Giant graviton interactions and M2-branes ending on multiple M5-branes

We study splitting and joining interactions of giant gravitons with angular momenta $N^{1/2}\ll J\ll N$ in the type IIB string theory on $AdS_5 \times S^5$ by describing them as instantons in the tiny graviton matrix model introduced by Sheikh-Jabbari. At large $J$ the instanton equation can be mapped to the four-dimensional Laplace equation and the Coulomb potential for $m$ point charges in an $n$-sheeted Riemann space corresponds to the $m$-to-$n$ interaction process of giant gravitons. These instantons provide the holographic dual of correlators of all semi-heavy operators and the instanton amplitudes exactly agree with the pp-wave limit of Schur polynomial correlators in ${\cal N}=4$ SYM computed by Corley, Jevicki and Ramgoolam. By making a slight change of variables the same instanton equation is mathematically transformed into the Basu-Harvey equation which describes the system of M$2$-branes ending on M$5$-branes. As it turns out, the solutions to the sourceless Laplace equation on an $n$-sheeted Riemann space correspond to $n$ M5-branes connected by M2-branes and we find general solutions representing M2-branes ending on multiple M5-branes. Among other solutions, the $n=3$ case describes an M2-branes junction ending on three M5-branes. The effective theory on the moduli space of our solutions might shed light on the low energy effective theory of multiple M5-branes.

supersymmetric ten-dimensional plane-wave background [21]. We refer to this matrix model as the IIB plane-wave matrix model in this paper.
In the lightcone gauge we fix x + = t while imposing h 0l = 0 and choose the spatial world-volume coordinates σ l such that the lightcone momentum density −p − is a constant. The lightcone Hamiltonian for the 3-brane is then given by [21,31] −P + = d 3 σ [σ] 2(−P − ) where I, J, K = 1, 2, · · · , 8 are transverse directions, x I = (x i , x a ) and p I = (p i , p a ) are the conjugate momenta of x I . P ± are the zero-modes of p ± and the conjugate momenta of x ± .
[σ] is the total volume in the σ-space defined as [σ] = d 3 σ . (2.6) The Nambu three-bracket in (2.5) is defined for real functions, f p (σ) with p = 1, 2, 3, as Since the constraints, h r0 = 0, can be recast as This should correspond to the generator of the residual local symmetry analogous to the area-preserving diffeomorphism of the membrane theory in the lightcone gauge. We further compactify the x − in the background (2.2) on a circle of radius R, resulting in the quantised total lightcone momentum: where J is an integer. We replace the functions by matrices, x I (σ) → X I , (2.11) where X I and Π I are J × J matrices, and implement the further replacements, where Υ 5 is a non-dynamical J ×J matrix explained in Appendix C and the quantum Nambu four-bracket is defined among matrices, F p with p = 1, 2, 3, 4, as In (2.13) the parameter l is analogous to in quantum mechanics and given by 4 With these replacements (2.11) -(2.14) we finally obtain the bosonic part of the lightcone Hamiltonian of the IIB plane-wave matrix model [21], 5 H B = R tr 1 2 (Π I ) 2 + 1 2 µ R 2 (X I ) 2 + (2π 2 T ) 2 2 · 3! [X I , X J , X K , (2.17) The full supersymmetric IIB plane-wave matrix model with P SU (2|2) × P SU (2|2) × U (1) symmetry is given by the following lightcone Hamiltonian [21]: , (2.18) where H B is given by (2.17). The J × J matrices Ψ are spinors of two SU (2)'s and each spinor carries two kinds of indices in which each index is the Weyl index of one of two SO(4)'s under the isomorphism, SO(4) ∼ = SU (2) × SU (2). There exist the constraints which would be a matrix regularisation of the supersymmetric extension of (2.9) in the continuum theory: on the physical states [21]. The bracket [ , ] denotes the matrix commutator. The lightcone Hamiltonian (2.18) can be derived from a Lagrangian of the corresponding supersymmetric matrix quantum mechanics with U (J) gauge symmetry in which the component of the gauge field A 0 is set to zero. In order to maintain this gauge condition along the lightcone time flow, one has to impose the Gauss-law constraints which are nothing but (2.19). The P SU (2|2) × P SU (2|2) × U (1) superalgebra in the plane-wave background can be realised by the J × J matrices [21]. The plane-wave background (2.2) can be obtained from AdS 5 × S 5 : One starts with the global AdS 5 × S 5 spacetime where R S denotes the AdS 5 and S 5 radius of curvature. One then zooms into the trajectory of a particle moving along a great circle in S 5 at large angular momentum J and sitting at the centre ρ = 0 of AdS 5 . To see what happens, one introduces rescaled coordinates, where i = 1, 2, 3, 4 and a = 5, 6, 7, 8 and further introduces the lightcone coordinates with µ 0 being a dimensionless parameter. Due to the strong centrifugal force, at large angular momentum J = −i∂ θ 9 the trajectory of a particle is confined to the region close to the great circle in the 56 plane of R 6 where S 5 is embedded. This implies that (2.23) Since the particle at the centre of ρ = 0 of AdS 5 , we also have In this region of spacetime, (2.23) and (2.24), the global AdS 5 × S 5 spacetime (2.20) is approximated by the plane-wave background (2.2) with the identification The relation between R and R S is given by In this paper, the plane-wave background is the approximation of the AdS 5 × S 5 geometry near the observer with large angular momentum J. Thus the matrix size J in the IIB plane-wave matrix model is considered to be very large for our purposes.

