Table Of ContentHU-EP-17/03
Yangian Symmetry and Integrability
of Planar N = 4 Super-Yang–Mills Theory
Niklas Beisert1, Aleksander Garus1 and Matteo Rosso2
1Institut fu¨r Theoretische Physik,
7
Eidgen¨ossische Technische Hochschule Zu¨rich,
1
0 Wolfgang-Pauli-Strasse 27, 8093 Zu¨rich, Switzerland
2
{nbeisert,agarus}@itp.phys.ethz.ch
n
a
J 2Institut fu¨r Physik,
1 Humboldt Universit¨at zu Berlin,
3
Zum Grossen Windkanal 6, D-12489 Berlin, Germany
]
h [email protected]
t
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p
e
h
[
1
v
2
6
1
Abstract
9
0
. In this letter we establish Yangian symmetry of planar N = 4 super-
1
0 Yang–Mills theory. We prove that the classical equations of motion
7 of the model close onto themselves under the action of Yangian gen-
1
erators. Moreover we propose an off-shell extension of our statement
:
v
which is equivalent to the invariance of the action and prove that it is
i
X
exactly satisfied. We assert that our relationship serves as a criterion
r
a for integrability in planar gauge theories by explicitly checking that it
applies to integrable ABJM theory but not to non-integrable N = 1
super-Yang–Mills theory.
1 Introduction
The assumption that planar N = 4 super-Yang–Mills (sYM) theory is integrable (see [1]
for a review) has unlocked an enormous body of data – not only perturbatively but also
at strong and intermediate coupling – thanks to advanced methods of integrable systems.
Two prime examples are the finite-coupling computations of the cusp dimension [2] by
meansofanintegralequation[3]andofthescalingdimensionoftheKonishioperator[4]by
means of the thermodynamic Bethe ansatz [5]. Their smooth interpolations between weak
and strong coupling are viewed as strong confirmations of the AdS/CFT correspondence.
While the feature of integrability in this model is now supported by an overwhelming
amount of evidence, it largely remains a conjecture except for certain observables in
certain corners of parameter space. The main obstacle in proving integrability is the lack
of a proper definition for this feature within a planar gauge theory.
A key property of integrable systems is the existence of a large amount of hidden
symmetries. For this model, the relevant algebra has been identified as the Yangian
Y[psu(2,2|4)] [6] which is an infinite-dimensional quantum algebra based on the Lie su-
peralgebra psu(2,2|4) of superconformal symmetries. The original formulation, however,
merely addressed the spectrum of one-loop anomalous dimensions for which Yangian sym-
metry is largely broken due to the pertinent cyclic boundary conditions. Sometime later,
Yangian invariance has been established for colour-ordered scattering amplitudes at tree
level [7]. Unfortunately, at loop level this symmetry is severely affected by infra-red sin-
gularities inherent to scattering of massless particles. Only recently, the Yangian has been
found to be a proper symmetry of the expectation values of certain non-singular Wilson
loops [8]. However, the Yangian has never been shown to be a symmetry of the model
itself.
The goal of the present letter is to fill this gap and to establish Yangian symmetry
for planar N = 4 sYM. The obvious strategy would be to show that the action of the
model is invariant under the Yangian generators. However, there are several difficulties
in this approach: First, the large-N limit must play a decisive role for integrability is
c
clearly restricted to the planar limit. However, it is not a priori evident how to define the
planar limit of the Lagrangian or of the Yangian generators. Another difficulty lies in the
apparent incompatibility of the Yangian with the cyclicity of the Lagrangian. Finally, one
needstodefinehowexactlytheYangiangeneratorsactonthefields. Thisturnsouttobea
rather subtle issue within a gauge theory because symmetries necessarily act non-linearly
in the sense that one field is mapped to a single or to several fields. On the one hand,
non-linearity complicates the analysis, especially in combination with the above issues.
