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

Journal of High Energy Physics, May 2018

Abstract We study splitting and joining interactions of giant gravitons with angular momenta N 1/2 ≪ J ≪ N in the type IIB string theory on AdS5 × S5 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 \( \mathcal{N} \) = 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 M2-branes ending on M5-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.

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

Accepted: May Giant graviton interactions and M2-branes ending on multiple M5-branes Shinji Hirano 0 1 2 3 5 6 Yuki Sato 0 1 2 4 6 Jan Smuts Avenue 0 1 2 6 Johannesburg 0 1 2 6 South Africa 0 1 2 6 0 Kitashirakawa-Oiwakecho , Kyoto 606-8502 , Japan 1 University of the Witwatersrand 2 DST-NRF Centre of Excellence in Mathematical and Statistical Sciences , CoE-MaSS 3 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University 4 Department of Physics, Faculty of Science, Chulalongkorn University 5 School of Physics and Mandelstam Institute for Theoretical Physics & 6 Thanon Phayathai , Pathumwan, Bangkok 10330 , Thailand We study splitting and joining interactions of giant gravitons with angular momenta N 1=2 AdS-CFT Correspondence; D-branes; M(atrix) Theories; M-Theory - J N in the type IIB string theory on AdS5 S5 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 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 M2-branes ending on M5-branes. As it turns out, the solutions to the sourceless Laplace equation on an nsheeted Riemann space correspond to n M5-branes connected by M2-branes and we nd 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 e ective theory on the moduli space of our solutions might shed light on the low energy e ective theory of multiple M5-branes. 1 Introduction 2 IIB plane-wave matrix model 2.1 Vacua 2.2 Instanton equations 3 Four-dimensional Laplace equation in Riemann spaces General m-to-n functions of sphere giants The Basu-Harvey equation M2-branes stretched between two M5-branes | funnel solution M2-branes ending on multiple M5-branes U(N ) SYM [6]. On the CFT side, their interactions correspond to multi-point correlators of Schur polynomial operators and have been computed exactly for half-BPS giants in [5]. S5 which is dual to N = 4 { 1 { However, on the gravity side, being extended objects (spherical D3-branes), it is rather challenging to go beyond kinematics and study their dynamical interaction process except for so-called heavy-heavy-light three point interactions. This is the problem we tackle in the most part of this paper and we report modest but nontrivial progress on this issue. Instead of attempting to solve the issue once and for all, we consider a certain subset of giant gravitons, namely, those whose angular momentum J are relatively small, i.e. in the range N 1=2 J N . These giants can be studied in the plane-wave background [7{9]: for an observer moving fast in the sphere, the spacetime looks approximately like a plane-wave geometry.1 Thus if the size of giants is small enough,2 the observer moving along with the giants can study them in the plane-wave background [7{9]. This strategy was inspired by the recent work of one of the authors which studied splitting and joining interactions of membrane giants in the M-theory on AdS4 S7=Zk at nite k by zooming into the plane-wave background [13, 14]. Since the M-theory on the planewave background is described by the BMN plane-wave matrix model [7], small membrane giants can be studied by this matrix quantum mechanics. Their idea is that since the vacua of the BMN matrix model represent spherical membranes, instantons interpolating among them correspond to the process of membrane interactions. They explicitly constructed these instantons by mapping the BPS instanton equation [15] to Nahm's equation [16, 17] in the limit of large angular momenta where Nahm's equation becomes equivalent to the 3d Laplace equation [18, 19]. The crux of their construction is to consider the Laplace equation not in the ordinary 3d Euclidean space but in a 3d analog of 2d Riemann surfaces, dubbed Riemann space [20]. In our case of the type IIB string theory on AdS5 S5, as it turns out, the most e ective description of giant gravitons with the angular momentum N 1=2 J N is provided by the tiny graviton matrix model proposed by Sheikh-Jabbari [21, 22] rather than BMN's type IIB string theory on the pp-wave background.3 The description of giant graviton interactions is similar to the above M-theory case, and in the large J limit the instanton equation in this matrix quantum mechanics can be mapped to the Laplace equation but in four dimensions instead of three. As we will see, the 4d Coulomb potential for m point charges in an n-sheeted Riemann space corresponds to the m-to-n interaction process of giant gravitons. An advantage over the M-theory case is that we can compare our description of giant graviton interactions to that of N = 4 SYM. Indeed, we nd that the instanton amplitude exactly agrees with the pp-wave limit of Schur polynomial correlators in N = 4 SYM computed by Corley, Jevicki and Ramgoolam [5]. This also implies that these instantons successfully provide the holographic dual of correlators of all semi-heavy operators. Last but not the least, as a byproduct of this study we are led to nd new results on elusive M5-branes. By a slight change of variables, the instanton equation of the type IIB plane-wave matrix model 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 1The plane-wave geometry can be obtained from AdSp Sq by taking the Penrose limit [10{12]. 2Small giants are an oxymoron. They are small in the sense that their size is much smaller than the AdS radius, but they are not point-like and much larger than the Planck length. 