The S-Matrix of AdS/CFT and Yangian Symmetry
arXiv:0704.0400v4  [nlin.SI]  27 Mar 2008
arxiv:0704.0400
AEI-2007-019
The S-Matrix of AdS/CFT
and Yangian Symmetry
Niklas Beisert
Max-Planck-Institut für Gravitationsphysik
Albert-Einstein-Institut
Am Mühlenberg 1, 14476 Potsdam, Germany
nbeisert@aei.mpg.de
Abstract
We review the algebraic construction of the S-matrix of AdS/CFT. We
also present its symmetry algebra which turns out to be a Yangian of
the centrally extended su(2|2) superalgebra.
1 Introduction and Overview
Bethe’s ansatz [1] for solving a one-dimensional integrable model was and remains a
powerful tool in contemporary theoretical physics: 75 years ago it solved one of the
first models of quantum mechanics, the Heisenberg spin chain [2]; today it provides
exact solutions for the spectra of certain gauge and string theories and thus helps us
understand their duality [3] better. Since the discovery of integrable structures in planar
N = 4 supersymmetric gauge theory [4] and in planar IIB string theory on AdS5×S5 [5]
the tools for computing and comparing the spectra of both models have evolved rapidly.
We now have complete asymptotic Bethe equations [6, 7] which interpolate smoothly
between the perturbative regimes in gauge and string theory and which agree with all
available data.
In this note we will focus on the S-matrix [8] in the excitation picture above a ferro-
magnetic ground state. We start by reviewing the algebraic construction of the S-matrix
in Sec. 2. In Sec. 3 we subsequently show that this S-matrix has indeed a larger symmetry
algebra: a Yangian.
http://arxiv.org/abs/0704.0400v4
2 The Universal Enveloping Algebra U(su(2|2)⋉R2)
In this section the results on the S-matrix of AdS/CFT shall be reviewed from an al-
gebraic point of view. The applicable symmetry is a central extension h of the Lie
superalgebra su(2|2) which we consider first. We continue by presenting the Hopf al-
gebra structure of its universal enveloping algebra and its fundamental representation.
Finally, we comment on the S-matrix and its dressing phase factor.
Lie Superalgebra. The symmetry in the excitation picture for light cone string theory
on AdS5×S5 and for single-trace local operators in N = 4 supersymmetric gauge theory
is given by two copies of the Lie superalgebra [9, 10]
h := su(2|2)⋉ R2 = psu(2|2)⋉ R3. (2.1)
It is a central extension of the standard Lie superalgebras su(2|2) or psu(2|2), see [11].
It is generated by the su(2)× su(2) generators Rab, Lαβ, the supercharges Qαb, Saβ and
the central charges C, P, K. The Lie brackets of the su(2) generators take the standard
[Rab,R
d] = δ
d − δadRcb, [Lαβ ,Lγδ] = δ
Lαδ − δαδ Lγβ,
[Rab,Q
d] = −δadQγb + 12δ
d, [L
d] = +δ
d − 12δ
[Rab,S
δ] = +δ
δ − 12δ
δ, [L
δ] = −δαδ Scβ + 12δ
δ. (2.2)
The Lie brackets of two supercharges yield
{Qαb,Scδ} = δcbLαδ + δαδ Rcb + δcbδαδ C,
{Qαb,Qγd} = εαγεbdP,
{Saβ,Scδ} = εacεβδK. (2.3)
The remaining Lie brackets vanish.
Where appropriate, we shall use the collective symbol JA for the generators. The Lie
brackets then take the standard form
[JA, JB] = fABC J
C . (2.4)
For simplicity of notation, we shall pretend that all generators are bosonic; the general-
isation to fermionic generators by insertion of suitable signs and graded commutators is
straightforward.
Hopf Algebra. Next we consider the universal enveloping algebra U(h) of h. The
construction of the product is standard, and one identifies the Lie brackets (2.4) with
graded commutators. For the coproduct one can introduce a non-trivial braiding [12,13]
∆JA = JA ⊗ 1 + U [A] ⊗ JA (2.5)
∆Rab = R
b ⊗ 1 + 1⊗Rab,
∆Lαβ = L
β ⊗ 1 + 1⊗ Lαβ,
∆Qαb = Q
b ⊗ 1 + U+1 ⊗Qαb,
∆Saβ = S
β ⊗ 1 + U−1 ⊗Saβ,
∆C = C⊗ 1 + 1⊗ C,
∆P = P⊗ 1 + U+2 ⊗P,
∆K = K⊗ 1 + U−2 ⊗ K,
∆U = U ⊗ U .
Table 1: The coproduct of the braided universal enveloping algebra U(h).
with some abelian1 generator U (a priori unrelated to the algebra) and the grading
[R] = [L] = [C] = 0, [Q] = +1, [S] = −1, [P] = +2, [K] = −2. (2.6)
The coproduct is spelt out in Tab. 1 for the individual generators. The above grading
is derived from the Cartan charge of the sl(2) automorphism [11] of the algebra h and
therefore the coproduct is compatible with the algebra relations.
We should define the remaining structures of the Hopf algebra: the antipode S and
the counit ε [12,13]. The antipode is an anti-homomorphism which acts on the generators
S(1) = 1, S(U) = U−1, S(JA) = −U−[A]JA. (2.7)
The counit acts non-trivially only on 1 and U
ε(1) = ε(U) = 1, ε(JA) = 0. (2.8)
Cocommutativity. This coproduct is in general not quasi-cocommutative as can eas-
ily be seen by considering the central charges P, K in Tab. 1. To make it quasi-cocommu-
tative we have to satisfy the constraints [12]
1− U+2
1− U+2
⊗P, K⊗
1− U−2
1− U−2
⊗ K. (2.9)
They are solved by identifying the central charges P, K with the braiding factor U as
follows [13]
P = gα
1− U+2
, K = gα−1
1− U−2
. (2.10)
This leads to the following quadratic constraint
PK− gα−1P− gαK = 0. (2.11)
It was furthermore shown in [14] that the coproduct is quasi-triangular, at least at the
level of central charges, see also [15].
1Curiously, we can include the supersymmetric grading (−1)F in the generator U to manually impose
the correct statistics. This is helpful for an implementation within a computer algebra system. In this
case U would anticommute with fermionic generators.
Fundamental Representation. The algebra h has a four-dimensional representation
[10] which we will call fundamental. The corresponding multiplet has two bosonic states
|φa〉 and two fermionic states |ψα〉. The action of the two sets of su(2) generators has to
be canonical
Rab|φc〉 = δcb |φa〉 − 12δ
b |φc〉,
Lαβ|ψγ〉 = δγβ |ψα〉 − 12δ
β |ψγ〉. (2.12)
The supersymmetry generators must also act in a manifestly su(2)×su(2) covariant way
Qαa|φb〉 = a δba|ψα〉,
Qαa|ψβ〉 = b εαβεab|φb〉,
Saα|φb〉 = c εabεαβ|ψβ〉,
Saα|ψβ〉 = d δβα|φa〉. (2.13)
We can write the four parameters a, b, c, d using the parameters x±, γ and the constants
g, α as
g γ, b =
, c =
, d =
. (2.14)
The parameters x± (together with γ) label the representation and they must obey the
constraint
− x− − 1
. (2.15)
The three central charges C,P,K and U are represented by the values C, P,K and U
which read
1 + 1/x+x−
1− 1/x+x− , P = gα
, K =
, U =
. (2.16)
They furthermore obey the quadratic relation C2−PK = 1
. Note that the corresponding
quadratic combination of central charges C2−PK is singled out by being invariant under
the sl(2) external automorphism.
Fundamental S-Matrix. In [10,14] an S-matrix acting on the tensor product of two
fundamental representations was derived. It was constructed by imposing invariance
under the algebra h
[∆JA,S] = 0. (2.17)
We will not reproduce the result here, it is given in [14]. Note that we have to fix the
parameters ξ = U =
x+/x− in order to make the action of the generators compatible
with the coproduct (2.5).2
2This identification removes all braiding factors from the S-matrix in [14] which will thus satisfy the
standard Yang-Baxter (matrix) equation, see also [10, 16, 17].
This S-matrix has several interesting properties. Firstly, it is not of difference form;
it cannot be written as a function of the difference of some spectral parameters. Sec-
ondly, the S-matrix could be determined uniquely up to one overall function merely by
imposing a Lie-type symmetry (2.17) [10]. This unusual fact is related to an unusual
feature of representation theory of the algebra h: The tensor product of two fundamental
representations is irreducible in almost all cases [14].
Intriguingly this S-matrix is equivalent to Shastry’s R-matrix [18] of the one-dimen-
sional Hubbard model [19]. Furthermore the Bethe equations [10] contain two copies of
the Lieb-Wu equations for the Hubbard model [20]. These observations of [14] estab-
lish a link between an important model of condensed matter physics and string theory
(complementary to the one in [21]).
Finally, let us note that one can derive (asymptotic) Bethe equations from the S-
matrix and thus confirm the conjecture in [6]. So far this step has been performed in
two different ways: by means of the nested coordinate [10] and the algebraic [17] Bethe
ansatz.
Phase Factor. The remaining overall phase factor of the S-matrix clearly cannot be
determined by demanding invariance under h. The phase factor was computed to some
approximation from gauge theory [22] and from string theory [23]. The problem of an
algebraically undetermined phase factor is in fact generic. Usually one imposes a further
crossing symmetry relation to obtain a constraint on it. Indeed the known string phase
factor is consistent with crossing symmetry [24] as was shown in [25]. By substituting a
suitable ansatz [26] for the phase factor into the crossing symmetry relation a conjecture
for the all-orders phase factor at strong coupling was made in [27].
A corresponding all-orders expansion at weak coupling was presented in [7]. The
latter conjecture was obtained by a sort of analytic continuation in the perturbative
order of the series. Let us illustrate this principle by means of a very simple example:
Consider the rational function f(x) = 1/(1−x). It has the following expansions at x = 0
and at x = ∞
n, f(x)
−n (2.18)
with an = 1 and bn = −1. When we consider an and bn as analytic functions of the
index, we can make the observation (“reciprocity”)
an = −b−n. (2.19)
Of course there are various ways in which the two functions +1 and −1 could be related,
but the choice (2.19) appears to work for a surprisingly large class of functions!3 It was
proved in [30] that it does apply for the conjectured expansion of the phase factor. Very
useful integral expressions for the phase have recently appeared in [31]. The analytic
expression of the dressing phase can formally be obtained from the psu(2, 2|4) Bethe
3Among other physical examples, we have identified circular Maldacena-Wilson loops [28] and non-
critical string theory [29] where this reciprocity can be applied. Furthermore, summation by the Euler-
MacLaurin formula (also known as zeta-function regularisation) is consistent with it. I thank Curt
Callan, Marcos Mariño and Tristan McLoughlin for discussions of this principle.
equations [32] (see however appendix D in [33]) in analogy to the covariant approach
of [34, 21, 35]. While this proposal may seem to be encouraging in general, it is at the
same time strange from the Hopf algebra point of view to use an S-matrix which does
not obey the crossing relation [32]. This calls for further investigations.
Several tests of the phase have recently appeared, they are based on four-loop unitary
scattering methods [36], numerical evaluation [37, 38], analytic methods [37, 30, 39] and
on taking a certain highly non-trivial limit [40].
3 The Yangian Y(su(2|2) ⋉ R2)
In the section we investigate Yangian symmetry [41,42] for the above S-matrix. We will
start with a very brief review of Yangian symmetry for generic S-matrices (see [43] for
more extensive reviews), and then we apply the framework to the S-matrix discussed
above.
Yangians and S-Matrices. Typically the symmetries of rational S-matrices are of
Yangian type. The Yangian Y(g) of a Lie algebra g is a deformation of the universal
enveloping algebra of half the affine extension of g.
More plainly, it is generated by the g-generators JA and the Yangian generators ĴA.
Their commutators take the generic form
[JA, JB] = fABC J
[JA, ĴB] = fABC Ĵ
C , (3.1)
and they should obey the Jacobi and Serre relations
J[A, [JB, JC]]
J[A, [JB, ĴC]]
Ĵ[A, [ĴB, JC]]
2fAGD f
F fGHKJ
{DJEJF}. (3.2)
The symbol fABC = gADgBEf
C represents the structure constants f
C with two indices
lowered by means of the inverse of the Cartan-Killing forms gAD and gBE . The brackets
{ } and [ ] at the level of indices imply total symmetrisation and anti-symmetrisation,
respectively. Finally, ~ is a scale parameter whose value plays no physical role. The first
two relations lead to a constraint on the structure constants fABC . The third relation
a deformation of the Serre relation for affine extensions of Lie algebras.
The Yangian is a Hopf algebra and the coproduct takes the standard form
∆JA = JA ⊗ 1 + 1⊗ JA,
∆ĴA = ĴA ⊗ 1 + 1⊗ ĴA + 1
~fABCJ
B ⊗ JC . (3.3)
where fABC = gBDf
C . The antipode S is defined by
S(JA) = −JA, S(ĴA) = −ĴA + 1
~fABCf
D, (3.4)
4For g = su(2) it has to be replaced by a quartic relation.
and the counit ε takes the standard form
ε(1) = 1, ε(JA) = ε(ĴA) = 0. (3.5)
For the study of integrable systems, the evaluation representations of the Yangian
are of special interest. For these the action of the Yangian generators ĴA is proportional
to the Lie generators
ĴA|u〉 = ~uJA|u〉. (3.6)
Here |u〉 is some state of the evaluation module with spectral parameter u. This Yangian
representation is finite-dimensional if the g-representation is. One merely has to ensure
that the Serre relation (3.2) is satisfied. This is indeed not the case for all representations
of all Lie algebras. The power of the Yangian symmetry lies in the fact that tensor
products of evaluation representations are typically irreducible (except for special values
of their spectral parameters). This allows for simple proofs (e.g. for the Yang-Baxter
relation) by representation theory arguments.
Let us finally consider the connection to the S-matrix. The S-matrix is a permutation
operator; it acts by interchanging two modules of the algebra
S : V1 ⊗ V2 → V2 ⊗ V1. (3.7)
In particular, for the tensor product of two evaluation modules one has
S|u1, u2〉 ∼ |u2, u1〉. (3.8)
Invariance of the S-matrix under the Yangian means
[∆JA,S] = [∆ĴA,S] = 0 (3.9)
for all generators JA, ĴA. The existence of such an S-matrix is equivalent to quasi-
cocommutativity of Y(g). Note that only the difference of spectral parameters appears in
the invariance condition: We can write the action of the coproduct of Yangian generators
on the evaluation module |u1, u2〉 as
∆ĴA ≃ (u1 − u2)JA ⊗ 1 + u2∆JA + ~fABCJB ⊗ JC . (3.10)
Here the first equation in (3.9) ensures that the term proportional to u2 drops out from
the second equation. Therefore the S-matrix typically depends on the difference u1 − u2
of spectral parameters only.
Yangians in AdS/CFT. Yangian symmetries for planar AdS/CFT have been inves-
tigated in [44], both for classical string theory and for gauge theory at leading order,
see also [45] Yangian symmetry also persists to higher perturbative orders in both mod-
els [22, 46] and it is likely that it also exists at finite coupling. This Yangian can be
understood as a symmetry of the Hamiltonian on an infinite world sheet or as an expan-
sion of the full monodromy matrix. The Lie symmetry in this picture is psu(2, 2|4) and
the Yangian would be Y(psu(2, 2|4)).
Here we consider a different picture of well-separated excitations on a ferromagnetic
ground state and of their scattering matrix. In this picture the Lie symmetry reduces to
two copies of h and the corresponding Yangian would be Y(h). Our Yangian should arise
as a subalgebra of the full Yangian Y(psu(2, 2|4)) when acting on asymptotic excitation
states.
Hopf Algebra. Let us now consider Y(h). We have already studied the universal
enveloping algebra U(h). All we still need to do is to introduce one generator ĴA for each
JA obeying the relations (3.1,3.2), and define its coproduct, antipode as well as counit.
In (2.5) we have defined a braided coproduct for the universal enveloping algebra.
For consistency with the Serre relations, we also have to apply an analogous braiding to
the standard Yangian coproduct
∆ĴA = ĴA ⊗ 1 + U [A] ⊗ ĴA + ~fABCJBU [C] ⊗ JC . (3.11)
Note that lowering an index requires to use the inverse Cartan-Killing form of the algebra.
In the case of h the Cartan-Killing form is degenerate and we need to extend the algebra
by the sl(2) outer automorphism, see [14]. Effectively, lowering an index leads to an
interchange of the automorphism generators with the central charges. We refrain from
spelling out the Cartan-Killing form or the modified structure constants. Instead we
present the complete set of coproducts of Yangian generators in Tab. 2, where we also
fix the value of ~.
For the sake of completeness we state the antipode5 and the counit
S(ĴA) = −U−[A]ĴA, ε(ĴA) = 0. (3.12)
Cocommutativity. An important question is if this coproduct can be quasi-cocom-
mutative.6 A first step is to consider the central generators Ĉ, P̂, K̂. For that purpose
it is favourable to choose suitable combinations
Ĉ′ = Ĉ+ 1
gα−1P− 1
P̂′ = P̂+ C
P− 2gα
K̂′ = K̂− C
K− 2gα−1
, (3.13)
for whom the coproduct almost trivialises
∆Ĉ′ = Ĉ′ ⊗ 1 + 1⊗ Ĉ′,
∆P̂′ = P̂′ ⊗ 1 + U+2 ⊗ P̂′,
∆K̂′ = K̂′ ⊗ 1 + U−2 ⊗ K̂′. (3.14)
The combination Ĉ′ is already cocommutative, and in order to make the generators P̂′,
K̂′ cocommutative we have to set as above in (2.9,2.10)
P̂′ = iguPP, K̂
′ = iguKK (3.15)
with two universal constants uP and uK. With this choice, Ĉ, P̂, K̂ also become cocom-
mutative because they differ from Ĉ′, P̂′, K̂′ only by central elements.
5Note that fA
= 0 here, so there is no contribution from the Lie generators.
6The braiding factors in (3.11) turn out to be very important for the Yangian. It can easily be
seen that without them the coproduct cannot be quasi-cocommutative. This is in contradistinction
to the universal enveloping algebra where the braided as well as the unbraided coproduct are quasi-
cocommutative.
∆R̂ab = R̂
b ⊗ 1 + 1⊗ R̂ab
Rac ⊗Rcb − 12R
b ⊗Rac
SaγU+1 ⊗Qγb − 12Q
bU−1 ⊗Saγ
δab S
γU+1 ⊗Qγd + 14δ
dU−1 ⊗Sdγ ,
∆L̂αβ = L̂
β ⊗ 1 + 1⊗ L̂αβ
Lαγ ⊗ Lγβ + 12L
β ⊗ Lαγ
QαcU−1 ⊗Scβ + 12S
βU+1 ⊗Qαc
δαβ Q
cU−1 ⊗Scδ − 14δ
δU+1 ⊗Qδc,
∆Q̂αb = Q̂
b ⊗ 1 + U+1 ⊗ Q̂αb
LαγU+1 ⊗Qγb + 12Q
b ⊗ Lαγ
RcbU+1 ⊗Qαc + 12Q
c ⊗Rcb
CU+1 ⊗Qαb + 12Q
b ⊗ C
εαγεbdPU−1 ⊗Sdγ − 12ε
αγεbdS
γU+2 ⊗P,
∆Ŝaβ = Ŝ
β ⊗ 1 + U−1 ⊗ Ŝaβ
RacU−1 ⊗Scβ − 12S
β ⊗Rac
LγβU−1 ⊗Saγ − 12S
γ ⊗ Lγβ
CU−1 ⊗Saβ − 12S
β ⊗ C
εacεβδKU+1 ⊗Qδc + 12ε
acεβδQ
cU−2 ⊗ K,
∆Ĉ = Ĉ⊗ 1 + 1⊗ Ĉ
PU−2 ⊗ K− 1
KU+2 ⊗P,
∆P̂ = P̂⊗ 1 + U+2 ⊗ P̂
− CU+2 ⊗P+P⊗ C,
∆K̂ = K̂⊗ 1 + U−2 ⊗ K̂
+ CU−2 ⊗ K− K⊗ C.