Vacua
Similar to the plane-wave matrix model for M-theory [7], the IIB plane-wave matrix model has abundant static zero energy configurations [21]. Since the bosonic Hamiltonian (2.17) can be expressed as a sum of squares, there exist three kinds of vacua [21]: The solutions to (2.29) and (2.30) preserve a half of the supersymmetries and represent concentric fuzzy S 3 classified by J × J representations of Spin(4) = SU (2) L × SU (2) R [21,22] We denote this J × J irreducible representation by J. As for reducible representations, the matrices are block-diagonal and each size is, say, J l with l = 1, 2, · · · , n and J = J 1 +J 2 +· · ·+J n , which can be expressed as J 1 ⊕J 2 ⊕· · ·⊕J n . This configuration corresponds to the concentric n fuzzy S 3 's and the block of size J l has the radius, In order for the plane-wave approximation to be valid, the radius of each giant graviton r l should be much smaller than R S . This leads to the condition Quantum corrections are well controlled if the length scale r l is much larger than the 10d Planck length l p = g 1/4 s l s . This yields another condition Combining the two (2.35) and (2.36), we obtain the bound for J l : In the following, we study the tunnelling processes which interpolate various vacua (corresponding to giant gravitons) classified by the representation of Spin(4), i.e. (anti-)instanton solutions of the IIB plane-wave matrix model. As will be elaborated further, the (anti-)instantons describe splitting or joining interactions of concentric giants. 6 Similar (anti-)instantons have been discussed in the BMN matrix model [15,13] and our analysis will be analogous to theirs.

Instanton equations
In order to find (anti-)instanton solutions, we consider the Euclidean IIB plane-wave matrix model. Hereafter we ignore the fermionic matrices by setting Ψ = 0. The Euclidean action for the bosonic IIB plane-wave matrix model is where t is now the Euclidean time. One can show that the Euclidean action (2.38) can be rewritten as sum of squares and boundary terms: (2.39) Therefore, the Euclidean action is bounded by the boundary terms and (anti-)instantons are configurations which saturate the bound. In this manner, the (anti-)instanton equations can be obtained: and the same equations with the replacement, (i, j, k, l) ↔ (a, b, c, d). We will focus on the (anti-)instanton equation (2.40) associated with AdS 5 , but the S 5 case can be obtained from the AdS 5 case by interchanging the indices. One notices that the (anti-)instanton equation (2.40) implies the equation: where the double sign is correlated with the one in (2.40) and where X j 0 (±∞) are matrices forming static concentric fuzzy S 3 's and U is an arbitrary unitary matrix introduced to maintain the gauge condition, A 0 = 0.
Using the equation (2.41), one can show that the (anti-)instanton action is non-negative: In particular, we are going to consider instantons interpolating between the vacuum of m giant gravitons, J 1 ⊕J 2 ⊕· · ·⊕J m , at t = −∞ and that of n giant gravitons, J 1 ⊕J 2 ⊕· · ·⊕J n , at t = +∞, where J = J 1 + J 2 + · · · + J m = J 1 + J 2 + · · · + J n . The Euclidean action in this case becomes Since this condition always holds if m ≤ n, we mostly focus on splitting interactions by setting m ≤ n unless otherwise stated. Joining interactions, i.e. m ≥ n, can be obtained via anti-instantons. Note that the condition (2.46) is a necessary condition for instantons to exist and the necessary and sufficient condition will be discussed in the end of Section 3.
In the dual CFT it is expected that this type of giant graviton interactions corresponds to (m + n)-point functions of antisymmetric Schur operators (for sphere giants) and symmetric Schur operators (for AdS giants) [5,4]. In fact, the pp-wave limit of 3pt functions of (anti-)symmetric Schur operators has been discussed in [30]: where O S 5 J and O AdS 5 J are antisymmetric and symmetric Schur operators, respectively. These correspond to the 2-to-1 process; two giants with J 1 ⊕ J 2 at t = −∞ joining into one giant with J at t = +∞.
We thus find the exact agreement within our approximation between the 3pt function of antisymmetric Schur operators (2.47) and the instanton amplitude, since we found (2.49) for J = J 1 + J 2 in (2.45). Note that this is exponentially small in the range N 1/2 J N but remains finite at large N . The 3pt function of symmetric Schur operators (2.48), however, cannot correspond to instantons since it grows exponentially as opposed to damping, whereas the instanton action was proven to be always positive. We will not resolve this puzzle concerning AdS giants raised in [30] and only focus on interactions of sphere giants in the rest of our paper.
As we will show later, this agreement for antisymmetric Schur operators persists to generic (m + n)-point functions, i.e. to the instantons interpolating m sphere giants at t = −∞ and n sphere giants at t = +∞.

