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Published for SISSA by Springer

Received: November 2, 2015

Accepted: January 20, 2016

Published: February 8, 2016

On membrane interactions and a three-dimensional

analog of Riemann surfaces

Stefano Kovacs,a,b Yuki Satoc and Hidehiko Shimadad

aDublin Institute for Advanced Studies,

10 Burlington Road, Dublin 4, Ireland
bICTP South American Institute for Fundamental Research, IFT-UNESP,

São Paulo, SP 01440-070, Brazil
cNational Institute for Theoretical Physics,

School of Physics and Mandelstam Institute for Theoretical Physics,

University of the Witwartersrand,

Wits 2050, South Africa
dOkayama Institute for Quantum Physics,

Okayama, Japan

E-mail: skovacs@stp.dias.ie, Yuki.Sato@wits.ac.za,

shimada.hidehiko@googlemail.com

Abstract: Membranes in M-theory are expected to interact via splitting and joining

processes. We study these effects in the pp-wave matrix model, in which they are asso-

ciated with transitions between states in sectors built on vacua with different numbers of

membranes. Transition amplitudes between such states receive contributions from BPS

instanton configurations interpolating between the different vacua. Various properties of

the moduli space of BPS instantons are known, but there are very few known examples of

explicit solutions. We present a new approach to the construction of instanton solutions

interpolating between states containing arbitrary numbers of membranes, based on a con-

tinuum approximation valid for matrices of large size. The proposed scheme uses functions

on a two-dimensional space to approximate matrices and it relies on the same ideas behind

the matrix regularisation of membrane degrees of freedom in M-theory. We show that

the BPS instanton equations have a continuum counterpart which can be mapped to the

three-dimensional Laplace equation through a sequence of changes of variables. A descrip-

tion of configurations corresponding to membrane splitting/joining processes can be given

in terms of solutions to the Laplace equation in a three-dimensional analog of a Riemann

surface, consisting of multiple copies of R3 connected via a generalisation of branch cuts.

We discuss various general features of our proposal and we also present explicit analytic

solutions.

Keywords: Penrose limit and pp-wave background, Solitons Monopoles and Instantons,

M(atrix) Theories, M-Theory

ArXiv ePrint: 1508.03367

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP02(2016)050

mailto:skovacs@stp.dias.ie
mailto:Yuki.Sato@wits.ac.za
mailto:shimada.hidehiko@googlemail.com
http://arxiv.org/abs/1508.03367
http://dx.doi.org/10.1007/JHEP02(2016)050


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Contents

1 Introduction 2

2 BPS instantons 4

2.1 Instanton equations 5

2.2 Some properties of BPS instantons and their moduli spaces 8

3 Mapping to three-dimensional Laplace equation 10

3.1 Continuum approximation 10

3.2 Classical membrane perspective 12

4 Solutions 13

4.1 Stable sphere 14

4.2 Single membrane splitting and Riemann space 15

4.2.1 Qualitative picture: introduction of the concept of Riemann space 15

4.2.2 Analytic solution 20

4.3 General Riemann spaces 28

4.3.1 More solutions 31

4.3.2 Comments on the moduli space of solutions 33

5 Conclusions and discussion 42

A Geometric proof of the equivalence between Laplace and continuum Nahm

equations 48

B Behaviour at the splitting point 50

C Other solutions 54

C.1 BHP solution 54

C.2 Solution with positive and negative point charges 55

D Relation between Hobson’s and Sommerfeld’s solutions 56

E Boundary conditions at branch disks and reformulation in terms of inte-

gral equations 58

F Equations for general pp-wave matrix model 61

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1 Introduction

The best candidate for a microscopic description of M-theory is the matrix model origi-

nally proposed in [1] as a regularisation of the supermembrane theory and subsequently

conjectured in [2] to describe the full dynamics of the theory in light-front quantisation

when the size of the matrices is sent to infinity. The matrix model provides a formula-

tion of M-theory which is capable of describing multi-membrane configurations, arising as

block-diagonal matrices. Membranes in M-theory should interact via splitting and joining

processes and therefore the matrix model should capture such effects. However, no concrete

proposal for the description of these splitting/joining interactions in the matrix model has

been formulated. More generally there is no quantitative prescription for the study of this

type of membrane interactions, which would make it possible to evaluate the associated

transition amplitudes. This makes it difficult to test any results for these amplitudes that

may be obtained from the matrix model. This is of course a general difficulty with all

predictions of the matrix model, which so far have been mostly tested at the level of the

low energy supergravity approximation.

The AdS/CFT correspondence — and specifically the duality proposed in [3], relating

M-theory in AdS4×S7/Zk to an N = 6 Chern-Simons theory — provides in principle new

means of testing the matrix model predictions by comparing them to a dual CFT. In the

context of the duality of [3] it was recently shown in [4] that the matrix model description of

M-theory in AdS4×S7/Zk can be quantitatively compared to the dual gauge theory without

relying on the supergravity approximation or compactification to type IIA string theory

in ten dimensions. The crucial observation of [4] was that, focussing on large angular

momentum states in M-theory and the dual CFT sector involving monopole operators,

natural approximation schemes arise on the two sides of the duality, so that a systematic,

quantitative comparison is possible.

On the gravity side, M-theory states with large angular momentum, J , along a great

circle in S7 can be studied using the pp-wave approximation. The associated matrix model

uses J × J matrices and its action was constructed in [5]. We will use basic properties

of the model which were further studied in [6]. Multi-membrane states in the pp-wave

matrix model consist of concentric membranes and their fluctuations. More precisely the

vacua of the theory consist of spherical membranes which extend in AdS4 directions and

are point-like in S7. They are classified by a set of integers, Ji, i = 1, . . . , n, corresponding

to a partition of the total angular momentum among n membranes. The fluctuations of

the spherical membranes described by the pp-wave matrix model are associated, in the

dual gauge theory, with certain monopole operators. The latter are characterised by their

integer GNO charges, which are in one-to-one correspondence with the angular momenta,

Ji, of the membranes. The sector of monopole operators with large GNO charges can be

reliably studied using a weakly-coupled effective low-energy approximation.

The AdS4/CFT3 duality in this M-theoretic regime relates correlation functions of

monopole operators in the ABJM theory to processes on the gravity side in which the dual

states interact in the bulk and propagate to the boundary. The simplest such process in-

volves a single membrane splitting into two — with the associated three states propagating

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to the boundary — and it corresponds to a three-point correlation function of monopole op-

erators. Similar processes have been studied in the case of the AdS5/CFT4 correspondence,

in which three-point functions of gauge-invariant operators are related to cubic interaction

vertices in the bulk associated with the splitting of strings in the pp-wave approximation.

A method to compare the cubic vertices in string theory to the corresponding conformal

three-point functions in the CFT was discussed in [7], following a spacetime interpretation

first proposed in [8]. Studies using other approaches can be found in [9–14]. Similar aspects

have been extensively studied beyond the pp-wave approximation taking advantage of the

integrability appearing in the analysis of the system. For recent interesting developments

and up-to-date references, see [15, 16].

The three-point correlation functions of monopole operators, in particular those with

large charge J , which are relevant in the present case, are well defined in the ABJM theory

and one expects that they should be computable within the CFT framework. Although

in this paper we focus on the gravity side of the correspondence, it is important that in

principle the results of the matrix model calculation can be verified by independent means.

Successfully comparing the physical transition amplitudes associated with a membrane

splitting or joining process to the gauge theory results for three-point functions would

provide such an independent test. The agreement would not only represent a highly non-

trivial test of the AdS4/CFT3 correspondence, but it would also provide strong evidence

indicating that the matrix model captures the aspects of the dynamics of membranes related

to splitting/joining interactions.

In the context of the pp-wave matrix model, processes of joining or splitting of mem-

branes correspond to transitions between states built on vacua with different numbers

of membranes. A semi-classical description of these transitions can be given in terms of

tunnelling amplitudes associated with instanton configurations which interpolate between

states containing different numbers of membranes. In this paper we initiate a study of these

interpolating configurations. The relevant instantons are BPS solutions to the Euclidean

equations of motion of the pp-wave matrix model [17], with boundary conditions specify-

ing the single or multiple membrane states between which they interpolate. The equations

obeyed by these BPS instantons, with the same boundary conditions, were studied in a

different context in [18], where they were found to arise as the equations describing domain

walls interpolating between distinct isolated vacua of the so-called N = 1∗ SYM theory. A

number of properties of these instanton equations and the associated moduli spaces were

studied in [18]. An important ingredient in the analysis presented in [18] is the observation

that the relevant instanton equations can be mapped to the Nahm equations [19, 20] which

arise in the construction of BPS monopoles, although with different boundary conditions.

The BPS instantons of the pp-wave matrix model are saddle points representing the

dominant contribution to the tunnelling processes. Physical transition amplitudes associ-

ated with the splitting/joining processes can be computed using a semi-classical approxi-

mation around the instanton solutions. This involves evaluating the determinants arising

from the non-zero mode fluctuations as well as the integration over the collective coordi-

nates associated with zero modes. Although there may be alternative approaches to the

computation of the splitting/joining transition amplitudes, in order to carry out the stan-

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dard semi-classical calculations it is necessary to obtain the explicit form of the classical

instanton solutions. However, although various general features of the instanton equations

— including conditions for the existence of solutions and properties of their moduli spaces

— were studied in [18], little is known about explicit solutions. In particular, no solution

is known for the most elementary process we are interested in, i.e. the splitting of a single

membrane into two.

In this paper we show that an efficient approach to the construction of solutions to the

instanton equations as well as to the study of their moduli spaces can be developed using

a continuum approximation. Such an approximation scheme is valid in the case of large

matrices and, therefore, it is applicable in the context of the M-theoretic regime of the

AdS4/CFT3 duality considered in [4]. In this approximation the original instanton equa-

tions can be mapped to a continuum version of the Nahm equations, which in turn can be

shown to be equivalent to the three-dimensional Laplace equation, using results in [21–23].

In this formulation equipotential surfaces for the solution to the Laplace equation directly

represent the profile of the membranes at different times. Using this approach we present

explicit analytic solutions describing the splitting of membranes.

A remarkable feature of our construction is that, in order to obtain a solution repre-

senting the splitting of one membrane into two, we are led to consider the Laplace equation

not in the ordinary three-dimensional Euclidean space, but rather in a three-dimensional

generalisation of a two-sheeted Riemann surface, which we refer to as a Riemann space

following [24]. The emergence of Riemann spaces, which can be naturally motivated in

the context of the construction of our solutions, is quite intriguing. The possibility that in

the description of membrane interactions Riemann spaces may play a central role, similar

to that played by Riemann surfaces in string perturbation theory, is very interesting and

deserves further investigation.

This paper is organised as follows. In section 2 we describe the instanton equations

and we recall some general properties of their solutions and of the associated moduli space

which were obtained in [18]. In section 3 we discuss the continuum approximation scheme

used in our analysis and the mapping of the instanton equations to the three-dimensional

Laplace equation. In section 4 we present explicit solutions to the Laplace equation which

are relevant for the pp-wave matrix model and we describe the corresponding solutions to

the BPS instanton equation within our continuum approximation. The most interesting

solution — which utilises a two-sheeted Riemann space to describe the splitting of a single

membrane into two — is considered in section 4.2. We conclude with a discussion of our

results in section 5. Technical details are discussed in various appendices.