On the other hand, exact non-linear invariance is a very strong statement which typically
extends to the quantum theory unless the symmetry is anomalous. The combination of
these difficulties has arguably been a show-stopper in proving Yangian symmetry of the
model in the past. Here we start with the somewhat more moderate goal to establish
Yangian symmetry for the classical equations of motion of N = 4 sYM. We first show
that the equations of motion close onto themselves under an appropriately defined action
of the Yangian generators. This alone is not sufficient to establish the Yangian as a proper
symmetry of the model. We therefore propose an off-shell extension of the statement and
prove its validity. We claim that our relationship is equivalent to Yangian invariance of
the action, and thus serves as a suitable criterion for integrability in planar gauge theories.
In order to substantiate these claims, we consider similar superconformal gauge theory
models: on the one hand, we show that evidently non-integrable N = 1 sYM theory
2
does not satisfy our relationship. On the other hand, we prove Yangian symmetry of
Aharony–Bergman–Jafferis–Maldacena (ABJM) theory [9] for which signs of integrability
have been found following the work [10].
2 N = 4 super-Yang–Mills
The N = 4 sYM theory in four dimensions consists of the gauge field A , four Weyl
µ
fermions Ψ together with their conjugates Ψ¯a and six real scalar fields Φ . We denote
aα α˙ m
spacetime vector indices by µ,ν,... = 0,...,3 and spinor indices of the two chiralities
˙
by α,β,... = 1,2 and α˙,β,... = 1,2, respectively. Moreover, m,n,... = 1,...,6 and
a,b,... = 1,2,3,4 denote indices for the vector and spinor representations of SO(6),
respectively. All matter fields transform in the adjoint representation of the gauge group.
The covariant derivative of a generic covariant field Z is D Z := ∂ Z −i[A ,Z], and the
µ µ µ
field strength equals F := ∂ A −∂ A −i[A ,A ].
µν µ ν ν µ µ ν
The Lagrangian of the theory takes the form
L = −1 trF Fµν +triΨ¯aσαα˙DµΨ
4 µν α˙ µ aα
− i tr(cid:0)Ψ σab(cid:15)αβ[Φm,Ψ ]−Ψ¯aσm(cid:15)α˙β˙[Φ ,Ψ¯b](cid:1)
2 aα m bβ α˙ ab m β˙
− 1 trDµΦmD Φ + 1 tr[Φm,Φn][Φ ,Φ ]. (1)
2 µ m 4 m n
Here, σµ and σm denote the generalisations of the Pauli matrices to (3+1)D and to 6D,
αα˙ ab
respectively. Furthermore, (cid:15)αβ, (cid:15)α˙β˙ and (cid:15)abcd denote the totally antisymmetric tensors. As
already alluded to in the introduction, we are mainly going to work with the equations of
motion of the theory. To that end, we introduce a short-hand notation for the variation of
the action with respect to a generic field Z (the indices A,B,... enumerate the various
A
fields) Z˘A := δS/δZ , so that the equations of motion simply read Z˘A = 0. We have
A
explicitly that
Ψ˘¯α˙ = iσαα˙DµΨ +i(cid:15)α˙γ˙σm[Φ ,Ψ¯b],
a µ aα ab m γ˙
Ψ˘aα = iσα˙αDµΨ¯a +i(cid:15)αβσab[Φm,Ψ ],
µ α˙ m βb
Φ˘ = D DµΦ +(cid:2)Φ ,[Φn,Φ ](cid:3)+ iσab(cid:15)αβ{Ψ ,Ψ }+ iσ (cid:15)α˙γ˙{Ψ¯a,Ψ¯b},
m µ m n m 2 m aα bβ 2 m,ab α˙ γ˙
A˘ = DνF +i[Φ ,D Φn]+σα˙α{Ψ¯a,Ψ } (2)
µ νµ n µ µ α˙ aα
N = 4 sYM theory is a superconformal theory; its action is invariant under the
four-dimensional extended superconformal algebra psu(2,2|4), and quantum effects do
not spoil such invariance. In the following, we will need the action of the generators of
dilatations D, translations P and Lorentz transformation M on a generic field Z
µ µν
DZ = i(cid:0)xσD +∆ (cid:1)Z,
σ Z
PρZ = iDρZ,
M Z = i(cid:0)x D −x D +Σ (cid:1)Z. (3)
µν µ ν ν µ µν
In the above equations, ∆ is the conformal dimension of the field Z, and Σ is a
Z µν
spin-specific part of the transformation. To make transformations consistent for the non-
covariant gauge field A , we have to make the peculiar definitions ∆ := 0 and D A :=
µ A µ ν
3
F . Supersymmetry generators Qa act as
µν α
QaΦ = σabΨ ,
α m m bα
QaΨ = −1σµ (cid:15)(cid:15)˙γ˙σν F δa + i(cid:15) σmσca[Φ ,Φn],
α bβ 2 β(cid:15)˙ αγ˙ µν b 2 βα bc n m
QaΨ¯b = iσabσµ D Φm,
α γ˙ m αγ˙ µ
QaAµ = iσµ (cid:15)γ˙(cid:15)˙Ψ¯a. (4)
α αγ˙ (cid:15)˙
Analogous expressions hold for the conjugate generator Q¯ , with the roles of Ψ and Ψ¯a
aα˙ aα α˙
interchanged.