3In this paper we refer to the tiny graviton matrix model as the type IIB plane-wave matrix model. { 2 { solutions might shed light on the low energy e ective theory of multiple M5-branes [26{ instanton equation and nd the (anti-)instanton action for the m-to-n joining and splitting process of giant gravitons. As the rst check of our proposal we show that the instanton amplitude e SE in the case of the 2-to-1 interaction agrees with the 3-point correlators of antisymmetric Schur operators in the dual CFT, i.e. N = 4 SYM. In section 3, we transform the instanton equation to the Basu-Harvey equation by a suitable change of variables and show that in the large J limit it is further mapped locally to the 4d Laplace equation. We then solve the 4d Laplace equation in multi-sheeted Riemann spaces and nd the solutions which describe the generic m-to-n joining and splitting process of (concentric) sphere giants. In section 4, we discuss the pp-wave limit of correlators of antisymmetric Schur operators in the dual CFT and show that they exactly agree with the instanton amplitudes obtained in section 3. In section 5, we study the Basu-Harvey equation in the original context, namely, as a description of the M2-M5 brane system. In the large J limit corresponding to a large number of M2-branes, we nd the solutions to the 4d Laplace equation which describe M2-branes ending on multiple M5-branes. Section 6 is devoted to summary and discussions. In the appendices A, B and C we elaborate further on some technical details. 2 IIB plane-wave matrix model The tiny graviton matrix model was proposed by Sheikh-Jabbari as a candidate for the discrete lightcone quantisation (DLCQ) of the type IIB string theory on the maximally supersymmetric ten-dimensional plane-wave background [21]. We refer to this matrix model as the IIB plane-wave matrix model in this paper. Here we outline the derivation of the IIB plane-wave matrix model. The bosonic part of the IIB plane-wave matrix model can be obtained by a matrix regularisation of the e ective action for a 3-brane [21]: S = T Z dtd3 q j det(h )j + C^^^^ ; (2.1) ; 8. The background metric is the plane-wave geometry: with +; ; 1; where T = 1=((2 )3gsls4) is the D3-brane tension with gs and ls being the string coupling constant and string length, respectively. The world-volume coordinates are = (t; l) = 0; 1; 2; 3 and l = 1; 2; 3. The indices for the target space are hatted, ^; ^; ^; ^ = g^^dx^dx^ = 2dx+dx 2(xixi + xaxa)dx+dx+ + dxidxi + dxadxa ; (2.2) { 3 { with i = 1; 2; 3; 4 and a = 5; 6; 7; 8. The induced metric on the 3-brane is h and C^^^^ is the Ramond-Ramond 4-form with nonvanishing components C+ijk = ijklxl ; C+abc = abcdxd : The parameter in (2.2) and (2.4) is the mass parameter. In the lightcone gauge we x x+ = t while imposing h0l = 0 and choose the spatial world-volume coordinates l such that the lightcone momentum density p is a constant. The lightcone Hamiltonian for the 3-brane is then given by [21, 36] P+ = Z d 3 [ ] is the total volume in the -space de ned as ; 8 are transverse directions, xI = (xi; xa) and pI = (pi; pa) are the are the zero-modes of p and the conjugate momenta of x . Z [ ] ( p ) pI J R ; { 4 { (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) The Nambu three-bracket in (2.5) is de ned for real functions, fp( ) with p = 1; 2; 3, as Since the constraints, hr0 = 0, can be recast as the dynamics of x can be determined by that of the transverse directions. The con= 0, can be rewritten as This should correspond to the generator of the residual local symmetry analogous to the area-preserving di eomorphism 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, where XI and I are J J matrices, and implement the further replacements, xI ( ) pI ( ) XI ; J The full supersymmetric IIB plane-wave matrix model with PSU(2j2) PSU(2j2) symmetry is given by the following lightcone Hamiltonian [21]: H = HB + R tr y R _ _ _ _ _ _ ( ij ) _ _[Xi; Xj ; y _ _ ; 5] + _ _ ( ab) _ _[Xa; Xb; y _ _ ; 5 ] ; where 5 is a non-dynamical J J matrix explained in appendix C and the quantum Nambu four-bracket is de ned among matrices, Fp with p = 1; 2; 3; 4, as In (2.13) the parameter l is analogous to ~ in quantum mechanics and given by4 (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) U(1) (2.18) With these replacements (2.11){(2.14) we nally obtain the bosonic part of the lightcone Hamiltonian of the IIB plane-wave matrix model [21],5 ! ! 1 J 1 1 4! where HB is given by (2.17). The J J matrices 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 4We explain how to x the parameter l in appendix C. 5The bosonic lightcone Hamiltonian (2.17) becomes the one in [21] by choosing the unit, 4 ls4 = 1, and changing ! . This sign di erence originates from that in the replacement (2.13). where RS denotes the AdS5 and S5 radius of curvature. One then zooms into the trajectory of a particle moving along a great circle in S5 at large angular momentum J and sitting at the centre = 0 of AdS5. To see what happens, one introduces rescaled coordinates, = x 0 RS ; p(xi)2 RS ; 9 = x 9 RS ; a = 2 + x a RS ; where i = 1; 2; 3; 4 and a = 5; 6; 7; 8 and further introduces the lightcone coordinates x + = x0 + x9 ; x = 0(x0 x9) ; with 0 being a dimensionless parameter. Due to the strong centrifugal force, at large angular momentum J = to the great circle in the 56 plane of R6 where S5 is embedded. This implies that Since the particle at the centre of = 0 of AdS5, we also have would be a matrix regularisation of the supersymmetric extension of (2.9) in the continuum theory: i[Xi; i] + i[Xa; a] + 2 y + 2 y _ _ _ _ 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 eld A0 is set to zero. In order to maintain this gauge condition along the lightcone time ow, one has to impose the Gauss-law constraints which are nothing but (2.19). The U(1) superalgebra in the plane-wave background can be realised jxaj RS 1; jxij RS jx j RS approximated by the plane-wave background (2.2) with the identi cation S5 spacetime (2.20) is The relation between R and RS is given by R 2 HB = tr ( I )2 + there exist three kinds of vacua [21]: Xi = Xa = 3! 3! 2 2RT ijkl[Xj ; Xk; Xl; 5] 6= 0 ; 2 2RT abcd[Xb; Xc; Xd; 5] 6= 0 ; Xa = Xi = 0 : The solutions to (2.29) and (2.30) preserve a half of the supersymmetries and represent concentric fuzzy S3 classi ed by J J representations of Spin(4) = SU(2)L SU(2)R [21, 22]. (See appendix C for more details.) These fuzzy S3's are identi ed with giant gravitons and in particular the solutions to (2.29) and (2.30) are called AdS and sphere giants, respectively. For irreducible representations of Spin(4), the solutions to (2.29) and (2.30) become a single giant graviton with the radius because 0 : In this paper, the plane-wave background is the approximation of the AdS5 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. (2.27) S5 geometry Similar to the plane-wave matrix model for M-theory [7], the IIB plane-wave matrix model has abundant static zero energy con gurations [21]. Since the bosonic Hamiltonian (2.17) can be expressed as a sum of squares, (2.28) (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) which can be inferred from (2.25), (2.26) and We denote this J J irreducible representation by J. As for reducible representations, the matrices are block-diagonal and each size is, say, Jl with l = 1; 2; ; n and J = J1 + J2 + + Jn, which can be expressed as J1 J2 Jn. This con guration corresponds to the concentric n fuzzy S3's and the block of size Jl has the radius, r = r J 2 2RT = RS r J N ; RS4 = 4 N gsls4 : rl = r J l 2 2RT = RS r J l N : { 7 { In order for the plane-wave approximation to be valid, the radius of each giant graviton rl should be much smaller than RS. This leads to the condition Planck length lp = gs1=4ls. This yields another condition Quantum corrections are well controlled if the length scale rl is much larger than the 10d (2.35) (2.36) (2.37) Combining the two (2.35) and (2.36), we obtain the bound for Jl: HJEP05(218)6 In the following, we study the tunnelling processes which interpolate various vacua (corresponding to giant gravitons) classi ed 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 [13, 15] and our analysis will be analogous to theirs. 2.2 Instanton equations In order to nd (anti-)instanton solutions, we consider the Euclidean IIB plane-wave matrix model. Hereafter we ignore the fermionic matrices by setting = 0. The Euclidean action for the bosonic IIB plane-wave matrix model is { 8 { 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: dXi dt + + dXa dt (2 2RT )2 2 d dt d dt 6These vacua are 1=2-BPS and marginally stable. Nonetheless, the instanton and anti-instanton amplitudes corresponding, respectively, to splitting and joining interactions are nonvanishing. However, they are equal and there is an equilibrium of splitting and joining processes. Xa 2 2RT abcd[Xb; Xc; Xd; 5 ] 2 Therefore, the Euclidean action is bounded by the boundary terms and (anti-)instantons are con gurations which saturate the bound. In this manner, the (anti-)instanton equations can be obtained: 3! d dt 2 ; dXi dt Xi 2 2RT ijkl[Xj ; Xk; Xl; 5] = 0 ; 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 AdS5, but the S5 case can be obtained from the AdS5 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 W [X] = (Xi)2 + 12 2 2RT ijklXi[Xj ; Xk; Xl; 5] : The equation (2.41) implies that the functional W [X] monotonically decreases or increases in progress of the Euclidean time depending on a choice of the double sign. We call solutions such that W [X] decreases (increases) instantons (anti-instantons). These tunnelling processes would be governed by the path integral with boundary conditions: Xj ( 1) = X0 ( j 1) ; j Xj (1) = U X0 (1)U 1 ; where X0 ( j negative: 1) are matrices forming static concentric fuzzy S3's and U is an arbitrary unitary matrix introduced to maintain the gauge condition, A0 = 0. Using the equation (2.41), one can show that the (anti-)instanton action is nongiant gravitons, J1 J2 at t = +1, where J = J1 + J2 + this case becomes Jm, at t = 1 and that of n giant gravitons, J01 J02 + Jm = J10 + J20 + + J n0. The Euclidean action in When deriving the second equality, we have used (2.25), (2.26) and (2.33). From (2.45) together with the non-negativity of the Euclidean action (2.44), one nds the condition for the partition of J : 1 2R In particular, we are going to consider instantons interpolating between the vacuum of m n X J 02 i i=1 m X Jj : 2 j=1 { 9 { 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 su cient 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) [4, 5]. In fact, the pp-wave limit of 3pt functions of (anti-)symmetric Schur operators has been discussed in [35]: hOJS5 OJS15 OJS25 i = hOJAdS5 OJA1dS5 OJA2dS5 i = s s (N where OS5 and OAdS5 are antisymmetric and symmetric Schur operators, respectively. These correspond to the 2-to-1 process; two giants with J1 J2 at t = 1 joining into one giant with J at t = +1. We thus nd the exact agreement within our approximation between the 3pt function of antisymmetric Schur operators (2.47) and the instanton amplitude, since we found e SE = e 2N ; J1J2 for J 0 = J1 +J2 in (2.45). Note that this is exponentially small in the range N 1=2 but remains nite 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 [35] 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 = 3 1 and n sphere giants at t = +1. Four-dimensional Laplace equation in Riemann spaces We wish to nd 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 gure 3: [# of giants at t = [# of giants at t = +1] = [# of sheets of Riemann space] : We begin with making a change of variables, and the instanton equation (2.40) can be rewritten in terms of the new variables, dZi ds = 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 nd the solutions describing giant graviton interactions, they have to asymptote to the vacua (static giant gravitons) at the in nite past and future:7 Zi(s) = Zi(s) = s RT s RT 2 s X0i ( 1) + 2 s X0i (1) + ; ; for s ! 1 ; for s ! 0 ; where the ellipses indicate subleading terms and X0i ( Spin(4) satisfying (2.29) corresponding to the clusters of giants. X0i ( 1) are J J representations of 1) also need to satisfy the necessary and su cient 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 Zi can be approximated by the functions zi(s; ) and the quantum Nambu 4-bracket [ ; ; ; 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 [ ] ijklfzj ; zk; zlg = J 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: (z1; z2; z3; z4) $ (s; 1; 2; 3) : This means solving s as a function of zi: s = (zi) : 4 X i=1 = 0 : (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) Using this hodograph transformation the equation (3.7) is mapped to the 4d Laplace equation (see appendix A for details): 7These boundary conditions can be shifted by identify matrices Zi(s) ! Zi(s) aiIJ J . We will then nd solutions to the Laplace equation (3.10) corresponding to splitting interactions of concentric giants. The equipotential surface provides the pro le 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 S3 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 X0i . By the change of variables (3.3) we can map X0i to the matrices Z0i representing the spatial coordinates of giants: When the matrix size J is very large, we replace the matrices X0i and Z0i , by functions xi and zi, and accordingly, (3.11) is approximated by Z0i = RT s X0i : zi = xi ; (3.11) where xi form a three-sphere of radius (2.32): xi = rni ; Here ni is the unit vector normal to the three-sphere (see appendix C for details). One can solve s as a function of zi by (3.12): 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 = 1 and n (concentric) giants at t = +1. 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 R4. The 4d Riemann spaces are a four-dimensional analogue of 2d Riemann surfaces, and the precise de nition will be given in 3.1. The use of Riemann spaces has been rst 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. = \AP B, = log jAP j=jBP j and jAOj = jBOj = a. 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 ( ; ) in the bipolar coordinates relating to the two-dimensional Cartesian coordinates ( ; ) as = cosh cos ; = a sin cosh cos : (3.15) (3.16) The de nition of and is given as follows. We call two points in the two-dimensional Cartesian coordinates, ( a; 0) and (a; 0), A and B, respectively (see gure 1). The angle \AP B is denoted by de ned to be in the interval [ ; ]; = log jAP j ; jBP j where jAP j and jBP j are the lengths of segments, AP and BP , respectively and by de nition 2 ( If we extend the interval of from [ ; 3 ], 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 R2's by the cut AB such that if 2 [ ; ], the space belongs to an R 2 and if 2 [ ; 3 ], it does to the other R2. 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 to be [ ; + 2 n] with n being positive integer. In that case we prepare (n + 1) copies of R2 such that each R2 is speci ed by the di erent interval of , [ 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 (z1; z2; z3; z4) = r(cos ; sin cos '; sin sin ' cos !; sin sin ' sin !) ; (3.17) 8A 3d extension of the bipolar coordinates is called the toroidal coordinates or the peripolar coordinates. line stands for a three-ball of radius a embedded in R3. This three-ball plays a role analogous to a branch cut in a 2d Riemann space once we extend the interval of . as where (z1; z2; z3; z4) = ( ; cos '; sin ' cos !; sin ' sin !) ; a sin cosh cos r cos = = ; r sin = = cosh 2 [ cos The gure 2 is a graphical expression of the hypertoroidal coordinates in which r = jOP j. Since and are the same as (3.15), the interval of and is ; ] and 2 ( 1; 1), respectively. The angles, ' and !, are respectively de ned to be in the intervals, [0; ] and Extending the interval of from [ ; +2 n] 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 = + 2 m with m = 0; 1; ; n (see gure 2). We call this three-ball a branch three-ball. As before we prepare (n + 1) copies of R4 such that each R4 is designated by the di erent interval of , [ three-ball, we can construct a 4d (n + 1)-sheeted Riemann space. It goes back to 1896 when Sommerfeld rst considered the three-dimensional Laplace equation in Riemann spaces [20]. We shall extend his idea to the four-dimensional space for the purpose of nding solutions describing splitting interactions of concentric giant gravitons, following the success of [13] in their application of [37, 38] to membrane interactions. 3.2 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, are the branch three-balls and point electric charges, respectively. The branch three-balls are all identi ed. The number of point charges m corresponds to the number of giant gravitons at t = and the number of sheets of the Riemann space n to the number of giant gravitons at t = +1. 1 the snapshots of giant gravitons at time s in the z-space are the equipotential surfaces s = (zi), and a single giant with angular momentum J corresponds to the Coulomb potential created by a point charge J . From (3.3) the in nite past and future correspond to s = +1 and s = 0, respectively. The construction of our solutions goes as follows. (See gure 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 J1 J2 and the other J01 J0 2 J0n at t = 1 with the constraints J1 + +Jm = J10 + +J n0 = J and m n. Simply put, the correspondence is [# of giants at t = +1] = [# of sheets of Riemann space] : (3.20) (3.21) The number of point charges equals the number of giants at t = 1, and the number of sheets is the number of giants at t = +1. This is because the in nite past s = +1 corresponds to the diverging potential at the locations of m point charges and the in nite future s = 0 to the asymptotic in nities, zi ! 