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Published for SISSA by Springer

Received: January 23, 2016

Revised: February 20, 2016

Accepted: March 11, 2016

Published: March 22, 2016

Anomalous dimensions of heavy operators from

magnon energies

Robert de Mello Koch, Nirina Hasina Tahiridimbisoa and Christopher Mathwin

National Institute for Theoretical Physics,

School of Physics and Mandelstam Institute for Theoretical Physics, University of Witwatersrand,

Wits, 2050, South Africa

E-mail: robert@neo.phys.wits.ac.za,

NirinaMaurice.HasinaTahiridimbisoa@students.wits.ac.za,

christopher.mathwin@students.wits.ac.za

Abstract: We study spin chains with boundaries that are dual to open strings suspended

between systems of giant gravitons and dual giant gravitons. Motivated by a geometrical

interpretation of the central charges of su(2|2), we propose a simple and minimal all loop

expression that interpolates between the anomalous dimensions computed in the gauge

theory and energies computed in the dual string theory. The discussion makes use of a

description in terms of magnons, generalizing results for a single maximal giant graviton.

The symmetries of the problem determine the structure of the magnon boundary reflec-

tion/scattering matrix up to a phase. We compute a reflection/scattering matrix element

at weak coupling and verify that it is consistent with the answer determined by symmetry.

We find the reflection/scattering matrix does not satisfy the boundary Yang-Baxter equa-

tion so that the boundary condition on the open spin chain spoils integrability. We also

explain the interpretation of the double coset ansatz in the magnon language.

Keywords: 1/N Expansion, AdS-CFT Correspondence, Gauge-gravity correspondence

ArXiv ePrint: 1506.05224

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP03(2016)156

mailto:robert@neo.phys.wits.ac.za
mailto:NirinaMaurice.HasinaTahiridimbisoa@students.wits.ac.za
mailto:christopher.mathwin@students.wits.ac.za
http://arxiv.org/abs/1506.05224
http://dx.doi.org/10.1007/JHEP03(2016)156


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Contents

1 Introduction 1

2 Giants with open strings attached 5

3 Action of the dilatation operator 8

4 Large N diagonalization: asymptotic states 10

5 String theory description 11

6 From asymptotic states to exact eigenstates 15

7 S-matrix and boundary reflection matrix 17

8 Links to the double coset Ansatz and open spring theory 23

9 Conclusions 25

A Large N eigenstates 26

B Two loop computation of boundary magnon energy 32

C The difference between simple states and eigenstates vanishes at

large N 33

D Review of dilatation operator action 34

E One loop computation of bulk/boundary magnon scattering 36

F No integrability 38

1 Introduction

In this article we will connect two distinct results that have been achieved in the context

of gauge/gravity duality. The first result, which is motivated by the Penrose limit in the

AdS5×S5 geometry [1], is the natural language for the computation of anomalous dimen-

sions of single trace operators in the planar limit provided by integrable spin chains (see [2]

for a thorough review). For the spin chain models we study, using only the symmetries

of the system, one can determine the exact large N anomalous dimensions and the two

magnon scattering matrix. Using integrability one can go further and determine the com-

plete scattering matrix of spin chain magnons [3, 4]. The second results which we will

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use are the powerful methods exploiting group representation theory, which allow one to

study correlators of operators whose classical dimension is of order N . In this case, the

large N limit is not captured by summing the planar diagrams. Our results allow a rather

complete understanding of the anomalous dimensions of gauge theory operators that are

dual to giant graviton branes with open strings suspended between them. These results

generalize the analysis of [5] to systems that include non-maximal giant gravitons and dual

giant gravitons. The boundary magnons of an open string attached to a maximal giant

graviton are fixed in place — they can not hop between sites of the open string. In the case

of non maximal giant gravitons and dual giant gravitons there are non-trivial interactions

between the open string and the brane, allowing the boundary magnons to move away from

the string endpoints.

The operators we focus on are built mainly out of one complex U(N) adjoint scalar

Z, and a much smaller number M of impurities given by a second complex scalar field

Y , which are the “magnons” that hop on the lattice of the Zs. The dilatation operator

action on these operators matches the Hamiltonian of a spin chain model comprising of

a set of defects that scatter from each other. The spin chain models enjoy an SU(2|2)2

symmetry. The symmetries of the system determine the energies of impurities, as well as

the two impurity scattering matrix [3, 4]. The SU(2|2) algebra includes two sets of bosonic

generators (Rab and Lαβ) that each generate an SU(2) group. The action of the generators

is summarized in the relations

[Rab, T
c] = δcbT

a − 1

2
δabT

c , [Lαβ , T
γ ] = δγβT

α − 1

2
δαβT

γ (1.1)

where T is any tensor transforming as advertised by its index. The algebra also includes

two sets of super charges Qαa and Sbβ . These close the algebra

{Qαa , Sbβ} = δbaL
α
β + δαβR

b
a + δbaδ

α
βC , (1.2)

where C is a central charge, and

{Qαa , Q
β
b } = 0 , {Saα, Sbβ} = 0. (1.3)

We will realize this algebra on states that include magnons. When the magnons are well

separated, each magnon transforms in a definite representation of su(2|2) and the full state

transforms in the tensor product of these individual representations. Acting on the ith

magnon we can have a centrally extended representation [3, 4]

{Qαa , Sbβ} = δbaL
α
β + δαβR

b
a + δbaδ

α
βCi , (1.4)

{Qαa , Q
β
b } = εαβεab

ki
2
, {Saα, Sbβ} = εαβε

abk
∗
i

2
. (1.5)

The total multimagnon state must be in a representation for which the central charges

ki, k
∗
i vanish. Thus the multi magnon state transforms under the representation with

C =
∑
i

Ci ,
∑
i

ki = 0 =
∑
i

k∗i . (1.6)

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A key ingredient to make use of the su(2|2) symmetry entails determining the central

charges ki, k
∗
i and hence the representations of the individual magnons. There is a natural

geometric description of the system, first obtained by an inspired argument in [6] and later

put on a firm footing in [7], which gives an elegant and simple description of these central

charges. The two dimensional spin chain model that is relevant for planar anomalous

dimensions is dual to the worldsheet theory of the string moving in the dual AdS5×S5

geometry. This string is a small deformation of a 1
2 BPS state. A convenient description of

the 1
2 -BPS sector (first anticipated in [8]) is in terms of the LLM coordinates introduced

in [9], which are specifically constructed to describe 1
2−BPS states built mainly out of

Zs. In the LLM coordinates, there is a preferred LLM plane on which states that are

built mainly from Zs orbit with a radius r = 1 (in convenient units). Consider a closed

string state dual to a single trace gauge theory operator built mainly from Zs, but also

containing a few magnons M . The closed string solution looks like a polygon with vertices

on the unit circle. The sides of the polygon are the magnons. The specific advantage of

these coordinates is that they make the analysis of the symmetries particularly simple and

allow a perfect match to the SU(2|2)2 superalgebra of the gauge theory described above.

Matching the gauge theory and gravity descriptions in this way implies a transparent

geometrical understanding of the ki and k∗i , as we now explain. The commutator of two

supersymmetries in the dual gravity theory contains NS-B2 gauge field transformations. As

a consequence of this gauge transformation, strings stretched in the LLM plane acquire a

phase which is the origin of the central charges ki and k∗i . It follows that we can immediately

read off the central charges for any particular magnon from the sketch of the closed string

worldsheet on the LLM plane: the straight line segment corresponds to a complex number

which is the central charge [7].

The gauge theory operators that correspond to closed strings have a bare dimension

that grows, at most, as
√
N . We are interested in operators whose bare dimension grows

as N when the large N limit is taken. These operators include systems of giant graviton

branes. The key difference as far as the sketch of the state on the LLM plane is concerned,

is that the giant gravitons can orbit on circles of radius r < 1 while dual giant gravitons

orbit on circles of radius r > 1. The magnons populating open strings which are attached

to the giant gravitons can be divided into boundary magnons (which sit closest to the ends

of the open string) and bulk magnons. The boundary magnons will stretch from a giant

graviton located at r 6= 1 to the unit circle, while bulk magnons stretch between points on

the unit circle. We will also consider the case below that the entire open string is given by

a single magnon, in which case it will stretch between two points with r 6= 1.

The computation of correlators of the corresponding operators in the field theory is

highly non-trivial. Indeed, as a consequence of the fact that we now have order N fields in

our operators, the number of ribbon graphs that can be drawn is huge. These enormous

combinatoric factors easily overpower the usual 1
N2 suppression of non-planar diagrams so

that both planar and non-planar diagrams must be summed to capture even the leading

large N limit of the correlator [10]. This problem can be overcome by employing group

representation theory techniques. The article [11] showed that it is possible to compute

the correlation functions of operators built from any number of Zs exactly, by using the

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Schur polynomials as a basis for the local operators of the theory. In [12] these results

were elegantly explained by pointing out that the organization of operators in terms of

Schur polynomials is an organization in terms of projection operators. Completeness and

orthogonality of the basis follows from the completeness and orthogonality of the underlying

projectors. With these insights [11, 12], many new directions opened up. A basis for

the local operators which organizes the theory using the quantum numbers of the global

symmetries was given in [13, 14]. Another basis, employing projectors related to the Brauer

algebra was put forward in [15] and developed in a number of interesting works [16–22].

For the systems we are interested in, the most convenient basis to use is provided by the

restricted Schur polynomials. Inspired by the Gauss Law which will arise in the world

volume description of the giant graviton branes, the authors of [23] suggested operators

in the gauge theory that are dual to excited giant graviton brane states. This inspired

idea was pursued both in the case that the open strings are described by an open string

word [24–26] and in the case of minimal open strings, with each open string represented

by a single magnon [27, 28]. The operators introduced in [24, 27] are the restricted Schur

polynomials. Further, significant progress was made in understanding the spectrum of

anomalous dimensions of these operators in the studies [25, 26, 29–34]. Extensions which

consider orthogonal and symplectic gauge groups and other new ideas, have also been

achieved [35–40].

In this paper we will connect the string theory description and the gauge theory de-

scription of the operators corresponding to systems of excited giant graviton branes. Our

study gives a concrete description of the central charges ki and some of the consequences of

the su(2|2) symmetry. We will see that the restricted Schur polynomials provide a natural

description of the quantum brane states. For the open strings we find a description in

terms of open spin chains with boundaries and we explain precisely what the boundary

interactions are. The double coset ansatz of the gauge theory, which solves the problem

of minimal open strings consisting entirely of a single magnon, also has an immediate and

natural interpretation in the same framework.

There are closely related results which employ a different approach to the questions

considered in this article. A collective coordinate approach to study giant gravitons with

their excitations has been pursued in [41–45]. This technique employs a complex collective

coordinate for the giant graviton state, which has a geometric interpretation in terms of

the fermion droplet (LLM) description of half BPS states [8, 9]. The motivation for this

collective coordinate starts from the observation that within semiclassical gravity, we think

of the D-branes as being localized in the dual spacetime geometry. It might seem however,

that since in the field theory the operators we write down have a precise R-charge and a

fixed energy, they are dual to a delocalized state. Indeed, since gauge/gravity duality is a

quantum equivalence it is subject to the uncertainty principle of quantum mechanics. The

R-charge of an operator is the angular momentum of the dual states in the gravity theory, so

that by the uncertainty principle, the dual giant graviton-branes must be fully delocalized

in the conjugate angle in the geometry. The collective coordinate parametrizes coherent

states, which do not have a definite R-charge and so may permit a geometric interpretation

of the position of the D-brane as the value of the collective coordinate. With the correct

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choice for the coherent states, mixing between different states of a definite R-charge would

be taken into account and so when diagonalizing the dilatation operator (for example)

the mixing between states with different choices of the values of the collective coordinate

might be suppressed. This computation would be, potentially, much simpler than a direct

computation utilizing operators with a definite R-charge. Of course, by diagonalizing the

dilatation operator for operators dual to giant graviton brane plus open string states, one

would expect to recover the collective coordinates, but this may only be possible after

a complicated mixing problem in degenerate perturbation theory is solved. Some of the

details that have emerged from our study do not support this semiclassical reasoning.

Specifically, we find that the brane states are given by restricted Schur polynomials and

these do not receive any corrections when the perturbation theory problem is solved, so that

there does not seem to be any need to solve a mixing problem which constructs localized

states from delocalized ones. Our large N eigenstates do have a definite R-charge. The

nontrivial perturbation theory problem involves mixing between operators corresponding

to the same giant graviton branes, but with different open string words attached. Thus, it

is an open string state mixing problem, solved with a discrete Fourier transform, as it was

for the closed string. However, there is general agreement between the approaches: the

Fourier transform solves a collective coordinate problem which diagonalizes momentum,

rather than position.

For an interesting recent study of anomalous dimensions, at finite N , using a very

different approach, see [46].