Four-dimensional Laplace equation in Riemann spaces
We wish to find solutions to the instanton equation (2.40) when the matrix size J is very large. In the case of the BMN matrix model, the instanton equation analogous to (2.40) can be mapped to the 3d Laplace equation and various solutions, such as one membrane splitting into two membranes, have been found [13]. In this section we show that the instanton equation (2.40) can be mapped to the 4d Laplace equation following the procedure laid out in [13]. As will be shown later, the key observation in [13] is the following relations, as illustrated in Fig. 3: We begin with making a change of variables, We note that this is mathematically the same as the Basu-Harvey equation [23] which describes M2 branes ending on M5 branes. This connection to the Basu-Harvey equation will be exploited in the later section.
In order to find the solutions describing giant graviton interactions, they have to asymptote to the vacua (static giant gravitons) at the infinite past and future: 7 where the ellipses indicate subleading terms and X i 0 (±∞) are J ×J representations of Spin(4) satisfying (2.29) corresponding to the clusters of giants. X i 0 (±∞) also need to satisfy the necessary and sufficient condition for the existence of instantons discussed in the end of Section 3. These set the boundary conditions for the solutions we are after.
When the matrix size is very large, the matrices Z i can be approximated by the functions z i (s, σ µ ) and the quantum Nambu 4-bracket [ * , * , * , Υ 5 ] by the Nambu 3-bracket. This is the "classicalisation" of the brackets, reversing the procedure (2.11) -(2.14). Then the Basu-Harvey equation (3.4) can be approximated by which can be locally mapped to the 4d Laplace equation as shown in Appendix A. Essentially, this map can be made by interchanging the role of dependent and independent variables: This means solving s as a function of z i : Using this hodograph transformation the equation (3.7) is mapped to the 4d Laplace equation (see Appendix A for details): We will then find solutions to the Laplace equation (3.10) corresponding to splitting interactions of concentric giants. The equipotential surface provides the profile of giant gravitons for a given s in the z-space.
Let us see how a single fuzzy three-sphere can be described by a solution to the Laplace equation. A single fuzzy S 3 corresponds to the J × J irreducible representation of Spin(4) which is a static solution to the instanton equation (2.29) and denoted by the matrices X i 0 .
By the change of variables (3.3) we can map X i 0 to the matrices Z i 0 representing the spatial coordinates of giants: When the matrix size J is very large, we replace the matrices X i 0 and Z i 0 , by functions x i and z i , and accordingly, (3.11) is approximated by where x i form a three-sphere of radius (2.32): Here n i is the unit vector normal to the three-sphere (see Appendix C for details). One can solve s as a function of z i by (3.12): (3.14) This is nothing but the Coulomb potential in four dimensions with charge J at the origin, which, of course, solves the Laplace equation (3.10). Through this simple example, we have learned that a single giant graviton with angular momentum J can be described by the 4d Coulomb potential for point charge J. We shall generalise this to the instantons interpolating between m (concentric) giants at t = −∞ and n (concentric) giants at t = +∞. As will be explained in Section 3.2, these splitting processes of concentric giants are described by the solutions to the 4d Laplace equation in multi-sheeted Riemann spaces rather than the ordinary 4d Euclidean space R 4 . The 4d Riemann spaces are a four-dimensional analogue of 2d Riemann surfaces, and the precise definition will be given in 3.1.
The use of Riemann spaces has been first emphasised in the study of membrane interactions [13]: They considered splitting interactions of concentric spherical membranes with large angular momenta as instantons in the BMN matrix model. It was found that the instanton equation in their case can be locally mapped to the 3d Laplace equation and the splitting interactions correspond to the Coulomb potentials in multi-sheeted 3d Riemann spaces.