2 BPS instantons

In our study of configurations describing membrane splitting/joining processes we focus on

the pp-wave matrix model [5]. As discussed in [4] this model provides a good approximation

to the dynamics of M-theory in AdS4 × S7/Zk in a sector containing membranes with

large angular momentum, J . M-theory in this background is dual to the N = 6 Chern-

Simons theory with gauge group U(N)×U(N) and level k constructed in [3]. The region of

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applicability of the pp-wave approximation is defined by the conditions 1 � J � (Nk)1/2.

Moreover for J � (Nk)1/3 the model is weakly coupled.

In this section we review the BPS instanton equations of the pp-wave matrix model and

recall some of their properties. These equations were first studied in a different context

in [18], where they were shown to describe supersymmetric domain walls interpolating

between vacua of the N = 1∗ SYM theory. Their relevance in the pp-wave matrix model

was pointed out in [17], where they were first shown to describe instanton configurations

interpolating between different vacua.

2.1 Instanton equations

The Euclidean action describing the M-theory side of the AdS4/CFT3 duality in the large

J sector takes the form [4–6]

SE =

∫
dt tr

{
k

2R

(
DY i

Dt

)2

+
k

2R

(
DXm

Dt

)2

+ (2πT )2
R

4k

(
(i[Xm, Xn])2 + 2(i[Xm, Y i])2

)
+

k

2R3
(Xm)2 + (2πT )2

R

2k

(
i

2
εijk[Y

j , Y k] +
1

2πT

2k

R2
Y i

)2
(2.1)

+
1

2
ΨT DΨ

Dt
+ 2πT

R

k

1

2

(
ΨTγm[Xm,Ψ] + ΨTγi[Y i,Ψ]

)
− 3i

4

1

R
ΨTγ123Ψ

}
,

where T = 1/[(2π)2l3P ] is the membrane tension, R is the radius of S7 (and twice the

radius of AdS4) and γα (α = 1, 2, . . . , 9) are SO(9) gamma matrices, with γ123 = γ1γ2γ3.

Following the notation in [4], we have denoted by Y i (i = 1, 2, 3) and Xm (m = 4, 5, . . . , 9)

matrices originating from the membrane coordinates in AdS4 and S7 respectively. Ψ is a

16 component matrix valued spinor. The Y i’s, Xm’s and the components of Ψ are K ×K
matrices, with K = J/k. The covariant derivative in (2.1) is defined by

DX

Dt
=

dX

dt
− i[A0, X] (2.2)

where A0 is the gauge potential associated with the invariance of the model under time

dependent unitary transformations. In the following we will choose the gauge A0 = 0,

which is compatible with the boundary conditions relevant for the tunnelling processes we

are interested in.

The matrix model defined by the action (2.1) is a special case of a family of pp-wave

matrix models, which are usually parameterised by a mass scale µ. In the following we use

the model (2.1), which is the one relevant for the large J sector of the AdS/CFT duality

of [3] and corresponds to a specific choice of µ. To simplify notation and equations, we

also focus on the k = 1 case. The generalisation is straightforward and in appendix F

we present the main formulae for the case of a general pp-wave background. Using the

equations in the appendix it is easy to recover the results for arbitrary k.

The potential term in the action (2.1) is written as a sum of squares and thus it is

manifestly non-negative. This makes it easy to identify all the vacua of the model, i.e. the

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zero-energy minima of (2.1). They correspond to the configurations [5, 6]

Y i
0 =

2

(2πT )R2
Li , with [Li, Lj ] = iεijkLk , (2.3)

Xm
0 = 0 , Ψ0 = 0 (2.4)

and therefore they are classified by (generally reducible) J dimensional representations of

SU(2). The solution in which the Y i’s are proportional to the J dimensional irreducible rep-

resentation corresponds to a single spherical membrane. It extends in the AdS4 directions,

with radius

r =

√
J2 − 1

(2πT )R2
≈ J

(2πT )R2
, (2.5)

and carries momentum J/R along a great circle of S7. Generic vacua are given by block

diagonal Y i matrices corresponding to reducible SU(2) representations. The blocks have

size Ji, i = 1, 2, . . . , n, with
∑

i Ji = J and they represent n concentric membranes of radii

ri ≈
Ji

(2πT )R2
, (2.6)

with angular momenta Ji along the same great circle in S7.

We will denote the irreducible J dimensional SU(2) representation by J and the re-

ducible representation which is the direct sum of irreducible representations of dimension

J1, J2, . . . , Jn by J1 ⊕ J2 ⊕ · · · ⊕ Jn.

We are interested in the tunnelling processes corresponding to classical solutions inter-

polating between a vacuum associated with the SU(2) representation Li(−∞) in the infinite

past (t = −∞) and another vacuum associated with Li(+∞) at t = +∞. These processes

are governed by the path integral with boundary conditions

Y i(−∞) =
2

(2πT )R2
Li(−∞)

Y i(+∞) =
2

(2πT )R2
ULi(+∞)U

−1 . (2.7)

In (2.7) we have chosen the generators Li(±∞) to correspond to the standard embedding of

SU(2) consisting of block diagonal J × J matrices. U denotes an arbitrary unitary matrix,

which we need to include to take into account the gauge choice, A0 = 0.

In order to obtain the equations obeyed by the classical interpolating configurations

we rewrite the bosonic part of the Euclidean action (2.1) in the form of a sum of squares

plus a boundary term,

SE =
1

2R

∫
dt tr

[(
dXm

dt

)2

+
1

R2
(Xm)2 +

(2πT )2R2

2

{
(i[Xm, Xn])2 + (i[Xm, Y i])2

}
+

(
dY i

dt
± 2

R
Y i ± i(2πT )

R

2
εijk[Y j , Y k]

)2

∓ d

dt

(
2

R
Y iY i + i(2πT )

R

3
εijkY i[Y j , Y k]

)
+ fermions

]
. (2.8)

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From (2.8) it follows that, on configurations interpolating between vacua at t = ±∞, the

bosonic part of the Euclidean action obeys the bound

SE ≥ ∓ tr

(
1

R2
Y iY i + i

(2πT )

6
εijkY i[Y j , Y k]

)∣∣∣∣+∞
−∞

. (2.9)

Using the explicit form of the vacuum configurations at t = ±∞ the bound (2.9) becomes

SE ≥ ∓
1

2R
W [Y ]

∣∣∣∣+∞
−∞

= ∓ 1

3R2

(
2

(2πT )R2

)2
tr
(
Li(+∞)L

i
(+∞) − L

i
(−∞)L

i
(−∞)

)
, (2.10)

where following [18] we defined the functional

W [Y ] = tr

(
2

R
Y iY i + i(2πT )

R

3
εijkY i[Y j , Y k]

)
. (2.11)

The BPS (anti-)instantons are the configurations [17, 18] which saturate the bound (2.10)

and therefore satisfy the equations

dY i

dt
± 2

R
Y i ± i(2πT )

R

2
εijk[Y j , Y k] = 0, (2.12)

with Xm = 0 and Ψ = 0. In this paper, we refer to solutions of (2.12) in which one chooses

the upper or lower signs as instantons or anti-instantons, respectively. We note that the

Gauss law constraints are satisfied as a consequence of (2.12),[
Y i,

dY i

dt

]
= ∓i(2πT )

R

2
εijk[Y i, [Y j , Y k]] = 0. (2.13)

Since the (bosonic part of) the Euclidean action (2.1) is non-negative, so should be the

expressions on the right hand side of (2.9) and (2.10), when the bounds are saturated.

Therefore W [Y ] should decrease from t = −∞ to t = +∞ in the case of instantons (upper

signs in (2.9)–(2.12)) and increase in the case anti-instantons. This can be seen explicitly

noticing that the instanton equations imply

d

dt
W [Y ] = ∓1

2

∣∣∣∣∂W [Y ]

∂Y i

∣∣∣∣2 , (2.14)

where the signs are correlated with those in (2.12).

In the following we focus on instanton configurations and all the equations that we

present correspond to the choice of upper signs in (2.12). Anti-instanton solutions can be

simply obtained from the corresponding instanton by changing t to −t.
A detailed discussion of the general conditions for the existence of solutions to the

instanton equations (2.12) can be found in [18]. We will revisit these issues in section 4.3.2.

Here we only recall a necessary condition. For a BPS solution connecting two vacua to

exist, the vacuum with the larger value of W [Y ] should correspond to a representation

which does not contain more irreducible blocks than the one with the smaller value of

W [Y ]. This allows us to identify instanton configurations as corresponding to membrane

splitting processes, while anti-instantons describe joining processes.

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Evaluating W [Y ] on a vacuum configuration (2.3) containing m membranes with an-

gular momenta Ji, i = 1, . . . ,m, one gets

W [Y0] =
2

3R

(
2

(2πT )R2

)2
tr
(
LiLi

)
=

2

3R

(
2

(2πT )R2

)2 1

4

m∑
i=1

tr
[(
J2
i − 1

)
1Ji×Ji

]
. (2.15)

Therefore on a generic instanton interpolating between the representations J1⊕J2⊕· · ·⊕Jm
and J ′1 ⊕ J ′2 ⊕ · · · ⊕ J ′n (with

∑m
i=1 Ji =

∑n
i=1 J

′
i = J , n ≥ m) the Euclidean action is

SE =
−1

2R

(
tr (W [Y0(+∞)])− tr (W [Y0(−∞)])

)
(2.16)

=
−4

3(2πT )2R6

(
tr
(
Li(+∞)L

i
(+∞) − L

i
(−∞)L

i
(−∞)

))
=

1

3(2πT )2R6

(
m∑
i=1

Ji
3−

n∑
i=1

J ′i
3

)
≥ 0

In the following sections we will focus on the most elementary process in which a single

membrane splits into two. This corresponds to an instanton solution in which Li(−∞) is

taken to be the irreducible representation J and Li(+∞) is the representation J1⊕J2, where

J1+J2 = J as required by angular momentum conservation. For this process the Euclidean

action is

SE =
1

4R2

(
2

(2πT )R2

)2
JJ1J2 =

1

8

JJ1J2
N

, (2.17)

where in the final equality we have used the relations T = 1/[(2π)2l3P ] and (R/lP )6 =

25π2N .

The (anti-)instantons described above are local minima of the Euclidean action of

the pp-wave matrix model. Their contribution to physical transition amplitudes associ-

ated with membrane splitting/joining processes can be evaluated using a standard semi-

classical approximation. The latter involves the integration over the bosonic and fermionic

collective coordinates associated with zero modes in the instanton background. From the

semi-classical calculation of transition amplitudes it should be possible to extract effective

interaction vertices for membranes. In the present paper we focus on the construction of

classical instanton solutions, leaving the semi-classical evaluation of the corresponding am-

plitudes for a future publication. Among the questions that we will not address is whether

the BPS instantons discussed here give the dominant contributions to the amplitudes,

or instead other saddle points associated with non-BPS configurations need to be taken

into account.