This article is organized as follows: in section 2 we recall the relevant facts about the

restricted Schur polynomials. The action of the dilatation operator on these restricted

Schur polynomials is studied in section 3 and the eigenstates of the dilatation operator

are constructed in section 4. Section 5 provides the dual string theory interpretation of

these eigenstates and perfect agreement between the energies of the string theory states

and the corresponding eigenvalues of the dilatation operator is demonstrated. In sections 6

and 7 we consider the problem of magnon scattering, both in the bulk and off the boundary

magnons. We have checked that the magnon scattering matrix we compute is consistent

with scattering results obtained in the weak coupling limit of the theory. One important

conclusion is that the spin chain is not integrable. In section 8 we review the double coset

ansatz and describe the dual string theory interpretation of these results. Our conclusions

and some discussion is given in section 9. The appendices collect some technical details.

2 Giants with open strings attached

In this section we will review the gauge theory description of the operators dual to giant

graviton branes with open string excitations. In this description, each open string is de-

scribed by a word with order
√
N letters. Most of the letters are the Z field. There are

however M ∼ O(1) impurities which are the magnons of the spin chain. For simplicity

we will usually take all of the impurities to be a second complex matrix Y . This idea was

first applied in [47] to reproduce the spectrum of small fluctuations of giant gravitons [48].

The description was then further developed in [49–53]. The articles [51–53] in particular

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developed this description to the point where interesting dynamical questions1 could be

asked and answered. The open string words are then inserted into a sea of Zs which make

up the giant graviton brane(s). Concretely, the operators we consider are

O
(
R,Rk1 , R

k
2 ; {ni}1, {ni}2, · · · , {ni}k

)
=

1

n!

∑
σ∈Sn+k

χR,Rk1 ,Rk2
(σ)Zi1iσ(1) · · ·Z

in
iσ(n)

(Wk)
in+1

iσ(n+1)
· · · (W2)

in+k−1

iσ(n+k−1)
(W1)

in+k
iσ(n+k)

(2.1)

where the open string words are

(WI)
i
j = (Y Zn1Y Zn2−n1Y · · ·Y ZnMI−nMI−1Y )ij . (2.2)

We have used the notation {ni}I in (2.1) to describe the integers {n1, n2, · · · , nMI
} which

appear in the Ith open string word. This is a lattice notation, which lists the number of Zs

appearing to the left of each of the Y s, starting from the second Y : the Zs form a lattice

and the ni give a position in this lattice. This notation is particularly convenient when

we discuss the action of the dilatation operator. We will also find an occupation notation

useful. The occupation notation lists the number of Zs between consecutive Y s, and is

indicated by placing the ni in brackets. Thus, for example O(R,R1
1, R

1
2, {n1, n2, n3}) =

O(R,R1
1, R

1
2, {(n1), (n2 − n1), (n3 − n2)}). R is a Young diagram with n + k boxes. A

bound state of ps giant gravitons and pa dual giant gravitons is described by a Young

diagram R with pa rows, each containing order N boxes and ps columns, each containing

order N boxes. χR,Rk1 ,Rk2
(σ) is a restricted character [24] given by

χR,Rk1 ,Rk2
(σ) = TrRk1 ,Rk2

(ΓR(σ)) . (2.3)

Rk is a Young diagram with n boxes, that is, it is a representation of Sn. The irreducible

representation R of Sn+k is reducible if we restrict to the Sn subgroup. Rk is one of

the representations that arise upon restricting. In general, any such representation will

be subduced more than once. Above we have used the subscripts 1 and 2 to indicate

this. We have in mind a Gelfand-Tsetlin like labeling to provide a systematic way to

describe the possible Rk we might consider. In this labeling, we use the transformation

of the representation under the chain of subgroups Sn+k ⊃ Sn+k−1 ⊃ Sn+k−2 ⊃ · · · ⊃ Sn.

This is achieved by labeling boxes in R. Dropping the boxes with labels ≤ i, we obtain

the representation of Sn+k−i to which Rk belongs. We have to spell out how this chain of

subgroups are embedded in Sn+k. Think of Sq as the group which permutes objects labeled

1, 2, 3, · · · , q. Here we have q = n+ k and the objects we have in mind are the Z fields or

the open string words. We associate an integer to an object by looking at the upper indices

in (2.1); as an example, the open string described by W2 is object number n + k − 1. To

go from Sn+k−i to Sn+k−i−1, we keep only the permutations that fix n+ k− i. We can put

the states in Rk1 and Rk2 into a 1-to-1 correspondence. The trace TrRk1 ,Rk2
sums the column

index over Rk1 and the row index over Rk2 . If we associate the row and column indices with

the endpoints of the open string, we can associate the endpoints of the open string I with

1For example, one could consider the force exerted by the string on the giant.

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Figure 1. A cartoon illustrating the R,Rk1 , R
k
2 labeling for an example with k = 4 open strings

and 3 giant gravitons. The shape of the strings stretching between the giants is not realistic — only

the locations of the end points of the open strings is accurate. The giant gravitons are orbiting on

the circles shown; the radius shown for each orbit is accurate. They wrap an S3 which is transverse

to the plane on which they orbit. The smaller the radius of the giant’s orbit, the larger the S3 it

wraps. The size of the S3 that the giant wraps is given by its momentum, which is equal to the

number of boxes in the column which corresponds to the giant. The numbers appearing in the

boxes of R4
1 tell us where the open strings start and the numbers appearing in the boxes of R4

2

where they end.

the box labeled I in Rk1 and Rk2 . The numbers appearing in the boxes of Rk1 literally tell

us where the k open strings start and the numbers in Rk2 where the k open strings end.

See figure 1 for an example of this labeling. Each Y in an open string word is a magnon.

We will take the number of magnons MI = O(1) ∀I. The Z
ij
iσ(j)

with 1 ≤ j ≤ n belong to

the system of giants and the Z’s appearing in WI belong to the Ith open string. It is clear

that n ∼ O(N).

Each giant graviton is associated with a long column and each dual giant graviton with

a long row in the Young diagrams labeling the restricted Schur polynomial. Our notation

for the Young diagrams is to list row lengths. Thus a Young diagram that has two columns,

one of length n1 and the second of length n2 with n2 < n1 is denoted (2n2 , 1n1−n2), while

a Young diagram with two rows, one of length n1 and one of length n2 (n1 > n2) is

denoted (n1, n2).

We want to use the results of [24–26] to study correlation functions of these operators.

The correlators are obtained by summing all contractions between the Zs belonging to

the giants, and by grouping the open string words in pairs and summing only the planar

diagrams between the fields in each pair of the open string words. To justify the planar

approximation for the open string words we take ni ≥ 0 and
∑L

i=1 ni ≤ O(
√
N). For a nice

careful discussion of related issues, see [54].

We can put these operators into correspondence with normalized states

O(R,Rk1 , R
k
2 ; {ni}1, {ni}2, · · · , {ni}k)↔ |R,Rk1 , Rk2 ; {ni}1, {ni}2, · · · , {ni}k〉 (2.4)

by using the usual state-operator correspondence available for any conformal field theory.

In what follows we will mainly use the state language.

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3 Action of the dilatation operator

The one loop dilatation operator, in the SU(2) sector, is [55, 56]

D = −
g2

YM

8π2
Tr

(
[Y,Z]

[
d

dY
,
d

dZ

])
. (3.1)

Our goal in this section is to review the action of this dilatation operator on the restricted

Schur polynomials, which was constructed in general in [25, 26]. When we act with D

on O(R,Rk1 , R
k
2 ; {ni}1, {ni}2, · · · , {ni}k) the derivative with respect to Y will act on a Y

belonging to a specific open string word. Thus, in the large N limit we can decompose

the action of D into a sum of terms, with each individual term being the action on a

specific open string. If we act on a magnon belonging to the bulk of the open string

word, then the only contribution comes by acting with the derivative respect to Z on a

field that is immediately adjacent to the magnon. We act only on the adjacent Z fields

because to capture the large N limit we should use the planar approximation for the open

string word contractions. To illustrate the action on a bulk magnon, consider the operator

corresponding to a single giant graviton with a single open string attached. The giant

has momentum n so that R is a single column with n + 1 boxes: R = 1n+1. Further,

R1
1 = R1

2 = 1n. The open string has three magnons and hence we can describe the

corresponding state as |1n+1, 1n, 1n; {n1, n2}〉. The action on the bulk magnon at large N is

Dbulk magnon|1n+1, 1n, 1n; {(n1), (n2)}〉 =
g2

YMN

8π2

[
2|1n+1, 1n, 1n; {(n1), (n2)}〉

− |1n+1, 1n, 1n; {(n1 − 1), (n2 + 1)}〉 − |1n+1, 1n, 1n; {(n1 + 1), (n2 − 1)}〉
]
. (3.2)

If we act on a magnon which occupies either the first or last position of the open string

word, we realize one of the four possibilities listed below.

1. The derivative with respect to Z acts on the Z adjacent to the Y , belonging to the

open string and the coefficient of the product of derivatives with respect to Y and Z

replaces these fields in the same order. None of the labels of the state change. This

term has a coefficient of 1 [25, 26].

2. The derivative with respect to Z acts on the Z adjacent to the Y , belonging to the

open string word and the coefficient of the product of derivatives with respect to Y

and Z replaces these fields in the opposite order. In this case, a Z has moved out

of the open string word and into its own slot in the restricted Schur polynomial —

a hop off interaction in the terminology of [25]. In the process the Young diagrams

labeling the excited giant graviton grows by a single box. If the string is attached

to a giant graviton, the column the endpoint of the relevant open string belongs to

inherits the extra box. If the string is attached to a dual giant graviton, the row the

endpoint of the relevant open string belongs to inherits the extra box. The coefficient

of this term is given by minus one times the square root of the factor associated with

the open string box divided by N [25, 26]. We remind the reader that a box in row

i and column j is assigned the factor N − i+ j.

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3. The derivative with respect to Z acts on a Z belonging to the giant and the coefficient

of the product of derivatives with respect to Y and Z replaces these fields in the

opposite order. In this case, a Z has moved from its own slot in the restricted Schur

polynomial and onto the open string word — a hop on interaction in the terminology

of [25]. In the process the Young diagrams labeling the giant graviton shrinks by

a single box. The details of which column/row shrinks is exactly parallel to the

discussion in point 2 above. The coefficient of this term is given by minus one times

the square root of the factor associated with the open string box divided by N [25, 26].

4. The derivative with respect to Z acts on a Z belonging to the giant and the coefficient

of the product of derivatives with respect to Y and Z replaces these fields in the same

order. This is a kissing interaction in the terminology of [25]. None of the labels of

the state change. The coefficient of this term is given by the factor associated with

the open string box divided by N [25, 26].

For the example we are considering the dilatation operator has the following large N action

on the magnons closest to the string endpoints

Dfirst magnon|1n+1, 1n, 1n; {(n1), (n2)}〉 =
g2

YMN

8π2

[(
1 + 1− n

N

)
|1n+1, 1n, 1n; {(n1), (n2)}〉

−
√

1− n

N

(
|1n+2, 1n+1, 1n+1; {(n1−1), (n2)}〉+|1n, 1n−1, 1n−1; {(n1+1), (n2)}〉

) ]
(3.3)

and

Dlast magnon|1n+1, 1n, 1n; {(n1), (n2)}〉 =
g2

YMN

8π2

[(
1 + 1− n

N

)
|1n+1, 1n, 1n; {(n1), (n2)}〉

−
√

1− n

N

(
|1n+2, 1n+1, 1n+1; {(n1), (n2 − 1)}〉+|1n, 1n−1, 1n−1; {(n1), (n2+1)}〉

) ]
. (3.4)

There are a few points worth noting: the complete action of the dilatation operator

can be read from the Young diagram labels of the operator. The factors of the boxes in

the Young diagram for the endpoints of a given open string determine the action of the

dilatation operator on that open string. When the labels Rk1 6= Rk2 , the string end points

are on different giant gravitons and the two endpoints are associated with different boxes

in the Young diagram so that the action of the dilatation operator on the two boundary

magnons is distinct. To determine these endpoint interactions we must go beyond the

planar approximation. Notice that for a maximal giant graviton we have n = N . In this

case, most of the boundary magnon terms in the Hamiltonian vanish and the boundary

magnons are locked in place at the string endpoints. The giant graviton brane is simply

supplying a Dirichlet boundary condition for the open string. For non-maximal giants, all

of the boundary magnon terms are non-zero and, for example, Z fields that belong to the

open string can wander into slots describing the giant. Alternatively, since the split between

open string and brane is probably not very sharp, we might think that the magnons can

wander from the string endpoints into the bulk of the open string. The coefficient of these

hopping terms is modified by the presence of the giant graviton, so that the boundary

magnons do not behave in the same way as the bulk magnons do.

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As a final example, consider a dual giant graviton which carries momentum n. In this

case, R is a single row of n boxes and we have

Dfirst magnon|n+ 1, n, n; {(n1), (n2)}〉 =
g2

YMN

8π2

[(
1 + 1 +

n

N

)
|n+ 1, n, n; {(n1), (n2)}〉

−
√

1+
n

N
(|n+2, n+1, n+1; {(n1−1), (n2)}〉+ |n, n−1, n−1; {(n1+1), (n2)}〉)

]
. (3.5)

In the appendix B we discuss the action of the dilatation operator at two loops.

4 Large N diagonalization: asymptotic states

We are now ready to construct eigenstates of the dilatation operator. We will not construct

exact large N eigenstates. Rather, we focus on states for which all magnons are well sep-

arated. From these states we can still obtain the anomalous dimensions. In section 6 we

will describe how one might use these asymptotic states to construct exact eigenstates, fol-

lowing [3, 4]. In the absence of integrability however, this can not be carried to completion

and our states are best thought of as very good approximate eigenstates.

The Zs in the open string word define a lattice on which the Y s hop. Our construction

entails taking a Fourier transform on this lattice. The boundary interactions allow Zs to

move onto and out of the lattice, so the lattice size is not fixed. It is not clear what the

Fourier transform is, if the size of the lattice varies. The goal of this section is to deal with

these complications. With each application of the one-loop dilatation operator, a single Z

can enter or leave the open string word. At γ loops at most γ Zs can enter or leave. At

any finite loop order (γ) the change in length ∆L = γ of the lattice is finite while the total

length L of the lattice is
√
N . Thus, at large N the ratio ∆L

L → 0 and we can treat the

lattice length as fixed. To implement this idea, we introduce the phases

qa = e
i2πka
J (4.1)

with ka = 0, 1, . . . , J − 1, as well as a cut off function whose form is shown in figure 2. The

eigenstate with two magnons is then given by

|ψ(q1)〉 =

n+J∑
m2=0

m2∑
m1=0

f(m2)qm1−m2
1 |1n+J+m1−m2+1, 1n+J+m1−m2 , 1n+J+m1−m2 ; {m2 −m1}〉

+

J+m2∑
m1=0

n∑
m2=0

f(m1)f(J −m1 +m2)

× qm1−m2
1 |1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J −m1 +m2}〉 . (4.2)

For a detailed discussion of the construction, we refer the reader to appendix A. At large

N it is now simple to show that

D|ψ(q1)〉 = 2×
Ng2

YM

8π2

(
1 +

[
1− n

N

]
−
√

1− n

N
(q1 + q−1

1 )

)
|ψ(q1)〉

= 2g2

(
1 +

[
1− n

N

]
−
√

1− n

N
(q1 + q−1

1 )

)
|ψ(q1)〉 . (4.3)

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Figure 2. The cutoff function used in constructing large N eigenstates.

Since both magnons are boundary magnons, the above formula shows that boundary

magnons carry momentum and it characterizes their anomalous dimension. The analy-

sis for the dual giant graviton of momentum n leads to

D|ψ(q1)〉 = 2×
Ng2

YM

8π2

(
1 +

[
1 +

n

N

]
−
√

1 +
n

N
(q1 + q−1

1 )

)
|ψ(q1)〉

= 2g2

(
1 +

[
1 +

n

N

]
−
√

1 +
n

N
(q1 + q−1

1 )

)
|ψ(q1)〉 . (4.4)

For the generalizations to states with more magnons and further details, the reader should

consult appendix A. This completes our discussion of the large N asymptotic eigenstates.

We will now consider the dual string theory description of these states.

5 String theory description

The string theory description of the gauge theory operators is most easily developed using

the limit introduced by Maldacena and Hofman [7], in which the spectrum on both sides

of the correspondence simplifies. The limit considers operators of large R charge J and

scaling dimension ∆ holding ∆ − J and the ’t Hooft coupling λ fixed. Both sides of the

correspondence enjoy an SU(2|2) × SU(2|2) supersymmetry with novel central extensions

as realized by Beisert in [3, 4]. Once the central charge of the spin-chain/worldsheet exci-

tations have been determined, their spectrum and constraints on their two body scattering

are determined. A powerful conclusion argued for in [7] using the physical picture devel-

oped in [6] is that there is a natural geometric interpretation for these central charges in

the classical string theory. This geometric interpretation also proved useful in the analysis

of maximal giant gravitons in [5]. In this section we will argue that it is also applicable to

the case of non-maximal giant and dual giant gravitons.

Giant gravitons carry a dipole moment under the RR five form flux F5. When they

move through the spacetime, the Lorentz force like coupling to F5 causes them to expand

in directions transverse to the direction in which they move [57]. The giant graviton orbits

on a circle inside the S5 and wraps an S3 transverse to this circle but also contained in the

S5. Using the complex coordinates x = x5 + ix6, y = x3 + ix4 and z = x1 + ix2 the S5 is

described by

|z|2 + |x|2 + |y|2 = 1 (5.1)

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Figure 3. The giant is orbiting on the smaller circle shown. Each red segment is a magnon. The

arrows in the figure simply indicate the orientation of the central charge ki of the ith magnon. The

LLM disk is shaded in this and subsequent figures. This is done to distinguish the rim of the LLM

disk from the orbits of the giant gravitons.

in units with the radius of the S5 equal to 1. The giant is orbiting in the 1 − 2 plane on

the circle |z| = r. The size to which the giant expands is determined by canceling the force

causing them to expand, due to the coupling to the F5 flux, against the D3 brane tension,

which causes them to shrink. Since the coupling to the F5 flux depends on their velocity,

the size of the giant graviton is determined by its angular momentum n as [58–60]

|x|2 + |y|2 =
n

N
. (5.2)

Using (5.1) we see that the giant graviton orbits on a circle of radius [58]

r =

√
1− n

N
< 1 . (5.3)

Consider now the worldsheet geometry for an open string attached to a giant graviton.

Following [7], we will describe this worldsheet solution using LLM coordinates [9]. The

worldsheet for this solution, in these coordinates, is shown in figure 3. The figure shows

an open string with 6 magnons. Each magnon corresponds to a directed line segment

in the figure. The first and last magnons connect to the giant which is orbiting on the

smaller circle shown. Between the magnons we have a collection of O(
√
N) Zs. These are

pushed by a centrifugal force to the circle |z| = 1 giving the string worldsheet the shape

shown in figure 3.

In the limit that the magnons are well separated, each magnon transforms in a definite

SU(2|2)2 representation. The open string itself transforms as the tensor product of the

individual magnon representations. The representation of each individual magnon is spec-

ified by giving the values of the central charges ki, k
∗
i appearing in (1.5). Regarding the

plane shown in figure 3 as the complex plane, k is given by the complex number determined

by the vector describing the directed segment corresponding to the magnon. In particu-

lar, the magnitude of k is given by the length of the line corresponding to the magnon.

The energy of the magnon, which transforms in a short representation, is determined by

supersymmetry to be [3, 4]

E =
√

1 + 2λ|k|2 = 1 + λ|k|2 − 1

2
λ2|k|4 + . . . (5.4)

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Figure 4. A bulk magnon subtending an angle θ has a length of 2 sin θ
2 .

For a magnon which subtends an angle θ as shown in figure 4, we find [7]

E = 1 + 4λ sin2 θ

2
+O(λ2) = 1 + λ(2− eiθ − e−iθ) +O(λ2) . (5.5)

This is in perfect agreement with the field theory answer (A.12) if we set λ = g2 and

q = ei
2πk
J = eiθ ⇒ θ =

2πk

J
. (5.6)

Thus the angle that is subtended by the magnon is equal to its momentum, which is the well

known result obtained in [7]. Consider now the boundary magnon, as shown in figure 5.

The circle on which the giant orbits has a radius given by

r =

√
1− n

N
. (5.7)

The large circle has a radius of 1 in the units we are using. Thus, the length of the boundary

magnon is given by the length of the diagonal of the isosceles trapezium shown in figure 5.

Consequently

E = 1 + λ

(
(1− r)2 + 4r sin2 θ

2

)
+O(λ2)

= 1 + λ
(

1 + r2 − r(eiθ + e−iθ)
)

+O(λ2) . (5.8)

This is again in complete agreement with (A.12) after we set θ = 2πk
J and recall

that r =
√

1− n
N . This is a convincing check of the boundary terms in the dilatation

operator and of our large N asymptotic eigenstates. In the description of maximal giant

gravitons, the boundary magnon always stretches from the center of the disk to a point on

the circumference of the circle |z| = 1. Consequently, for the maximal giant the boundary

magnon subtends an angle of zero and it never has a non-zero momentum. For submaximal

giants we see that the boundary magnons do in general carry non-zero momentum. This is

completely expected: in the case of a maximal giant graviton, the boundary magnons are

locked in the first and last position of the open string lattice. As we move away from the

maximal giant graviton, the coefficients of the boundary terms which allow the boundary

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Figure 5. A boundary magnon subtending an angle θ has a length of
√

(1− r)2 + 4r sin2 θ
2 .

Figure 6. A two strings attached to two giant gravitons state. To distinguish the two strings,

one of them has been indicated with dashed lines. Both giants are submaximal and so are moving

on circles with a radius |z| < 1. One of the strings has only two boundary magnons. The second

string has two boundary magnons and three bulk magnons. Notice that each open string has a

non-vanishing central charge. It is only for the full state that the central charge vanishes. See [45]

for closely related observations.

magnons to hop in the lattice, increase from zero, allowing the boundary magnons to move

and hence, to carry a non-zero momentum. In the appendix B we have checked that the

two loop answer in the field theory agrees with the O(λ2) term of (5.4).

Notice that the vector sum of the directed line segments vanishes. This is nothing

but the statement that our operator vanishes unless q−1
M = q1q2 · · · qM−1. This condition

ensures that although each magnon transforms in a representation of su(2|2)2 with non-

zero central charges, the complete state enjoys an su(2|2)2 symmetry that has no central

extension. It is for this reason that the central charges must sum to zero and hence that

the vector sum of the red segments must vanish. This is achieved in an interesting way for

certain multi-string states: each open string can transform under an su(2|2)2 that has a

non-zero central charge and it is only for the full state of all open strings plus giants that

the central charge vanishes. An example of this for a two string state is given in figure 6.

To conclude this section, we will consider an example involving a dual giant graviton.

In this case, the giant graviton orbits on a circle [59, 60]

r =

√
1 +

n

N
> 1 . (5.9)

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Figure 7. A boundary magnon subtending an angle θ has a length of
√

(r − 1)2 + 4r sin2 θ
2 .

The length of the line segment corresponding to the boundary magnon is again given by

the length of the diagonal of an isosceles trapezium, as shown in figure 7. Consequently

E = 1 + λ

(
(r − 1)2 + 4r sin2 θ

2

)
+O(λ2)

= 1 + λ
(

1 + r2 − r(eiθ + e−iθ)
)

+O(λ2) (5.10)

which is in perfect agreement with (4.4) after we set θ = 2πk
J and r =

√
1 + n

N .

6 From asymptotic states to exact eigenstates

The states we have written down above are asymptotic states in the sense that we have

implicitly assumed that all of the magnons are well separated. In this case the excitations

can be treated individually and the symmetry algebra acts as a tensor product represen-

tation. However, the magnons can come close together and even swap positions. When

they swap positions, we get different asymptotic states that must be combined to obtain

the exact eigenstate. The asymptotic states must be combined in a way that is compatible

with the algebra, as explained in [3]. This requirement ultimately implies a unique way to

complete the asymptotic states to obtain the exact eigenstate.

When two bulk magnons swap positions, the corresponding asymptotic states are com-

bined using the two particle S-matrix. The relevant two particle S-matrix has been de-

termined in [3, 4]. It is also possible for a bulk magnon to reflect/scatter off a boundary

magnon. For maximal giant gravitons [5], the reflection from the boundary preserves the

fact that the boundary magnon has zero momentum and it reverses the sign of the mo-

mentum of the bulk magnon. In this section we would like to investigate the scattering of

a bulk magnon off a boundary magnon for a non-maximal giant graviton.

We must require that the total central charge k of the state vanishes. Thus, after the

scattering the directed line segments must still sum to zero. Further the central charge

C of the state must remain unchanged. Taken together, these conditions uniquely fix the

momentum of both bulk and boundary magnon after the scattering.

In figure 8 the process of scattering a bulk magnon off the boundary magnon is shown.

After the scattering the magnons that have a different momentum, corresponding to line

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Figure 8. A bulk magnon scatters with a boundary magnon. In the process the direction of the

momentum of the bulk magnon is reversed.

segments that have changed and these are shown in green. In this case the giant graviton is

close enough to a maximal giant that the momentum of the boundary magnon is reversed,

so this is a reflection-like scattering. Before and after the scattering the line segments

line up to form a closed circuit, so that the central charge k of the state before and after

scattering is zero. To analyze the constraint arising from fixing the central charge C, we

parameterize the problem as shown in figure 9. There is a single parameter θ which is fixed

by requiring√
1 + 8λ sin2 ϕ2

2
+

√
1 + 8λ

(
[1 + r]2 + 4r sin2 ϕ1

2

)
=

√
1 + 8λ sin2 θ

2
+

√
1 + 8λ

(
[1 + r]2 + 4r sin2

(
ϕ1 + ϕ2 + θ

2

))
(6.1)

which is the condition that the state has the correct central charge C. In the above formula

we have

r =

√
1− b0

N
. (6.2)

The equation (6.1) has two solutions, one of which is negative θ = −ϕ2 and describes the

state before the scattering. We need to choose the solution for which θ 6= −ϕ2. Notice that

for b0 = N this condition implies that θ = ϕ2 which is indeed the correct answer [5]. In

this case, the bulk magnon reflects off the boundary with a reverse in the direction of its

momentum but no change in its magnitude. The momentum of the bulk magnon remains

zero. When b0 = 0 the momenta of the two magnons is exchanged which is again the correct

answer [3, 4]. When 0 < b0 < N we find the solution to (6.1) for the momentum of the

bulk magnon interpolates between reflection like scattering (when the momentum of the

magnon is reversed) and magnon like scattering (when the momenta of the two magnons

are exchanged). In this case though, in general, the magnitude of the momenta of the bulk

and the boundary magnons are not preserved by the scattering — the scattering is inelastic.

The fact that the scattering between boundary and bulk magnons is not elastic has

far reaching consequences. First, the system will not be integrable. In the case of purely

elastic scattering for all magnon scatterings, the number of asymptotic states that must

be combined to construct the exact energy eigenstate is roughly (M − 1)! for M magnons.

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Figure 9. A bulk magnon scatters with a boundary magnon. In the process the direction of the

momentum of the bulk magnon is reversed. Before the scattering the boundary magnon subtends

an angle ϕ1 and the bulk magnon subtends an angle ϕ2. After the scattering the boundary magnon

subtends an angle ϕ1 + ϕ2 + θ and the bulk magnon subtends an angle −θ.

This is the number of ways of arranging the magnons (distinguished by their momentum)

up to cyclicity. There are M magnon momenta appearing and these momenta are the

same for all the asymptotic states. The exact eigenstates can then be constructed using a

coordinate space Bethe ansatz. For the case of inelastic scattering, the momenta appearing

depend on the specific asymptotic state one considers and there are many more than

(M − 1)! asymptotic states that must be combined to construct the exact eigenstate. In

this case constructing the exact eigenstates from the asymptotic states appears to be a

formidable problem.

7 S-matrix and boundary reflection matrix

We have a good understanding of the symmetries of the theory and the representations

under which the states transform. Following Beisert [3, 4], this is all that is needed to

obtain the magnon scattering matrix. In this section we will carry out this analysis.

Each magnon transforms under a centrally extended representation of the SU(2|2)

algebra

{Qαa , Q
β
b } = εαβεab

ki
2
, {Saα, Sbβ} = εabεαβ

k∗i
2
, (7.1)

{Saα, Q
β
b } = δabL

β
α + δβαR

a
b + δab δ

β
αCi . (7.2)

There are also the usual commutators for the bosonic su(2) generators. There are three

central charges ki, k
∗
i , Ci for each SU(2|2) factor. Following [5] we set the central charges

of the two copies to be equal. It is useful to review how the bosonic part of the SU(2|2)2

symmetry acts in the gauge theory. N = 4 super Yang-Mills theory has 6 hermitian adjoint

scalars φi that transform as a vector of SO(6). We have combined them into the complex

fields as follows

X = φ1 + iφ2 , X̄ = φ1 − iφ2 ,

Y = φ3 + iφ4 , Ȳ = φ3 − iφ4 ,

Z = φ5 + iφ6 , Z̄ = φ5 − iφ6 . (7.3)

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The bosonic subgroup of SU(2|2)2 is SU(2) × SU(2) = SO(4) that rotates φ1, φ2, φ3, φ4

as a vector. In terms of complex fields, Y,X and Ȳ , X̄ transform under different SU(2|2)

groups. Z, Z̄ do not transform. To specify the representation that each magnon transforms

in, following [3, 4] we specify parameters ak, bk, ck, dk for each magnon, where

Qαa |φb〉 = akδ
b
a|ψα〉 , Qαa |ψβ〉 = bkε

αβεab|φb〉 , (7.4)

Saα|φb〉 = ckεαβε
ab|ψβ〉 , Saα|ψβ〉 = dkδ

β
α|φa〉 , (7.5)

for the kth magnon. We are using the non-local notation of [4]. Using the representation

introduced above

Q1
1Q

2
2|φ2〉 = akQ

1
1|ψ2〉 = bkakε

12ε12|φ2〉 , Q2
2Q

1
1|φ2〉 = 0 , (7.6)

so that kk = 2 ak bk. An identical argument using the Saα supercharges gives k∗k = 2 ck dk.

Consider next a state with a total of K magnons. If we are to obtain a representation

without central extension, we must require that the central charges vanish

k

2
=

K∑
k=1

kk
2

=

K∑
k=1

akbk = 0 ,

k∗

2
=

K∑
k=1

k∗k
2

=
K∑
k=1

ckdk = 0 . (7.7)

To obtain a formula for the central charge C consider

QαaS
b
β |φc〉 = ckQ

α
a ε
bcεβγ |ψγ〉 = ckbkε

bcεβγε
αγεad|φd〉 . (7.8)

Now set a = b and α = β and sum over both indices to obtain

QαaS
a
α|φc〉 = 2bkck|φc〉 . (7.9)

Very similar manipulations show that

SaαQ
α
a |φc〉 = 2akdk|φc〉 (7.10)

so that we learn the value of the central charge Ck

{Qαa , Saα}|φc〉 = 4C|φc〉 = 2(akdk + bkck)|φc〉 , ⇒ Ck =
1

2
(akdk + bkck) . (7.11)

Using

{S1
2 , Q

1
1} = L1

2 L1
2|ψ2〉 = |ψ1〉 (7.12)

we easily find

{S1
2 , Q

1
1}|ψ2〉 = (akdk − bkck)|ψ1〉 ⇒ akdk − bkck = 1 . (7.13)

This is also the condition to get an atypical representation of su(2|2) [4].

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Following [3], a useful parametrization for the parameters of the representation is

given by

ak =
√
gηk , bk =

√
g

ηk
fk

(
1−

x+
k

x−k

)
, (7.14)

ck =

√
giηk

fkx
+
k

, dk =

√
gx+

k

iηk

(
1−

x−k
x+
k

)
. (7.15)

The parameters x±k are set by the momentum pk of the magnon

ei
2πpk
J =

x+
k

x−k
. (7.16)

The parameter fk is a pure phase, given by the product
∏
j e

ipj , where j runs over all

magnons to the left of the magnon considered. To ensure unitarity |ηk|2 = i(x−k − x
+
k ).

The condition akdk − bkck = 1 to get an atypical representation implies that

x+
k +

1

x+
k

− x−k −
1

x−k
=
i

g
. (7.17)

This equation will be very useful in verifying some of the S-matrix formulas given below.

A useful parametrization for the parameters specifying the representation for a boundary

magnon is given by

ak =
√
gηk , bk =

√
g

ηk
fk

(
1− r

x+
k

x−k

)
, (7.18)

ck =

√
giηk

fkx
+
k

, dk =

√
gx+

k

iηk

(
1− r

x−k
x+
k

)
, (7.19)

where r =
√

1− n
N is the radius of the path on which the giant graviton of momentum n

orbits2 and the parameters x±k are again set by the momentum carried by the boundary

magnon according to (7.16). For the boundary magnon, fk is again a phase as described

above and now |ηk|2 = i(rx−k −x
+
k ). For a maximal giant graviton r = 0 and the boundary

magnon carries no momentum and |ηk|2 = −ix+
k . For the boundary magnon, the condition

akdk − bkck = 1 to get an atypical representation implies that

x+
k +

1

x+
k

− rx−k −
r

x−k
=
i

g
(7.20)

This equation will again be useful below. Equation (7.20) interpolates between (7.17) for

r = 1, which is the correct condition for a bulk magnon and the condition obtained for r = 0

x+
k +

1

x+
k

=
i

g
(7.21)

which was used in [5] for the boundary magnon attached to a maximal giant graviton.

2For an open string attached to a dual giant graviton, we would have r =
√

1 + n
N

where n is the

momentum of the dual giant graviton.

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Following [3, 4] one can check that the above parametrization obeys (7.7). Finally,

akbkckdk = g2(e−ipk − 1)(eipk − 1) = 4g2 sin2 pk
2

=
1

4

[
(akdk + bkck)

2 − (akdk − bkck)2
]

=
1

4

[
(2Ck)

2 − 1
]

(7.22)

so that

Ck = ±
√

1

4
+ 4g2 sin2 pk

2
. (7.23)

The components of an energy eigenstate in different asymptotic regions are related by

the bulk-bulk and boundary-bulk magnon scattering matrices S and R. S and R must

commute with the su(2|2) group. The labels of the representations of individual magnons

can change under the scattering but they must do so in a way that preserves the central

charges of the total state. In the picture of the energy eigenstates provided by the LLM

plane, the central charges are given by the directed line segments (which are vectors and

hence can also be viewed as complex numbers), one for each magnon. The fact that these

line segments close into polygons is the statement that the central charges k and k∗ of our

total state vanishes. The sum of the lengths squared of these line segments determines the

central charge C. By scattering these segments can rearrange themselves as long as the

sums
∑

i

√
1 + 2λl2i with li the length of segment i is preserved and so long as they still

form a closed polygon.

Implementing the consequences of invariance under SU(2|2)2 is exactly parallel to the

analysis of [3–5]. For the S-matrix describing the scattering of two bulk magnons, the

reader is referred to [3, 4]. When considering the equations for the reflection/scattering

matrix describing the reflection/scattering of a bulk magnon from a boundary magnon,

we need to pay attention to the fact that the central charges of the representation are no

longer swapped between the two magnons. Rather, the central charges after the reflection

are determined by solving (6.1). Denote the central charge of the boundary magnon before

the reflection by pB. Denote the central charge of the bulk magnon before the reflection

by pb. Denote the central charge of the boundary magnon after the reflection by kB.

Denote the central charge of the bulk magnon after the reflection by kb. Denote the

reflection/scattering matrix by R. Since the S-matrix has to commute with the bosonic

su(2) generators Schur’s Lemma implies that it must be proportional to the identity in

each given irreducible representation of su(2). This immediately implies that

R|φapBφ
b
pb
〉 = AR12|φ

{a
kB
φ
b}
kb
〉+BR

12|φ
[a
kB
φ
b]
kb
〉+

1

2
CR12ε

abεαβ |ψαkBψ
β
kb
〉 (7.24)

R|ψαpBψ
β
pb
〉 = DR

12|ψ
{α
kB
ψ
β}
kb
〉+ ER12|ψ

[α
kB
ψ
β]
kb
〉+

1

2
FR12εabε

αβ |φakBφ
b
kb
〉 (7.25)

R|φapBψ
β
pb
〉 = GR12|ψ

β
kB
φakb〉+HR

12|φakBψ
β
kb
〉

R|ψαpBφ
b
pb
〉 = KR

12|ψαkBφ
b
kb
〉+ LR12|φbkBψ

α
kb
〉 . (7.26)

The analysis now proceeds as in [3, 4]. Demanding the S-matrix commutes with the

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supercharges implies

AR12 = S12
0

η1η2x
′+
1 x

+
1 (x−1 − x

+
2 )
(
(x+

2 − rx
−
2 )(rx′+2 − x

′−
2 )x+

2 + (x−2 − rx
+
2 )(x′+2 − rx

′−
2 )x′+2

)
η′1η
′
2x
′+
2 x

+
2 (x−1 − x

+
1 )(x+

1 − x
′+
1 )(x+

1 (rx+
2 − x

−
2 ) + x−2 (rx−2 − x

+
2 ))

BR
12 = AR12

[
1 +

2x′−2 (x′−1 − x
′+
1 )

x′+1 (x−1 − x
+
2 )(x′−1 x

′−
2 − rx

′+
1 x
′+
2 )

B1

B2

]
B1 = x−2 x

′+
1

[
(x−1 −x

+
1 )(2x−1 −x

′−
1 )(x+

2 x
′+
1 −x

+
1 x

+
2 )−x′+1 x

−
1 (x+

2 −rx
−
2 )(x−1 −x

+
2 )
]rx′+2 −x′−2
rx′−2 −x

′+
2

+
[
x+

1 x
′+
1 (x−1 − x

+
2 )(x−2 − rx

+
2 ) + (x−1 − x

+
1 )x−2 x

+
2 (x′+1 − x

+
1 )
]
x′−1 x

′−
2

B2 = (rx−2 − x
+
2 )

[
x+

1 x
′−
2 x
′−
1

rx+
2 − x

−
2

rx−2 − x
+
2

− x′+1 x
−
1 x
−
2

rx′+2 − x
′−
2

rx′−2 − x
′+
2

]
CR12 = S0

12

2η2η1C1

fx+
2 (x+

1 − x
′+
1 )(x+

1 (rx+
2 − x

−
2 ) + x−2 (rx−2 − x

+
2 ))(x′−1 x

′−
2 − rx

′+
1 x
′+
2 )

C1 = x′+1
x−1 − x

+
2

x−1 − x
+
1

(
x′+1 x

−
1 x
−
2 (x+

2 − rx
−
2 )(rx′+2 − x

′−
2 ) + x+

1 x
′−
1 x
′−
2 (x−2 − rx

+
2 )(x′+2 − rx

′−
2 )
)

+ x−2 x
+
2 (x+

1 −x
′+
1 )
(
x−1 (rx′+1 x

′+
2 +x′−1 x

′−
2 −2x′+1 x

′−
2 )+x′−1 x

′−
2 (rx′−2 −x

′−
1 +x′+1 −x

′+
2 )
)

DR
12 = −S0

12

ER12 = −S0
12

1−2x+
1 x
′−
2

x′−1
x−1

(x′−1 −x
′+
1 +x′+2 −rx

′−
2 )−(x′−1 −x

′+
1 )− x′+1 x−2

x+1 x
′−
2

x+2−rx
−
2

x−2−rx
+
2

(x′−2 −rx
′+
2 )[

x+
1 +x−2

x+2−rx
−
2

x−2−rx
+
2

][
rx+′

1 x
+′
2 − x

−′
1 x
−′
2

]


FR12 = S0
12

2x+
1 x
′+
1 f(x−′1 − x

+′
1 )(x′−2 − rx

′+
2 )(x−2 − rx

+
2 )

η′1η
′
2x
−
1 x
−′
1

[
x+

1 (x−2 − rx
+
2 ) + x−2 (x+

2 − rx
−
2 )
][
x−′1 x

−′
2 − rx

+′
1 x

+′
2

]
×

[
x−1 − x

′−
1 +

rx−2 − x
+
2

x−2 − rx
+
2

x−2 x
−
1

x+
1

+
x′+2 − rx

′−
2

x′−2 − rx
′+
2

x′−1 x
′−
2

x′+1

]

GR12 = S0
12

η1x
+
1

[
x+

2 (rx−2 − x
+
2 )(rx′+2 − x

′−
2 ) + x′+2 (rx+

2 − x
−
2 )(x′+2 − rx

′−
2 )
]

η′2x
′+
2 (x−1 − x

+
1 )
[
x+

1 (x−2 − rx
+
2 ) + x−2 (x+

2 − rx
−
2 )
]

HR
12 = S0

12

η1(x−′1 − x
+′
1 )
[
x−1 x

−
2 (rx−2 − x

+
2 ) + x+

1 x
−′
1 (rx+

2 − x
−
2 )
]

η′1x
−′
1 (x−1 − x

+
1 )
[
x+

1 (x−2 − rx
+
2 ) + x−2 (x+

2 − rx
−
2 )
]

KR
12 = S0

12

η2x
−
2

[
x−1 x

+′
1 (rx+′

2 − x
−′
2 ) + x−′1 x

−′
2 (rx−′2 − x

+′
2 )
]

η′2x
−′
1 x
−′
2

[
x+

1 (x−2 − rx
+
2 ) + x−2 (x+

2 − rx
−
2 )
]

LR12 = S0
12

η2x
−
2 (x−1 − x

−′
1 )(x−′1 − x

+′
1 )

η′1x
−′
1

[
x+

1 (x−2 − rx
+
2 ) + x−2 (x+

2 − rx
−
2 )
] (7.27)

where

x+
1

x−1
= eipb

x+
2

x−2
= eipB , (7.28)

x+
1′

x−1′
= eikb

x+
2′

x−2′
= eikB . (7.29)

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Figure 10. A bulk magnon scatters with a boundary magnon. The sum of the momenta of the

two magnons is π. Here we only show two of the magnons; we indicate them in red before the

scattering and in green after the scattering. In the process the direction of the momentum of both

magnons is reversed.

Thus, the S-matrix is determined up to an overall phase. Here we have simply chosen

D12 = −S0
12 which specifies the overall phase. This overall phase is constrained by crossing

symmetry [61]. It is simple to verify that this R matrix is unitary for any value of r and

any momenta, and further that it reproduces the bulk S matrix for r = 1 and the reflection

matrix for scattering from a maximal giant graviton for r = 0. In performing this check

we compared to the expressions in [62]. To provide a further check of these expressions, we

have considered the case that the boundary and the bulk magnons have momenta that sum

to π, as shown in figure 10. In this situation it is very simple to compute the final momenta

of the two magnons — the final momenta are minus the initial momenta. In appendix E

we have computed the value of 1
2

(
1 +

BR12
AR12

)
at one loop. We find this agrees perfectly with

the answer obtained from (7.27). To perform this check, one needs to express x± in terms

of p by solving x+ = x−eip and (7.20) for the boundary magnon or (7.17) for the bulk

magnon. Doing this we find

x− = e−i
p
2

(
1

2g sin p
2

+ 2g sin
p

2

)
+O(g2), (7.30)

for a bulk magnon and

x− = − i

g(r − eip)
+ ige−ip(r − eip)re

ip − 1

r + eip
+O(g2) (7.31)

for a boundary magnon. Inserting these expansions into (7.27) and keeping only the leading

order (which is g0) at small g, we reproduce (E.13) for any allowed value of r.

It is a simple matter to verify that the boundary Yang-Baxter equation is not satisfied

by this reflection matrix, indicating that the system is not integrable. This conclusion

follows immediately upon verifying that changing the order in which the bulk magnons

scatter with the boundary magnon leads to final states in which the magnons have different

momenta. Consequently, the integrability is lost precisely because the scattering of the

boundary and bulk magnons, for boundary magnons attached to a non-maximal giant

graviton, is inelastic.

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8 Links to the double coset Ansatz and open spring theory

There is an interesting limiting case that we can consider, obtained by taking each open

string word to simply be a single Y , i.e. each open string is a single magnon. In this case one

must use the correlators computed in [27, 28] as opposed to the correlators computed in [24].

The case with distinguishable open strings is much simpler since when the correlators are

computed, only contractions between corresponding open strings contribute; when the

open strings are identical, it is possible to contract any two of them. In this case one must

consider operators that treat these “open strings” symmetrically, leading to the operators

constructed in [27]. In a specific limit, the action of the dilatation operator factors into an

action on the Zs and an action on the Y s [31, 32]. The action on the Y s can be diagonalized

by Fourier transforming to a double coset which describes how the magnons are attached to

the giant gravitons [32, 33]. For an operator labeled by a Young diagram R with p long rows

or columns, the action on the Zs then reduces to the motion of p particles along the real line

with their coordinates given by the lengths of the Young diagram R, interacting through

quadratic pair-wise interaction potentials [34]. For interesting related work see [63]. Our

goal in this section is to explain the string theory interpretation of these results.

The conclusion of [32, 33] is that eigenstates of the dilatation operator given by opera-

tors corresponding to Young diagrams R that have p long rows or columns can be labeled

by a graph with p vertices and directed edges. The number of directed edges matches the

number of magnons Y used to construct the operator. These graphs have a natural inter-

pretation in terms of the Gauss Law expected from the worldvolume theory of the giant

graviton branes [23]. Since the giant graviton has a compact world volume, the Gauss

Law implies the total charge on the giant’s world volume vanishes. Each string end point

is charged, so this is a constraint on the possible open string configurations: the number

of strings emanating from the giant must equal the number of strings terminating on the

giant. Thus, the graphs labeling the operators are simply enumerating the states consistent

with the Gauss Law. To stress this connection we use the language “Gauss graphs” for

the labels, we refer to the vertices of the graph as branes since each one is a giant graviton

brane and we identify the directed edges as strings since each is a magnon. The action

of the dilatation operator is nicely summarized by the Gauss graph labeling the operator.

Count the number nij of strings (of either orientation) stretching between branes i and j

in the Gauss graph. The action of the dilatation operator on the Gauss graph operator is

then given by

DOR,r(σ) = −
g2

YM

8π2

∑
i<j

nij(σ)∆ijOR,r(σ) . (8.1)

The operator ∆ij is defined in appendix D. For a proof of this, see [32, 33]. To obtain

anomalous dimensions one needs to solve an eigenproblem on the R, r labels, which has

been accomplished in [34] in complete generality.

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For three open strings stretched between three giant gravitons we have to solve the

following eigenvalue problem

g2
YM

8π2

[
(2N − c1 − c2 + 3)O(c1, c2, c3)−

√
(N − c1 + 1)(N − c2 + 1)O(c1 + 1, c2 − 1, c3)

−
√

(N − c1)(N − c2 + 2)O(c1 − 1, c2 + 1, c3)
]

+
g2

YM

8π2

[
(2N − c2 − c3 + 5)O(c1, c2, c3)−

√
(N − c2 + 1)(N − c3 + 3)O(c1, c2−1, c3+1)

−
√

(N − c2 + 2)(N − c3 + 2)O(c1, c2 + 1, c3 − 1)
]

+
g2

YM

8π2

[
(2N − c1 − c3 + 4)O(c1, c2, c3)−

√
(N − c3 + 2)(N − c1 + 1)O(c1+1, c2, c3−1)

−
√

(N − c3 + 3)(N − c1)O(c1−1, c2, c3+1)
]

= γO(c1, c2, c3) (8.2)

where c1, c2 and c3 are the lengths of the columns = momenta of the three giant gravitons

and γ is the anomalous dimension. At large N , approximating for example O(c1, c2, c3) =

O(c1 +1, c2, c3−1) which amounts to ignoring back reaction on the giant gravitons, we have

g2
YMN

8π2

[√
1− c1

N
−
√

1− c2

N

]2

O(c1, c2, c3) +
g2

YMN

8π2

[√
1− c2

N
−
√

1− c3

N

]2

O(c1, c2, c3)

+
g2

YMN

8π2

[√
1− c3

N
−
√

1− c1

N

]2

O(c1, c2, c3) = γO(c1, c2, c3) . (8.3)

The Gauss graph associated with this operator has a string stretching between the brane

of momentum c1 and the brane of momentum c3, a string stretching between the brane of

momentum c1 and the brane of momentum c2 and a string stretching between the brane

of momentum c2 and the brane of momentum c3.

On the string theory side, since our magnons don’t carry any momentum, we have three

giants moving in the plane with magnons stretched radially between them. Identifying the

central charges, we find they are radial vectors with length equal to the distance between

the giants. With these central charges we can write down the energy

E =
√

1 + 2λ(r1 − r2)2 +
√

1 + 2λ(r1 − r3)2 +
√

1 + 2λ(r3 − r2)2 . (8.4)

Using the usual translation between the momentum of the giant graviton and the radius

of the circle it moves on

ri =

√
1− ci

N
i = 1, 2, 3 (8.5)

we find that the order λ term in the expansion of (8.4) precisely matches the gauge theory

result (8.3).

If we don’t ignore back reaction on the giant graviton, we find that (8.2) leads to

a harmonic oscillator eigenvalue problem. In this case, we are keeping track of the Zs

slipping past a magnon, from one giant onto the next. In this way, one of the giants will

grow and one will shrink thereby changing the radius of their orbits and hence the length

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of the magnon stretched between them. In this process we would expect the energy to vary

continuously, which is exactly what we see at large N . A specific harmonic oscillator state

(see [34] for details) corresponds to two giant gravitons executing a periodic motion. In

one period, the giants first come towards each other and then move away from each other

again. Exciting these oscillators to any finite level, we find an energy that is of order the

’t Hooft coupling divided by N . These very small energies translate into motions with a

huge period.

There is an important point worth noting. The harmonic oscillator problem that arises

from (8.2) is obtained by expanding (8.2) assuming that c1 − c2 is order
√
N and c1, c2

are of order N . The oscillator Hamiltonian then arises as a consequence of (and depends

sensitively on) the order 1 shifts in the coefficients of the terms in (8.2). Thus to really

trust the oscillator Hamiltonian we find we must be sure that (8.2) is accurate enough that

we can expand it and the order 1 term we obtain is accurate. This is indeed the case, as

we discuss in appendix D.

