p (cid:17) = 1 (cid:0) n − q − n (cid:1) n (cid:88) j =0 p − j (cid:18) xj (cid:19)(cid:18) n − q − n − j (cid:19) (C.1)which satisfies the recursion relation0 = n (1 − p ) K n − ( x ; p, q ) + { x − n (1 − p ) − p ( q − n ) } K n ( x ; p, q ) + p ( q − n ) K n +1 ( x ; p, q ) . (C.2)The function F ( a, b ; c ; 2) is related to F ( a, b ; c ; − 1) by F ( a, b ; c ; z ) = Γ( c ) Γ( c − a − b )Γ( c − a ) Γ( c − b ) F ( a, b ; a + b + 1 − c ; 1 − z )+ Γ( c ) Γ( a + b − c )Γ( a ) Γ( b ) (1 − z ) c − a − b F ( c − a, c − b ; 1 + c − a − b ; 1 − z ) . (C.3)As a corollary, F ( − a, b ; c ; z ) = ( c − b ) a ( c ) a F ( − a, b ; b − c − a + 1; 1 − z ) , ( a = 0 , , , . . . ) (C.4)where ( x ) a = Γ( x + a ) / Γ( x ). We also have F ( a, b ; 1 + a − b ; − 1) = Γ(1 + a − b ) Γ(1 + a )Γ(1 + a ) Γ(1 + a − b ) (C.5)Now we can rewrite the rotation matrix (B . 22) as (cid:104) j, j (cid:12)(cid:12) j, j (cid:105) = ( − j + j j (cid:115)(cid:18) jj + j (cid:19)(cid:18) jj + j (cid:19) × ( j − j ) j + j ( − j ) j + j F ( − j − j , − j − j ; − j − j + 1; − . (C.6)and (B . 29) as F m ( l , l ) = ( − m − − N c + ( n Z − (cid:115)(cid:18) N c − n Z + 1 m − (cid:19)(cid:18) N c − n Z + 1 N c − l (cid:19) × ( l − N c − m − ( n Z − N c − m − F ( − m + 1 , − N c + l ; N c − l − m + 3; − . (C.7)Note that the original expression in terms of F ( a, b ; c ; 2) is more suitable for numerical evaluation.By combining the above identities, we find F m (cid:48) ( s, s + 1) = 2 − N c − m (cid:48) +3 s (cid:115)(cid:18) N c − sN c − s (cid:19)(cid:18) N c − s m (cid:48) − (cid:19) × Γ (cid:0) s − N c + (cid:1) Γ ( − N c − m (cid:48) + s ) Γ ( N c + 2 − m (cid:48) − s )Γ(1 − m (cid:48) )Γ ( − N c − m (cid:48) + 2 s ) Γ (cid:0) N c + − m (cid:48) − s (cid:1) (C.8)which vanishes at m (cid:48) = 0 , , , . . . due to Γ(1 − m (cid:48) ) = ∞ .35 Details of strong coupling analysis D.1 Geometry We parametrize S by using the embedding coordinates as X = R/ √ ρ cos η cos θ X = R/ √ ρ cos η sin θ X = R/ √ ρ sin η cos θ X = R/ √ ρ sin η sin θ (D.1) X = R (cid:112) − /ρ cos φ X = R (cid:112) − /ρ sin φ where ρ ≥ , η ∈ [0 , π/ 2] and φ, θ , θ ∈ [0 , π ). The polar coordinates on S spanned by( X , X , X , X ) make the symmetry of SO (4) = SU (2) × SU (2) manifest. The metric on R t × S is ds = R (cid:26) − dt + dρ ρ − ρ + ( ρ − dφ ρ + dη + cos η dθ + sin η dθ ρ (cid:27) (D.2)and the Laplacian is∆ = 1 R (cid:40) − ∂ ∂t + 4( ρ − ρ ∂ ∂ρ + 4 ρ ∂∂ρ + ρρ − ∂ ∂φ + ρR ∆ S (cid:41) ∆ S = ∂ ∂η + 2 cot(2 η ) ∂∂η + 1cos η ∂ ∂θ + 1sin η ∂ ∂θ . (D.3)The Jacobian is d Ω = R sin(2 η )4 ρ dρ dφ dη dθ dθ (D.4)and thus T (cid:90) Σ R sin(2 η )2 ρ dφ dη dθ dθ = N c (cid:90) R dt ∂ t φρ (D.5)where (4 . 3) is used. The RR four-form satisfies dC ∝ ( d Ω + ∗ d Ω ), and the normalization is chosenso that the BPS configuration satisfies E = J . D.2 Spherical harmonics The scalar spherical harmonics on S is [59, 60]∆ S Φ k,m ,m = − k ( k + 2) Φ k,m ,m (D.6)where ∆ S is given in (D . 3) andΦ k,m ,m ( η, θ , θ ) = C k,m ,m (cid:0) e iθ cos η (cid:1) m − m (cid:0) e iθ sin η (cid:1) m + m P m + m ,m − m k/ − m (cos 2 η ) C k,m ,m = (cid:114) k + 12 π (cid:115) ( k/ m )! ( k/ − m )!( k/ m )! ( k/ − m )! , k − m i = 0 , , . . . , k, k ∈ Z ≥ (D.7)36nd P a,bn ( x ) is the Jacobi polynomial. Note that { m i } are half-integers when k is odd, and Φ k,m ,m is complex-valued, (Φ k,m ,m ) ∗ = ( − m + m Φ k, − m , − m . (D.8)The spherical harmonics satisfy the orthogonality (cid:90) π dθ (cid:90) π dθ (cid:90) π/ dη sin 2 η − m + m Φ k,m ,m Φ l, − n , − n = δ l,k δ m ,n δ m ,n (D.9)and the integrated spherical harmonics satisfy (cid:90) π dθ (cid:90) π dθ (cid:90) π/ dη sin 2 η (cid:110) ( ∂ η Φ k,m ,m ) ( ∂ η Φ l, − n , − n ) + ( ∂ θ Φ k,m ,m ) ( ∂ θ Φ l, − n , − n )cos η + ( ∂ θ Φ k,m ,m ) ( ∂ θ Φ l, − n , − n )sin η (cid:111) = k ( k + 2) δ l,k δ m ,n δ m ,n . (D.10)The first few eigenfunctions areΦ , , = 1 √ π , Φ , , = e iθ sin ηπ , Φ , , − = e iθ cos ηπ , (D.11)Φ , , = (cid:114) π cos(2 η ) , Φ , , = (cid:114) π e i ( θ + θ ) sin(2 η ) , Φ , , = (cid:114) π e i ( θ − θ ) sin(2 η ) . The functions Φ (cid:96), , are equal to the Legendre polynomial P (cid:96) (cos 2 η ). D.3 Classical solutions at j = 0 , Consider the classical D3-brane solutions around j = 0 , 1, or equivalently J = 0 , N c /g s . For thispurpose, we use the coordinate r = 1 / √ ρ with 0 ≤ r ≤ 1. The metric (D . 2) becomes ds = R (cid:26) − dt + dr − r + (1 − r ) dφ + r (cid:16) dη + cos η dθ + sin η dθ (cid:17)(cid:27) . (D.12)We expand the equations of motion around r = (cid:112) j + (cid:15) r ( t, η ) , φ = t + (cid:15) φ ( t, η ) , ( j = 0 , . (D.13)First, consider the case j = 1. The EoM for φ becomes ∂ t r = 0. If we write r ≡ − e − s ( η ) , theEoM for r becomes0 = 14 (cid:0) s (cid:48) − η ) s (cid:48) − s (cid:48)(cid:48) (cid:1) − (cid:15) (cid:26) ∂ t φ − e − s ( η ) (cid:0) s (cid:48)(cid:48) ( s (cid:48) + 2) − s (cid:48) ( s (cid:48) + 4) + 4 cot(2 η ) s (cid:48) ( s (cid:48) + 6) (cid:1)(cid:27) + O ( (cid:15) ) . (D.14) Our convention is same as JacobiP[n,a,b,x] in Mathematica . O ( (cid:15) ) is e − s ( η ) = (cid:18) c (2) + c (1) log cot η (cid:19) . (D.15)To maintain r ∈ [0 , 1] we need c (1) = 0. Then, the equation (D . 14) produces ∂ t φ = 0. Thus thereis no non-trivial solution.Next, consider the case j = 0. The EoM’s give us0 = r (cot(2 η ) r + ∂ η r ) = r ∂ η φ . (D.16)The non-trivial solution of the first equation is r = c √ sin 2 η (D.17)for a constant c . This function diverges around η = 0 , π/ 2, and is inconsistent with r ∈ [0 , E On non-abelian DBI The symmetry of the effective U ( p ) theory becomes non-abelian if some of the column lengthsbecome equal, l i = l i +1 for some i in (A . p giant graviton branescoincide at strong coupling. Let us make a short digression about non-abelian DBI action to examinethis situation. The DBI action is a low energy effective action of closed and open string massless modes on thebrane. As a worldvolume theory, the DBI action without U (1) flux is also an example of 4d conformaltheory. In AdS × S , the conformal symmetry of the target spacetime is nonlinearly realized [61,62].The DBI action can be made supersymmetric in the sense of κ symmetry [40, 63, 64] and of theworldvolume symmetry [65, 66]. The fundamental strings can be coupled to multiple coincidentD-branes by introducing non-abelian flux F µν [67]. The addition of the CS term to the non-abelianDBI induces dielectric effects [44].One way to define non-abelian DBI is to expand the DBI action in a formal series of F ,det( G + 2 πα (cid:48) F ) = det G (cid:18) − (2 πα (cid:48) ) G ab F bc G cd F da + . . . (cid:19) . (E.1)Then we promote F to a non-abelian field, and take the trace. This procedure suffers from theordering ambiguity, which should be fixed by the consistency with the open string amplitude [68–71].In Section 4, we want to find classical solutions continuously connected to the spherical giantgraviton. From the above prescription for the non-abelian DBI in (E . 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