2.2 Some properties of BPS instantons and their moduli spaces

A key observation used in the analysis of [18] is that (2.12) can be mapped to the Nahm

equations arising in the construction of BPS multi-monopole configurations [19, 20]. In

order to reduce (2.12) to the Nahm equations one considers the change of variables [18]

Zi = C e2t/RY i, s = e−2t/R, (2.18)

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where C is a constant, which we take to be real, so that the Zi’s are Hermitian matrices.

Substituting into (2.12) gives

dZi

ds
= i(2πT )

R2

4C
εijk[Zj , Zk] , (2.19)

which by a suitable choice of the constant C can be brought to the canonical form of the

Nahm equations.

In terms of the original variables, we are interested in solutions Y i(t) approaching

constant configurations proportional to different SU(2) representations at t = ±∞. After

the change of variables (2.18) we therefore need to consider Nahm equations defined on a

semi-infinite interval with boundary conditions [18]

Zi(s) ∼
Li(−∞)

s
+ · · · for s→∞ , Zi(s) ∼

Li(+∞)

s
+ · · · for s→ 0 , (2.20)

where the ellipses indicate sub-leading terms.

The Nahm equations have been extensively studied for their central role in the physics

of monopoles and a great deal is known about their properties. The particular boundary

conditions (2.20) relevant in the present case are non-standard and less studied, although

they have been considered in a different context in [25]. In general, little is known about

explicit solutions to (2.19)–(2.20). The only known examples appear to be solutions pre-

sented in [18]. These include a particular configuration interpolating between a vacuum

associated with the representation Li(−∞) and the trivial vacuum corresponding to the J-

dimensional representation 1 ⊕ 1 ⊕ · · · ⊕ 1. The explicit form of this solution, in terms of

the original Y i(t) matrices, is

Y i(t) =
2

(2πT )R2

1

1 + e2(t−t0)/R
Li(−∞) . (2.21)

In the following we will refer to (2.21) as the Bachas-Hoppe-Pioline (BHP) solution. The

only other known explicit solutions are slight variations of this one, obtained by means

of tensor products. They interpolate between different specific pairs of representations

and were also presented in [18]. However, no systematic approach to the construction of

solutions has been proposed and no concrete examples are known for the class of processes

we are interested in, i.e. those in which all the membranes involved in the splitting or

joining transition carry large angular momenta. In the following sections we propose a

strategy for the construction of instanton configurations in this large angular momentum

sector. For the important case of the splitting of one membrane into two we also present

an explicit solution, which represents an essential step in the calculation of the associated

physical transition amplitude.

Various properties of the moduli space of BPS instantons were discussed in [18], where,

in particular, the dimension of the moduli space of a general instanton configuration was

determined. For an instanton describing the splitting process with m membranes at t =

−∞ and nmembranes at t = +∞— i.e. connecting the two representations J1⊕J2⊕· · ·⊕Jm
and J ′1 ⊕ J ′2 ⊕ · · · ⊕ J ′n (with

∑m
i=1 Ji =

∑n
i=1 J

′
i = J and n ≥ m and assuming that this

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is an allowed process according to the criterion in [18]) — the (complex) dimension of the

moduli space, M, was found to be

dimC (M) =

n∑
i=1

(2i− 1)J ′i −
m∑
i=1

(2i− 1)Ji , (2.22)

where in the sums the representations are ordered by decreasing dimension, i.e. J1 ≥ J2 ≥
· · · ≥ Jm and J ′1 ≥ J ′2 ≥ · · · ≥ J ′n. For the splitting of one membrane into two (2.22) gives

a (real) dimension

dimR (M) = 2(J ′1 + 3J ′2 − J1) = 4J ′2 , (2.23)

where we used J1 = J ′1 + J ′2 = J .

An interesting feature of the moduli space of BPS instantons is an additivity rule [18].

Given three vacua associated with the SU(2) representations L(A), L(B) and L(C), the

moduli spaces M(A,B), M(B,C) and M(A,C) of configurations connecting A and B, B and

C and A and C respectively satisfy

dim
(
M(A,B)

)
+ dim

(
M(B,C)

)
= dim

(
M(A,C)

)
. (2.24)

In section 4.3 we will comment on how this property is reflected in the description of the

instanton moduli space arising from our reformulation in terms of the Laplace equation.

The instanton configurations discussed in the previous subsection interpolate between

vacua of the pp-wave matrix model containing different numbers of membranes. In [26]

it was shown, based on properties of the representations of the relevant supersymmetry

algebra, that these vacua are non-perturbatively protected and their energies are exactly

zero in the full quantum theory. This means that different vacua do not mix and tunnelling

transitions should be possible only if excited states are involved. It is natural to expect that

the mechanism forbidding transitions between vacua should be associated with selection

rules induced by the integration over fermion zero modes in the instanton background. This

was shown explicitly in [17] in the case of the special solution (2.21) and similar selection

rules should exist for more general tunnelling amplitudes such as those that we study in

this paper.

3 Mapping to three-dimensional Laplace equation

In this section we present our general strategy for the construction of instanton solutions

describing tunnelling configurations interpolating between vacua containing large mem-

branes.

3.1 Continuum approximation

The central ingredient in the approach that we develop is a continuum approximation valid

for large J , i.e. for large matrix size. In this approximation scheme, the matrices, Y i(t),

are replaced by functions, yi(t, σ1, σ2), of two spatial coordinates, (σ1, σ2), and (Euclidean)

time, t. The mathematics underlying this replacement is essentially the same that is

utilised in the derivation of the matrix model as a regularisation of the supermembrane

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theory [1, 27, 28]. In the matrix regularisation, the starting point is the continuum theory

which is formulated in terms of functions on the membrane 2+1 dimensional world-volume.

At fixed time, for any function, f , defined on the membrane world-space parametrised by

the coordinates (σ1, σ2), one introduces a corresponding matrix, ρ(f). The map ρ between

functions and matrices satisfies the properties1

ρ(fg) ≈ 1

2

(
ρ(f)ρ(g) + ρ(g)ρ(f)

)
, (3.1)

ρ ({f, g}) ≈ 2πJ

i[σ]
[ρ(f), ρ(g)] , (3.2)

1

[σ]

∫
f d2σ ≈ 1

J
tr
(
ρ(f)

)
, (3.3)

where the two sides are understood to be equal up to terms of higher order in 1/J . The

Lie bracket, {f, g}, between two functions, f(σ1, σ2) and g(σ1, σ2), is defined as

{f, g} =
∂f

∂σ1
∂g

∂σ2
− ∂f

∂σ2
∂g

∂σ1
(3.4)

and the conventional constant [σ] denotes the area of the base space

[σ] =

∫
d2σ. (3.5)

In the present case we use a substitution of this type in reverse order. The starting point are

time-dependent matrices, which we approximate by functions depending on two auxiliary

continuous coordinates. Such an approach, based on the approximation of discrete objects

by functions of continuous variables is of course quite standard. An elementary example

of this type of approximation is the description of sound waves (phonons) in a solid, i.e. a

discrete lattice, by a continuum wave equation.

Using the rules (3.1)–(3.3), the instanton equations (2.12) obeyed by the matrices Y i(t)

get mapped to non-linear partial differential equations for a set of functions yi(t, σ2, σ2),

∂yi

∂t
± 2

R
yi ∓ (2πT )R

4π

[σ]

J
εijk{yj , yk} = 0 . (3.6)

The boundary conditions for the matrices Y i(t) at t = ±∞ determine the corresponding

boundary conditions for the functions yi(t, σ1, σ2). The latter should describe families of

concentric spheres at t = ±∞. The approximate identities (3.1)–(3.3) imply that, for large

J , solutions to the original equations (2.12) are well approximated by functions satisfying

these equations with the appropriate boundary conditions.

The linear term in the continuum version (3.6) of the BPS equations can be eliminated

by the change of variables

zi =
(2πT )R2

4π
e2t/Ryi , s = e−2t/R, (3.7)

1For a discussion of the matrix regularisation emphasising its interpretation as an approximation between

discrete and continuum objects, see [29].

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which is analogous to (2.18) that was used in [18].2 Substituting into (3.6) yields the

equations
∂zi

∂s
= − [σ]

2J
εijk{zj , zk} , (3.8)

for the functions zi(s, σ1, σ2), i = 1, 2, 3, where the variable s is defined on a semi-infinite

interval with s = +∞ corresponding to t = −∞ and s = 0 corresponding to t = +∞. As

in the case of the yi’s the boundary conditions for the zi’s at s = +∞ and s = 0 follow

from those required for the corresponding matrices (2.20). We will return to the analysis

of the relevant boundary conditions for the equations (3.8) in the next section where we

discuss various explicit solutions.

The equations (3.8), sometimes referred to as the SU(∞) Nahm equations (after an

appropriate rescaling to normalise the coefficients), are known to be (locally) equivalent

to the three-dimensional Laplace equation [21, 22].3 In order to map (3.8) to the Laplace

equation one uses a so-called hodograph transformation — a change of variables in which

the roles of dependent and independent variables are exchanged,

(z1, z2, z3) ←→ (s, σ1, σ2) . (3.9)

A fundamental feature of such a transformation is that it allows one to map a non-linear

differential equation to a linear one.

The change of variables (3.9) means, in particular, that one considers the originally

independent variable s as a function, φ(zi), of the new independent variables zi, i = 1, 2, 3.

One can then show that if the zi’s satisfy the equations (3.8) as functions of s, σ1, σ2, then

s = φ(zi) obeys the Laplace equation

∇2φ =
∂

∂zi
∂

∂zi
φ = 0 . (3.10)

A geometric derivation of this result is presented in appendix A. Alternative proofs can be

found in [21, 22].

This reformulation of the continuum version of the Nahm equations has a simple and

very intuitive interpretation. The time variable s becomes a ‘potential’ function obeying the

Laplace equation. The associated equipotential surfaces, φ(zi)=const., for different values

of the potential represent implicit equations describing the membrane profile at the given

time. Therefore a sequence of equipotential surfaces for values of s ranging from s = +∞
to s = 0 directly captures the evolution of the membrane configurations in Euclidean time.

3.2 Classical membrane perspective

In the previous subsection we have obtained the equations (3.6) as an approximation,

valid for large matrices, to the original instanton equations (2.12). More specifically the

approximation requires the dimension of all the irreducible representations contained in

2This change of variables is similar to that used in [7] to relate the Poincaré coordinates of the AdS space

to the coordinates of the pp-wave space. It is also reminiscent of the coordinate transformation involved in

the radial quantisation of the CFT on the boundary.
3The continuum Nahm equations were also considered in connection with membrane theory in [30].