Hypertoroidal coordinates and Riemann spaces
We introduce the coordinates which are particularly useful for studying the solutions to the 4d Laplace equation in multi-sheeted Riemann spaces. In this paper we call them the hypertoroidal coordinates.
To set up, we consider a point P designated by (ρ, θ) in the bipolar coordinates relating to the two-dimensional Cartesian coordinates (ξ, η) as The definition of ρ and θ is given as follows. We call two points in the two-dimensional Cartesian coordinates, (−a, 0) and (a, 0), A and B, respectively (see Fig. 1). The angle ∠AP B is denoted by θ defined to be in the interval [−π, π]; where |AP | and |BP | are the lengths of segments, AP and BP , respectively and by definition ρ ∈ (−∞, ∞). If we extend the interval of θ from [−π, π] to [−π, 3π], the bipolar coordinates become multi-valued. To make the coordinates single-valued, we introduce a cut, say the segment AB, and stitch two copies of R 2 's by the cut AB such that if θ ∈ [−π, π], the space belongs to an R 2 and if θ ∈ [π, 3π], it does to the other R 2 . This space is a 2d (two-sheeted) Riemann space, which can be easily extended to an (n + 1)-sheeted Riemann space if one considers the interval of θ to be [−π, π +2πn] with n being positive integer. In that case we prepare (n+1) copies of R 2 such that each R 2 is specified by the different interval of θ, [−π + 2πm, π + 2πm] with m = 0, 1, · · · , n. We then stitch the (n + 1) R 2 's together by the cut AB, resulting in an (n + 1)-sheeted Riemann space.
We introduce the hypertoroidal coordinates as a 4d extension of the bipolar coordinates. 8 This can be constructed by rewriting the 4d spherical coordinates (z 1 , z 2 , z 3 , z 4 ) = r(cos λ, sin λ cos ϕ, sin λ sin ϕ cos ω, sin λ sin ϕ sin ω) , (3.17) as (z 1 , z 2 , z 3 , z 4 ) = (η, ξ cos ϕ, ξ sin ϕ cos ω, ξ sin ϕ sin ω) , The Fig. 2 is a graphical expression of the hypertoroidal coordinates in which r = |OP |. The region inside the circle drown by the heavy line stands for a three-ball of radius a embedded in R 3 . This three-ball plays a role analogous to a branch cut in a 2d Riemann space once we extend the interval of θ.
Since ξ and η are the same as (3.15), the interval of ρ and θ is θ ∈ [−π, π] and ρ ∈ (−∞, ∞), respectively. The angles, ϕ and ω, are respectively defined to be in the intervals, [0, π] and [0, 2π]. Extending the interval of θ from [−π, π] to [−π, π + 2πn] with n being positive integer as in the case of the bipolar coordinates, the hypertoroidal coordinates become multi-valued. In order to make the coordinates single-valued, we need to introduce an object analogous to a cut in a 2d Riemann space which is a three-ball of radius a located at θ = −π + 2πm with m = 0, 1, · · · , n (see Fig. 2). We call this three-ball a branch three-ball. As before we prepare (n + 1) copies of R 4 such that each R 4 is designated by the different interval of θ, [−π + 2πm, π + 2πm] with m = 0, 1, · · · , n. Gluing the (n + 1) R 4 's at the branch three-ball, we can construct a 4d (n + 1)-sheeted Riemann space.
It goes back to 1896 when Sommerfeld first considered the three-dimensional Laplace equation in Riemann spaces [19,20]. We shall extend his idea to the four-dimensional space for the purpose of finding solutions describing splitting interactions of concentric giant gravitons, following the success of [13] in their application of [32,33] to membrane interactions.

Splitting interactions of giant gravitons
We now discuss in detail the construction of solutions to the 4d Laplace equation in Riemann spaces which describe splitting interactions of concentric giant gravitons with large angular momenta. As we have seen in Section 3, after the map to the Laplace equation, the snapshots of giant gravitons at time s in the z-space are the equipotential surfaces s = φ(z i ), and a single giant with angular momentum J corresponds to the Coulomb potential created by a point charge J. From (3.3) the infinite past and future correspond to s = +∞ and s = 0, respectively.
The construction of our solutions goes as follows. (See Fig. 3): In a 4d Riemann space with n-sheets we place m point charges but only allow at most one charge per a single sheet. This corresponds to the instanton interpolating one vacuum J 1 ⊕ J 2 ⊕ · · · ⊕ J m at t = −∞ and the other J 1 ⊕J 2 ⊕· · ·⊕J n at t = ∞ with the constraints J 1 +· · ·+J m = J 1 +· · ·+J n = J and m ≤ n. Simply put, the correspondence is The number of point charges equals the number of giants at t = −∞, and the number of sheets is the number of giants at t = +∞. This is because the infinite past s = +∞ corresponds to the diverging potential at the locations of m point charges and the infinite future s = 0 to the asymptotic infinities, z i → ∞, in the Riemann space. The electric flux runs through the branch three-ball to different sheets and escapes to the asymptotic infinities. By construction the necessary condition (2.46), or equivalently, the condition m ≤ n is automatically satisfied. This construction is the 4d analog of the one for membrane interactions [13]. The above construction can be worked out explicitly by applying Sommerfeld's extended image technique [19,20]: To begin with, we consider the 4d Coulomb potential where J is a point charge placed at z i = z i 0 in R 4 . Using the hypertoroidal coordinates (ρ, θ, ϕ, ω) defined in (3.18) and (3.19), the distance squared from the charge is expressed as where we defined cosh α := cosh ρ cosh ρ 0 − sinh ρ sinh ρ 0 (cos ϕ cos ϕ 0 + sin ϕ sin ϕ 0 cos(ω − ω 0 )) .

(3.24)
As explained in Section 3.1, once we extend the interval of the angle θ from [−π, π] to [−π, π + 2π(n − 1)] with n ≥ 2, the hypertoroidal coordinates become multi-valued and we can construct an n-sheeted Riemann space by stitching n R 4 's at the branch three-ball of radius a and make the coordinates single-valued. Because the distance squared R 2 in (3.23) is periodic in the angle θ, so is the Coulomb potential (3.22) and there must be charges placed in every single sheet at the same location. In other words, the Coulomb potential is an n-charge solution where every single sheet has one charge at the same location in R 4 . In order to find generic m ≤ n charge solutions, we first look for the electrostatic potential created by a single charge placed in only one of the n sheets in the Riemann space. This is going to serve as the building block for the construction of more general potentials. One can distill a single charge contribution from the Coulomb potential (3.22) by expressing it as a contour integral and deforming the contour [19,20].