3 Yangian invariance of N = 4 sYM
WewanttostudyclassicalYangiansymmetryofplanarN = 4sYM;ordinarily, onewould
show the invariance of the action S. This works well for the superconformal generators
JK at level zero of the Yangian (the indices K,L,... enumerate a basis of the level-zero
algebra psu(2,2|4)), namely JKS = 0. As outlined in the introduction, there are several
difficulties in formulating invariance of the action to higher-level generators. Gladly, most
of them disappear when acting on the equations of motion instead. In the following, we
will show how this can be achieved and how to promote their invariance to a powerful
off-shell statement.
A level-one generator (cid:98)JK has a bilocal contribution determined completely by the
level-zero generators
(cid:98)JK = 1fK JM ∧JN. (5)
biloc 2 MN
Explicitly, the bilocal term acts in the following way on a sequence Z Z ...Z of fields
1 2 n
n
(cid:98)JK (Z ...Z ) = fK (cid:88) Z ...(cid:0)JMZ (cid:1)...(cid:0)JNZ (cid:1)...Z . (6)
biloc 1 n MN 1 i j n
1=i<j
Notice that this is where the planar limit is relevant for Yangian symmetry: Only in
the planar limit, the ordering of fields within a matrix product is universally and unam-
biguously defined because there are no identities between matrix polynomials when N is
c
sufficiently large.
Let us apply the simplest level-one generator P(cid:98)ρ, also known as the dual superconfor-
mal generator [7], to the equations of motion (2). We will work with the easiest of them,
the Dirac equation. The bilocal part of the level-one momentum P(cid:98)ρ takes the form
P(cid:98)ρ = D∧Pρ −Lρ ∧Pµ − iσρ,α˙βQ¯ ∧Qa. (7)
biloc µ 4 aα˙ β
Applying it to the Dirac equation we get
P(cid:98)µ Ψ˘¯α˙ = −i(cid:15)α˙γ˙σm{Φ ,DµΨ¯b}− i(cid:15)α˙γ˙σm{DµΦ ,Ψ¯b}
biloc a ab m γ˙ 2 ab m γ˙
− i(cid:15)α˙γ˙σmσµ σα(cid:15)˙{DνΦ ,Ψ¯b}− iσµ,ρα˙σmσcb(cid:8)Ψ ,[Φ ,Φn](cid:9). (8)
2 ab αγ˙ ν m (cid:15)˙ 2 ac n ρb m
Itisusefultoobservethattheexplicitx-dependenceduetosomeofthebosonicgenerators
in (3) drops out completely. This is related to the triviality of [Pµ,P(cid:98)ρ].
A level-one generator can also have some local contributions, which act on a single
field at a time just like the level-zero generators. These contributions are not determined
4
by level-zero symmetry, and we may adjust them according to our needs. We now ask
whether there is a single-field action of P(cid:98)ρ ensuring the invariance of the equations of
motion. With the following choice for the single-field action of P(cid:98)µ
P(cid:98)µΦ = 0,
m
P(cid:98)µΨ = 1σµ (cid:15)γ˙(cid:15)˙σm{Φ ,Ψ¯b},
aα 2 αγ˙ ab m (cid:15)˙
P(cid:98)µΨ¯a = 1σµ (cid:15)βγσab{Φm,Ψ },
α˙ 2 βα˙ m γb
P(cid:98)µAρ = iηµρ{Φ ,Φm}, (9)
2 m
the combination of local and bilocal terms in the action of P(cid:98) gives
P(cid:98)µΨ˘¯α˙ = −1σµ σm(cid:15)γ˙α˙{Φ ,Ψ˘αb}. (10)
a 2 αγ˙ ba m
The r.h.s. is proportional to Ψ˘ and vanishes on shell. Hence P(cid:98)ρ is an on-shell symmetry
of the N = 4 sYM Dirac equation.
TheinvarianceoftheactionS underasuperconformalgeneratorJK impliesastronger,
off-shell relationship for the equations of motion. To that end, consider the invariance of
the action, JKS = Z˘A(cid:0)JKZ (cid:1) = 0. Now vary this equation with respect to a generic field
A
Z to obtain an equivalent relation which holds off-shell
C
δ(cid:0)JKZ (cid:1)
JKZ˘C = −Z˘A A . (11)
δZ
C
The r.h.s. is now a specific combination of the equations of motion Z˘A given by the action
of the generators JK on the fields Z of the theory. Getting inspiration from the structure
A
of the bilocal term (5) as well as from the level-zero formula (11) we propose an analogous
formula for the level-one Yangian generators (cid:98)JK:
(cid:98)JKZ˘C = −Z˘A δ(cid:0)(cid:98)JKZA(cid:1) +fK Z˘A(cid:18)JM ∧ δ (cid:19)(cid:0)JNZ (cid:1). (12)
δZ MN δZ A
C C
Let us explicitly demonstrate how the two terms on the r.h.s. are to be understood. The
former is the direct counterpart of the r.h.s. of (11). Assuming that, e.g., (cid:98)JKZ = Z Z ,
A 1 2
it evaluates to
−Z˘A δ(cid:0)(cid:98)JKZA(cid:1) = −(cid:2)δCZ˘AZ +δCZ Z˘A(cid:3). (13)
δZ 1 2 2 1
C
Concerning the second term, we first observe that it vanishes for the linear contribution
of JN acting on Z . Assuming that JNZ = Z Z , it evaluates to
A A 1 2
(cid:18) (cid:19)
δ
fK Z˘A JM ∧ (cid:0)JNZ (cid:1) = fK (cid:2)δC(cid:0)JMZ (cid:1)Z˘A −δCZ˘A(cid:0)JMZ (cid:1)(cid:3). (14)
MN δZ A MN 2 1 1 2
C
It is now a straightforward exercise to verify that the formula (12) indeed reproduces
(10) correctly. Similar checks can also be made for all the other equations of motion.
As the supersymmetry generators Qa and Q¯ map the Dirac equation to the remaining
α aα˙
equations of motion, already the algebraic relations guarantee their invariance under P(cid:98)ρ.
Mostimportantly, theyalsoimplyinvarianceundertheremaininginfinitelymanyYangian
generators.
We conclude that the relationship (12) holds in classical planar N = 4 sYM theory.
Furthermore, we will demonstrate in [11], that it can be lifted to an invariance statement
for the action of N = 4 sYM. In that sense, classical planar N = 4 sYM is invariant
under Yangian symmetry.
5
4 Correlation Functions
The novel symmetry relationship (12) should have implications for observables. A fun-
damental class of observables in a quantum field theory is given by correlators of fields.
Although not gauge-invariant on their own, they are building blocks for observables like
Wilson loops, scattering amplitudes (via the LSZ reduction), correlators of local oper-
ators or form factors. An ordinary symmetry results in Ward–Takahashi identities for
correlators. The goal is thus to derive Ward–Takahashi identities for level-one Yangian
symmetries, and subsequently use them to prove symmetries of the aforementioned gauge-
invariant observables. In the following we shall briefly sketch the level-one symmetry for
planar correlators of the fields; a detailed treatment can be found in the follow-up pa-
per [11].