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the vacua at t = ±∞ to be large. The same equations can also be understood as classical

(Euclidean) equations of motion for the membrane theory in the AdS4×S7 background in

the pp-wave approximation. In the notations of [4] the Euclidean action for the pp-wave

membrane theory (in the A0 = 0 gauge) is

SE =

∫
dt d2σ

[
1

2R

J

[σ]

(
∂yi

∂t

)2

+
1

2R

J

[σ]

(
∂xm

∂t

)2

+
1

2R3

J

[σ]
(xm)2 +

2

R3

J

[σ]
(yi)2

+
RT 2

4

[σ]

J

(
{xm, xn}2 + 2{xm, yi}2 + {yi, yj}2

)
− T

R
εijkyi{yj , yk}

+
1

R

J

[σ]
θT
∂θ

∂t
+ iT

(
θTγi{yi, θ}+ θTγm{xm, θ}

)
− 3i

2

J

[σ]

1

R2
θTγ123θ

]
. (3.11)

As in the case of the matrix model, the bosonic part of this action can be rewritten as a

sum of squares plus a boundary term,

1

2R

J

[σ]

∫
dtd2σ

[(
∂xm

∂t

)2
+

1

R2
(xm)2 +

R2T 2

2

[σ]2

J2

(
{xm, xn}2 + 2{xm, yi}2

)
(3.12)

+

(
∂yi

∂t
± 2

R
yi ∓ RT

2

[σ]

J
εijk{yj , yk}

)2
∓ ∂

∂t

(
2

R
(yi)2 − RT

3

[σ]

J
εijkyi{yj , yk}

)]
.

This formula shows that (3.6) can be obtained minimising the membrane Euclidean action

and therefore these equations describe the BPS instanton configurations of the membrane

theory.

This is of course consistent and not surprising. The matrix model contains membrane

degrees of freedom and the matrix configurations we focussed on represent regularised mem-

brane states. However, we prefer to emphasise the point of view presented in the previous

subsection in which the continuum equations (3.6) are viewed as an approximation to the

corresponding matrix model instanton equations. This is because the matrix model itself

is more fundamental as a candidate for a microscopic formulation of quantum M-theory.

Moreover the calculation of physical transition amplitudes in semi-classical approximation

should be carried out in the matrix model. This in particular should allow one to compute

tunnelling amplitudes between states corresponding to non-Abelian degrees of freedom as-

sociated with excitations in off-diagonal blocks, which have no simple counterpart in the

continuum.

4 Solutions

In this section we discuss solutions to the Laplace equation which correspond to membrane

splitting processes. We will see that, in order to construct solutions with the required

properties, we need to introduce the concept of Riemann space. After explaining the

general features of our proposal, we present exact solutions to the Laplace equation (3.10)

with appropriate boundary conditions, which, based on the arguments in the previous

section, provide approximate solutions to the BPS instanton equations (2.12).

We first discuss the simplest solution corresponding to a static single spherical mem-

brane, i.e. the simplest stable vacuum of the pp-wave matrix model. This case (which can

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be thought of as a solution in the zero instanton sector) allows us to illustrate the effects

of the sequence of changes of variables that we use to map the original instanton equations

to the Laplace equation. In section 4.2 we then discuss the solution corresponding to the

splitting of a single membrane into two membranes, which requires the introduction of the

notion of Riemann space. More general splitting processes, which involve more complex

examples of Riemann spaces, are discussed in section 4.3. A reformulation of the solu-

tion (2.21) discussed in [17, 18] in terms of the Laplace equation is presented, together

with other simple new solutions, in appendix C.

4.1 Stable sphere

In order to understand the asymptotic behaviour of non-trivial instanton solutions and the

corresponding required boundary conditions, it is instructive to first consider the simplest

solution to the BPS instanton equations (2.12), namely the static configuration correspond-

ing to the irreducible representation J ,

Y i(t) =
2

(2πT )R2
Li , ∀ t ∈ (−∞,+∞) , (4.1)

where [Li, Lj ] = iεijkLk. The continuum counterpart of this solution is

yi(t, σ1, σ2) = r ni(σ1, σ2) , (4.2)

where r = J/[(2πT )R2] and ni(σ1, σ2) is a unit vector in the radial direction. Using the

standard parametrisation of the sphere we can take

ni(σ1, σ2) = (sin θ cosϕ, sin θ sinϕ, cos θ) , with σ1 = 1− cos θ , σ2 = ϕ . (4.3)

Equations (4.2)–(4.3) provide a static solution to the continuum version of the BPS instan-

ton equations (3.6), as can be verified using the definition of the Lie bracket (3.4). In (4.3)

we have chosen a parameterisation suitable for the North pole patch of the unit sphere. In

general there is freedom in the choice of the (σ1, σ2) variables associated with the invariance

of the membrane theory under area preserving diffeomorphisms. The (σ1, σ2) coordinates

should be such that the area element dσ1dσ2 coincides with the natural SO(3) symmetric

area element on a round sphere.

Using the change of variables (3.7) we get a solution to the continuum Nahm equa-

tions (3.8)

zi(s, σ1, σ2) =
J

4π

1

s
ni(σ1, σ2) . (4.4)

This represents a sphere with radius increasing as s varies from +∞ to 0. Inverting (4.4) to

obtain s = φ(zi) we immediately deduce that the potential solving the associated Laplace

equation satisfies

φ(z) =
J

4π|z|
, (4.5)

where z = (z1, z2, z3). Therefore the solution to the Laplace equation corresponding to

a single static membrane configuration is simply the Coulomb potential generated by a

positive point charge J located at the origin.

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We note that a static spherical membrane in the (t,y) variables becomes a time-

dependent spherical membrane in the (s, z) variables, with radius changing in time as 1/s.

Using the transformation (3.7) in reverse order, one finds indeed that (4.5) corresponds to

a membrane whose distance from the origin is constant in time t,

|y| = J

(2πT )R2
, with y = (y1, y2, y3) , (4.6)

which of course represents a static sphere of radius J/(2πTR2).

As a simple generalisation of (4.5) we can consider the case of a point charge located

away from the origin. The potential is

φ(z) =
J

4π|z − z0|
, z0 6= 0 . (4.7)

In this case the equipotential surfaces are spheres centred at z0. In terms of the original y

variables we have

|y| = J

(2πT )R2

|z|
|z − z0|

. (4.8)

This configuration corresponds to a spherical membrane moving in towards the origin from

infinity. As (Euclidean) time evolves from t = −∞, corresponding to s = +∞ where

|z − z0| → 0, to t = +∞, corresponding to s = 0, |y| decreases from ∞ to a constant.

This run-away behaviour of the solution in the infinite past makes it non-physical, because

the solution does not satisfy the required boundary conditions. This is a general result,

which implies that the potentials relevant for the description of configurations interpolating

between stable vacua of the pp-wave matrix model can only involve positive charges located

at the origin.4

4.2 Single membrane splitting and Riemann space

In this section we consider the most interesting solution, which describes a single membrane

with angular momentum J splitting into two membranes with angular momenta J1 and J2
(with J1 + J2 = J). This corresponds to a configuration interpolating between a spherical

membrane of radius J/(2πTR2) and two concentric membranes of radii Ji/(2πTR
2), i =

1, 2. It is the most elementary example of a splitting process and it allows us to illustrate

general properties which are common to all splitting/joining transitions.

4.2.1 Qualitative picture: introduction of the concept of Riemann space

We will now argue that it is not possible to describe a membrane splitting process in

terms of a solution to the Laplace equation formulated in a standard Euclidean space R3.

Instead, we are led to introduce the concept of Riemann space explained in detail below.

In order to motivate our proposal, let us first discuss the qualitative features of a typical

instanton solution describing a splitting process. The transition from a spherical membrane

to two concentric membranes can be qualitatively described as involving the steps depicted

4The location of negative charges can be arbitrary. In appendix C we will present explicit examples of

physically acceptable solutions involving negative charges away from the origin.

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in figure 1. Starting with a single membrane of radius J/(2πTR2) a small perturbation

forms and grows until the membrane splits into two, resulting in the creation of a second

disconnected membrane inside the first one, as shown in steps (i) to (iii) in the figure.5

To understand the subsequent steps it is important to notice that membranes are

charged and thus oriented objects. Membranes and anti-membranes are distinguished by

their orientation. In figure 1 the orientation is indicated by the relative position of the con-

tinuous and dashed contour lines: membranes have an outer continuous contour, whereas

in anti-membranes the outer contour is dashed. In the pp-wave background only spherical

membranes of a definite orientation are stable. This is the result of a balance between the

tension and the effect of the quadratic potential associated with the gravitational back-

ground on one side and the force due to the three-form flux on the other. The first two

forces tend to shrink the membranes, while the latter forces them to expand. These two

effect add up in the case of anti-membranes, which, as a result, are unstable.

As shown in figure 1, after the splitting takes place the internal membrane that is

formed has opposite orientation compared to the initial one and the resulting configuration

is not stable. To complete the transition to a stable vacuum consisting of two concentric

membranes it is necessary for the internal membrane to flip its orientation. This process

requires intermediate steps in which the membrane self-intersects after developing cusps,

see (iv)-(vi) in figure 1.

The process shown in the figure has axial symmetry, but this is of course not a re-

quirement. In general the details of the intermediate configurations can vary, however,

an essential element, common to all splitting processes, is the fact that a flipping of the

orientation is inevitable. In the illustrative example shown in the figure the sequence of

steps leading to the final two-membrane state involves a splitting resulting in the forma-

tion of a second internal anti-membrane followed by the flipping of the orientation of the

latter. However, as we will see in explicit examples in the next subsection, the splitting

and flipping steps can also take place in reverse order, or simultaneously.

Applying the approach described in section 3, our goal is to construct a solution,

φ(z), to the Laplace equation such that the associated equipotential surfaces reproduce

the qualitative behaviour shown in figure 1.

The crucial feature of any splitting process, which emerges from the qualitative argu-

ments we presented above, is the fact that necessarily there exists a finite interval of values

of the potential for which the equipotential surfaces are self-intersecting. Such a behaviour

is not compatible with the potential φ(z) being a solution to the Laplace equation in the

ordinary R3 space, because the associated gradient, ∇φ(z), would be ill-defined, since the

normal direction differs on the two portions of a self-intersecting equipotential surface. In

the case of the intersections occurring during the flipping of the membrane orientation, this

issue arises not just at isolated singular points, but over a finite region corresponding to an

interval of values of φ(z). This motivates us to consider multi-valued potential functions.

5The point where the splitting occurs is singular from the point of view of the partial differential equation

governing the evolution of the membranes. The nature of the singularity and the properties of the instanton

solution in the vicinity of a generic splitting point are universal and can be studied using the description

in terms of the Laplace equation, as discussed in appendix B.

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(i) (ii) (iii)

(v)(iv) (vi)

Figure 1. Qualitative depiction of the splitting process. The process necessarily involves both

splitting and flipping.

More concretely the above observations lead us to propose that the potential relevant

for the representation of a membrane splitting process should be a solution to the Laplace

equation in a space, which we refer to as a Riemann space, that is a three-dimensional

generalisation of a two-sheeted Riemann surface. Such a space consists of two copies of

R
3 connected by a branch surface, bounded by a branch curve or loop. For simplicity

we will assume the surface connecting the two R3’s to always have the topology of a disk

and in the following we will use the expression branch disk without necessarily implying a

circular planar shape.6 Deformations of the branch disk which leave the branch loop fixed

yield physically equivalent Riemann spaces. This is analogous to properties of branch cuts

and branch points of Riemann surfaces. A space defined in this way is locally equivalent

to R
3, but differs globally. The use of a Riemann space makes it possible to have a

potential which locally solves the Laplace equation, while avoiding the problems caused by

self-intersecting equipotential surfaces. The solution is such that different portions of self-

intersecting surfaces live on different sheets of the Riemann space, so that no intersections

take place within the same copy of R3.

The description of a stable spherical membrane in section 4.1 provides an indication of

what the asymptotic behaviour of the potential should be in the case of a splitting process.

6In appendix E we present a reformulation of the Laplace equation in a Riemann space as an integral

equation and we discuss in more detail the boundary conditions at the branch disk. To construct solutions

relevant for the description of membrane splitting processes we require the potential to be finite at the

branch loop and also to decay sufficiently rapidly at infinity.

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The infinite past, t → −∞, corresponds to s → +∞, i.e. large values of the potential. In

this region the equipotential surfaces should be small spheres with radius proportional to J

and approaching zero. The infinite future, t→ +∞, corresponds to s→ 0, i.e. small values

of the potential approaching zero from above. In this region the equipotential surfaces

should be two large concentric spheres with radii proportional to J1 and J2 and growing

indefinitely. Using a Riemann space we can construct a potential which provides a concrete

realisation of this type of asymptotic behaviour and is also well-defined in the intermediate

region where the membranes self-intersect. More precisely, in order to account for the

behaviour at large φ we consider a point charge J at the origin of the first R3, so that

close to the charge we have a Coulomb potential, whose equipotential surfaces are small

spheres. The behaviour in the two R3’s for small s, on the other hand, can be understood

as resulting from the splitting of the initial flux J of ∇φ, with a fraction J1 going to infinity

in the first space and a fraction J2 = J − J1 passing into the second space through the

branch disk.

Figure 2 illustrates qualitatively the splitting process as rendered in the Riemann space,

with the six rows in the picture corresponding to the same stages of the splitting process

shown in figure 1. The first two columns depict the two sheets of the Riemann space, with

each copy of R3 represented as a rectangle. The horizontal slit corresponds to the branch

disk, with the dots at the end points indicating the branch loop. The third column shows

the two copies of R3 superposed. Membranes, or portions of membranes, are represented

as continuous lines in the first R3 and as dashed lines in the second R3. The same notation

will be used in the figures throughout the paper. As can be seen comparing figure 1 and

the third column of figure 2, the membrane profiles in the individual panels of the latter

(obtained joining the continuous and dashed lines from the two sheets of the Riemann

space) have qualitatively the same shapes as those shown at stages (i)-(vi) of the former.7

Re-analysing the splitting process as presented in figure 2 allows us to clarify how the

issues associated with the flipping of the membrane orientation and the self-intersections

in the equipotential surfaces can be addressed using a Riemann space. As the value of

the potential decreases, away from the point charge, the equipotential surfaces intersect

the branch disk and therefore extend partially into the second copy of R3. This is shown

at step (iii), where the splitting has already taken place. Crucially the use of a Riemann

space allows us to have self-intersecting surfaces which result from the superposition of

portions of surfaces without self-intersections in each sheet. This is illustrated in row (v) in

figure 2. Notice that in the third column intersections occur only between a continuous and

a dashed line, never between two continuous or two dashed lines. The final configuration

in the splitting process is represented by two membranes, each contained entirely in one

copy of R3, with the superposition resulting in two concentric surfaces.

As already pointed out, a feature of the solution is that during the flipping process

the equipotential surfaces develop cusps. These special points occur as the equipotential

7Notice that in figure 2 the membrane profiles are depicted using the zi coordinates, whereas figure 1

shows the qualitative evolution in terms of the yi coordinates. Because of the time dependent rescaling

relating the yi and zi coordinates, the size of the membranes increases through the (i)-(vi) steps in figure 2,

while the same does not happen in figure 1.

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(i)

(ii)

(iii)

(iv)

(v)

(vi)

Figure 2. Qualitative depiction of splitting in Riemann space.

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surfaces intersect the branch loop, as shown at step (iv) in figure 2. From the explicit

solutions discussed in the next subsection one can verify that the potential itself is regular

over the entire Riemann space (except for the origin of the first R3 where the point charge

is located), including the branch loop where the cusps arise.8 At these points, however, the

gradient of the potential diverges. This allows us to physically characterise the branch loop

in the Riemann space in terms of the behaviour of the solution to the continuum Nahm

equations. Recalling that the potential, s = φ(z), is obtained inverting the functions zi(s),

we deduce that the branch loop, where ∇φ(z) diverges, corresponds to points where the

velocity of the front of the membrane, as described by zi(s) in Euclidean time, vanishes.

Various other properties of the instanton solutions corresponding to membrane split-

ting processes and their moduli spaces can be given an intuitive interpretation using a

description in terms of Riemann spaces. We will discuss some of these aspects in sec-

tion 4.3.2.

4.2.2 Analytic solution

In section 3 we have shown that the problem of solving the continuous version of the instan-

ton equations can be mapped to that of finding solutions to the three-dimensional Laplace

equation. Then in the previous subsection we have discussed the boundary conditions that

we propose to consider in order to obtain a potential with the required properties. We

now present analytic examples showing explicitly that solutions obeying such boundary

conditions can be constructed.

Based on the examples of static solutions presented in section 4.1, we expect the

potential for large s (corresponding to t → −∞ in the original Euclidean time variable)

to have a Coulomb-like singularity. The considerations in the previous subsection about

the expected qualitative behaviour of the potential then lead us to consider the Laplace

equation in a Riemann space made of two copies of R3, with boundary conditions associated

with the presence of a single positive point charge at the origin of the first R3.

The idea of studying the Laplace equation in a Riemann space has actually been

considered long ago by Sommerfeld in a 1896 paper [24], which introduces the idea of

Riemann spaces to develop a generalisation of the standard method of images for the

solution of electrostatics problems. The examples studied in [24] involve multiple copies

of R3 with branch curves consisting of straight lines. Following Sommerfeld’s proposal, an

analytic solution to the Laplace equation with boundary conditions precisely of the type

we are interested in was constructed by Hobson in [31]. This paper considers a Riemann

space consisting of two copies of R3 connected by a flat circular branch disk and computes

the potential generated by a point charge located in the first R3. More recently, Riemann

spaces have been considered in [32–35]. It can be shown that Sommerfeld’s solution with

a straight branch line and Hobson’s solution with a circular branch loop can be related by

an inversion transformation, see appendix D.

Hobson’s solution is written in terms of so-called peripolar coordinates, which are

particularly suited to the specific geometry under consideration. The peripolar coordinates

8This is true in general and not only in the cases with axial symmetry for which we present plots.

– 20 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

a

A B

r ’

r

z

a

P
3

z 1

z 2 N

N

Figure 3. Relation between peripolar and Cartesian coordinates.

of a point P are designated by (ρ, θ, ϕ). Their definition in R3 and their relation to the

Cartesian coordinates, (z1, z2, z3), can be given as follows. One starts with a circle of

radius a, which will later be identified with the branch loop and we conventionally take to

lie in the (z1, z2) plane. A plane containing the point P and the axis of the circle, which

we identify with the z3 axis, intersects the circle at two diametrically opposite points, A

and B. Denoting the distances of P from A and B respectively by r and r′, the coordinate

ρ of P is defined as

ρ = log
r

r′
, (4.9)

while θ is the ÂPB angle. The angle ϕ is the standard polar angle for the projection, N ,

of P onto the (z1, z2) plane. Figure 3 illustrates the definition of ρ, θ and ϕ and their

relation to the Cartesian coordinates.

The angle ϕ is taken to vary between 0 and 2π. From the definition ρ ∈ (−∞,+∞)

and θ is defined to be in the interval [−π,+π]. The angle θ goes to 0 when the distance

of P from the origin goes to ∞ and in the region of the (z1, z2) plane outside the circle.

It has a discontinuity as one passes through the interior of the disk bounded by the circle

of radius a. It approaches +π or −π if one approaches a point inside the disk from above

(z3 > 0) or below (z3 < 0) respectively. Points on the z3 axis correspond to ρ = 0 and

points on the circle of radius a in the (z1, z2) plane have ρ = ±∞.

Denoting by ξ the distance of the point N from the origin, the peripolar and Cartesian

coordinates are related by

z1 = ξ cosϕ , z2 = ξ sinϕ , z3 =
a sin θ

cosh ρ− cos θ
, (4.10)

where

ξ =
a sinh ρ

cosh ρ− cos θ
. (4.11)

– 21 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

To describe (in peripolar coordinates) a Riemann space consisting of two copies of R3

connected by a branch disk coinciding with the disk bounded by the circle of radius a in

the (z1, z2) plane, one simply allows the range of θ to extend from −π to 3π. The intervals

[−π, π] and [π, 3π] correspond to the first and second R3 respectively. Moving in the first

R
3 from infinity towards the origin along the positive z3 axis (or any other curve in the

z3 > 0 region) θ is positive and increases from 0 to π. Crossing the disk at z3 = 0, one

crosses into the second R
3, while θ varies continuously. Moving along the negative z3

away from the origin in the second space, θ continues to increase from π, reaching 2π as

z3 → −∞. Approaching the origin along the positive z3 axis in the second space the angle

θ varies from 2π to 3π, which is reached on the upper side of the disk. Passing through

the disk one crosses back into the first space (with z3 < 0) and θ goes back to −π.

The Laplacian in peripolar coordinates takes the form

∇2 =

(
cosh ρ− cos θ

a2

)[
(cosh ρ− cos θ)

∂2

∂ρ2
+

(1− cos θ cosh ρ)

sinh ρ

∂

∂ρ

+(cosh ρ− cos θ)
∂2

∂θ2
− sin θ

∂

∂θ
+

(cosh ρ− cos θ)

sinh2 ρ

∂2

∂ϕ2

]
. (4.12)

In [31] Hobson computed the potential solving the Laplace equation in the two-sheeted

Riemann space with a circular branch loop for an arbitrary relative position of the point

charge in the first space relative to the branch disk. Denoting by z0 the location of the

point charge, with peripolar coordinates (ρ0, θ0, ϕ0), the potential at a generic point z of

coordinates (ρ, θ, ϕ) is

φ(z, z0) =
J

4π|z − z0|

[
1

2
+

1

π
arcsin

(
cos

(
θ − θ0

2

)√
2

coshα+ 1

)]
, (4.13)

where

coshα = cosh ρ cosh ρ0 − cos(ϕ− ϕ0) sinh ρ sinh ρ0 (4.14)

and

|z − z0| = a
√

2

√
coshα− cos(θ − θ0)

(cosh ρ− cos θ)(cosh ρ0 − cos θ0)
. (4.15)

Notice that the potential (4.13) is periodic in θ with period 4π, as appropriate for the two-

sheeted Riemann space. One can verify that φ(z, z0) is well-defined and finite everywhere,

including the branch loop, except for the location of the point charge, i.e. z = z0 in the

first space, where it has a Coulomb-like divergence.

As anticipated in the previous subsection, the flux of ∇φ provides a measure of the

angular momentum carried by the different membranes, which is also proportional to their

respective radius. In the case under consideration we have a two-sheeted Riemann space

with one point charge in the first copy of R3. We will see that the equipotential surfaces

for large values of the potential are small spheres centred at the location of the point

charge. They represent the single membrane with angular momentum J at large negative

values of the original Euclidean time t. The flux of ∇φ through these surfaces is J . The

asymptotic equipotential surfaces for small values of the potential approaching zero will be

– 22 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

shown to be two (approximately) concentric spheres with diverging radii, one in each copy

of R3. The flux of ∇φ through these spheres (coinciding with the flux at infinity in the

respective R3) equals the angular momentum of the corresponding membrane, Ji, i = 1, 2.

Conservation of the flux, which corresponds to conservation of angular momentum for the

membranes, implies J = J1 + J2. The way in which the flux J is split between the two

spaces is controlled by the size of the branch disk and by its position relative to the point

charge. For a disk of given radius, if the point charge is located very far from the disk only

a small fraction of the flux passes through the disk into the second space and thus we have

J2 � J1, corresponding to a final state with one membrane much larger than the other. If

we reduce the distance between the point charge and the branch disk, the fraction of flux

passing into the second space increases. When the distance of the charge from the disk

tends to zero, we approach the case in which the flux is equally spit between the two spaces,

which in turn corresponds to having two membranes of equal radius in the final state.

These general considerations can be made more precise by studying the asymptotic

behaviour of φ(z, z0) at a large distance from both the point charge and the branch disk

in each of the two spaces. For this purpose we analyse φ(z, z0) in the region defined by

ρ → 0, θ → 0 in the first space and ρ → 0, θ → 2π in the second space. In both cases we

have coshα ≈ cosh ρ0. The asymptotic behaviour at infinity in the first space is

φ(z, z0) ≈
J

4π|z − z0|

1

2
+

1

π
arcsin

 cos
(
θ0
2

)
cosh

(ρ0
2

)
 (4.16)

and in the second space it is

φ(z, z0) ≈
J

4π|z − z0|

1

2
− 1

π
arcsin

 cos
(
θ0
2

)
cosh

(ρ0
2

)
 . (4.17)

The factors in square brackets in (4.16) and (4.17), multiplying the Coulomb potential

J/(4π|z−z0|), represent the fractions of the total flux staying in the first space or escaping

into the second space, respectively. They determine the angular momenta J1 and J2 of the

two membranes in the final state as fractions of the total angular momentum J .

In the axially symmetric case in which the point charge is located on the axis of the

branch disk, corresponding to ρ0 = 0, the asymptotic formulae (4.16) and (4.17) simplify

and we get

φ(z, z0) ≈
J

4π|z − z0|

(
1− |θ0|

2π

)
, (4.18)

in the first space and

φ(z, z0) ≈
J

4π|z − z0|
|θ0|
2π

, (4.19)

in the second space. Notice that for θ0 → ±π, i.e. when the location of the point charge

approaches the disk, the ratio goes to 1/2 as it should be.

In the following we set a = 1 and we present plots of the equipotential surfaces of the

potential (4.13) in the axially symmetric case, ρ0 = 0. We can also set ϕ0 = 0 without loss

– 23 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

4 2 0 2 4

6

4

2

0

2

4

4 2 0 2 4

6

4

2

0

2

4

Figure 4. Contour plots showing families of equipotential surfaces in each of the two copies

of R3 which constitute the Riemann space. The contours shown are for θ0 = − 4π
5 and cor-

respond to unequally spaced values of φ̂. The values of the potential used here are φ̂ =

{3.113, 1.613, 0.878, 0.593, 0.368, 0.323, 0.310, 0.278, 0.173, 0.128} and include those used in figure 5.

of generality, so that the only remaining parameter in the solution is the angle θ0, whose

value controls the distance of the branch loop from the point charge.9 As will be shown in

the explicit examples below, the value of θ0 also affects the order in which the splitting and

flipping of the membrane orientation take place. Therefore the sequence in which these

steps occur is correlated with the way the angular momentum gets divided between the

two membranes in the final state.

Using the relations (4.10)–(4.11) in (4.13), one can obtain the form of the potential in

Cartesian coordinates, φ = φ(z1, z2, z3), which is the expression used to produce the plots

presented below. In all the following figures we use a rescaled potential, φ̂, related to (4.13)

by φ̂ = (4π/J)φ.

In figure 4 we show contour plots for the potential (4.13) with θ0 = −4π/5. The figure

shows families of equipotential surfaces separately in the two copies of R3. Each R
3 is

represented by a square, with the vertical direction being the direction of the z3 axis. The

branch disk is indicated by a horizontal slit, the point charge (contained in the first R3,

which is on the left) is denoted by a small crossed circle below the disk. The contours are

displayed as continuous lines in the first space and as dashed lines in the second space.

The following figures depict the evolution of the profile of the membranes throughout

the splitting process for different values of θ0. They show how the equipotential surfaces

for the potential in the analytic solution (4.13) reproduce the steps that were qualitatively

discussed in the previous subsection. In these figures the two copies of R3 are superposed.

The (portions of) membranes living in the first or second space are depicted as continuous

9We note that solutions with different θ0 can be related by a conformal transformation.

– 24 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

or dashed lines respectively. The figures display axially symmetric solutions, therefore the

three-dimensional shape of the membranes can be generated rotating the contours about

the vertical axis.

It is interesting to notice that the equipotential surfaces deviate significantly from a

spherical shape only for a rather small range of the potential around the value where the

splitting takes place. This is the region in which the equipotential surfaces cross the branch

disk. The surfaces then quickly revert to Coulomb-like behaviour outside this region for

both larger and smaller values of φ̂.

Figure 5 shows the membrane profiles corresponding to the choice θ0 = −4π/5 in (4.13).

In this case the point charge is quite close to the branch disk and correspondingly the

flux/angular momentum gets split almost evenly between the two spaces. The sequence

of plots shows the formation of cusps, which in these examples with axial symmetry occur

simultaneously at all points on the branch loop. The membrane is then seen self-intersecting

before it splits into two. As a result, with this choice of boundary conditions when the

splitting occurs the second membrane has already the correct orientation, but the two

membranes are still intersecting. Notice that, as anticipated in the qualitative description

of the previous subsection, all the intersections between equipotential surfaces in the figure

(in the third, fourth and fifth panel) involve a dashed and a continuous line, i.e. they

always occur between (portions of) equipotential surfaces belonging to different sheets of

the Riemann space. The same feature can be observed in all the subsequent figures in this

and the next subsection.

Figure 6 shows the evolution of the membrane profiles for θ0 = −π/6, which cor-

responds to a point charge much further from the branch disk. In this case the splitting

process follows a sequence of steps similar to those depicted qualitatively in figures 1 and 2.

The initial membrane splits into a membrane and an anti-membrane, which subsequently

flips its orientation. The cusps are formed after the splitting and the subsequent flipping

of the orientation of the internal membrane requires the latter to self-intersect. This takes

place in the fifth panel in the figure and is shown more clearly in figure 7, which shows an

enlargement of the rectangular area marked in figure 6. This example shows a concrete

realisation, in an exact analytic solution, of the crucial step — illustrated at stage (v) in

figures 1 and 2 — which led us to the concept of Riemann space. Notice the relative size

of the two membranes in the final state; in this case (with the point charge further away

from the disk) the flux is split less evenly than in the previous case and correspondingly

the second membrane is much smaller.

Figure 8 shows the splitting process for θ0 = −π/3, i.e. a value intermediate between

the cases shown in the previous two figures. In this case the splitting and flipping phases

occur simultaneously. At the splitting point the second membrane is still self-intersecting

and its orientation not completely flipped. The relative size of the two membranes in the

final state reflects the fact that in this case the value of θ0 (which controls the distance

between point charge and branch disk) is intermediate between those in the two previous

examples. It corresponds to a splitting of the flux/angular momentum J into J1 and J2
such that the difference |J1 − J2| is larger than in the first case and smaller than in the

second case.

– 25 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

4 2 0 2 4

6

4

2

0

2

4

=3.113

4 2 0 2 4

6

4

2

0

2

4

=0.593

4 2 0 2 4

6

4

2

0

2

4

=0.323

4 2 0 2 4

6

4

2

0

2

4

=0.310

4 2 0 2 4

6

4

2

0

2

4

=0.278

4 2 0 2 4

6

4

2

0

2

4

=0.128

Figure 5. Evolution of the membrane profiles, showing the different phases of the splitting process

in Euclidean time in the case θ0 = −4π/5.

– 26 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

15 10 5 0 5 10 15

20

15

10

5

0

5

10

=0.311

15 10 5 0 5 10 15

20

15

10

5

0

5

10

=0.146

15 10 5 0 5 10 15

20

15

10

5

0

5

10

=0.141

15 10 5 0 5 10 15

20

15

10

5

0

5

10

=0.135

15 10 5 0 5 10 15

20

15

10

5

0

5

10

=0.126

15 10 5 0 5 10 15

20

15

10

5

0

5

10

=0.062

Figure 6. Evolution of the membrane profiles, showing the different phases of the splitting process

in Euclidean time in the case θ0 = −π/6.

– 27 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

Figure 7. Enlargement of the area marked by a rectangle in the fifth panel of figure 6.

If the point charge is not located on the axis of the disk the qualitative behaviour of

the solutions and the associated equipotential surfaces remains the same. The processes

display similar sequences of steps involving splitting and flipping. In this case, however,

the cusp singularities where ∇φ diverges do not form simultaneously on the whole branch

loop, but rather at pairs of points.

The equipotential surfaces for the potential (4.13), once expressed in Cartesian coor-

dinates, represent implicit equations of the membrane profiles as described by the zi(s)

variables. The evolution of the shape of the membranes during a splitting process in terms

of the original coordinates, yi(t), can be reconstructed using the change of variables (3.7)

in the potential φ(z1, z2, z3).10 The qualitative features of the equipotential surfaces pa-

rameterised by the yi coordinates are similar. The only significant difference is that the

size of the membranes does not grow steadily as seen in the previous figures. This is the

same difference that was observed in section 4.1 in the case of the description of the static

spherical solution in terms of the yi and zi variables. Using the yi variables one can instead

verify immediately that the solutions describe a process in which the final membranes have

radii which add up to the radius of the single membrane in the initial state. In the next

subsection we will present plots showing the evolution of the membrane profiles, using the

parametrisation in terms of the yi variables, in the case of solutions interpolating between

configurations with two membranes in the initial state and two in the final state.

4.3 General Riemann spaces

In the previous subsections we presented explicit solutions to the Laplace equation describ-

ing the most elementary splitting process with a single membrane in the initial state and

two membranes in the final state. It is natural to ask whether a similar approach can be

used to construct other solutions corresponding to (anti-)instantons of the pp-wave matrix

model interpolating between states with different numbers of membranes. A large class of

solutions connecting multi-membrane states was shown to exist in [18]. For our reformula-

tion in terms of the Laplace equation to be equivalent to the original (continuum) instanton

equations, solutions should exist for all the allowed instanton configurations. Therefore it

10In addition one also needs to redefine the yi variables by constant shifts to bring the location of point

charge to the origin and avoid the run-away behaviour mentioned at the end of section 4.1.

– 28 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

5 0 5

10

5

0

5

=0.605

5 0 5

10

5

0

5

=0.258

5 0 5

10

5

0

5

=0.226

5 0 5

10

5

0

5

=0.217

5 0 5

10

5

0

5

=0.213

5 0 5

10

5

0

5

=0.108

Figure 8. Evolution of the membrane profiles, showing the different phases of the splitting process

in Euclidean time in the case θ0 = −π/3.

