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JHEP05(2018)065

Published for SISSA by Springer

Received: March 14, 2018

Accepted: May 1, 2018

Published: May 9, 2018

Giant graviton interactions and M2-branes ending on

multiple M5-branes

Shinji Hirano

a,b

and Yuki Sato

School of Physics and Mandelstam Institute for Theoretical Physics &

DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS),

University of the Witwatersrand,

1 Jan Smuts Avenue, Johannesburg, South Africa

Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University,

Kitashirakawa-Oiwakecho, Kyoto 606-8502, Japan

Department of Physics, Faculty of Science, Chulalongkorn University,

Thanon Phayathai, Pathumwan, Bangkok 10330, Thailand

E-mail: shinji.hirano@wits.ac.za, ysato.phys@gmail.com

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 AdS

× S

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 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.

Keywords: AdS-CFT Correspondence, D-branes, M(atrix) Theories, M-Theory

ArXiv ePrint: 1803.04172

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP05(2018)065

JHEP05(2018)065

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.

Thus if the size of giants is small enough,

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 split-

ting and joining interactions of membrane giants in the M-theory on AdS

×S

at ﬁnite

k by zooming into the plane-wave background [13, 14]. Since the M-theory on the plane-

wave 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 equa-

tion 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 AdS

×S

, as it turns out, the most eﬀec-

tive 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.

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 gi-

ant 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 ﬁnd 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 instan-

tons 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 ﬁnd 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

The plane-wave geometry can be obtained from AdS

× S

by taking the Penrose limit [10–12].

Small 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.

In this paper we refer to the tiny graviton matrix model as the type IIB plane-wave matrix model.

– 2 –

JHEP05(2018)065

the Basu-Harvey context, to a large number of M2-branes, we ﬁnd the solutions describing

M2-branes ending on multiple M5-branes, including the funnel solution [24, 25] 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 sur-

prisingly, multiple M5-branes solutions are constructed from a trivial constant electrostatic

potential. Upon further generalisations, the eﬀective theory on the moduli space of our

solutions might shed light on the low energy eﬀective theory of multiple M5-branes [26–

28, 31–34].

This paper is organised as follows: in section 2, we review the IIB plane-wave matrix

model and its BPS vacua which contain concentric fuzzy three-spheres. We then discuss the

instanton equation and ﬁnd the (anti-)instanton action for the m-to-n joining and splitting

process of giant gravitons. As the ﬁrst check of our proposal we show that the instanton

amplitude e

−S

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 ﬁnd 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 ﬁnd 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 su-

persymmetric 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

eﬀective action for a 3-brane [21]:

S = −T

dtd

|det(h

µν

)| + C

ˆµˆν ˆρ

∂x

ˆµ

∂t

∂x

ˆν

∂σ

∂x

ˆρ

∂σ

∂x

∂σ

, (2.1)

where T = 1/((2π)

) is the D3-brane tension with g

and l

being the string coupling

constant and string length, respectively. The world-volume coordinates are σ

= (t, σ

)

with µ = 0, 1, 2, 3 and l = 1, 2, 3. The indices for the target space are hatted, ˆµ, ˆν, ˆρ,

λ =

+, −, 1, ··· , 8. The background metric is the plane-wave geometry:

ˆµˆν

ˆµ

ˆν

= −2dx

−

− µ

+ x

)dx

+ dx

, (2.2)

– 3 –

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