arXiv:1104.2474v1 [hep-th] 13 Apr 2011
Yangians, S-matrices and AdS/CFT
Department of Mathematics, University of York,
Heslington, York, YO10 5DD
dimensional quantum group (specifically, Yangian) symmetry lying behind the inte-
grability of the AdS/CFT spectral problem. In passing, the chance is taken to give a
concise anthology of basic facts concerning Yangians and integrable systems, and to
store a series of remarks, observations and proofs the author has collected in a five-
year span of research on the subject. We hope this exercise will be useful for future
attempts to study Yangians in field and string theories, with or without supersymme-
This review is meant to be an account of the properties of the infinite-
AdS/CFT, Integrable Systems, Exact S-matrices, Quantum Groups,
Yangians, Lie Superalgebras, Representation Theory
1This is an author-created, un-copyedited version of an article (invited topical review) accepted
for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not
responsible for any errors or omissions in this version of the manuscript or any version derived from it.
The definitive publisher authenticated version will be available online at [details to follow].
“What makes you think that the theory will still be integrable?”
(M. Staudacher, replying to A. A. Migdal at the Itzykson Meeting, Paris, 2007)
Gauge theories play a dominant role in our current understanding of the nature of
fundamental interactions at very short distances. A prominent example of such a the-
ory is the Standard Model of elementary particles, which is remarkably successful in
describing the physics up to the currently available energy scale. This description is,
however, to a significant extent restricted to the perturbative regime. The derivation
of analytical results when the coupling constant is large is an extremely challeng-
ing task. This represents an obstacle to the complete understanding of interesting
nonperturbative phenomena, like, for instance, confinement.
The revolutionary discovery of integrable structures in Quantum Chromodynam-
ics (QCD) , and, more recently, in planar N = 4 Supersymmetric Yang-Mills
(SYM) theory and AdS/CFT , has changed this situation2.
For a Hamiltonian system with 2n-dimensional phase space, complete integrabil-
ity stands for the existence of n independent integrals of motion, written as integrals
of local densities, in involution (i.e. Poisson-commuting with each other). One of
these integrals of motion is the Hamiltonian itself, while the other ones are some-
times referred to as higher Hamiltonians. According to the Liouville-Arnold theo-
rem, the equations of motion can then be solved by quadratures. This means that
there exists a set of canonical coordinates (‘action-angle’) such that the action vari-
2According to the AdS/CFT correspondence [3–7], the scaling dimension of gauge-invariant com-
posite operators should match the energy of the corresponding closed string states. In particular, wewill
be focusing our attention on string states with large values of some spin or angular momentum quantum
number Q, corresponding to composite operators containing a large (order Q) number of fields. The
energy of these states / dimension of these operators can be expressed as E = Q+ε(Q,λ), with ε going
to zero at weak ’t Hooft coupling λ ≡ g2
reduces to the bare dimension Q (see, for instance, ). The anomalous dimension ε is a dynamical
quantity which should interpolate between the two sides of the correspondence, and which will be our
main object of interest .
YMN (gYMbeing the Yang-Mills coupling) where the dimension
ables (momenta) are constants of motion, and the angles (coordinates) are linear in
time and parameterize a torus. For a field theory, the number of degrees of freedom
is normally infinite, and one associates integrability with the existence of an infinite
number of independent local conserved charges in involution. In scattering theory,
integrability implies pure reshuffling of momenta (‘diffractionless’ scattering). In
general, flavour degrees of freedom can be transformed in a complicated way during
the scattering. One has ‘transmission’ if the flavours are unchanged, ‘reflection’ if
they are exchanged. We recommend [10–12] for classical references on integrable
systems (see also the excellent ).
A link with the Yang-Mills Millennium prize problem3has been also advertised.
The situation in AdS/CFT is quite peculiar because of conformal invariance. More-
over, ’t Hooft’s limit N → ∞, with λ = g2
butions, according to the standard argument that the action for such configurations
scales in this limit asN
one single interacting four-dimensional gauge theory in this special limit will be im-
portant for progress in the Yang-Mills problem as well. For a relatively recent report,
underlying the potential role of AdS/CFT and integrability, see .
The N = 4 theory is a quantum conformal field theory (CFT). The information
on its spectrum is encoded in the short-distance power-law behavior of (2-point) cor-
relators of composite operators. In determining this behavior for all operators of the
theory one encounters a non-trivial operator-mixing, which makes the calculations
notoriously difficult. The observation of  is that, in the planar limit, the problem
translates into the equivalent problem of finding the spectrum of certain spin-chain
Hamiltonians. This spectrum consists ofspin-wave excitations and their bound states,
and the dynamics (S-matrix4) describing their scattering turns out to be completely
integrable [15,16]. Planarity is probably a crucial ingredient for the appearance of
YMN fixed, suppresses instanton contri-
λ[finite]. However, one hopes that the understanding of even
3For any compact gauge group G, one is to show that quantum Yang-Mills theory on R4exists and
has a mass gap ∆ > 0 (i.e. the lightest particle has strictly positive mass squared).
4We take a chance and clarify that, whenever we will be talking of S-matrices in this review, it will
always be referred to the two-dimensional scattering of excitations in the integrable models effectively
describing the SYM spectral problem in various regimes (spin-chain, sigma model). Never will we be
talking of a spacetime SYM S-matrix (also because, in that case, conformal invariance would be an
obstacle to the definition of asymptotics states).
integrability. It would be overwhelming to give here a comprehensive list of the rele-
vant references. They can be found in many of the available reviews (just to mention
some of the most recent ones, see [9,17–20]).
The result strictly applies to infinitely long chains, which are related to gauge
theory operators composed of an infinite number of fields. When the spin-chains are
of finite length, certain corrections occur that go under the name of ‘wrapping effects’
[21–23], since the range of the interactions exceeds the length of the spin-chain.
Recently [24,25], these effects have been shown to be calculable for very specific
operators and at the first few significant orders in perturbation theory, by techniques
of finite-volume integrability5. The first confirmation that one has obtained from
these impressive results is that the ingredients used in the mirror theory approach
, i.e. the mirror bound states, are all one needs to sum over in order to reproduce
the field theory result. In other words, no excitation is missing.
The technology developed so far has been impressive, see for instance [28–37].
Both gauge perturbation theory for short operators and string perturbation theory in
the form of L¨ uscher corrections have proceeded to a tremendous degree of sophis-
tication. A very convincing matching has been shown6. This remarkable result has
strengthened the expectation that the entire planar sector of the theory may in fact
be integrable, and accessible via the so-called Thermodynamic Bethe Ansatz (TBA)
method. The latter consists in obtaining a set of master equations, whose solutions
encode the spectral data of the theory. This program has the potential of providing a
set of exact analytic results for an interacting four-dimensional quantum field theory,
and, with it, a new insight in our understanding of strongly-coupled nonperturbative
phenomena in gauge theories. Once more, the study of two-dimensional models is
showing its power in modelling our understanding of four-dimensional theories (cf.
, Introduction, lines 37-58). Currently, a remarkable effort is being put into the
construction and test of such a TBA system of equations [43–45].
5These techniques involve the use of the so-called L¨ uscher corrections. Such corrections do not
assume integrability, but, if the theory is integrable, they are expected to complete to a set of exact
integral equations for the spectrum (see also ).
6Notably, the issue concerning some mismatches , which were still announced to affect the
strong coupling regime, has very recently been resolved [39–41].
Despite the progress obtained, several fundamental questions are still left unan-
swered. First of all, a systematic way of taking into account the above-mentioned
wrapping corrections has not yet been provided, due to their highly complicated na-
ture . Furthermore, no rigorous proof of integrability is yet available, and the
quantum Hamiltonian of the system is not known in closed form, but only to a cer-
tain order in perturbation theory. Instead, so far the approach has been (in the philos-
ophy of the inverse scattering method) to assume integrability and S-matrix factor-
ization, deduce the entire integrable structure, and a posteriori check the validity of
the assumptions (see also ). However, with long-range Hamiltonians (as the one
emerging from gauge perturbation theory actually is) even setting up an asymptotic
scattering theory is problematic, and it is still a challenge to rigorously prove the inte-
grability of the asymptotic problem. Perhaps, with the help of the algebraic methods
we are going to describe in this review, the knowledge of the complete Hamiltonian
will eventually become accessible7. The full algebraic structure is still, in many re-
spects, mysterious, and higher correlation functions of the theory are just starting to
be explored from the point of view of integrability. Three-point functions8are still
quite a virgin territory, and it is still unclear if the power of integrability will provide
a systematic way of computing them. When appropriately normalized, these three-
point functions scale as the two-point functions in the planar limit, and one would like
to compute them with spin-chain techniques. In this respect, the universal R-matrix
of quantum groups has been used in the past  to encode the braiding relations of
quantum field multiplets in an integrable 1+1-dimensional QFT, thereby extending
“off-shell” the “on-shell” quantum-group symmetry of the S-matrix. Along the same
lines, correlation functions and form factors9could be studied with the help of the
Not fully understood is also the nature of certain fascinating dualities that have
7The so-called ‘dressing phase’ (see formula (74) and subsequent text) is essential for the Hamilto-
nian. In , the presence of this phase has been connected to boosts and general twist transformations
for the long-range spin-chain, see also section 3.1 and references therein.
8Because of quantum conformal invariance, one-, two- and three-point functions contain all the
information one needs.
9Form factors are matrix elements of field operators. They satisfy algebraic relations, called form-
factor axioms [49,50], depending locally on the fields and their sectors.
been observed in Wilson loops and n-point functions. These dualities have recently
been related to algebraic structures very similar to those responsible for the integra-
bility of the spectral problem, in particular to an infinite-dimensional symmetry of
the so-called Yangian type . It is plausible that all the Yangians we will pro-
gressively encounter in this review (sigma model, spin-chain, S-matrix, spacetime
n-point functions) all share a common origin deeply inside the integrable structure of
Hopf algebras and quantum groups provide a suitable mathematical framework
where to study these properties. Quantum groups were co-discovered by Drinfeld
and by Jimbo (notice also the earlier work ). For standard textbook-references
on Hopf algebras / quantum groups, see for instance [53–57]. The algebraic reason
for integrability can often be singled out in the existence of an infinite-dimensional
non-abelian symmetry algebra (such as the Yangian) that severely constrains the dy-
namics. Like the angular momentum in quantum mechanics, a non-abelian algebra
commuting with the Hamiltonian generates the subspaces of equal-energy states, and
the spectrum re-organizes itself in terms of the corresponding irreducible represen-
tations. The S-matrix is nearly fixed purely by the symmetry algebra, and it dis-
plays very specific features . For a review on how Hopf algebras systematize the
scattering problem in integrable systems, we refer to . According to an idea of
Zamolodchikov’s, the infinite dimensional quantum group symmetry of massive inte-
grable field theories plays the same role in their exact solution as that of the Virasoro
algebra for conformal field theories.
An accurate knowledge of the quantum algebra governing the integrability of the
asymptotic problem might reveal crucial insights into the structure of the finite-size
corrections as well (see, for instance, ). The almost miraculous results described
earlier for short operators in N = 4 SYM are a strong motivation for the search of
deep algebraic structures responsible for such a matching. These structures should
ideally take over the job of completing the proof of spectral equivalence to an arbi-
trary loop order, where the direct computation will be challenged.
The Yangian has already turned out to be very useful to derive some results and
check others, which would have otherwise taken a perhaps prohibitive amount of
work. Even before the explicit derivation of all bound state S-matrices , Yan-
gian symmetry had been used to derive the bound state Bethe equations  without
the need of an explicit diagonalization  of the corresponding transfer matrices10.
Such diagonalization also makes use of the Yangian, and turns out to be essential
to prove important conjectures put forward in the literature . These conjectures,
in turn, play a very important role in deriving equations for the finite-size problem
(Thermodynamic Bethe Ansatz and Y-system), and one may wonder if the Yangian
could play a role in a possible group-theoretical proof of the proposals that have been
so far advanced in the literature , and in describing the system even atfinite length
One will then be able to see if it is possible to apply this algebraic framework to
the quantization of the (dual) two-dimensional sigma model, a formidable problem
where all conventional methods have failed so far. On the other hand, its understand-
ing is believed to be instrumental in order to clarify the relationship between strings
and nonperturbative phenomena in gauge field theories. This fascinating connection
has been long sought-for through the work of many generations of theoreticians.
The point of view we would like the reader to take away from the present exposi-
tion is that there is a deep and beautiful algebraic structure, not entirely understood,
which underlies the integrability of the AdS/CFT system. Fully understanding this
structure will most likely provide not only a way of testing the proposals put forward
so far for an exact solution, and possibly deriving them from first principles (see also
), but may also represent a significant progress in Mathematics. The quantum
group behind the complicated beauty of this integrable system most probably repre-
sents a new structure mathematicians have not come across so far11.
The review is structured as follows. In section 2, we briefly display two of the
traditional realizations of the Yangian algebra, namely Drinfeld’s first and second
realization, as those that have been mostly used in the AdS/CFT context so far. In
section 3, we review the Yangian symmetry of the perturbative super Yang-Mills
spin-chain (section 3.1), and of the classical string sigma model (section 3.2), both
10One striking features of these Bethe equations is that, when expressed in terms of the appropriate
bound state variables, they basically assume the same form as the Bethe equations for fundamental
11P. Etingof, private communication.
related to the superconformal symmetry algebra psu(2,2|4). We also discuss general
features of classical integrability, higher charges and Lax pairs, using as a toy model
the theory of the principal chiral field (14). Starting from section 4, we enter the
core of the topic of this review, i.e. the quantum group structure of the AdS/CFT
S-matrix, based on the centrally-extended psl(2|2) Lie superalgebra. In section 4.1,
we describe in detail the relevant quasi-triangular Hopf algebra and how it emerges
from the spin-chain and from the string sigma model picture, together with some
general notions of Lie superalgebras. In section 4.2, we describe the psl(2|2) Yan-
gian symmetry of the S-matrix. In section 5, we focus on the semiclassical limit
of the quasi-triangular Hopf algebra. Section 5.1 contains standard notions related to
classical r-matrices, Belavin-Drinfeld theorems, quantum doubles and loop-algebras,
and various technology connected to the classical Yang-Baxter equation. Sections
5.2 to 5.5 describe the corresponding AdS/CFT case, and highlight the main simi-
larities and the important new phenomena one encounters, such as the presence of
the so-called secret symmetry (section 5.4). In section 6, we describe bound state
representations, providing details about the differential-operator formalism of ,
and show how to construct the corresponding S-matrices. We also briefly discuss
the issue of ‘fusion’. This discussion is then expanded upon in section 7, where
long (i.e. typical) representations are treated. After recalling some notions of the
representation theory of Lie superalgebras, we display the construction of long rep-
resentations for the centrally-extended psl(2|2) case (section 7.2), and discuss their
reducibility properties. In the same section, we study the quasi-triangular structure
in these representations and discuss general rectangular Young tableaux. In section
8, we quickly mention recent progress connected to Yangian symmetry in spacetime
n-point amplitudes, where structures similar to those presented in this review for the
spectral problem are being observed right now. In fact, very recent is the discovery
of the above-mentioned ‘secret symmetry’ also in this context , with the role of
the secret Yangian generator played, in perfect analogy with the spectral problem we
will be treating here (see section 5.4), by the helicity generator of u(2,2|4). Section
9 contains a list of conclusions that one can draw in the light of the results obtained
so far, in particular for what concerns deriving general character formulas, finding
the universal R-matrix and elucidating the role of the secret symmetry. All these are
 D. A. Leites and V. V. Serganova, Solutions of the Classical Yang-Baxter
Equation for Simple Superalgebras, Theor. Math. Phys. 58 (1984) 16.
 R.-B. Zhang, M. D. Gould, and A. J. Bracken, Solutions of the graded
classical Yang-Baxter equation and integrable models, J. Phys. A24 (1991)
 G. Karaali, Constructing r-matrices on simple Lie superalgebras, J. Algebra
282 (2004) 83.
 G. Karaali, A New Lie Bialgebra Structure on sl(2,1), Contemp. Math. 413
 C.-N. Yang, Some exact results for the many body problems in one dimension
with repulsive delta function interaction, Phys. Rev. Lett. 19 (1967) 1312.
 S. M. Khoroshkin and V. N. Tolstoy, Universal R-matrix for quantized
(super)algebras, Commun. Math. Phys. 276 (1991) 599.
 A. A. Belavin and V. G. Drinfeld, Classical Yang-Baxter equation for simple
Lie algebras, Funct. Anal. Appl. 17 (1983) 220.
 G. Arutyunov and S. Frolov, On AdS5×S5string S-matrix, Phys. Lett. B639
(2006) 378, [hep-th/0604043].
 J. Cai, S. Wang, K. Wu, and C. Xiong, Universal R-matrix Of The Super
Yangian Double DY(gl(1|1)), Comm. Theor. Phys. 29 (1998) 173,
 N. Beisert, An SU(1|1)-invariant S-matrix with dynamic representations,
Bulg. J. Phys. 33S1 (2006) 371, [hep-th/0511013].
 G. Arutyunov, M. de Leeuw, and A. Torrielli, Universal blocks of the
AdS/CFT Scattering Matrix, JHEP 05 (2009) 086, [arXiv:0903.1833].
 A. Rej and F. Spill, The Yangian of sl(n|m) and the universal R-matrix,
 S. Moriyama and A. Torrielli, A Yangian Double for the AdS/CFT Classical
r-matrix, JHEP 06 (2007) 083, [0706.0884].
 M. de Leeuw, The S-matrix of the AdS5xS5superstring, Based on PhD thesis,
 N. Beisert and F. Spill, The Classical r-matrix of AdS/CFT and its Lie
Bialgebra Structure, Commun. Math. Phys. 285 (2009) 537,
 M. de Leeuw, Bound States, Yangian Symmetry and Classical r-matrix for the
AdS5×S5Superstring, JHEP 06 (2008) 085, [arXiv:0804.1047].
 N. Dorey and B. Vicedo, A symplectic structure for string theory on
integrable backgrounds, JHEP 03 (2007) 045, [hep-th/0606287].
 S. Aoyama, Classical Exchange Algebra of the Superstring on S5with the
 A. Mikhailov and S. Schafer-Nameki, Algebra of transfer-matrices and
Yang-Baxter equations on the string worldsheet in AdS(5) x S(5), Nucl. Phys.
B802 (2008) 1, [arXiv:0712.4278].
 B. Vicedo, Semiclassical Quantisation of Finite-Gap Strings, JHEP 06
(2008) 086, [arXiv:0803.1605].
 B. Vicedo, Finite-g Strings, J.Phys.A 44 (2011) 124002,
 M. Magro, The Classical Exchange Algebra of AdS5 x S5, JHEP 01 (2009)
 B. Vicedo, The classical R-matrix of AdS/CFT and its Lie dialgebra structure,
Lett.Math.Phys. 95 (2011) 249–274, [arXiv:1003.1192].
 M. Magro, Review of AdS/CFT Integrability, Chapter II.3: Sigma Model,
Gauge Fixing, arXiv:1012.3988.
 N. Beisert, The Classical Trigonometric r-Matrix for the Quantum-Deformed
Hubbard Chain, arXiv:1002.1097.
 T. Matsumoto, S. Moriyama, and A. Torrielli, A Secret Symmetry of the
AdS/CFT S-matrix, JHEP 09 (2007) 099, [arXiv:0708.1285].
 F. Spill and A. Torrielli, On Drinfeld’s second realization of the AdS/CFT
su(2|2) Yangian, J. Geom. Phys. 59 (2009) 489, [arXiv:0803.3194].
 V. K. Dobrev, Note on Centrally Extended su(2|2) and Serre Relations,
Fortsch. Phys. 57 (2009) 542, [arXiv:0903.0511].
 J. F. Cornwell, Group theory in physics, volume III: Supersymmetries and
infinite-dimensional algebras, Academic Press, New York (1989).
 A. Torrielli, Structure of the string R-matrix, J. Phys. A42 (2009) 055204,
 F. Spill, Weakly coupled N=4 Super Yang-Mills and N=6 Chern-Simons
theories from u(2|2) Yangian symmetry, JHEP 03 (2009) 014,
 I. Heckenberger, F. Spill, A. Torrielli, and H. Yamane, Drinfeld second
realization of the quantum affine superalgebras of D(1)(2,1 : x) via the Weyl
groupoid, RIMS Kokyuroku Bessatsu B8 (2008) 171, [arXiv:0705.1071].
 T. Matsumoto and S. Moriyama, An Exceptional Algebraic Origin of the
AdS/CFT Yangian Symmetry, JHEP 04 (2008) 022, [arXiv:0803.1212].
 T. Matsumoto and S. Moriyama, Serre Relation and Higher Grade
Generators of the AdS/CFT Yangian Symmetry, JHEP 0909 (2009) 097,
 A. B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models.
Scaling Three State Potts and Lee-Yang Models, Nucl. Phys. B342 (1990)
 J. Ambjorn, R. A. Janik, and C. Kristjansen, Wrapping interactions and a
new source of corrections to the spin-chain / string duality, Nucl. Phys. B736
(2006) 288, [hep-th/0510171].
 A. B. Zamolodchikov and A. B. Zamolodchikov, Factorized S-matrices in
two dimensions as the exact solutions of certain relativistic quantum field
models, Annals Phys. 120 (1979) 253.
 P. P. Kulish, N. Y. Reshetikhin, and E. K. Sklyanin, Yang-Baxter Equation
and Representation Theory. 1, Lett. Math. Phys. 5 (1981) 393.
 G. Arutyunov and S. Frolov, The Dressing Factor and Crossing Equations,
J.Phys.A A42 (2009) 425401, [arXiv:0904.4575].
 C. Ahn and R. I. Nepomechie, Yangian symmetry and bound states in
AdS/CFT boundary scattering, JHEP 05 (2010) 016, [arXiv:1003.3361].
 N. MacKay and V. Regelskis, On the reflection of magnon bound states,
JHEP 08 (2010) 055, [arXiv:1006.4102].
 L. Palla, Yangian symmetry of boundary scattering in AdS/CFT and the
explicit form of bound state reflection matrices, JHEP 1103 (2011) 110,
 H.-Y. Chen, N. Dorey, and K. Okamura, On the scattering of magnon
boundstates, JHEP 11 (2006) 035, [hep-th/0608047].
 R. Roiban, Magnon bound-state scattering in gauge and string theory, JHEP
04 (2007) 048, [hep-th/0608049].
 N. Beisert, R. Hernandez, and E. Lopez, A crossing-symmetric phase for
AdS5×S5strings, JHEP 11 (2006) 070, [hep-th/0609044].
 N. Beisert, B. Eden, and M. Staudacher, Transcendentality and crossing, J.
Stat. Mech. 0701 (2007) P021, [hep-th/0610251].
 S. R. Coleman and H. J. Thun, On the prosaic origin of the double poles in
the Sine-Gordon S matrix, Commun. Math. Phys. 61 (1978) 31.
 V. G. Drinfeld, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990) 1419.
 H. Pfeiffer, Factorizing twists and the universal R-matrix of the Yangian
Y(sl(2)), J. Phys. A 33 (2000) 8929, [math-ph/0006032].
 N. Gromov and V. Kazakov, Review of AdS/CFT Integrability, Chapter III.7:
Hirota Dynamics for Quantum Integrability, arXiv:1012.3996.
 A. Kuniba, T. Nakanishi, and J. Suzuki, T-systems and Y-systems in integrable
systems, J. Phys. A44 (2011) 103001, [arXiv:1010.1344].
 F. H. L. Essler, H. Frahm, F. Goehmann, A. Kluemper, and V. E. Korepin, The
One-Dimensional Hubbard Model, Cambridge University Press (2005).
 G. Arutyunov and S. Frolov, String hypothesis for the AdS5 x S5 mirror,
JHEP 03 (2009) 152, [arXiv:0901.1417].
 I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, Quantum integrable
models and discrete classical Hirota equations, Commun. Math. Phys. 188
(1997) 267–304, [hep-th/9604080].
 V. Kazakov, A. Sorin, and A. Zabrodin, Supersymmetric Bethe ansatz and
Baxter equations from discrete Hirota dynamics, Nucl. Phys. B790 (2008)
 G. Arutyunov, M. de Leeuw, and A. Torrielli, On Yangian and Long
Representations of the Centrally Extended su(2|2) Superalgebra, JHEP 06
(2010) 033, [arXiv:0912.0209].
 Y.-Z. Zhang and M. D. Gould, A unified and complete construction of all
finite dimensional irreducible representations of gl(2|2), Journal of
Mathematical Physics 46 (2005) 013505.
 A. H. Kamupingene, N. A. Ky, and T. D. Palev, Finite-dimensional
representations of the Lie superalgebra gl(2|2) in a gl(2)⊕gl(2) basis. I.
Typical representations, J. Math. Phys. 30 (1989) 553.
 T. D. Palev and N. I. Soilova, Finite dimensional representations of the Lie
superalgebra gl(2|2) in a gl(2)×gl(2) basis. 2. Nontypical representations,
J. Math. Phys. 31 (1990) 953.
 G. Gotz, T. Quella, and V. Schomerus, Tensor products of psl(2|2)
 G. Gotz, T. Quella, and V. Schomerus, Representation theory of sl(2|1), J.
Algebra 312 (2007) 829, [hep-th/0504234].
 V. Kazakov and P. Vieira, From Characters to Quantum (Super)Spin Chains
via Fusion, JHEP (2008) 050, [arXiv:0711.2470].
 A. Baha Balantekin and I. Bars, Dimension and Character Formulas for Lie
Supergroups, J. