Giant graviton interactions and M2branes ending on multiple M5branes
Accepted: May
Giant graviton interactions and M2branes ending on multiple M5branes
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 KitashirakawaOiwakecho , Kyoto 6068502 , Japan
1 University of the Witwatersrand
2 DSTNRF Centre of Excellence in Mathematical and Statistical Sciences , CoEMaSS
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
AdSCFT Correspondence; Dbranes; M(atrix) Theories; MTheory

J
N in the type IIB string theory on AdS5
S5 by describing them
as instantons in the tiny graviton matrix model introduced by SheikhJabbari. At large J
the instanton equation can be mapped to the fourdimensional Laplace equation and the
Coulomb potential for m point charges in an nsheeted Riemann space corresponds to the
mton interaction process of giant gravitons. These instantons provide the holographic
dual of correlators of all semiheavy operators and the instanton amplitudes exactly agree
with the ppwave 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 BasuHarvey equation which describes the system of M2branes ending
on M5branes. As it turns out, the solutions to the sourceless Laplace equation on an
nsheeted Riemann space correspond to n M5branes connected by M2branes and we nd
general solutions representing M2branes ending on multiple M5branes. Among other
solutions, the n = 3 case describes an M2branes junction ending on three M5branes. The
e ective theory on the moduli space of our solutions might shed light on the low energy
e ective theory of multiple M5branes.
1 Introduction
2 IIB planewave matrix model
2.1
Vacua
2.2 Instanton equations
3
Fourdimensional Laplace equation in Riemann spaces
General mton functions of sphere giants
The BasuHarvey equation
M2branes stretched between two M5branes  funnel solution
M2branes ending on multiple M5branes
U(N ) SYM [6]. On the CFT side, their interactions correspond to multipoint correlators
of Schur polynomial operators and have been computed exactly for halfBPS giants in [5].
S5 which is dual to N = 4
{ 1 {
However, on the gravity side, being extended objects (spherical D3branes), it is rather
challenging to go beyond kinematics and study their dynamical interaction process except
for socalled heavyheavylight 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 planewave background [7{9]: for
an observer moving fast in the sphere, the spacetime looks approximately like a planewave
geometry.1 Thus if the size of giants is small enough,2 the observer moving along with the
giants can study them in the planewave 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 Mtheory on AdS4
S7=Zk at nite
k by zooming into the planewave background [13, 14]. Since the Mtheory on the
planewave background is described by the BMN planewave 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 SheikhJabbari [21, 22] rather than BMN's
type IIB string theory on the ppwave background.3
The description of giant graviton
interactions is similar to the above Mtheory 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 nsheeted Riemann space corresponds to the mton interaction process of
giant gravitons. An advantage over the Mtheory 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 ppwave 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 semiheavy operators.
Last but not the least, as a byproduct of this study we are led to
nd new results
on elusive M5branes. By a slight change of variables, the instanton equation of the type
IIB planewave matrix model is identical to the BasuHarvey equation which describes the
system of M2branes ending on M5branes [23]. In the large J limit which corresponds, in
1The planewave 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 pointlike and much larger than the Planck length.
3In this paper we refer to the tiny graviton matrix model as the type IIB planewave matrix model.
{ 2 {
solutions might shed light on the low energy e ective theory of multiple M5branes [26{
instanton equation and nd the (anti)instanton action for the mton 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 2to1 interaction agrees with the 3point correlators of
antisymmetric Schur operators in the dual CFT, i.e. N = 4 SYM. In section 3, we transform
the instanton equation to the BasuHarvey 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 multisheeted Riemann spaces and
nd the solutions
which describe the generic mton joining and splitting process of (concentric) sphere giants.
In section 4, we discuss the ppwave 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 BasuHarvey equation in the original context,
namely, as a description of the M2M5 brane system. In the large J limit corresponding
to a large number of M2branes, we nd the solutions to the 4d Laplace equation which
describe M2branes ending on multiple M5branes. Section 6 is devoted to summary and
discussions. In the appendices A, B and C we elaborate further on some technical details.
2
IIB planewave matrix model
The tiny graviton matrix model was proposed by SheikhJabbari as a candidate for the
discrete lightcone quantisation (DLCQ) of the type IIB string theory on the maximally
supersymmetric tendimensional planewave background [21]. We refer to this matrix model
as the IIB planewave matrix model in this paper.
Here we outline the derivation of the IIB planewave matrix model. The bosonic part
of the IIB planewave matrix model can be obtained by a matrix regularisation of the
e ective action for a 3brane [21]:
S =
T
Z
dtd3
q
j det(h )j + C^^^^
;
(2.1)
; 8. The background metric is the planewave geometry:
with
+; ; 1;
where T = 1=((2 )3gsls4) is the D3brane tension with gs and ls being the string coupling
constant and string length, respectively. The worldvolume 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 3brane is
h
and C^^^^ is the RamondRamond 4form 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
worldvolume coordinates l such that the lightcone momentum density
p is a constant.