In order to set up the quantum theory, we should first fix the gauge and thus make
correlation functions of the fields well-defined observables. This step can potentially break
symmetries, but gladly it leaves the level-one Yangian symmetry intact: We show in [11]
that there exists a (trivial) extension of the symmetry representation on the Faddeev–
Popov ghosts such that the invariance statement (12) continues to hold. Note that it
must be supplemented by suitable BRST-exact terms which are irrelevant for physical
observables.
Ward–Takahashi identities for Yangian symmetry are obtained by acting with the
non-linear level-one momentum P(cid:98) on some collection of fields. Symmetry implies that
the planar quantum expectation value of the resulting expression vanishes. This amounts
to some extra differential constraints on planar correlation functions. Note, however, the
following few caveats: In a gauge-fixed theory, one should expect some residual terms
due to the unphysical modes which may violate the symmetry. At leading order, these
should take the form of total derivatives of the external fields. Furthermore, the resulting
identity can receive divergences at loop level. If the level-one symmetry can be renor-
malised appropriately to make the identity hold at higher orders in perturbation theory,
it is a proper quantum symmetry. Otherwise, the level-one Yangian symmetry would be
anomalous.
In [11] we demonstrate that (discarding issues related to gauge fixing) the level-one
relationship (12) combined with superconformal symmetry results in additional Ward–
Takahashi identities at tree level at least up to four external points. Therefore Yangian
symmetry indeed has concrete implications for planar correlators.
5 Other superconformal theories
It is tempting to consider the relationship eq. (12) as a criterion for integrability in other
planar gauge theories. The criterion can be applied to all models whose global symmetries
form a semi-simple Lie (super)algebra. In the following we support our proposal by
showing that the criterion agrees with our expectations for two sample gauge theories
with classical superconformal symmetry.
First we consider N = 1 pure sYM theory in four dimensions which is non-integrable.
This theory is classically superconformal, its global symmetry algebra being su(2,2|1); it
contains a single vector field and a single Weyl fermion. The Lagrangian of this theory
can be obtained from N = 4 sYM by dropping all the scalar fields and retaining one
6
single Weyl fermion; it reads
L = −1 trF Fµν +triΨ¯ σα˙βDµΨ . (15)
4 µν α˙ µ β
Similarly, the action of the superconformal generators can be obtained from the corre-
sponding expressions for N = 4 sYM, eq. (3,4).
As before, we can check our criterion for Yangian symmetry using the level-one gen-
erator P(cid:98)ρ of Y[su(2,2|1)] acting on the Dirac equation of this theory. Considering the
quantum numbers, one finds no admissible terms for the single-field action of P(cid:98)ρ,
P(cid:98)ρAµ = P(cid:98)ρΨα = P(cid:98)ρΨ¯β˙ = 0. (16)
The computation of the l.h.s. of eq. (12) in this case yields a term proportional to
¯
{Ψ ,F }, whereas the r.h.s. vanishes identically. Our criterion therefore states that
α˙ µν
N = 1 sYM does not possess Yangian symmetry, in accordance with the fact that this
theory, as a whole, is not integrable. Importantly, this demonstrates that it is not merely
an artefact due to other properties such as gauge or superconformal symmetry.
The second example we study is the so-called ABJM theory, a Chern–Simons-matter
theory with gauge group U(N )×U(N ). The matter sector consists of four chiral multi-
c c
plets Φa, Ψ transforming in the bifundamental of the gauge groups, together with their
aα
conjugate multiplets Φ¯ , Ψ¯a transforming in the conjugate bifundamental; the gauge fields
a α
˜
associated with the two components of the gauge group are A and A . ABJM is a super-
µ µ
conformal theory with N = 6 supersymmetry, its global symmetry algebra being osp(6|4).
Similarly to N = 4 sYM, this theory appears to be exactly integrable in the planar limit,
andalsointhiscaseitsintegrabilityisencodedintheexistenceofaYangiansymmetry[12]
based on its superconformal algebra.
InordertostudytheYangiansymmetryofplanarABJM,westartfromtheLagrangian
L = L −L +Ψ¯aαD γµ Ψβ +D ΦaDµΦ¯ +.... (17)
CS C(cid:102)S µ αβ a µ a
Here L and L are the Chern–Simons terms, and we omitted the Yukawa couplings
CS C(cid:102)S
and scalar potential.