9 Conclusions

In this study we have used the descriptions of the action of the dilatation operator derived

using an approach which relies heavily on group representation theory techniques, to study

the anomalous dimensions of operators with a bare dimension that grows as N , as the

large N limit is taken. For these operators, even just to capture the leading large N

limit, we are forced to sum much more than just the planar diagrams and this is precisely

what the representation theoretic approach manages to do. We have demonstrated an

exact agreement with results coming from the dual gravity description, which is convincing

evidence in support of this approach. It gives definite correct results in a systematic large

N expansion, demonstrating that the representation theoretic methods provide a useful

language and calculational framework with which to tackle the kinds of large N but non-

planar limits we have studied in this article. Of course, we have mainly investigated the

leading large N limit and the computation of 1
N corrections is an interesting problem that

we hope to return to in the future.

The progress that was made in understanding the planar limit of N = 4 super Yang-

Mills theory is impressive (see [2] for a comprehensive review). Of course, much of the

progress is thanks to integrability. There are however results that do not rely on inte-

grability, only on the symmetries of the theory. In our study we clearly have a genuine

extension of methods (giant magnons, the SU(2|2) scattering matrix) that worked in the

planar limit, into the large N but non-planar setting. Further, even though integrability

does not persist, it is present when the radius r of the circle on which the graviton moves

is r = 0 (maximal giant graviton) or r = 1 (point-like giant graviton). If we perturb about

these two values of r, we are departing from integrability in a controlled way and hence we

might still be able to exploit integrability. For more general values of r, we have managed

to find asymptotic eigenstates in which the magnons are well separated and we expect these

to be very good approximate eigenstates. Indeed, anomalous dimensions computed using

these asymptotic eigenstates exactly agree with the dual string theory energies. Without

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the power of integrability it does not seem to be easy to patch together asymptotic states

to obtain exact eigenstates.

We have a clearer understanding of the non-planar integrability discovered in [29–34].

The magnons in these systems remain separated and hence free, so they are actually non-

interacting. One of the giants would need to lose all of its momentum before any two

magnons would scatter. It is satisfying that the gauge theory methods based on group

representation theory are powerful enough to detect this integrability directly in the field

theory. The results we have found here give the all loops prediction for the anomalous

dimensions of these operators. In the limit when we consider a very large number of fields

there would seem to be many more circumstances in which one could construct operators

that are ultimately dual to free systems. This is an interesting avenue that deserves careful

study, since these simple free systems may provide convenient starting points, to which

interactions may be added systematically.

A possible instability associated to open strings attached to giants has been pointed

out in [51]. In this case it seems that the spectrum of the spin chain becomes continuous,

the ground state is no longer BPS and supersymmetry is broken. The transition that

removes the BPS state is simply that the gap from the ground state to the continuum

closes. Of course, the spectrum of energies is discrete but this is only evident at subleading

orders in 1/N when one accounts for the back reaction of the giant graviton-branes. The

question of whether these BPS states with given quantum numbers exist or not has been

linked to a walls of stability type description [64] in [45]. It would be interesting to see if

these issues can be understood using the methods of this article.

Acknowledgments

We would like to thank David Berenstein, Sanjaye Ramgoolam, Joao Rodrigues and Costas

Zoubos for useful discussions and/or correspondence. This work is based upon research

supported by the South African Research Chairs Initiative of the Department of Science

and Technology and National Research Foundation. Any opinion, findings and conclusions

or recommendations expressed in this material are those of the authors and therefore the

NRF and DST do not accept any liability with regard thereto.

A Large N eigenstates

In section 4 we explained that at any finite loop order (γ) the change in length ∆L = γ

of the open string word lattice is finite while the total length L of the lattice is
√
N .

This implies that at large N the ratio ∆L
L → 0 and we can treat the lattice length as

fixed. This observation is most easily used by first introducing “simple states” that have a

definite number of Zs, in the lattice associated to each open string. This is accomplished

by relaxing the identification of the open string word with the lattice. The dilatation

operator’s action now allows magnons to move off the open string, mixing simple states

with states that are not simple. However, by modifying these simple states we can build

states that are closed under the action of the dilatation operator. Our simple states are

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defined by taking a “Fourier transform” of the states (2.4). The simplest system to consider

is that of a single giant, with a single string attached, excited by only two magnons (i.e.

only boundary magnons — no bulk magnons). The string word is composed using J Z

fields and the complete operator using J + n Zs. Introduce the phases

qa = e
i2πka
J (A.1)

with ka = 0, 1, . . . , J−1. As a consequence of the fact that the lattice is a discrete structure,

momenta are quantized with the momentum spacing set by the inverse of the total lattice

size. This explains the choice of phases in (A.1). The simple states we consider are thus

given by

|q1, q2〉 =
J−1∑
m1=0

m1∑
m2=0

qm1
1 qm2

2 |1
n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J −m1 +m2}〉 (A.2)

+
J−1∑
m2=0

m2∑
m1=0

qm1
1 qm2

2 |1
n+J+m1−m2+1, 1n+J+m1−m2 , 1n+J+m1−m2 ; {m2 −m1}〉 .

This Fourier transform is a transform on the lattice describing the open string worldsheet.

The two magnons sit at positions m1 and m2 on this lattice. If m2 > m1, there are m2−m1

Zs between the magnons. If m1 > m2, there are J + m2 −m1 Zs between the magnons.

The Zs before the first magnon of the string and after the last magnon of the string, are

mixed up with the Zs of the giant — they do not sit on the open string word. All of

the terms in (A.2) are states with different positions for the two magnons, but each is

a giant that contains precisely n Zs with an open string attached, and the open string

contains precisely J Zs. We can’t distinguish where the string begins and where the giant

ends: the open string and giant morph smoothly into each other. This is in contrast to

the case of a maximal giant graviton, where the magnons mark the endpoints of the open

string.3 If this interpretation is consistent we must recover the expected inner product

on the lattice and we do: consider a giant with momentum n. An open string with a

lattice of J sites is attached to the giant. The string is excited by M magnons, at positions

n1, . . . , nM−1 and nM , with nj+1 > nj . The corresponding normalized states, denoted by

|n; J ;n1, n2, · · · , nk〉 will obey4

〈n; J ;n1,m2, · · · ,mM |n, J, n1, n2, · · · , nM 〉 = δm2n2 · · · δmMnM nk+1 > nk,mk+1 > mk .

(A.3)

This is the statement that, up to the ambiguity of where the open string starts, the magnons

must occupy the same sites for a non-zero overlap. It is clear that (G(x) ≡ 1x+1, 1x, 1x and

again, nj+1 > nj ,mj+1 > mj)

〈G(n+ J +m1 −m2); {m2, · · · ,mM}|G(n+J+n1−n2); {n2, · · · , nM}〉 = δm2n2 · · · δmknk
3For the maximal giant graviton, the boundary magnons are not able to hop and so sit forever at the

end of the open string. For a non-maximal giant graviton the boundary magnons can hop. Even if they are

initially placed at the string endpoint, they will soon explore the bulk of the string.
4As a consequence of the fact that it is not possible to distinguish where the open string begins and

where the giant ends, there is no delta function setting the positions of the first magnons to be equal to

each other — we have put this constraint in by hand in (A.3).

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reproducing the lattice inner product. The simple states are an orthogonal set of states.

To check this, compute the coefficient ca of the state |1n+a+1, 1n+a, 1n+a; {J−a}〉. Looking

at the two terms in (A.2) we find the following two contributions

ca =
J−1∑
m1=a

qm1
1 qm1−a

2 +

a−1∑
m1=0

qm1
1 qm1−a

2

=

{
Jq−a2 if k1 + k2 = 0

0 if k1 + k2 6= 0
. (A.4)

Thus, q1 = q−1
2 to get a non-zero result. We will see that this zero lattice momentum

constraint maps into the constraint that the su(2|2) central charges of the complete magnon

state must vanish. Our simple states are then given by setting q2 = q−1
1 and are labeled

by a single parameter q1; denote the simple states using a subscript s as |q1〉s.
The asymptotic large N eigenstates are a small modification of these simple states.

When we apply the dilatation operator to the simple states nothing prevents the boundary

magnons from “hopping past the endpoints of the open string”, so the simple states are not

closed under the action of the dilatation operator. We need to relax the sharp cut off on the

magnon movement, by allowing the sums that appear in (A.2) above to be unrestricted.

We accomplish this by introducing a “cut off” function, shown in figure 2. In terms of this

cut off function f(·) our eigenstates are

|ψ(q1)〉 =

n+J∑
m2=0

m2∑
m1=0

f(m2)qm1−m2
1 |1n+J+m1−m2+1, 1n+J+m1−m2 , 1n+J+m1−m2 ; {m2 −m1}〉

+

J+m2∑
m1=0

n∑
m2=0

f(m1)f(J −m1 +m2)

× qm1−m2
1 |1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J −m1 +m2}〉 . (A.5)

The dilatation operator can not arrange that the number of Zs between two magnons

becomes negative. Thus, any bounds on sums in the definition of our simple states enforcing

this are respected. On the other hand, the dilatation operator allows boundary magnons

to hop arbitrarily far beyond the open string endpoint. Bounds in the sums for simple

states enforcing this are not respected. Replace these bounds enforced as the upper limit

of a sum, by bounds enforced by the cut off function. From figure 2 we see that the cut

off function is defined using a parameter δJ . We require that δJ
J → 0 as N →∞, so that

at large N the difference between these eigenstates and the simple states |q1〉s vanishes, as

demonstrated in appendix C. We also want to ensure that

f(i) = f(i+ 1) + ε ∀i (A.6)

with ε→ 0 as N →∞. (A.6) is needed to ensure that we do indeed obtain an eigenstate.

It is straight forward to choose a function f(x) with the required properties. We could for

example choose δJ to be of order N
1
4 . Our large N answers are not sensitive to the details

of the cut off function f(x). When 1/N corrections to the eigenstates are computed f(x)

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may be more constrained and we may need to reconsider the precise form of the cut off

function and how we implement the bounds.

It is now straight forward to verify that, at large N , we have

D|ψ(q1)〉 = 2×
Ng2

YM

8π2

(
1 +

[
1− n

N

]
−
√

1− n

N
(q1 + q−1

1 )

)
|ψ(q1)〉

= 2g2

(
1 +

[
1− n

N

]
−
√

1− n

N
(q1 + q−1

1 )

)
|ψ(q1)〉 . (A.7)

For the dual giant graviton of momentum n we find

D|ψ(q1)〉 = 2×
Ng2

YM

8π2

(
1 +

[
1 +

n

N

]
−
√

1 +
n

N
(q1 + q−1

1 )

)
|ψ(q1)〉

= 2g2

(
1 +

[
1 +

n

N

]
−
√

1 +
n

N
(q1 + q−1

1 )

)
|ψ(q1)〉 . (A.8)

The generalization to include more magnons is straight forward. We will simply con-

sider increasingly complicated examples and for each simply quote the final results. The

discussion is most easily carried out using the occupation notation. For example, the simple

states corresponding to three magnons are

|q1, q2, q3〉 =

J−1∑
n3=0

n3∑
n2=0

n2∑
n1=0

qn1
1 qn2

2 qn3
3 |G(n+ J + n1 − n3); {(n2 − n1), (n3 − n2)}〉

+

J−1∑
n1=0

n1∑
n3=0

n3∑
n2=0

qn1
1 qn2

2 qn3
3 |G(n+ n1 − n3); {(J + n2 − n1), (n3 − n2)}〉

+

J−1∑
n2=0

n2∑
n1=0

n1∑
n3=0

qn1
1 qn2

2 qn3
3 |G(n+ n1 − n3); {(n2 − n1), (J + n3 − n2)}〉 (A.9)

where we have again lumped together the Young diagram labels G(x) = R,R1
1, R

1
2 =

1x+1, 1x, 1x. The coefficient of the ket |G(n+ J − a− b); {(a), (b)}〉 is given by the sum

J−1∑
n1=0

(q1q2q3)n1qa2q
a+b
3 (A.10)

which vanishes if k1 + k2 + k3 6= 0. Consequently we can set q3 = q−1
1 q−1

2 . Including the

cut off function, our energy eigenstates are given by

|ψ(q1, q2)〉 =
∞∑

n3=0

n3∑
n2=0

n2∑
n1=0

qn1−n3
1 qn2−n3

2 f(n3)|G(n+ J + n1 − n3); {(n2 − n1), (n3 − n2)}〉

+

J+n2∑
n1=0

∞∑
n3=0

n3∑
n2=0

qn1−n3
1 qn2−n3

2 f(n1)f(J+n3−n1)|G(n+n1−n3); {(J+n2−n1), (n3−n2)}〉

+

J+n3∑
n2=0

n2∑
n1=0

∞∑
n3=0

qn1−n3
1 qn2−n3

2 f(n2)f(J+n3−n1)|G(n+n1−n3); {(n2−n1), (J+n3−n2)}〉.