1, in the Riemann space. The electric ux runs through the branch three-ball to di erent sheets and escapes to the asymptotic in nities. By construction the necessary condition (2.46), or equivalently, the condition (3.22) (3.23) (3.24) ; ] to n is automatically satis ed. 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 [20]: to begin with, we consider the 4d Coulomb potential where J is a point charge placed at zi = z0i in R4. Using the hypertoroidal coordinates ( ; ; '; !) de ned in (3.18) and (3.19), the distance squared from the charge is expressed as HJEP05(218)6 As explained in section 3.1, once we extend the interval of the angle 2, the hypertoroidal coordinates become multi-valued and we can construct an n-sheeted Riemann space by stitching n R4's at the branch three-ball of radius a and make the coordinates single-valued. Because the distance squared R2 in (3.23) is periodic in the angle , 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 R4. n charge solutions, we rst 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 [20]. 3.2.1 Coulomb potential in two-sheeted Riemann space Let us rst consider the two-sheeted case. We complexify the angle and introduce the complex variable = ei =2 which covers the two-sheeted Riemann space. The Coulomb potential (3.22) can then be expressed as a contour integral C I d 0 R d 0 R 1 0 ei( ! ei =2 ! 1 ei( cosh cos ) cos( 0 0)) ; (3.25) cos )=(cosh 0 = i1. where the contour C is a unit circle surrounding = ei =2. The factor (cosh cos 0) in the integrand is inserted to ensure that the integrand vanishes at We now deform the contour C to a rectangle of width 4 and an in nite height while avoiding the poles at 0 = 0 i and 0 + 2 i (see gure 4). The contributions from the vertical edges cancel out owing to the periodicity, and those from the horizontal edges at in nity simply vanish. The single charge contribution is extracted as the residue of a pole and its pair in the lower-half plane. Note rst that at 0 ' 0 + 2k i we have cosh The relevant part of the contour comes in from in nity, encircles a pole clockwise and goes back to in nity, picking up the residue. The single charge potential is thus found to be J I The consistency requires ei( cos 0)(cosh cos ) cos( 0 0)) : (zi) = k=0(zi) + k=1(zi) ; where the second term is the contribution from a charge on the second sheet. Carrying out the contour integral (3.28), we nd that i : Besides the poles at 0 = + 4k with k 2 Z, the integrand in (3.25) has the poles at One can check that (3.29) holds by noting that k=1( ) = k=0( + 2 ). (3.26) (3.27) (3.28) (3.29) (3.30) It is straightforward to generalise the two-sheet case to the n-sheeted Riemann space. We start from cos 0)(cosh cos ) 1 ei( 0)=n (cosh 0)) ; and deform the contour in a similar manner to the two-sheet case. There are poles at 0 = + 2nk with k 2 Z and i : The single charge potential is thus given by 1 cos2 cos2 ei( 2 0 2n 0 cos ) By superposing di erent contributions, it is easy to construct the solution to the 4d Laplace equation describing general m giants J1 J2 J0 1 J0 2 J0n at t = 1 with the constraints J1 + Similar to the two-sheet case, the rectangle contour with width 2n picks up the residues from these poles. The Coulomb potential splits into (zi) = k=0(zi) + k=1(zi) + + k=n 1(zi) : A(J~; J1; ; Jm) : m;n(zi) = (nJl)(zi; 0 + 2 (l 1)) ; Drawing a horizontal line at J~ of Ji's below J~: where we de ned (nJ)(zi; 0) in (3.34). The Mathematica plot of the 2-to-3 splitting giant graviton interaction is shown in gure 5 for the potential 2;3(zi). Before closing this section, we discuss the necessary and su cient condition for the existence of instantons when J 1. In the case of instantons in the BMN matrix model, the necessary and su cient condition given in [41] 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 su cient condition in our case as well, and we just state the condition: we consider an instanton interpolating m giant gravitons at t = 1 and n giant gravitons at t = 1 characterised by J1 and m Jm and J01 J0n, respectively and satisfying J1 + n. Since the angular momenta are positive, one can consider a histogram of Ji's. 0 on the histogram, we de ne the area of the histogram (3.31) (3.32) HJEP05(218)6 (3.33) (3.34) (3.35) (3.36) and three nal concentric giants in chronological order: the thin solid, dashed and thick solid lines indicate (projections of) the equipotential surfaces in the 1st-, 2nd- and 3rd-sheets. The horizontal dashed segment and the black point are the branch three-ball and the point charge, respectively. The splitting interaction is happening from (ii) to (v) through the branch. 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 ) de ned by [4, 5]: ; where RJ is an irreducible representation of U(N ) expressed by a Young diagram with J boxes, RJ ( ) is the character of the symmetric group SJ in the representation RJ , the sum is over all elements of SJ and Z is an N N complex matrix with i1; i2; ; iJ = 1; 2; ; N . If the representation RJ is symmetric (antisymmetric), the operator (4.1) corresponds to an AdS giant (a sphere giant) [4, 5]. 