We want to check whether eq. (12) holds for the Dirac equation of ABJM
Ψ˘¯ = −γµ (cid:15)γβD Ψ +Ψ Φ¯ Φb −Φ Φ¯bΨ
aα αγ µ aβ aα b b aα
+2ΦbΦ¯ Ψ −2Ψ Φ¯ Φb +2(cid:15) ΦbΨ¯cΦd. (18)
a bα bα a abcd α
As before, we consider the level-one generator P(cid:98)ρ of Y[osp(6|4)], whose bilocal part is of
the same form described in eq. (7),
P(cid:98)ρ = D∧Pρ −Lρ ∧Pµ + 1 (cid:15)abcdγρ,αβQ ∧Q . (19)
biloc µ 16 abα cdβ
In order to compute the action of P(cid:98)ρ on Ψ˘¯ , we need the action of the supercharges on
aα
the fields
Q Φc = δcΨ −δcΨ ,
abα a bα b aα
Q Ψ = −(cid:15) γµ D Φd −2(cid:15) (cid:15) ΦdΦ¯ Φe −(cid:15) (cid:15) (cid:0)ΦdΦ¯ Φe −ΦeΦ¯ Φd(cid:1), (20)
abα cβ abcd αβ µ αβ abde c αβ abcd e e
Q Aµ = 2γµ (cid:15)βγ(cid:0)Ψ Φ¯ −Ψ Φ¯ +(cid:15) ΦcΨ¯d(cid:1),
abα αβ bγ a aγ b abcd γ
7
and similarly for Φ¯ , Ψ¯a and A˜ . We can now show that eq. (12) holds for P(cid:98)ρ acting on
a α µ
Ψ˘¯ , provided that the single-field action of P(cid:98)ρ is
aα
P(cid:98)ρΦa = 0,
P(cid:98)ρΨ = γρ (cid:15)βγ(cid:2)Ψ Φ¯ Φb +ΦbΦ¯ Ψ −2Ψ Φ¯ Φb −2ΦbΦ¯ Ψ (cid:3),
aα αβ aγ b b aγ bγ a a bγ
P(cid:98)ρAµ = 1(cid:15)ρµν(cid:0)D ΦaΦ¯ +ΦaD Φ¯ (cid:1)+2ηµρ(cid:0)ΦaΦ¯ ΦbΦ¯ −(cid:15)αβΨ Ψ¯a(cid:1), (21)
2 ν a ν a b a aα β
andsimilarlyforΦ¯ ,Ψ¯a andA˜ . Itissimilarlypossibletoshowthattheoff-shellinvariance
a γ µ
conditioneq.(12)holdsforalltheequationsofmotionofABJM,andthatthereforeplanar
ABJM is classically Yangian invariant. However, the compatibility with gauge fixing, cf.
the comments in the previous section, is subtler than for N = 4 sYM. At the moment we
do not understand how to compensate terms due to the non-linear action eq. (20) of the
supercharges on the gauge fields. We will return to this issue in [11].
6 Comments and conclusions
In this letter we showed that the equations of motion of both N = 4 super-Yang–Mills
and ABJM – two superconformal field theories which are apparently integrable in the
planar limit – are invariant under the Yangian of the relevant superconformal algebra.
Moreover, we derived an off-shell relationship (12) which serves as a formal statement of
invariance of the classical theory. It can therefore be seen as a criterion for integrability
in planar gauge theories.
In a companion paper [11] we will provide a more detailed account of our claims and
elaborate on some more advanced aspects which we can merely touch upon in the present
letter: Starting from (12) we will derive a definition for Yangian invariance of the action of
a planar gauge theory. Furthermore, we will show how Yangian symmetry survives gauge
fixing. Finally, we will establish novel Ward–Takahashi identities for planar correlators of
fields.