Table 2: The coproduct of the Yangian generators in Y(h).
Fundamental Evaluation Representation. For the fundamental evaluation repre-
sentation we make the ansatz7
ĴA|X 〉 = ig(u+ u0)JA|X 〉. (3.16)
By comparison with (3.13,3.15) we can infer that u has to be related to the parameters
of the fundamental representation by
u = x+ +
= x− +
(x+ + x−)(1 + 1/x+x−) . (3.17)
Furthermore uP and uK in (3.15) have to both coincide with the universal constant
u0 = uP = uK.
As an aside we state the eigenvalue of the quadratic combination
CĈ − 1
PK̂ − 1
KP̂ = 1
ig(u+ u0). (3.18)
Fundamental S-Matrix. Using the coproducts in Tab. 2 we have confirmed that the
S-matrix is also invariant under all of the Yangian generators
[∆ĴA,S] = 0. (3.19)
We have used a computer algebra system to evaluate the action of the Yangian gener-
ators and the S-matrix.9 To show invariance requires heavy use of the identity (2.15).
Superficially it is very surprising to find all these additional symmetries of the S-matrix.
The deeper reason however should be that the coproduct is quasi-cocommutative. We
have thus proved quasi-cocommutativity when acting on fundamental representations.
It is interesting to see that the S-matrix is based on standard evaluation represen-
tations of the Yangian. Nevertheless, it is not a function of the difference of spectral
parameters. This unusual property traces back to the link between the spectral param-
eter u and the h-representation parameters x± in (3.17). The latter is again related to
the braiding in the coproduct (3.11).
As our S-matrix is equivalent [14] to Shastry’s R-matrix, our Yangian is presumably
an extension of the su(2)×su(2) Yangian symmetry of the Hubbard model found in [47].
4 Conclusions and Outlook
In this note we have reviewed the construction of the S-matrix with centrally extended
su(2|2) symmetry that appears in the context of the planar AdS/CFT correspondence
and the one-dimensional Hubbard model. We have furthermore shown that the S-matrix
has an additional Yangian symmetry whose Hopf algebra structure we have presented.
This Yangian is not quite a standard Yangian, but its coproduct needs to be braided in
order to be quasi-cocommutative. This fact is intimately related to the existence of a
7We believe, but we have not verified that this is compatible with the Serre relations (3.2).
8It is conceivable that a further consistency requirement fixes the value of u0, presumably to zero.
9We have also confirmed the invariance of the singlet state found in [10].
triplet of central charges with non-trivial coproduct and leads to the wealth of unusual
features of the S-matrix.
In connection to the Yangian there are many points left to be clarified. Most im-
portantly the representation theory needs to be understood. Which representations of h
lift to evaluation representations of Y(h)? At what values of the spectral parameters do
their tensor products become reducible? This information could be used to prove that
the coproduct is quasi-cocommutative. Also the Yang-Baxter equation for the S-matrix
should follow straightforwardly. It might also give some further understanding of bound
states [48].
Then it would be highly desirable to construct a universal R-matrix for this Yangian
and show that it is quasi-triangular. This would put large parts of the integrable structure
for arbitrary representations of this algebra on solid ground much like for the case of
generic simple Lie algebras.
Some further interesting questions include: Is this Yangian the unique quasi-co-
commutative Hopf algebra based on h? Does the double Yangian [42] exist and what is
its structure? Can the sl(2) automorphism of the algebra be included at the Yangian
level such that the coproduct is quasi-cocommutative? What would the representations
be in this case?
Acknowledgements. I am grateful to C. Callan, D. Erkal, A. Kleinschmidt, P. Ko-
roteev, N. MacKay, M. Mariño, T. McLoughlin, J. Plefka, F. Spill and B. Zwiebel for
interesting discussions.
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