– 29 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

should be possible to identify suitable boundary conditions corresponding to all the pairs

of initial and final states for the original equations, which are permitted according to the

criteria in [18]. Indeed, we will show that the conditions for the existence of BPS instanton

solutions obtained in [18] can be reproduced using our formulation in terms of the Laplace

equation in a Riemann space.

An analysis of the general features of the solutions we discussed above provides hints

for the generalisation to more complicated processes. In the cases discussed in section 4.2

the two membranes in the final state correspond to equipotential surfaces in the asymptotic

regions at infinity, one in each of the two copies of R3 constituting the Riemann space.

On the other hand the single membrane in the infinite past corresponds to small spheres

centred at the location of the point charge. These considerations lead us to propose the

following general prescription. In order to describe a process with n membranes in the

final state one should consider a Riemann space made of n copies of R3. Moreover, if the

initial state contains m membranes one should consider m (positive) point charges at the

origin of m distinct R3’s. The values of the point charges, Ji, i = 1, 2, . . . ,m, correspond

to the angular momenta of the m membranes in the initial state. The values of the flux of

∇φ at infinity in each copy of R3, J ′i , i = 1, 2, . . . , n, correspond to the angular momenta

of the n membranes in the final state. We point out that an immediate consequence of

this prescription is that by construction the number, n, of membranes in the final state

is greater than or equal to the number, m, of membranes in the initial state. Therefore

the necessary condition, n ≥ m, for the existence of instanton solutions is automatically

satisfied.

In the case of a generic Riemann space the relation between the outgoing fluxes at

infinity, J ′i , and the values of the point charges at the origin, Ji, is non-trivial. This is

because the way in which the flux of ∇φ gets split among the different spaces depends in a

complicated way on the number, shape and size of the branch disks and on their positions

relative to the point charges. In the most elementary case of the solution (4.13) we gave

explicit formulae for the asymptotic fluxes associated with the two membranes in the final

state in (4.16)–(4.19).

General multi-sheeted Riemann spaces can involve different combinations of branch

surfaces. This is the case even in the most elementary example discussed in the previous

subsection, in which one membrane with angular momentum J splits into two membranes

with angular momenta J1 and J2. We described this process using a Riemann space made

of two copies of R3 with a single point charge, J , and we presented an exact solution

involving a circular branch disk. However, the same process receives contributions from

more complicated Riemann spaces in which the two R3’s are connected by multiple branch

disks, so long as the flux at infinity in the two spaces is split into the same fractions,

J1 and J2. Of course in order to compute a specific physical transition amplitude it is

necessary to sum all the contributions with the given initial and final states. The issue

of how the different copies of R3 are connected is related to the parameterisation of the

moduli spaces of the associated instanton solutions and we will briefly comment on this

point in section 4.3.2.

– 30 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

At the present stage we do not have a proof of existence of solutions to the three-

dimensional Laplace equation in arbitrary Riemann spaces, but we propose as a conjecture

that solutions should exist for all the boundary conditions which according to our pre-

scription correspond to allowed (anti-)instantons. The results of [24] and [31] provide a

starting point for the analysis of the issue of existence of solutions in general Riemann

spaces. One may view the explicit solutions in [24] and [31] as playing a role similar to

that played by the explicit solution for a spherical conductor obtained by the method of

images in connection with the theory of the electric potential for arbitrarily shaped con-

ductors. Moreover the Laplace equation in the Riemann space can be recast into the form

of an integral equation as discussed in appendix E. This reformulation may also provide a

useful approach to obtain a proof of existence for the solutions.

4.3.1 More solutions

Before discussing how some general properties of the instanton moduli space arise within

our formulation using the Laplace equation, we now present other exact solutions. We use

these specific examples to illustrate some interesting features arising more generally in the

description of instanton configurations in terms of equipotential surfaces.

The linearity of the Laplace equation of course implies that linear combinations of

solutions are also solutions. We can take advantage of this simple observation to obtain new

interesting examples. In particular, a solution describing a process with two membranes

with angular momenta J1 and J2 in the initial state and two in the final state — which

based on our prescription requires a two-sheeted Riemann space with a point charge in each

copy of R3 — can be obtained as a linear combination of the potential (4.13) with charge

J1 and the analogous potential with point charge J2 at z = z0 in the second space. To

obtain the potential due to a point charge at the point corresponding to z0 in the second

space, we recall that in peripolar coordinates the first sheet is parameterised by θ ∈ [−π, π]

and the second sheet by θ ∈ [π, 3π]. Therefore the potential induced by a charge located

at the point in the second R3 corresponding to z0 is obtained substituting θ0 → θ0 + 2π

in (4.13). The resulting solution in the two sheeted Riemann space with point charges in

both spaces is

φ(z, z0) =
J1

4π|z − z0|

[
1

2
+

1

π
arcsin

(
cos

(
θ − θ0

2

)√
2

coshα+ 1

)]

+
J2

4π|z − z0|

[
1

2
− 1

π
arcsin

(
cos

(
θ − θ0

2

)√
2

coshα+ 1

)]
(4.20)

=
1

4π|z − z0|

[
(J1 + J2)

2
+

(J1 − J2)
π

arcsin

(
cos

(
θ − θ0

2

)√
2

coshα+ 1

)]
,

where |z − z0| and coshα are given in (4.15) and (4.14) respectively and we have chosen

the radius of the branch loop to be a = 1. The sign difference between the first two lines

in (4.20) comes from the shift in θ0. Recalling that coshα is independent of θ0, we get

arcsin

[
cos

(
θ − (θ0 + 2π)

2

)√
2

coshα+ 1

]
=− arcsin

[
cos

(
θ − θ0

2

)√
2

coshα+ 1

]
, (4.21)

– 31 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

because the shift by π in the argument flips the sign of the cosine and the inverse sine

function is odd. Notice that if the two charges are equal, J1 = J2, (4.20) reduces to a

simple Coulomb potential in both sheets of the Riemann space. This is consistent with

our general prescription. With equal charges the fluxes of ∇φ from the first space into the

second and from the second space into the first are equal, irrespective of the position, z0,

of the charges. So the flux at infinity in the two spaces is the same and hence the final state

is the same as the initial state. Therefore this case corresponds simply to a stable vacuum

configuration with two membranes of the same size. Note that, up to an overall scale, the

potential (4.20) can also be obtained from the superposition of the solution (4.13) and a

simple Coulomb potential.

We now focus on the special case of axially symmetric solutions (corresponding to

ρ0 = ϕ0 = 0) with J1 = 2J , J2 = J for which (4.20) becomes

φ(z, z0) =
J

4π|z − z0|

[
3

2
+

1

π
arcsin

(
cos

(
θ − θ0

2

)√
2

cosh ρ+ 1

)]
, (4.22)

where we used the fact that coshα = cosh ρ when z0 is on the axis of the disk.

The final states in the process described by the potential (4.22) depend on the way

in which the total flux at infinity is divided between the two spaces, which in turn is

controlled by the value of θ0. Different choices for θ0 give rise to solutions displaying

various interesting features and below we present a few selected examples.

The following figures 9, 11 and 13 show the evolution of the two membranes described

by the solution (4.22) for θ0 = −2π/3, θ0 = −π/10 and θ0 = −π/6 respectively. As in

the previous subsection, the plots show equipotential surfaces for the rescaled potential

φ̂ = (4π/J)φ. The figures depict the equipotential surfaces for φ(y1, y2, y3) expressed in

terms of the coordinates, yi(t), in the original instanton equations (3.6). The conventions

used are the same as in previous figures. The (portions of) membranes in the first copy of

R
3 are shown as continuous contours and those in the second R3 as dashed contours. Since

here we are using the yi coordinates, the contours represent the evolution of the profiles

of the membranes from t = −∞ to t = +∞. Notice that with this parameterisation of

the solution the membranes do not expand steadily. However, their relative radii change

between the initial and final state because of the transfer of angular momentum. The use of

these variables makes manifest the conservation of angular momentum (J ′1 +J ′2 = J1 +J2),

which in the figures is reflected in the fact that the radii of the spheres in the initial and

final states add up to the same total. In the figures we have not displayed the branch loop

as in these coordinates its position is not constant.

In the case shown in figure 9 the value of θ0 is −2π/3 and thus the point charges are

relatively close to the branch disk. As a result there is a large fraction of flux passing from

the first R3 into the second. This corresponds to a large transfer of angular momentum

and thus a significant change in the relative radii of the two membranes. In the sequence

shown in the figure the second and third steps show both membranes extending across

both spaces. In the third panel the outer membrane develops a self-intersection. The small

region marked by a square is shown enlarged in figure 10, where the self-intersection is

more clearly visible. The membranes then split and reconnect in the way shown in the

– 32 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

fourth panel of figure 9 resulting in two equipotential surfaces each contained entirely in

one copy of R3.

In the case shown in figure 11 the point charges are at θ0 = −π/10, i.e. further from

the branch disk and this means that there is a smaller transfer of angular momentum.

With this choice of boundary conditions we observe another interesting feature: for some

intermediate values of Euclidean time there are three membranes involved in the process.

A small membrane detaches from the larger membrane in the first R3 and subsequently

gets absorbed by the other membrane. In the intermediate steps this third membrane

crosses the branch disk from the first R3 into the second, flipping its orientation. The

self-intersection involved in the flipping of the membrane orientation is shown in figure 12,

which is an enlargement of the small rectangle marked in the fourth panel of figure 11.

Figure 13 depicts a case intermediate between the previous two examples, correspond-

ing to θ0 = −π/6. In this case the transfer of angular momentum does not involve the

exchange of a third membrane, but the interplay between the equipotential surfaces is

rather interesting. In figure 14 we show a detailed view of the area within the rectangles

marked in the third and fourth panels in figure 13, presenting a sequence of contour plots

for values of the potential between φ̂ = 0.380 and φ̂ = 0.378. The fourth panel in figure 14

shows clearly that in this case the splitting takes place not at a point, but simultaneously

at points along a ring. This is of course not a generic feature. It is a degenerate case which

only occurs in axially symmetric solutions for certain values of the parameters. A similar

splitting ring is also present in the previously discussed case of figure 9.

It is quite remarkable that the rather striking behaviour of membranes illustrated by

the above examples may be reproduced by solutions of such a simple and well studied

equation as the Laplace equation.

4.3.2 Comments on the moduli space of solutions

In this subsection we discuss some features of the moduli space of instantons associated

with general Riemman spaces. As we previously noted, the number of copies of R3, n,

corresponds to the number of membranes in the final state. Positive charges, Ji ≥ 0

(i = 1, . . . , n), can only be placed at the origin of these spaces. The charge Ji corresponds

to the angular momentum of the i-th membrane in the initial state. Some of the Ji’s may

be zero and the number of non-zero Ji’s is the number of membranes in the initial state.

The sheets of the Riemann space are connected by branch disks, bounded by branch

loops. By definition when a branch disk connects two copies of R3 the disk is located at the

same place, with the same shape, in the two copies.11 Except for this condition, the number

of branch loops connecting the copies of R3, their positions and their shapes are arbitrary.