An important next goal is to analyse the validity of Yangian symmetry at the quantum
level. In other words, do the Ward–Takahashi identities continue to hold, at least at the
level of loop integrands? And does renormalisation and performing the loop integrals
render the symmetry anomalous? Our strong expectation is that the symmetry will hold
exactly at all values of the coupling constant because we know that integrability leads to
consistentresultsatweak,intermediateandstrongcoupling,see[1–5]. Notethat,likewise,
the AdS/CFT dual string theory model is integrable [13]. However, it is to be expected
that particular observables, e.g. Wilson loops with cusps, will break Yangian symmetry
to some extent. Therefore, it will be most important to apply our definition of Yangian
symmetry to observables and to understand where and how it is broken concretely.
Acknowledgements
The authors would like to thank N. Drukker and J. Plefka for interesting discussions.
The work of NB and AG is partially supported by grant no. 615203 from the European
Research Council under the FP7 and by the Swiss National Science Foundation through
the NCCR SwissMAP. The work of MR is supported by the grant PL 457/3-1 “Yangian
Symmetry in Quantum Gauge Field Theory” of the German Research Foundation.
8
References
[1] N. Beisert et al., “Review of AdS/CFT Integrability: An Overview”,
Lett. Math. Phys. 99, 3 (2012), arxiv:1012.3982.
[2] M. K. Benna, S. Benvenuti, I. R. Klebanov and A. Scardicchio, “A Test of the AdS/CFT
correspondence using high-spin operators”, Phys. Rev. Lett. 98, 131603 (2007),
hep-th/0611135.
[3] N. Beisert, B. Eden and M. Staudacher, “Transcendentality and Crossing”,
J. Stat. Mech. 0701, P01021 (2007), hep-th/0610251.
[4] N. Gromov, V. Kazakov and P. Vieira, “Exact Spectrum of Planar N = 4
Supersymmetric Yang-Mills Theory: Konishi Dimension at Any Coupling”,
Phys. Rev. Lett. 104, 211601 (2010), arxiv:0906.4240.
[5] N. Gromov, V. Kazakov and P. Vieira, “Exact Spectrum of Anomalous Dimensions of
Planar N = 4 Supersymmetric Yang-Mills Theory”, Phys. Rev. Lett. 103, 131601 (2009),
arxiv:0901.3753. • D. Bombardelli, D. Fioravanti and R. Tateo, “Thermodynamic Bethe
Ansatz for planar AdS/CFT: A Proposal”, J. Phys. A42, 375401 (2009),
arxiv:0902.3930. • G. Arutyunov and S. Frolov, “Thermodynamic Bethe Ansatz for the
AdS × S5 Mirror Model”, JHEP 0905, 068 (2009), arxiv:0903.0141.
5
[6] L. Dolan, C. R. Nappi and E. Witten, “A Relation between approaches to integrability in
superconformal Yang-Mills theory”, JHEP 0310, 017 (2003), hep-th/0308089.
[7] J. M. Drummond, J. M. Henn and J. Plefka, “Yangian symmetry of scattering amplitudes
in N = 4 super Yang-Mills theory”, JHEP 0905, 046 (2009), arxiv:0902.2987.
[8] D. Mu¨ller, H. Mu¨nkler, J. Plefka, J. Pollok and K. Zarembo, “Yangian Symmetry of
smooth Wilson Loops in N = 4 super Yang-Mills Theory”, JHEP 1311, 081 (2013),
arxiv:1309.1676.
[9] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, “N = 6 superconformal
Chern-Simons-matter theories, M2-branes and their gravity duals”,
JHEP 0810, 091 (2008), arxiv:0806.1218.
[10] J. A. Minahan and K. Zarembo, “The Bethe ansatz for superconformal Chern-Simons”,
JHEP 0809, 040 (2008), arxiv:0806.3951.
[11] N. Beisert, A. Garus and M. Rosso, “Yangian Symmetry and its Ward Identities in
Planar Gauge Theories”, to appear.
[12] T. Bargheer, F. Loebbert and C. Meneghelli, “Symmetries of Tree-level Scattering
Amplitudes in N = 6 Superconformal Chern-Simons Theory”,
Phys. Rev. D82, 045016 (2010), arxiv:1003.6120.
[13] I. Bena, J. Polchinski and R. Roiban, “Hidden symmetries of the AdS × S5 superstring”,
5
Phys. Rev. D69, 046002 (2004), hep-th/0305116.
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