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 AJ denotes the antisymmetric representation. For antisymmetric representations the dimension dAJ of the representation AJ is always 1. The dimension of the representation AJ of the unitary group is with C( ) being the number of cycles in the permutation . For anti-symmetric representations the character A( ) 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 As for a histogram of Ji0's, one can de ne the area A(J~; J10 ; ; J n0) in the same manner. The necessary and su cient condition for the existence of instantons can be given [13]: A(J~; J10 ; ; J n0) A(J~; J1; ; Jm); 8J~: 4 Giant graviton correlators in CFT (3.37) (4.1) (4.2) (4.3) (4.4) (4.5) h AJ (Z) AJ (Z)i = J !DimN (AJ ) ; dAJ DimN (AJ ) = AJ ( )N C( ) ; 1 J ! X 2SJ fR := n!DimN (R) dR = Y(N i;j i + j) ; fAJ = Y(N J i=1 i + 1) = 9For more recent progress in the understanding of Schur correlators beyond the 1=2-BPS sector, see [39, 40] and references therein. (4.6) (4.7) (4.8) Thus we have (N J )! This provides the normalisation of higher point functions. Now the 3pt function of sphere giants, corresponding to one giant with momentum J = J1 + J2 spliting into two giants with momenta J1 and J2, is given by the formula: h AJ1 (Z) AJ2 (Z) AJ (Z)i = g(AJ1; AJ2; AJ ) dAJ1 dAJ2 dAJ = J !DimN (AJ ) N ! (N J )! ; where g(AJ1; AJ2; AJ ) is a Littlewood-Richardson coe cient, an analogue of the ClebschGordan coe cient, and denotes the multiplicity of the representation AJ in the tensor product of representations AJ1 and AJ2, and we have used g(AJ1; AJ2; AJ ) = 1. This is incidentally identical to the two-point function. Thus the normalised three-point functions yield where jj AJ jj := pJ !DimN (AJ )=dAJ . 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 [4, 5] h R1(Z) Rn(Z) T1(Z) Tm(Z)i = X g(R1; R2; U Qin=1 dRi ; Rn; U ) nU !DimN (U ) g(T1; T2; ; Tm; U ) dU Qim=1 dTi : (4.9) We only consider the case where all R's and T 's are antisymmetric representations. The At large N and J the middle factor fU := nU !DimN (U) = Q numbers of boxes for Ri and Ti are Ji0 and Ji, respectively and J1 + i;j(N dU 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 rst order R's and T 's such that the number of boxes J 0 1 J 0 2 J n0 and J1 J2 Jm. Then the dominant Young diagrams U at large N and J are composed by rst 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 R1 R2 Rn. For these representations we have fU := nU ! DimN (U ) dU m J k = Y Y (N k=1 ik=1 ik + k) = Y n k=1 (N 1)! 1)! ; (4.10) where Jk is the number of boxes in the k-th column of the Young diagram U . For large Jk's and N we can approximate Jk 's by Jk's. Since the Littlewood-Richardson coe cients which exactly agrees with the instanton amplitude e SE for generic m-to-n instanton action (2.45). 5 The Basu-Harvey equation As we have seen in section 3, the instanton equation (2.40) in the IIB plane-wave matrix model can be mapped to the Basu-Harvey equation (3.4) by a change of variables (3.3). In order to conform to the original parameterisation in [23], we make a slight adjustment to the transformation (3.3), s = e 2 t ; 1 M11 where M11 is the eleven-dimensional Planck mass and constant. The instanton equation (2.40) then becomes10 is the dimensionless coupling are of order 1, their contributions are negligible at large N and J , and we nd that hQin=1 Ri (Z) Qm k=1 Tk (Z)i Qin=1 jj Ri (Z)jj Qkm=1 jj Tk (Z)jj ' (N = e 41N (Pin=1 Ji02 Pim=1 Ji2) J1)! (N Jm)!(N (N !)n+m J10 )! (N J n0)! Ym k=1 1)! 3!J [ ] ijklfzj ; zk; zlg ; J := 64 3J M131 : dZi ds 4! 8 M131 ijkl 1 [ 5; Zj ; Zk; Zl] = 0 : SM1 is the M-theory circle corresponding to describing a self-dual string soliton. This was proposed as an equation describing M2-branes ending on M5-branes by the M2-brane worldvolume theory. This is a natural generalisation of Nahm's equation describing monopoles or the D1-D3 system. The four scalars Zi'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, Rt fuzzy S3) SM1 , where Rt is the time, Rs+ a semi-in nite line s 2 [0; +1] and 5. In [24] this was called the ridge solution 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 where 10The constant matrix G5 introduced in [23] is slightly di erent from 5, but this fact does not spoil the main argument shown in this paper. (4.11) HJEP05(218)6 (5.1) (5.2) (5.3) (5.4) By the hodograph transformation (3.8) we solve s as a function of zi's as done before and the equation (5.3) can be locally mapped to the 4d Laplace equation. Note that the total ux in this case is not J but J (see appendix A for details). The aforementioned ridge or spike solution is simply a Coulomb potential in R4 which is a solution to the 4d Laplace equation: with ai being a constant vector. As remarked, this describes the space Rs+ S3 and the radius of the three-sphere varies along the semi-in nite line as 2 is the 4d Coulomb potential as previously de ned in section 3. We can add a constant c to the Coulomb potential R 02) ! R 02) + c ; since the constant potential solves the 4d Laplace equation. We now focus on the constant part of the potential Note that s = 0 corresponds to the location of the M5-brane at which the radius of S3 becomes in nite. 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{25, 43]. The semi-in nite line Rs+ must be replaced by a near the two M5-branes at s = s0 the solution behaves as nite interval Is = fsjs 2 [ s0; +s0]g and j z i aij ' 2 p p J s s0 : 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 zi = ai, s goes to in nity and thus any solution with point charges cannot represent a nite 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 multisheeted Riemann space. In this case we expect that the number of sheets corresponds to the number of M5-branes. To nd the solution which satis es the boundary condition (5.7), recall the contour integral expression of the electrostatic potential ei( 02) cosh 0)=2 cosh cos cos 0 ; s = J d 0 ei( cosh cos cos 0 { 23 { (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) Ci +C i sinh cos 2 cosh 2 cos # : 1 1 ei( 1 cosh cos cos 0 1 1 ei( i )=2 1 ei( +i )=2 (5.11) (5.12) (5.13) (5.14) (5.15) One can check that this solves the 4d Laplace equation. The contribution from the second sheet is 0k=1(zi) = c k=0(zi). Note that at the two asymptotic in nities ( ; ) ! (0; 0) 0 and ( ; ) ! (0; 2 ) where zi's go to in nity, the electrostatic potential 0k=0(~z) approaches di erent values, cJ =(4 2) and 0, respectively. By shifting the potential by a constant s0, these values can be shifted to s0 and potential 0k=0(~z) describes a nite interval of length 2s0. potentials de ned on each sheet by the contour deformation: In the n-sheeted Riemann space the trivial constant potential splits into nontrivial c = k=0(zi) + 0k=1(zi) + 0 + 0k=n 1(zi) : The explicit form of the potentials for higher k's can be found in the end of this section. 