The lightcone Hamiltonian for the 3brane 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 zeromodes 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 threebracket 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
areapreserving 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 planewave 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 nondynamical J
J matrix explained in appendix C and the quantum
Nambu fourbracket 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 planewave 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 Gausslaw constraints which are nothing but (2.19). The
U(1) superalgebra in the planewave background can be realised
jxaj
RS
1;
jxij
RS
jx j
RS
approximated by the planewave 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 planewave background is the approximation of the AdS5
near the observer with large angular momentum J . Thus the matrix size J in the IIB
planewave matrix model is considered to be very large for our purposes.
(2.27)
S5 geometry
Similar to the planewave matrix model for Mtheory [7], the IIB planewave 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 blockdiagonal 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 planewave 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 planewave 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 planewave matrix
model. Hereafter we ignore the fermionic matrices by setting
= 0. The Euclidean action
for the bosonic IIB planewave 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=2BPS and marginally stable. Nonetheless, the instanton and antiinstanton
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 (antiinstantons). 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 nonnegativity 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 antiinstantons. 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 ppwave 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 2to1 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.
Fourdimensional 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 BasuHarvey equation [23] which
describes M2 branes ending on M5 branes. This connection to the BasuHarvey 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 4bracket [ ; ; ; 5] by the Nambu 3bracket.
This is the \classicalisation" of the brackets, reversing the procedure (2.11){(2.14). Then
the BasuHarvey 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 zspace.
Let us see how a single fuzzy threesphere 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 threesphere of radius (2.32):
xi = rni ;
Here ni is the unit vector normal to the threesphere (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 multisheeted Riemann spaces rather than the ordinary 4d Euclidean space
R4. The 4d Riemann spaces are a fourdimensional 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 multisheeted 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 multisheeted 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 twodimensional 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 twodimensional
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
multivalued. To make the coordinates singlevalued, 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 (twosheeted)
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 threeball of radius a embedded in R3. This threeball 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 multivalued.
In order to make the coordinates singlevalued, we need to introduce an object analogous
to a cut in a 2d Riemann space which is a threeball of radius a located at
=
+ 2 m
with m = 0; 1;
; n (see gure 2). We call this threeball a branch threeball. As before
we prepare (n + 1) copies of R4 such that each R4 is designated by the di erent interval
of , [
threeball, we can construct a 4d (n + 1)sheeted Riemann space.
It goes back to 1896 when Sommerfeld rst considered the threedimensional Laplace
equation in Riemann spaces [20]. We shall extend his idea to the fourdimensional 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 threeballs and point electric charges, respectively. The branch threeballs 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 zspace 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 nsheets 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 threeball 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 multivalued and we
can construct an nsheeted Riemann space by stitching n R4's at the branch threeball of
radius a and make the coordinates singlevalued. 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 ncharge 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 twosheeted Riemann space
Let us rst consider the twosheeted case. We complexify the angle
and introduce the
complex variable
= ei =2 which covers the twosheeted 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 lowerhalf 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 twosheet case to the nsheeted 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 twosheet 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 twosheet 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 2to3 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 3rdsheets. The horizontal
dashed segment and the black point are the branch threeball 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 largeR 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
Threepoint 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 antisymmetric
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 antisymmetric 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=2BPS 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 LittlewoodRichardson 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 twopoint function. Thus the normalised threepoint functions yield
where jj AJ jj := pJ !DimN (AJ )=dAJ . In the ppwave limit, as we discussed in the end of
section 2.2, this exactly agrees with the instanton amplitude as in (2.47).
General mton 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 kth column of the Young diagram U . For large
Jk's and N we can approximate Jk 's by Jk's. Since the LittlewoodRichardson coe cients
which exactly agrees with the instanton amplitude e SE for generic mton instanton
action (2.45).
5
The BasuHarvey equation
As we have seen in section 3, the instanton equation (2.40) in the IIB planewave matrix
model can be mapped to the BasuHarvey 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 elevendimensional 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 Mtheory circle corresponding to
describing a selfdual string soliton.
This was proposed as an equation describing M2branes ending on M5branes by the
M2brane worldvolume theory. This is a natural generalisation of Nahm's equation
describing monopoles or the D1D3 system. The four scalars Zi's are U(J ) matrices and
the coordinates transverse to M5branes, and s is one of the worldvolume coordinates
of M2branes. In the largeJ limit a prototypical solution to the BasuHarvey
equation (5.2) is a spike made of a bundle of J M2branes on a single M5brane of topology,
Rt
fuzzy S3)
SM1 , where Rt is the time, Rs+ a semiin 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
4bracket is replaced by the (classical) Nambu 3bracket and the BasuHarvey 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 threesphere varies along the semiin 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 M5brane at which the radius of S3
becomes in nite. This is interpreted as an M2brane spike threading out from a single
M5brane.