In the following we collectively refer to the number of copies of R3 and the number, the

positions and the shapes of the branch loops as the geometry of the Riemann space. Our

conjecture is that for any geometry of the Riemann space (and the distribution of positive

charges at the origins) we have a unique solution to the Laplace equation satisfying the

boundary conditions.

11If this condition were not met, we would get unphysical discontinuities in the instanton solutions in the

y-space, reconstructed using (3.7).

– 33 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

2 1 0 1 2

2

1

0

1

2

=3.220

2 1 0 1 2

2

1

0

1

2

=1.348

2 1 0 1 2

2

1

0

1

2

=1.271

2 1 0 1 2

2

1

0

1

2

=1.259

2 1 0 1 2

2

1

0

1

2

=1.120

2 1 0 1 2

2

1

0

1

2

=0.112

Figure 9. Evolution of the membrane profiles for a process with two membranes in the initial state

and two in the final state (θ0 = −2π/3).

– 34 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

Figure 10. Enlargement of the area marked by a square in the third panel of figure 9.

A point on the instanton moduli space is characterised by specifying the geometry of

the Riemann space. This will give an interpretation of the moduli space of solutions of

the BPS instanton equation (2.12) discussed in [18] when J is large (but finite). Stated

differently, the moduli space associated with (2.12) can be considered as a regularisation

of the moduli space of branch loops.

The outgoing flux at infinity in the i-th copy of R3, which we call J ′i , corresponds to

the angular momentum of the i-th membrane in the final state.

A necessary and sufficient condition for the existence of solutions to the instanton

equations interpolating between the initial state characterised by Ji, i = 1, . . . , n, and the

final state characterised by J ′i , i = 1, . . . , n, was established in [18]. Below we will show

that the same condition can be derived from our approach taking advantage, in particular,

of the linearity of the Laplace equation.

The linearity implies that for a fixed geometry of the Riemann space, one has a linear

relationship between Ji and J ′i ,

J ′i =
∑
j

KijJj . (4.23)

The matrix K characterises the flow of the flux lines for a given Riemann space. (It is

reminiscent of the coefficients of capacity in the electrostatic theory of conductors, which

are determined by the shape and positions of the conductors.) In general the matrix K is

not symmetric. We will also use a matrix-vector notation

J ′ = KJ , (4.24)

where J = (J1, . . . , Jn), J ′ = (J ′1, . . . , J
′
n).

Some simple but important properties follow from the definition of Kij . The first

property is the positivity of the elements of the matrix K

Kij ≥ 0. (4.25)

In order to see this, we consider the case in which there is only one unit charge, located

at the origin of the j-th copy of R3, in the entire Riemann space. The flux flowing to the

i-th space from the j-th space equals Kij . Since there is a positive charge in the j-th space

– 35 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

2 1 0 1 2

2

1

0

1

2

=0.424

2 1 0 1 2

2

1

0

1

2

=0.242

2 1 0 1 2

2

1

0

1

2

=0.235

2 1 0 1 2

2

1

0

1

2

=0.236

2 1 0 1 2

2

1

0

1

2

=0.224

2 1 0 1 2

2

1

0

1

2

=0.060

Figure 11. Evolution of the membrane profiles for a process with two membranes in the initial

state and two in the final state (θ0 = −π/10).

– 36 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

Figure 12. Enlargement of the area marked by a rectangle in the fourth panel of figure 11.

and no charge in the i-th space, it is clear that the flux should always flow from the j-th

to the i-th space, not the other way around. This implies Kij ≥ 0.

Other important properties of K are the sum rules,∑
i

Kij = 1 ∀ j , (4.26)∑
j

Kij = 1 ∀ i . (4.27)

The property (4.26) simply follows from the conservation of the number of flux lines (i.e.

the Laplace equation and Gauss’ theorem). The property (4.27) can be deduced as follows.

We consider a special potential function in the Riemann space defined by the requirement

that in each copy of R3 it is identical to the Coulomb potential associated with a unit

charge at the origin. Such a potential solves the Laplace equation and satisfies all the

boundary conditions.12 The existence of this solution implies that if Ji = (1, 1, 1, . . . , 1),

then J ′i = (1, 1, 1, . . . , 1) in (4.23), which is equivalent to (4.27).

The positivity (4.25) and the sum rules (4.26)–(4.27) imply rather strong constraints

on the allowed combinations of Ji’s and J ′i ’s, including conditions equivalent to the criteria

given in [18] for the existence of solutions.

First, one can show that ∑
i

J ′i
p ≤

∑
i

Jpi (4.28)

for any p ≥ 1. By using the vector norm ||v||p = (
∑

i |vi|p)1/p, the matrix norm is defined

by [36]

||A||p = max
v 6=0

||Av||p
||v||p

. (4.29)

Hence in order to establish (4.28) it is sufficient to show

||K||p ≤ 1. (4.30)

Using an inequality for matrix norms (cf. formula (1.11) in [37]),

||A||p ≤ ||A||1/p1 ||A||
1−1/p
∞ (4.31)

12We note that this is a generalisation of the observation given below (4.21).

– 37 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

2 1 0 1 2

2

1

0

1

2

=0.649

2 1 0 1 2

2

1

0

1

2

=0.402

2 1 0 1 2

2

1

0

1

2

=0.380

2 1 0 1 2

2

1

0

1

2

=0.378

2 1 0 1 2

2

1

0

1

2

=0.361

2 1 0 1 2

2

1

0

1

2

=0.102

Figure 13. Evolution of the membrane profiles for a process with two membranes in the initial

state and two in the final state (θ0 = −π/6).

– 38 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

(i) (ii)

(iii) (iv)

(v) (vi)

Figure 14. Enlargement of the area marked by rectangles in the third and fourth panels of figure 13,

showing intermediate steps between the two.

and the well-known formulae [36]

||A||1 = max
1≤j≤n

n∑
i=1

|Aij |, ||A||∞ = max
1≤i≤n

n∑
j=1

|Aij |, (4.32)

we see that ||K||1 = 1 and ||K||∞ = 1 and hence (4.30) and (4.28) are established. For

p = 3, we have ∑
i

J ′i
3 ≤

∑
i

J3
i , (4.33)

– 39 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

J

Ji J(a)i J(b)i

(i) (ii) (iii)

J

Figure 15. The angular momenta of membranes J = (J1, . . . , Jn) can be represented as a

histogram as shown in (i). The area of the histogram J below a given height J is denoted byA(J ;J).

We define histograms J(a) and J(b) by cutting the original histogram J along the horizontal line

and taking the part above and below it, as shown in (ii) and (iii).

which gives a continuum counterpart of the necessary condition (2.16) for the existence of

instantons (which can also be derived directly by defining the continuum version of (2.11)).

One can introduce an “entropy” associated with the distribution of the total angular

momentum among the individual membranes defined as∑
i

(−Ji log Ji) and
∑
i

(
−J ′i log J ′i

)
(4.34)

for the initial and final states respectively. Then by taking p = 1+ε in (4.28) and using the

conservation law
∑
Ji =

∑
J ′i , one can show that the entropy is always non-decreasing,∑
i

(
−J ′i log J ′i

)
≥
∑
i

(−Ji log Ji) . (4.35)

Let us now deduce the necessary and sufficient condition for the existence of instantons

given in [18]. The condition can be phrased as follows (cf. figure 3 in [18]). Since the Ji’s

are non negative, they can be represented by a histogram as shown in figure 15 (i). A

similar histogram can be constructed with the J ′i ’s in the final state. We draw a horizontal

line at a height J ≥ 0 on the histogram and denote the area of the histogram below the

value J by

A(J ;J). (4.36)

The condition in [18] can then be expressed as

A(J ;J ′) ≥ A(J ;J) ∀ J. (4.37)

(We note that we do not assume J1 ≥ J2 ≥ . . . ≥ Jn or J ′1 ≥ J ′2 ≥ . . . ≥ Jn here.) Thus,

the area of the support of the histogram J ′i is always larger than or equal to that of the

histogram Ji. An interpretation of this condition is that the histogram J ′i should be more

smeared compared to the histogram Ji, which is natural in view of (4.35).13

13We note that the condition (4.37) is stronger than the condition (4.33). For example, the case with

J = (5, 5, 0) and J ′ = (6, 2, 2) is allowed by (4.33), but it is prohibited by (4.37).

– 40 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

In order to prove (4.37) we introduce two auxiliary histograms, J(a)i and J(b)i, generated

cutting the histogram Ji at a fixed height J . As shown in figure 15, we define J(a)i and

J(b)i as the histograms above and below the horizontal line at the height J , respectively.

By definition we have

J = J(a) + J(b). (4.38)

Moreover

J ′ = KJ = KJ(a) +KJ(b). (4.39)

We first focus on the histogram defined by KJ(b). All columns in this histogram have

height smaller than or equal to J , i.e. (KJ(b))i ≤ J . To see this we recall that (4.27)

implies that

K

 J...
J

 =

 J...
J

 . (4.40)

Since all elements of J(b) are smaller than or equal to J by definition and all elements of

K are positive (4.25), the elements of KJ(b) cannot be larger than J .

The sum rule (4.26) implies that area of the histogram KJ(b) is the same as the area of

the histogram J(b), which by definition equals A(J ;J). Now recall that J ′ = KJ(a)+KJ(b).

Since the height of the histogram KJ(b) is less than or equal to J , all of the histogram

KJ(b) contributes to A(J ;J ′). The area A(J ;J ′) has a further contribution from KJ(a),

which is always non-negative, because of the positivity of K and J(a). Thus, we have

A(J ;J ′) ≥ A(J ;KJ(b)) = A(J ;J), (4.41)

and (4.37) is proven.

Another interesting property of the moduli space of solutions to the BPS instanton

equation (2.12), which was established in [18], is the additivity rule (2.24) for the dimension

of the moduli spaces. A possible interpretation of this result, which is instructive if perhaps

not mathematically rigorous, is the following. We assume that we have two instanton

solutions, interpolating between vacua A and B, and between vacua B and C, such that

both of them are reasonably localised in time. It is natural to expect that one can construct

an instanton solution interpolating between vacuum A and vacuum C, by stitching together

the two original solutions taken to be well separated in time. The number of parameters

of the composite instanton thus constructed should be given by the sum of the numbers of

parameters of the two original instantons.

This construction of a composite instanton from two instantons well separated in time

has a natural counterpart in our description of the instantons using Riemann spaces. The

following argument relies on viewing a solution in a multi-sheeted Riemann space as a

composite of solutions built using pairs of connected copies of R3. Within this picture the

basic idea underlying the construction is that, at least in the asymptotic region at infinity, a

branch disk with incoming flux can be viewed as a point charge. Here we wish to point out

the essential features of this construction without going into the details. In the following

it is crucial that the time variable in the BPS instanton corresponds to the logarithm of

– 41 –



J
H
E
P
0
2
(
2
0
1
6
)
0
5
0

the distance from the origin in our description. We start with two instanton solutions,

each associated with the branch disks D1 and D2. Because of the scale invariance of the

Laplace equation, one can move one of the disks, e.g. D2, further away from the origin by

rescaling simultaneously its size and its distance from the origin. The physics associated

with the second disk, such as the number of flux lines going into the disk and the profile

of the potential function do not change under the rescaling. We re