Coulomb potential in two-sheeted Riemann space
Let us first consider the two-sheeted case. We complexify the angle θ and introduce the complex variable ζ = e iθ/2 which covers the two-sheeted Riemann space. The Coulomb potential (3.22) can then be expressed as a contour integral , (3.25) where the contour C θ is a unit circle surrounding ζ = e iθ/2 . The factor (cosh ρ−cos θ)/(cosh ρ− cos θ ) in the integrand is inserted to ensure that the integrand vanishes at θ = ±i∞.
Besides the poles at θ = θ + 4kπ with k ∈ Z, the integrand in (3.25) has the poles at We now deform the contour C θ to a rectangle of width 4π and an infinite height while Figure 4: The contour deformation avoiding the poles at θ = θ 0 ± iα and θ 0 + 2π ± iα (see Fig. 4). The contributions from the vertical edges cancel out owing to the periodicity, and those from the horizontal edges at infinity simply vanish. The single charge contribution is extracted as the residue of a pole and its pair in the lower-half plane. Note first that at θ θ 0 + 2kπ ± iα we have The relevant part of the contour comes in from infinity, encircles a pole clockwise and goes back to infinity, picking up the residue. The single charge potential is thus found to be . (3.28) The consistency requires where the second term is the contribution from a charge on the second sheet. Carrying out the contour integral (3.28), we find that (3.30) One can check that (3.29) holds by noting that φ k=1 (θ) = φ k=0 (θ + 2π).

Coulomb potential in n-sheeted Riemann space
It is straightforward to generalise the two-sheet case to the n-sheeted Riemann space. We start from and deform the contour in a similar manner to the two-sheet case. There are poles at θ = θ + 2nkπ with k ∈ Z and θ = θ 0 + 2kπ ± iα . (3.32) Similar to the two-sheet case, the rectangle contour with width 2nπ picks up the residues from these poles. The Coulomb potential splits into The single charge potential is thus given by By superposing different contributions, it is easy to construct the solution to the 4d Laplace equation describing general m giants J 1 ⊕ J 2 ⊕ · · · ⊕ J m at t = −∞ splitting into n giants J 1 ⊕ J 2 ⊕ · · · ⊕ J n at t = ∞ with the constraints J 1 + · · · + J m = J 1 + · · · + J n = J and m ≤ n: where we defined φ  Before closing this section, we discuss the necessary and sufficient condition for the existence of instantons when J 1. In the case of instantons in the BMN matrix model, the necessary and sufficient condition given in [35] has been reproduced by the linearity of the 3d Laplace equation and the positivity of angular momenta [13]. Since the proof concerning the condition does not depend on the dimensionality, we can apply it directly to our case. We conjecture that the condition derived in [13] coincides with the necessary and sufficient condition in our case as well, and we just state the condition: We consider an instanton interpolating m giant gravitons at t = −∞ and n giant gravitons at t = ∞ characterised by J 1 ⊕ · · · ⊕ J m and J 1 ⊕ · · · ⊕ J n , respectively and satisfying J 1 + · · · + J m = J 1 + · · · + J n = J and m ≤ n. Since the angular momenta are positive, one can consider a histogram of J i 's. Drawing a horizontal line atJ ≥ 0 on the histogram, we define the area of the histogram of J i 's belowJ: As for a histogram of J i 's, one can define the area A(J; J 1 , · · · , J n ) in the same manner. The necessary and sufficient condition for the existence of instantons can be given [13]:

Giant graviton correlators in CFT
We consider splitting interactions of (concentric) sphere giants in the dual CFT, i.e. N = 4 U (N ) SYM, in the large-R charge sector [7]. The CFT operators dual to giant gravitons with angular momentum J are Schur operators of degree J for the unitary group U (N ) defined by [5,4]: where R J is an irreducible representation of U (N ) expressed by a Young diagram with J boxes, χ R J (σ) is the character of the symmetric group S J in the representation R J , the sum is over all elements of S J and Z is an N ×N complex matrix with i 1 , i 2 , · · · , i J = 1, 2, · · · , N . If the representation R J is symmetric (antisymmetric), the operator (4.1) corresponds to an AdS giant (a sphere giant) [5,4]. We will discuss correlation functions of Schur operators in antisymmetric representations in order to compare them with the instanton results found in the previous section. 9

Three-point functions of sphere giants
The normalisation of higher point functions can be provided by the two point function: where A J denotes the antisymmetric representation. For antisymmetric representations the dimension d A J of the representation A J is always 1. The dimension of the representation A J of the unitary group is with C(σ) being the number of cycles in the permutation σ. For anti-symmetric representations the character χ A (σ) is either 1 or −1. It is known that where the indices i and j are the label of rows and columns, respectively, in the Young diagrams associated with the representation R. If R is an anti-symmetric representation, there is only one column and J rows, yielding Thus we have This provides the normalisation of higher point functions. Now the 3pt function of sphere giants, corresponding to one giant with momentum J = J 1 + J 2 spliting into two giants with momenta J 1 and J 2 , is given by the formula: where g(A J 1 , A J 2 ; A J ) is a Littlewood-Richardson coefficient, an analogue of the Clebsch-Gordan coefficient, and denotes the multiplicity of the representation A J in the tensor product of representations A J 1 and A J 2 , and we have used g(A J 1 , A J 2 ; A J ) = 1. This is incidentally identical to the two-point function. Thus the normalised three-point functions yield where ||χ A J || := J!Dim N (A J )/d A J . In the pp-wave limit, as we discussed in the end of Section 2.2, this exactly agrees with the instanton amplitude as in (2.47).

General m-to-n functions of sphere giants
The general m → n correlators are also known and given by the formula [5,4] We only consider the case where all R's and T 's are antisymmetric representations. The numbers of boxes for R i and T i are J i and J i , respectively and J 1 + · · · + J m = J 1 + · · · + J n . At large N and J the middle factor f U : is dominated by the representations U * which have the largest number of columns as j labels the columns. Thus U * must have min(n, m) columns since it has to be constructible both from R's and T 's. Without loss of generality we can assume that m ≤ n.
We first order R's and T 's such that the number of boxes J 1 ≥ J 2 ≥ · · · ≥ J n and J 1 ≥ J 2 ≥ · · · ≥ J m . Then the dominant Young diagrams U * at large N and J are composed by first gluing m columns of diagrams T 's in this order and then moving some of the boxes down to the left while keeping the number of columns to be m. The boxes have to be moved so that U * is also constructible from R 1 ⊗ R 2 ⊗ · · · ⊗ R n . For these representations we have where J * k is the number of boxes in the k-th column of the Young diagram U * . For large J k 's and N we can approximate J * k 's by J k 's. Since the Littlewood-Richardson coefficients are of order 1, their contributions are negligible at large N and J, and we find that which exactly agrees with the instanton amplitude e −S E for generic m-to-n instanton action (2.45).

The Basu-Harvey equation
As we have seen in Section 3, the instanton equation ( where M 11 is the eleven-dimensional Planck mass and λ is the dimensionless coupling constant. The instanton equation (2.40) then becomes 10 This was proposed as an equation describing M2-branes ending on M5-branes by the M2brane worldvolume theory. This is a natural generalisation of Nahm's equation describing monopoles or the D1-D3 system. The four scalars Z i 's are U (J) matrices and the coordinates transverse to M5-branes, and s is one of the worldvolume coordinates of M2-branes. In the large-J limit a prototypical solution to the Basu-Harvey equation (5.2) is a spike made of a bundle of J M2-branes on a single M5-brane of topology, s a semi-infinite line s ∈ [0, +∞] and S 1 M is the M-theory circle corresponding to Υ 5 . In [24] this was called the ridge solution describing a self-dual string soliton.
When the matrix size J is large, as outlined in (2.11) -(2.14), the quantum Nambu 4-bracket is replaced by the (classical) Nambu 3-bracket and the Basu-Harvey equation becomes with a i being a constant vector. As remarked, this describes the space R + s × S 3 and the radius of the three-sphere varies along the semi-infinite line as Note that s = 0 corresponds to the location of the M5-brane at which the radius of S 3 becomes infinite. This is interpreted as an M2-brane spike threading out from a single M5-brane. We next consider M2-branes stretched between two M5-branes discussed in [23,24,25,37]. The semi-infinite line R + s must be replaced by a finite interval I s = {s|s ∈ [−s 0 , +s 0 ]} and near the two M5-branes at s = ±s 0 the solution behaves as An important observation is that the solution with this boundary condition cannot be constructed from Coulomb potentials. The reason is that the presence of a point charge necessarily develops a spike as we can see in (5.5): At the location of the charge z i = a i , s goes to infinity and thus any solution with point charges cannot represent a finite interval. This implies that the solutions describing two or more M5-branes are not in the same class of solutions as those describing giant graviton interactions. However, similar to the giant graviton case, the idea is to look for solutions to the 4d Laplace equation in the multi-sheeted Riemann space. In this case we expect that the number of sheets corresponds to the number of M5-branes.
To find the solution which satisfies the boundary condition (5.7), recall the contour integral expression of the electrostatic potential where R −2 is the 4d Coulomb potential as previously defined in Section 3. We can add a constant c to the Coulomb potential since the constant potential solves the 4d Laplace equation. We now focus on the constant part of the potential Besides the poles at θ = θ + 4kπ with k ∈ Z, there are poles at We deform the contour C θ to a rectangle of width 4π (for the two-sheet case) and an infinite height while avoiding the poles at θ = ±iρ and ±iρ + 2π. Noticing that near the poles cosh ρ − cos θ ∼ ±i sinh ρ (θ − (±iρ + 2kπ)) , (5.12) similar to the Coulomb potential case, the contribution from the first sheet to the constant potential can be found as One can check that this solves the 4d Laplace equation. The contribution from the second sheet is φ k=1 0 (z i ) = c − φ k=0 0 (z i ). Note that at the two asymptotic infinities (ρ, θ) → (0, 0) and (ρ, θ) → (0, 2π) where z i 's go to infinity, the electrostatic potential φ k=0 0 ( z) approaches different values, cJ λ /(4π 2 ) and 0, respectively. By shifting the potential by a constant s 0 , these values can be shifted to s 0 and −s 0 with the choice s 0 = cJ λ /(8π 2 ). Hence, the potential φ k=0 0 ( z) describes a finite interval of length 2s 0 . In the n-sheeted Riemann space the trivial constant potential splits into nontrivial potentials defined on each sheet by the contour deformation: 14) The explicit form of the potentials for higher k's can be found in the end of this section.

M2-branes stretched between two M5-branes -funnel solution
As discussed above, the solution representing M2-branes stretched between two M5-branes can be constructed from a trivial constant electrostatic potential by distilling the contribution from one of the two Riemann sheets. 11 The M2-branes connecting the two M5-branes have the shape of a funnel: Let us examine this solution more in detail. Recalling the parametrisation of the coordinates this can be expressed as The midpoint of the funnel s = 0 corresponds to θ = π, 3π which implies η = 0 and |ξ| ≤ a. This is the brach ball B 3 and thus in terms of z i 's the midpoint s = 0 corresponds to a threeball of radius a. We plot the funnel solution in Fig. 6. The constant s hypersurfaces are squashed three-spheres and the radius blows up at the endpoints s = ±s 0 and the squashed S 3 collapses to a three-ball at s = 0. 12 This collapse of the funnel throat is similar to what happens to D1-branes stretched between two D3-branes [36]. Figure 6: The funnel solution: The two ends at s = ±s 0 are the locations of the two M5branes. Each ring is a constant s hypersurface and represents a squashed S 3 whose radius blows up at the ends and which collapses to a three-ball at the midpoint.
Note that at the two asymptotic infinities where z i 's are very large, the coordinates ξ and η become very large, since z 2 1 + z 2 2 + z 2 3 + z 2 4 = ξ 2 + η 2 . Thus the funnel at large z i 's behaves as satisfying the boundary condition (5.7).

M2-branes ending on multiple M5-branes
The power of this method, albeit only in the limit of an infinite number of M2-branes, is that the solution can be easily generalised to the cases with more than two M5-branes. We start from the contour integral for a constant potential: Besides the poles at θ = θ + 2nkπ with k ∈ Z, there are poles at θ = ±iρ + 2k . (5.20) 12 If it were in one less dimensions, a squashed S 2 would have collapsed or flattened to a D 2 .

Summary and discussions
We studied the dynamical process of giant gravitons, i.e. their splitting and joining interactions, in the type IIB string theory on AdS 5 × S 5 . It was made possible by restricting  ourselves to small size giants whose angular momenta are in the range N 1/2 J N for which the spacetime can be well approximated by the plane-wave background. We found that the most effective description was provided by the tiny graviton matrix model of Sheikh-Jabbari [21,22], which we referred to as the IIB plane-wave matrix model, rather than BMN's type IIB string theory on the pp-wave background.
We showed, in particular, that their splitting/joining interactions can be described by instantons/anti-instantons in the IIB plane-wave matrix model. They connect one vacuum, a cluster of m concentric (fuzzy) sphere giants, in the infinite past to another vacuum, a cluster of n concentric (fuzzy) sphere giants, in the infinite future. In the large J limit the instanton equation can be mapped locally to the 4d Laplace equation and the m-ton interaction corresponds to the Coulomb potential of m point charges on an n-sheeted Riemann space.
Giant graviton interactions are dual to correlators of Schur polynomial operators in N = 4 SYM. The latter have been calculated exactly by Corley, Jevicki and Ramgoolam [5]. We compared the instanton amplitudes to the CFT correlators and found an exact agreement for generic m and n within the validity of our approximation. This lends strong support for our description of giant graviton interactions. However, to be more precise, the agreements are only for the sphere giants which expand in S 5 and are dual to antisymmetric Schur operators and a puzzle, as pointed out in [30], remains for the AdS giants which expand in AdS 5 and are dual to symmetric Schur operators. The issue is that the correlators of symmetric Schur operators exponentially grow rather than damp in the pp-wave limit.
A next step would be going beyond the classical approximation and include fluctuations about (anti-)instantons in order to find N/J 2 corrections. This involves integrations over bosonic and fermionic zero modes and requires finding the moduli space of (anti-)instantons which includes geometric moduli associated with the Riemann space, i.e. the number of sheets and the number, positions and shapes of branch three-balls, as discussed in the case of membrane interactions [13]. This is not an easy problem.
As a byproduct of this study we also found new results on multiple M5-branes. We exploited the fact that the instanton equation is identical to the Basu-Harvey equation which describes the system of M2-branes ending on M5-branes [23]. In the large J limit which corresponds, in the Basu-Harvey context, to a large number of M2-branes, we found the solutions describing M2-branes ending on multiple M5-branes, including the funnel solution and an M2-branes junction connecting three M5-branes as simplest examples. The number n of M5-branes corresponds to the number of sheets in the Riemann space, and somewhat surprisingly, multiple M5-branes solutions are constructed from a trivial constant electrostatic potential. Upon further generalisations, for example, adding more branch balls, the effective theory on the moduli space of our solutions might shed light on the low energy effective theory of multiple M5-branes [26,27,28,29].
Finally, our technique is applicable to the well-known SU(∞) limit of Nahm's equation which describes (an infinite number of) D1-branes ending on D3-branes by mapping it locally to the 3d Laplace equation [17,18]. This might give us a new perspective on the moduli space of monopoles. and Seiji Terashima for discussions and comments. SH would like to thank the Graduate School of Mathematics at Nagoya University, Yukawa Institute for Theoretical Physics and Chulalongkorn University for their kind hospitality. YS would like to thank all members of the String Theory Group at the University of the Witwatersrand for their kind hospitality, where this work was initiated. The work of SH was supported in part by the National Research Foundation of South Africa and DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). The work of YS was funded under CUniverse research promotion project by Chulalongkorn University (grant reference CUAASC).

A A derivation of the Laplace equation
We are going to show that the following differential equation can be mapped to the ndimensional Laplace equation: where z p and z p i with p, p i = 1, 2, · · · , n are functions of (s, σ l ) with l = 1, 2, · · · , n − 1. On the RHS the Nambu (n − 1)-bracket is defined by and [σ] = d n−1 σ .
In order to find z p (s, σ l ) from solutions to the Laplace equation ∆φ(z 1 , · · · , z n ) = 0, we use (A.7) and (A.6). Namely, the equation (A.7) implies that the electric flux density in the (n − 1)-dimensional σ-space is the constant J [σ] at a given s. In other words, the Guassian surface of constant electric fields is tangent to the σ-space and normal to the time s: E · ∂ z ∂ σ = 0 and E · ∂ z ∂s = −1 using (A.6), where the electric field E(z 1 , · · · , z n ) = − ∂φ ∂ z . These n equations determine z p 's as functions of (s, σ l ).

B The Euclidean 3-brane theory
In this appendix we are going to show that the continuum version of the Basu-Harvey equation (3.7) can be obtained from the Euclidean 3-brane theory.
We start with the gauge-fixed lightcone Hamiltonian (2.5). Using Hamilton's equation, the action becomes By a Wick-rotation the Euclidean action yields where t is the Euclidean time. The Euclidean action (B.3) can be recast as a sum of squares and boundary terms: This is minimised when the first order BPS equations are satisfied 13 By a change of variables,

C Three-spheres and their quantisation
We give a brief review of the relation between three-spheres and the Nambu 3-bracket. Upon quantisation of this relation, S 3 's become fuzzy S 3 's and the Nambu 3-bracket is replaced by the quantum Nambu 4-bracket. The construction of fuzzy S 3 's will be given below. The parameter in the quantisation of the Nambu bracket is analogous to in quantum mechanics (2.16) and fixed by the requirement that the radius of S 3 coincides with that of fuzzy S 3 . We start with an S 3 of radius r We choose the spherical coordinates to be x i = rn i = r(cos λ, sin λ cos ϕ, sin λ sin ϕ cos ω, sin λ sin ϕ sin ω) .

(C.2)
We can then show that x i 's satisfy the following equation: This establishes the relation between S 3 's and the Nambu 3-bracket.

C.1 Fuzzy three-spheres
The fuzzy S 3 's can be constructed as a subspace of fuzzy S 4 's [38,39]. We only recapitulate the essential part of the construction and leave details to the original papers [38,39]. We introduce J × J matrices, where Γ i are the four-dimensional 4 × 4 Dirac matrices, Γ 5 is the SO(4) chirality operator, 1 is the 4 × 4 unit matrix, n is an odd integer and the suffix 'sym' denotes a symmetric n-fold tensor product. Here P R is the projector onto the J × J representation R of SO(4) ∼ = SU (2) L × SU (2) R given by R = n − 1 4 , n + 1 4 ⊕ n + 1 4 , n − 1 4 , (C. 8) where (j L , j R ) is an irreducible representation of Spin(4) = SU (2) L ×SU (2) R . The dimension of R specifies the size of matrices J: J = dim R = (n + 1)(n + 3) 2 . (C.9) Using Υ i and Υ 5 , one can construct a fuzzy S 3 of unit radius: where the quantum Nambu 4-bracket is defined in (2.15) and This can be easily generalised to a fuzzy S 3 of radius r F by which satisfy We denote the irreducible J × J representation (C.8) of Spin(4) by J. In the case of a reducible representation, J 1 ⊕ J 2 ⊕ · · · ⊕ J n (C.14) with J 1 + J 2 + · · · + J n = J, the solutions to the equation (C.13) form n concentric fuzzy S 3 's. This establishes the relation between fuzzy S 3 's and the Nambu 4-bracket.

C.2 Fixing the quantisation parameter
We elaborate on our choice of the quantisation parameter l in (2.13). Recall that the threebrane theory defined by the Hamiltonian (2.5) has the vacua obeying As a result the radius (C.16) of the S 3 is found as where we used (2.25), (2.26) and (2.33). Indeed, with the choice of in (2.16), the quantisation procedure (2.11) -(2.14) yields the radius of the fuzzy S 3 to be (2.32) which coincides with (C.18).