We next consider M2branes stretched between two M5branes discussed in [23{25, 43].
The semiin nite line Rs+ must be replaced by a
near the two M5branes 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 M5branes 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 M5branes.
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 nsheeted 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
M2branes stretched between two M5branes  funnel solution
As discussed above, the solution representing M2branes stretched between two M5branes
can be constructed from a trivial constant electrostatic potential by distilling the
contribution from one of the two Riemann sheets.11 The M2branes connecting the two M5branes
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 twosheet 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 M2M5 sytem in [24, 25, 43].
s0 are the locations of the two M5branes.
Each ring is a constant s hypersurface and represents a squashed S3 whose radius blows up at the
ends and which collapses to a threeball 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 threespheres and the radius blows up at the endpoints s =
s0 and the squashed
S3 collapses to a threeball at s = 0.12 This collapse of the funnel throat is similar to what
= 0 and j j
happens to D1branes stretched between two D3branes [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
M2branes ending on multiple M5branes
The power of this method, albeit only in the limit of an in nite number of M2branes, is
that the solution can be easily generalised to the cases with more than two M5branes. 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
M5brane at s = s0 and n
1 M5branes 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
nth sheet and 0 on the other sheets, corresponding to one M5brane at s = s1, another
M5brane at s = s2 and n
2 M5branes at s = 0.
We can construct the most general solution with all di erent asymptotic values
describing n separated M5branes:
0(zi) =
n 1
X
k=0 2n sinh
sk sinh n (cosh
where sk is the modulus representing the location of each M5brane (see gure 7). As an
example of the cases with more than two M5branes, we plot an M2branes junction ending
on three di erent M5branes 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 planewave 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 threeballs and are all identi ed.
M2branes ending on multiple M5branes 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. M2branes connecting M5branes all meet at the branch threeballs.
of SheikhJabbari [21, 22], which we referred to as the IIB planewave matrix model, rather
than BMN's type IIB string theory on the ppwave background.
We showed, in particular, that their splitting/joining interactions can be described by
instantons/antiinstantons in the IIB planewave 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
mton interaction corresponds to the Coulomb potential of m point charges on an nsheeted
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 ppwave 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 threeballs, 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 M5branes. We
exploited the fact that the instanton equation is identical to the BasuHarvey equation
which describes the system of M2branes ending on M5branes [23]. In the large J limit
which corresponds, in the BasuHarvey context, to a large number of M2branes, we found
the solutions describing M2branes ending on multiple M5branes, including the funnel
solution and an M2branes junction connecting three M5branes as simplest examples.
The number n of M5branes corresponds to the number of sheets in the Riemann space,
and somewhat surprisingly, multiple M5branes 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 M5branes [26{34].
Finally, our technique is applicable to the wellknown SU(1) limit of Nahm's equation
which describes (an in nite number of) D1branes ending on D3branes 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, ChongSun Chu, Masashi Hamanaka,
Satoshi Iso, Hiroshi Isono, Stefano Kovacs, Niels Obers, Shahin SheikhJabbari, 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 DSTNRF Centre of Excellence in
Mathematical and Statistical Sciences (CoEMaSS). Opinions expressed and conclusions
arrived at are those of the author and are not necessarily to be attributed to the NRF or
the CoEMaSS. 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 ndimensional Laplace equation.
First note that the ndimensional 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 ndimensional 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 BasuHarvey
equation (3.7) can be obtained from the Euclidean 3brane 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 Wickrotation 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 nonnegativity of the Euclidean action by constructing the equations analogous
to (2.41) in the IIB planewave 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 BasuHarvey equation (3.7) and by
a hodograph transformation they can be locally mapped to the 4d Laplace equation as
explained in appendix A.
C
Threespheres and their quantisation
We give a brief review of the relation between threespheres and the Nambu 3bracket.
Upon quantisation of this relation, S3's become fuzzy S3's and the Nambu 3bracket is
replaced by the quantum Nambu 4bracket. 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 3bracket. 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 3bracket.
(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 fourdimensional 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
nfold 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 4bracket 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. This establishes the relation between fuzzy S3's and the Nambu 4bracket.
Fixing the quantisation parameter
We elaborate on our choice of the quantisation parameter l in (2.13). Recall that the
threebrane theory de ned by the Hamiltonian (2.5) has the vacua obeying
where
xi =
1
3!r2
ijklfxj ; xk; xlg ;
d
r
r =
s
J
0
J
2 2RT
= RS
r J
N
;
The solution to (C.15) is given by (C.2) which forms an S3 of radius r. Since the
coordinates are chosen as in (C.4), we have
Z
d
d! sin2 sin ' = 2 2
:
As a result the radius (C.16) of the S3 is found as
(C.15)
(C.16)
(C.17)
(C.18)
HJEP05(218)6
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 S3 to be (2.32) which coincides
with (C.18).
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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