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It is a simple matter to see that

D|ψ(q1, q2)〉 = (E1 + E2 + E3)|ψ(q1, q2)〉 (A.11)

where

E1 = g2

(
1 +

[
1− n

N

]
−
√

1− n

N
(q1 + q−1

1 )

)
E2 = g2

(
2− q2 − q−1

2

)
E3 = g2

(
1 +

[
1− n

N

]
−
√

1− n

N
(q3 + q−1

3 )

)
. (A.12)

Now consider the extension to states containing many magnons: for an M magnon

state, consider all M cyclic orderings of the “magnon positions”

n1 ≤ n2 ≤ n3 ≤ · · · ≤ nM−2 ≤ nM−1 ≤ nM ≤ J − 1

nM ≤ n1 ≤ n2 ≤ n3 ≤ · · · ≤ nM−2 ≤ nM−1 ≤ J − 1

nM−1 ≤ nM ≤ n1 ≤ n2 ≤ n3 ≤ · · · ≤ nM−2 ≤ J − 1

...
...

...

n2 ≤ n3 ≤ · · · ≤ nM−2 ≤ nM−1 ≤ nM ≤ n1 ≤ J − 1 . (A.13)

Construct the differences {n2 − n1, n3 − n2, n4 − n3, · · · , nM − nM−1, n1 − nM}. Every

difference except for one is positive. Add J to the difference that is negative, i.e. the

resulting differences are {∆2,∆3,∆4, · · · ,∆M ,∆1} with

∆i =


ni − ni−1 if ni ≥ ni−1

J + ni − ni−1 if ni ≤ ni−1

. (A.14)

For each ordering in (A.13) we have a term in the simple state. This term is obtained by

summing over all values of {n1, n2, · · · , nL} consistent with the ordering considered, of the

following summand

qn1
1 qn2

2 · · · q
nM
L |1

n+∆1+1, 1n+∆1 , 1n+∆1 ; {(∆2), (∆3), · · · , (∆M )}〉 . (A.15)

Repeating the argument we outlined above, this term vanishes unless q−1
M = q1q2 · · · qM−1

so that the summand can be replaced by

qn1−nM
1 qn2−nM

2 · · · qnM−1−nM
M−1 |1n+∆1+1, 1n+∆1 , 1n+∆1 ; {(∆2), (∆3), · · · , (∆M )}〉 . (A.16)

Finally, consider the extension to many string states and an arbitrary system of giant

graviton branes. Each open string word is constructed as explained above. We add extra

columns (one for each giant graviton) and rows (one for each dual giant graviton) to R.

The labels Rk1 and Rk2 specify how the open strings are connected to the giant and dual

giant gravitons. When describing twisted string states, the strings describe a closed loop,

“punctuated by” the giant gravitons on which they end. As an example, consider a two

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giant graviton state, with a pair of strings stretching between the giant gravitons. The two

strings carry a total momentum of J . Notice that we are using the two strings to define a

single lattice of J sites. One might have thought that the two strings would each define an

independent lattice. To understand why we use the two strings to define a single lattice,

recall that we are identifying the zero lattice momentum constraint with the constraint

that the su(2|2) central charges of the complete magnon state must vanish. There is a

single su(2|2) constraint on the two string state, not one constraint for each string. We

interpret this as implying there is a single zero lattice momentum constraint for the two

strings, and hence there is a single lattice for the two strings. This provides a straight

forward way to satisfy the su(2|2) central charge constraints. The first giant graviton

has a momentum of b0 and the second a momentum of b1. The first string is excited by

M magnons with locations {n1, n2, · · · , nM−1, nM} and the second by M̃ magnons with

locations {ñ1, ñ2, · · · , ñM̃−1, ñM̃} where we have switched to the lattice notation. We need

to consider the M + M̃ orderings of the {ni} and {ñi}. Given a specific pair of orderings,

we can again form the differences

∆1 =

{
n1 − ñM if n1 ≥ ñM

J + n1 − ñM if n1 ≤ ñM

∆i =


ni − ni−1 if ni ≥ ni−1

i = 2, 3, · · · ,M
J + ni − ni−1 if ni ≤ ni−1

∆M+1 =

{
ñ1 − nM if nM ≤ ñ1

J + ñ1 − nM if nM ≥ ñ1

∆M+i =


ñi − ñi−1 if ñi ≥ ñi−1

i = 2, 3, · · · , M̃
J + ñi − ñi−1 if ñi ≤ ñi−1

. (A.17)

For each ordering we again have a term in the simple state, obtained by summing over all

values of {n1, n2, · · · , nM , ñ1, ñ2, · · · , ñM̃} consistent with the ordering considered, of the

following summand

qn1
1 · · · q

nM
M q̃ñ1

1 · · · q̃
ñM̃
M̃
|G(∆1,∆M+1); {(∆2), (∆3), · · · , (∆M )},

{(∆M+2), (∆M+3), · · · , (∆M+M̃ )}〉 (A.18)

where

G(x, y) ≡ ,

2

1 ,

1

2 . (A.19)

In the first Young diagram above there are b1 + y + 1 rows with 2 boxes in each row and

b0 + x − b1 − y − 1 rows with 1 box in each row. Repeating the argument we outlined

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above, this term vanishes unless q̃−1
M̃

= q1 · · · qM q̃1 · · · q̃M̃−1 so that the summand can be

replaced by

q
n1−ñM̃
1 q

n2−ñM̃
2 · · · q̃ñM̃−1−ñM̃

M̃−1
|G(∆1,∆M+1); {(∆2), (∆3), · · · , (∆M )},

{(∆M+2), (∆M+3), · · · , (∆M+M̃ )}〉 . (A.20)

B Two loop computation of boundary magnon energy

The dilatation operator, in the su(2) sector, can be expanded as [55, 56]

D =

∞∑
k=0

(
g2

YM

16π2

)k
D2k =

∞∑
k=0

g2kD2k , (B.1)

where the tree level, one loop and two loop contributions are

D0 = Tr

(
Z
∂

∂Z

)
+ Tr

(
Y

∂

∂Y

)
, (B.2)

D2 = −2 : Tr

(
[Z, Y ]

[
∂

∂Z
,
∂

∂Y

])
: , (B.3)

D4 = D
(a)
4 +D

(b)
4 +D

(c)
4 , (B.4)

D
(a)
4 = −2 : Tr

([
[Y,Z] ,

∂

∂Z

] [[
∂

∂Y
,
∂

∂Z

]
, Z

])
:

D
(b)
4 = −2 : Tr

([
[Y,Z] ,

∂

∂Y

] [[
∂

∂Y
,
∂

∂Z

]
, Y

])
:

D
(c)
4 = −2 : Tr

(
[[Y,Z] , T a]

[[
∂

∂Y
,
∂

∂Z

]
, T a

])
: . (B.5)

The boundary magnon energy we computed above came from D2. By computing the

contribution from D4 we can compare to the second term in the expansion of the string

energies. Since we are using the planar approximation when contracting fields in the open

string words, in the limit of well separated magnons, the action of D4 can again be written

as a sum of terms, one for each magnon. Thus, if we compute the action of D4 on a state

|1n+1, 1n, 1n; {n1, n2}}〉 with a single string and a single bulk magnon, its a trivial step to

obtain the action of D4 on the most general state.

A convenient way to summarize the result is to quote the action of D4 on a state for

which the magnons have momenta q1, q2, q3. Of course, we will have to choose the qi so

that the total central charge vanishes as explained in the article above. Thus we could

replace q3 → (q1q2)−1 in the formulas below. We will write the answer for a general giant

graviton system with strings attached. For the boundary terms, each boundary magnon

corresponds to an end point of the string and each end point is associated with a specific

box in the Young diagram. Denote the factor of the box corresponding to the first magnon

by cF and the factor of the box associated to the last magnon by cL. A straight forward

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but somewhat lengthy computation, using the methods developed in [25, 26] gives

(D4)first magnon|ψ(q1, q2, q3)〉 = −g
4

2

[(
1 +

cF
N

)2
− 2

(
1 +

cF
N

)√cF
N

(q1 + q−1
1 )

+
cF
N

(q2
1 + 2 + q−2

1 )

]
|ψ(q1, q2, q3)〉

= −g
4

2

[
1 +

cF
N
−
√
cF
N

(q1 + q−1
1 )

]2

|ψ(q1, q2, q3)〉

= −1

2

[
g2

(
1 +

cF
N
−
√
cF
N

(q1 + q−1
1 )

)]2

|ψ(q1, q2, q3)〉 (B.6)

in perfect agreement with (5.4). The term D
(b)
4 does not make a contribution to the action

on distant magnons, since we sum only the planar open string word contractions. The

remaining terms D
(a)
4 , D

(c)
4 both make a contribution to the action on distant magnons.

For completeness note that

(D4)bulk magnon|ψ(q1, q2, q3)〉 = −1

2

[
2g2

(
2− (q2 + q−1

2 )
)]2 |ψ(q1, q2, q3)〉 . (B.7)

C The difference between simple states and eigenstates vanishes at

large N

In this section we want to quantify the claim made in section 4 that the difference between

our simple states and our exact eigenstates vanishes in the large N limit. We will do this

by computing the difference between the simple states and eigenstates and observing this

difference has a norm that goes to zero in the large N limit.

For simplicity, we will consider a two magnon state. The generalization to many

magnon states is straight forward. Our simple states have the form

|q〉 = N

(
J−1∑
m1=0

m1∑
m2=0

qm1−m2 |1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J −m1 +m2}〉 (C.1)

+

J−1∑
m2=0

m2∑
m1=0

qm1−m2 |1n+J+m1−m2+1, 1n+J+m1−m2 , 1n+J+m1−m2 ; {m2 −m1}〉

)
.

Requiring that 〈q|q〉 = 1 we find

N =
1

J
√
J + 1

. (C.2)

With this normalization we find that the simple states are orthogonal

〈qa|qb〉 = δkakb +O

(
1

J

)
where qa = ei

2πka
J , qb = ei

2πkb
J . (C.3)

This is perfectly consistent with the fact that in the planar limit the lattice states, given by

|1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J −m1 + m2}〉 are orthogonal and our simple states

are simply a Fourier transform of these.

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Our eigenstates have the form (we will see in a few moments that the normalization

in the next equation below is the same as the normalization in (C.2))

|ψ(q)〉 = N

( ∞∑
m2=0

m2∑
m1=0

f(m2)qm1−m2 |1n+J+m1−m2+1, 1n+J+m1−m2 , 1n+J+m1−m2 ; {m2 −m1}〉

+

J+m2∑
m1=0

∞∑
m2=0

f(m1)f(J −m1 +m2)qm1−m2 |1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J−m1+m2}〉

)
≡ |q〉+ |δq〉 (C.4)

where

|δq〉= N

(
n+J+1∑
m2=J

m2∑
m1=0

f(m2)qm1−m2 |1n+J+m1−m2+1, 1n+J+m1−m2 , 1n+J+m1−m2 ; {m2 −m1}〉

+

J+m2∑
m1=J

n+m1∑
m2=0

f(J−m1+m2)f(m1)qm1−m2 |1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J−m1+m2}〉

+

J−1∑
m1=0

n+m1∑
m2=m1+1

f(J−m1+m2)qm1−m2 |1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J−m1+m2}〉

)

= N

(
J+δJ∑
m2=J

m2∑
m1=0

f(m2)qm1−m2 |1n+J+m1−m2+1, 1n+J+m1−m2 , 1n+J+m1−m2 ; {m2 −m1}〉

+

l−∑
m1=J

J+δJ∑
m2=0

f(J−m1+m2)f(m1)qm1−m2 |1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J−m1+m2}〉

+

J−1∑
m1=0

m1+δJ∑
m2=m1+1

f(J−m1+m2)qm1−m2 |1n+m1−m2+1, 1n+m1−m2 , 1n+m1−m2 ; {J−m1+m2}〉

)

and l− is the smallest of J +m2 and J + δJ . It is rather simple to see that |δq〉 is given by

a sum of O(J) terms and that each term has a coefficient of order δJ . Consequently, up to

an overall constant factor cδq which is independent of J , we can bound the norm of |δq〉 as

〈δq|δq〉 ≤ cδqJ(δJ)2N 2 = cδq
(δJ)2

J(J + 1)
(C.5)

which goes to zero in the large J limit, proving our assertion that the difference between

the simple states and the large N eigenstates vanishes in the large N limit.

D Review of dilatation operator action

The studies [29, 30] have computed the dilatation operator action without invoking the dis-

tant corners approximation. The only approximation made in these studies is that correla-

tors of operators with p long rows/columns with operators that have p long rows/columns

and some short rows/columns, vanishes in the large N limit. These results are useful since

they provide data against which the distant corners approximation could be compared.

Further, we have demonstrated that the action of the dilatation operator reduces to a set

of decoupled harmonic oscillators in [31–34]. However, to obtain this result we needed to

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expand one of the factors in the dilatation operator to subleading order. The agreement of

the resulting spectrum5 is strong evidence that the distant corners approximation is valid.

It is worth discussing these details and explaining why we do indeed obtain the correct

large N limit. This point is not made explicitly in [31–34].

In terms of operators belonging to the SU(2) sector and normalized to have a unit two

point function, the action of the one loop dilatation operator

DOR,(r,s)(Z, Y ) =
∑
T,(t,u)

NR,(r,s);T,(t,u)OT,(t,u)(Z, Y )

is given by

NR,(r,s);T,(t,u) = −g2
YM

∑
R′

cRR′dTnm

dR′dtdu(n+m)

√
fT hooksT hooksr hookss
fR hooksR hookst hooksu

×

× Tr
([

ΓR((n, n+ 1)), PR→(r,s)

]
IR′ T ′

[
ΓT ((n, n+ 1)), PT→(t,u)

]
IT ′ R′

)
.

The above formula is exact. After using the distant corners approximation to simplify the

trace and prefactor, this becomes

DOR,(r,s)µ1µ2 = −g2
YM

∑
uν1ν2

∑
i<j

δ~m,~nM
(ij)
sµ1µ2;uν1ν2∆ijOR,(r,u)ν1ν2 . (D.1)

Notice that we have a factorized action: the ∆ij (explained below) acts only on the Young

diagrams R, r and

M (ij)
sµ1µ2;uν1ν2 =

m√
dsdu

(
〈~m, s, µ2 ; a|E(1)

ii |~m, u, ν2 ; b〉〈~m, u, ν1 ; b|E(1)
jj |~m, s, µ1 ; a〉

+〈~m, s, µ2 ; a|E(1)
jj |~m, u, ν2 ; b〉〈~m, u, ν1 ; b|E(1)

ii |~m, s, µ1 ; a〉
)

(D.2)

where a and b are summed, acts only on the s, µ1, µ2 labels of the restricted Schur polyno-

mial. a labels states in the irreducible representation s and b labels states in the irreducible

representation t. To spell out the action of operator ∆ij it is useful to split it up into three

terms

∆ij = ∆+
ij + ∆0

ij + ∆−ij . (D.3)

Denote the row lengths of r by ri and the row lengths of R by Ri. Introduce the Young

diagram r+
ij obtained from r by removing a box from row j and adding it to row i. Similarly

r−ij is obtained by removing a box from row i and adding it to row j. In terms of these

Young diagrams we have

∆0
ijOR,(r,s)µ1µ2 = −(2N +Ri +Rj − i− j)OR,(r,s)µ1µ2 , (D.4)

∆+
ijOR,(r,s)µ1µ2 =

√
(N +Ri − i)(N +Rj − j + 1)OR+

ij ,(r
+
ij ,s)µ1µ2

, (D.5)

∆−ijOR,(r,s)µ1µ2 =
√

(N +Ri − i+ 1)(N +Rj − j)OR−ij ,(r−ij ,s)µ1µ2 . (D.6)

5One can also compare the states that have a definite scaling dimension. The states obtained in the

distant corners approximation are in perfect agreement with the states obtained in [29, 30] by a numerical

diagonalization of the dilatation operator.

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As a matrix ∆ij has matrix elements

∆R,r;T,t
ij =

√
(N +Ri − i)(N +Rj − j + 1)δT,R+

ij
δt,r+ij

(D.7)

+
√

(N +Ri − i+ 1)(N +Rj − j)δT,R+
ij
δt,r+ij

− (2N +Ri +Rj − i− j)δT,Rδt,r .

In terms of these matrix elements we can write (D.1) as

DOR,(r,s)µ1µ2 = −g2
YM

∑
T,(t,u)ν1ν2

∑
i<j

δ~m,~nM
(ij)
sµ1µ2;uν1ν2 ∆R,r;T,t

ij OT,(t,u)ν1ν2 . (D.8)

Although the distant corners approximation has been used to extract the large N value

of M
(ij)
sµ1µ2;uν1ν2 , the action of ∆R,r;T,t

ij is computed exactly. In particular, the coefficients

appearing in (D.7) are simply the factors associated with the boxes that are added or

removed by ∆R,r;T,t
ij , and hence in developing a systematic large N expansion for ∆R,r;T,t

ij

we can trust the shifts of numbers of order N by numbers of order 1.

The limit in which the dilatation operator reduces to sets of decoupled oscillators

corresponds to the limit in which the difference between the row (or column) lengths of

Young diagram R are fixed to be O(
√
N) while the row lengths themselves are order N .

The continuum variables are then

xi =
Ri+1 −Ri√

R1
i = 1, 2, · · · , p− 1 (D.9)

when R has p rows (or columns) and the shortest row (or column) is R1. In this case, the

leading and subleading (order N and order
√
N) contribution to ∆ijOR,(r,s)µ1µ2 vanish,

leaving a contribution of order 1. This contribution is sensitive to the exact form of the

coefficients appearing in (D.7), and it is with these shifts that we reproduce the numerical

results of [29, 30].

E One loop computation of bulk/boundary magnon scattering

In this appendix we will compute the scattering of a bulk and boundary magnon, to one

loop, using the asymptotic Bethe ansatz. See [65] where studies of this type were first

suggested and [66] for related systems. We can introduce a wave function ψ(l1, l2, · · · ) as

follows

O =
∑
l1,l2,···

ψ(l1, l2, · · · )O(R,Rk1 , R
k
2 ; {l1, l2, · · · }) . (E.1)

We assume that the boundary magnon (at l1) and the next magnon along the open string

(at l2) are very well separated from the remaining magnons. These magnons are both

assumed to be Y impurities. To obtain the scattering we want, we only need to focus on

these two magnons. The time independent Schrödinger equation following from our one

loop dilatation operator is

Eψ(l1, l2) =
(

3 +
c

N

)
ψ(l1, l2)−

√
c

N
(ψ(l1 − 1, l2) + ψ(l1 + 1, l2))

− (ψ(l1, l2 − 1) + ψ(l1, l2 + 1)) (E.2)

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where c is the factor of the box that the endpoint associated to the magnon at l1 belongs

to. The equation (E.2) is valid whenever the two magnons are not adjacent in the open

string word, i.e. when l2 > l1 + 1.6 In the situation that the magnons are adjacent, we find

Eψ(l1, l1 + 1) =
(

1 +
c

N

)
ψ(l1, l1 + 1)−

√
c

N
ψ(l1 − 1, l2)− ψ(l1, l1 + 2) . (E.3)

We make the following Bethe ansatz for the wave function

ψ(l1, l2) = eip1l1+ip2l2 +R12 e
ip′1l1+p′2l2 . (E.4)

It is straight forward to see that this ansatz obeys (E.2) as long as

E = 3 +
c

N
−
√

c

N
(eip1 + e−ip1)− (eip2 + e−ip2) (E.5)

and √
c

N
(eip1 + e−ip1) + eip2 + e−ip2 =

√
c

N
(eip

′
1 + e−ip

′
1) + eip

′
2 + e−ip

′
2 . (E.6)

Note that (E.5) is indeed the correct one loop anomalous dimension and (E.6) can be

obtained by equating the O(λ) terms on both sides of (6.1), as it should be. From (E.3)

we can solve for the reflection coefficient R. The result is

R12 = −
2eip2 −

√
c
N e

ip1+ip2 − 1

2eip
′
2 −

√
c
N e

ip′1+ip′2 − 1
(E.7)

Two simple checks of this result are

1. We see that R12R21 = 1.

2. If we set c = N we recover the S-matrix of [65].

We will now move beyond the su(2) sector by considering a state with a single Y

impurity and a single X impurity. The operator with a Y impurity at l1 and an X impurity

at l2 is denoted O(R,Rk1 , R
k
2 ; {l1, l2, · · · })Y X and the operator with an X impurity at l1

and a Y impurity at l2 is denoted O(R,Rk1 , R
k
2 ; {l1, l2, · · · })XY . We now introduce a pair

of wave functions as follows

O =
∑
l1,l2,···

[
ψY X(l1, l2, · · · )O(R,Rk1 , R

k
2 ; {l1, l2, · · · })Y X

+ψXY (l1, l2, · · · )O(R,Rk1 , R
k
2 ; {l1, l2, · · · })XY

]
. (E.8)

From the one loop dilatation operator we find the time independent Schrödinger equa-

tion (E.2) for each wave function, when the impurities are not adjacent. When the impu-

6Notice that we are associating a lattice site to every field in the spin chain and not just to the Zs.

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rities are adjacent, we find the following two time independent Schrödinger equations

EψY X(l1, l1 + 1) =
(

2 +
c

N

)
ψY X(l1, l1 + 1)−

√
c

N
ψY X(l1 − 1, l1 + 1)

− ψXY (l1, l1 + 1)− ψY X(l1, l1 + 2) (E.9)

EψXY (l1, l1 + 1) =
(

2 +
c

N

)
ψXY (l1, l1 + 1)−

√
c

N
ψXY (l1 − 1, l1 + 1)

− ψY X(l1, l1 + 1)− ψXY (l1, l1 + 2) . (E.10)

Making the following Bethe ansatz for the wave function

ψY X(l1, l2) = eip1l1+ip2l2 +Aeip
′
1l1+ip′2l2

ψXY (l1, l2) = Beip
′
1l1+ip′2l2 (E.11)

we find that the two equations of the form (E.2) imply that both ψXY (l1, l2) and ψY X(l1, l2)

have the same energy, which is given in (E.5). The equations (E.9) and (E.10) imply that

A =
eip
′
2 + eip2 − 1−

√
c
N e

ip′1+ip′2

1 +
√

c
N e

ip′1+ip′2 − 2eip
′
2

,

B =
eip2 − eip′2

1 +
√

c
N e

ip′1+ip′2 − 2eip
′
2
. (E.12)

It is straight forward but a bit tedious to check that |A|2 + |B|2 = 1 which is a consequence

of unitarity. To perform this check it is necessary to use the conservation of momentum

p1 + p2 = p′1 + p′2, as well as the constraint (E.6). We now finally obtain

A

R12
=
eip
′
2 + eip2 − 1−

√
c
N e

ip′1+ip′2

2eip2 −
√

c
N e

ip1+ip2 − 1
. (E.13)

This should be equal to
1

2

(
1 +

B12

A12

)
(E.14)

where A12 and B12 are the S-matrix elements computed in section 7, describing the scat-

tering between a bulk and a boundary magnon. This allows us to perform a non-trivial

check of the S-matrix elements we computed.

F No integrability

The (boundary) Yang-Baxter equation makes use of the boundary magnon (B) and two

bulk magnons (1 and 2). For our purposes, it is enough to track only scattering between

bulk and boundary magnons. The Yang-Baxter equation requires equality between the

scattering7 which takes B+1→ B′+1′ and then B′+2→ B̃′+2̃ and the scattering which

7There are some bulk magnon scatterings that we are ignoring as they don’t affect our argument.

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takes B + 2→ B′ + 2′ and then B′ + 1→ B̃′ + 1̃. For the first scattering, given the initial

momenta p1, p2, pB, we need to solve√
1 + 8λ sin2 p1

2
+

√
1 + 8λ

(
(1 + r)2 + 4r sin2 pB

2

)
=

√
1 + 8λ sin2 k1

2
+

√
1 + 8λ

(
(1 + r)2 + 4r sin2 q

2

)
(F.1)√

1 + 8λ sin2 p2

2
+

√
1 + 8λ

(
(1 + r)2 + 4r sin2 q

2

)
=

√
1 + 8λ sin2 k2

2
+

√
1 + 8λ

(
(1 + r)2 + 4r sin2 kB

2

)
(F.2)

for the final momenta k1, k2, kB. For the second scattering we need to solve√
1 + 8λ sin2 p2

2
+

√
1 + 8λ

(
(1 + r)2 + 4r sin2 pB

2

)
=

√
1 + 8λ sin2 l2

2
+

√
1 + 8λ

(
(1 + r)2 + 4r sin2 s

2

)
(F.3)√

1 + 8λ sin2 p1

2
+

√
1 + 8λ

(
(1 + r)2 + 4r sin2 s

2

)
=

√
1 + 8λ sin2 l1

2
+

√
1 + 8λ

(
(1 + r)2 + 4r sin2 lB

2

)
(F.4)

for the final momenta l1, l2, lB. It is simple to check that, in general, k1 6= l1, k2 6= l2 and

kB 6= lB, so the two scatterings can’t possibly be equal.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

References

[1] D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from

N = 4 super Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].

[2] N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99

(2012) 3 [arXiv:1012.3982] [INSPIRE].