5.1 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: s0 with the choice s0 = cJ =(8 2). Hence, the 0 = We deform the contour C to a rectangle of width 4 (for the two-sheet case) and an in nite height while avoiding the poles at 0 = i and i + 2 . Noticing that near the poles cosh cos 0 similar to the Coulomb potential case, the contribution from the rst sheet to the constant potential can be found as s = k=0(zi) 0 s0 = funnel(zi) : s0 cos 2 cosh 2 Let us examine this solution more in detail. Recalling the parametrisation of the coordinates = 1 2 ln ( + a)2 + 2 a)2 + 2 ; cos = 2 + 2 a 2 p(( + a)2 + 2) (( a)2 + 2) ; (5.16) 11The funnel solution has been constructed from di erent descriptions of the M2-M5 sytem in [24, 25, 43]. s0 are the locations of the two M5-branes. Each ring is a constant s hypersurface and represents a squashed S3 whose radius blows up at the ends and which collapses to a three-ball at the midpoint. this can be expressed as funnel(zi) = s = v s0uu1 t 4a2 p( + a)2 + 2 + p( a)2 + 2 2 : (5.17) (5.19) (5.20) and behaves as The midpoint of the funnel s = 0 corresponds to = ; 3 which implies This is the brach ball B3 and thus in terms of zi's the midpoint s = 0 corresponds to a threeball of radius a. We plot the funnel solution in gure 6. The constant s hypersurfaces are squashed three-spheres and the radius blows up at the endpoints s = s0 and the squashed S3 collapses to a three-ball at s = 0.12 This collapse of the funnel throat is similar to what = 0 and j j happens to D1-branes stretched between two D3-branes [42]. Note that at the two asymptotic in nities where zi's are very large, the coordinates become very large, since z12 + z22 + z32 + z42 = 2 + 2. Thus the funnel at large zi's satisfying the boundary condition (5.7). 5.2 M2-branes ending on multiple M5-branes The power of this method, albeit only in the limit of an in nite 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: 0(zi) = J 8n 3 I C d 0 1 c ei( cosh 0)=n cosh cos cos 0 : Besides the poles at 0 = + 2nk with k 2 Z, there are poles at s s0 ' s0a2 2jzij2 ( s0 s s0) ; (5.18) 0 = We deform the contour C to a rectangle of width 2n and an in nite height while avoiding the poles at 0 = i + 2k with k = 0; 1; ; n 1. Noticing that near the poles cosh cos 0 similar to the Coulomb potential case, the contribution from the rst sheet to the constant potential is given by cosh sinh s0 sinh n (cosh 2n sinh 1 cos cos ) cos2 2n 1 ; 1 ei( cosh cos cos 0 1 where s0 = cJ =(4 2). This asymptotes to s0 at ( ; ) = (0; 0) on the rst sheet k = 0 and 0 at ( ; ) = (0; 2k ) with k = 1; ; n 1 on the other sheets, corresponding to one M5-brane at s = s0 and n 1 M5-branes at s = 0. The general solutions are given by the superposition of the potentials from di erent sheets. For example, the superposition of the two 0k=0(zi) and 0k=1(zi) s1 sinh n (cosh 2n sinh s2 sinh n (cosh 2n sinh asymptotes to s1 at ( ; ) = (0; 0) on the rst sheet, s2 at ( ; ) = (0; 2(n n-th sheet and 0 on the other sheets, corresponding to one M5-brane at s = s1, another M5-brane at s = s2 and n 2 M5-branes at s = 0. We can construct the most general solution with all di erent asymptotic values describing n separated M5-branes: 0(zi) = n 1 X k=0 2n sinh sk sinh n (cosh where sk is the modulus representing the location of each M5-brane (see gure 7). As an example of the cases with more than two M5-branes, we plot an M2-branes junction ending on three di erent M5-branes corresponding to n = 3 with some choice of the locations (s1; s2; s3) in gure 8. 6 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 AdS5 S5. 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 e ective description was provided by the tiny graviton matrix model (5.22) (5.23) (5.24) ; n 1 labelling the sheets of the Riemann space. The thick line segments represent the branch three-balls and are all identi ed. M2-branes ending on multiple M5-branes correspond to the electrostatic potential distilled from a constant potential by means of contour deformation and there are no charges present in the Riemann space. M2-branes connecting M5-branes all meet at the branch three-balls. 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 in nite past to another vacuum, a cluster of n concentric (fuzzy) sphere giants, in the in nite 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. { 27 { 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 S5 and are dual to antisymmetric Schur operators and a puzzle, as pointed out in [35], remains for the AdS giants which expand in AdS5 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 uctuations about (anti-)instantons in order to nd N=J 2 corrections. This involves integrations over bosonic and fermionic zero modes and requires nding 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 e ective theory on the moduli space of our solutions might shed light on the low energy e ective theory of multiple M5-branes [26{34]. Finally, our technique is applicable to the well-known SU(1) limit of Nahm's equation which describes (an in nite number of) D1-branes ending on D3-branes by mapping it locally to the 3d Laplace equation [18, 19]. This might give us a new perspective on the moduli space of monopoles. Acknowledgments We would like to thank Robert de Mello Koch, Chong-Sun Chu, Masashi Hamanaka, Satoshi Iso, Hiroshi Isono, Stefano Kovacs, Niels Obers, Shahin Sheikh-Jabbari, Hidehiko Shimada 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). Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the NRF or the CoE-MaSS. The work of YS was funded under CUniverse research promotion project by Chulalongkorn University (grant reference CUAASC). We are going to show that the following di erential equation can be mapped to the ndimensional Laplace equation: pp1 pn 1 zp1 ; zp2 ; f ; zpn 1 g ; where zp and zpi with p; pi = 1; 2; ; n are functions of (s; l) with l = 1; 2; On the r.h.s. the Nambu (n 1)-bracket is de ned by f zp1 ; zp2 ; Z n 1 : The equation (A.1) describes an evolution of an (n 1)-dimensional hypersurface embedded n with time s. We can express this hypersurface at a constant time slice as a function (A.1) ; n (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) ; zn) satisfying the equation s = (z1; z2; ; zn) : =: ds ^ d p 1 = and in R (z1; z2; We now follow the proof in [13] given in the case of n = 3, extend it to general n and show that the electrostatic potential (z1; ; zn) satis es the n-dimensional Laplace equation. First note that the n-dimensional volume element can be expressed as dz1 ^ dz2 ^ ^ d n 1 where p = 1; 2; ; n and from (A.4) volume Vn = Qin=11 Ii ux conservation yields ^d n 1, the equation (A.1) can then be rewritten as J d 1 ^ d 2 Integrating (A.7) over the boundary hypersurface @Vn = Qin=11 Ii @Is of the in nitesimal Is where the intervals Ii = [ i; i + d i] and Is = [s; s + ds], the 0 = Z Z Vn dz1 ^ dz2 ^ ^ dzn : This is nothing but the n-dimensional Laplace equation. In order to nd zp(s; l) from solutions to the Laplace equation (z1; ; zn) = 0, we use (A.7) and (A.6). Namely, the equation (A.7) implies that the electric ux density in the (n 1)-dimensional -space is the constant [J] at a given s. In other words, the Guassian surface of constant electric elds is tangent to the -space and normal to the time s: E~ These n equations determine zp's as functions of (s; l). 1 using (A.6), where the electric eld E~ (z1; ; zn) = 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- xed lightcone Hamiltonian (2.5). Using Hamilton's equation, pI = p ; J the action becomes I = J dtd3 IE = J dtd3 By a Wick-rotation the Euclidean action yields + 3J 3J 1 3! 1 3! J J 2(xI )2 where t is the Euclidean time. The Euclidean action (B.3) can be recast as a sum of squares and boundary terms: IE = J dtd3 + + dt d dt x i x a RT [ ] 2 J xi 2 3!J RT [ ] ijklfxj ; xk; xlg 3!J RT [ ] abcdfxb; xc; xdg 2 2 fxi; xa; xbg2 + fxa; xi; xj g2 12 12 J RT [ ] ijklxifxj ; xk; xlg J RT [ ] abcdxafxb; xc; xdg : (B.1) (B.2) (B.3) HJEP05(218)6 (B.5) (B.6) (B.7) This is minimised when the rst order BPS equations are satis ed13 3!J 3!J RT [ ] ijklfxj ; xk; xlg = 0 ; RT [ ] abcdfxb; xc; xdg = 0 ; xi = xa = 0 : xi = 0 ; xi 6= 0 ; xa 6= 0 ; 13One can show the non-negativity of the Euclidean action by constructing the equations analogous to (2.41) in the IIB plane-wave matrix model. By a change of variables, the BPS equations (B.5) and (B.6) transform to r 2 RT xI (t; ) = e tzI (s; ) ; s = e 2 t ; 3!J [ ] ijklfzj; zk; zlg = 3!J [ ] abcdfzb; zc; zdg = J J : These equations are the continuum version of the Basu-Harvey equation (3.7) and by a hodograph transformation they can be locally mapped to the 4d Laplace equation as explained in appendix A. 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, S3's become fuzzy S3's and the Nambu 3-bracket is replaced by the quantum Nambu 4-bracket. The construction of fuzzy S3's will be given below. The parameter ` in the quantisation of the Nambu bracket is analogous to ~ in quantum mechanics (2.16) and xed by the requirement that the radius of S3 coincides with that of fuzzy S3. We start with an S3 of radius r (B.8) (B.9) (B.10) X(xi)2 = r2 : We choose the spherical coordinates to be xi = rni = r(cos ; sin cos '; sin sin ' cos !; sin sin ' sin !) : We can then show that xi's satisfy the following equation: xi = 3!r2 where l (l = 1; 2; 3) are the coordinates on the S3 and have the volume element Here f ; ; g is the Nambu 3-bracket. For a unit S3, in particular, we have 3 = sin2 sin ' d d'd! : ni = 3! 1 ijklfnj; nk; nlg = 1 sin2 sin ' This establishes the relation between S3's and the Nambu 3-bracket. (C.1) (C.2) (C.3) (C.4) (C.5) The fuzzy S3's can be constructed as a subspace of fuzzy S4's [44, 45]. We only recapitulate the essential part of the construction and leave details to the original papers [44, 45]. We introduce J J matrices, i = P ( i 5 = P ( 5 R R 1 n 1)symPR ; 1 n 1)symPR ; (C.6) (C.7) (C.8) (C.9) (C.10) (C.11) (C.12) (C.13) (C.14) 4 ijkl[Xj ; Xk; Xl; 5] ; X(Xi)2 = rF2 1J J : J representation (C.8) of Spin(4) by J. In the case of a where i are the four-dimensional 4 4 Dirac matrices, 5 is the SO(4) chirality operator, 1 is the 4 4 unit matrix, n is an odd integer and the su x `sym' denotes a symmetric n-fold tensor product. Here PR is the projector onto the J J representation R of SO(4) = SU(2)L SU(2)R given by HJEP05(218)6 R = 4 n 1 n + 1 4 n + 1 n 1 where (jL; jR) is an irreducible representation of Spin(4) = SU(2)L SU(2)R. The dimen sion of R speci es the size of matrices J : J = dim R = (n + 1)(n + 3) 2 N i = p 1 J Xi = rF N i = p i 5 ; i 5 : rF J 4 i=1 J1 J2 Jn Using i and 5, one can construct a fuzzy S3 of unit radius: N i = 3! J ijkl[N j ; N k; N l; 5] ; 4 i=1 X(N i)2 = 1J J ; where the quantum Nambu 4-bracket is de ned in (2.15) and This can be easily generalised to a fuzzy S3 of radius rF by which satisfy Xi = J 3!rF2 We denote the irreducible J reducible representation, with J1 + J2 + + Jn = J , the solutions to the equation (C.13) form n concentric fuzzy S3's. 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Shinji Hirano, Yuki Sato. Giant graviton interactions and M2-branes ending on multiple M5-branes, Journal of High Energy Physics, 2018, 65, DOI: 10.1007/JHEP05(2018)065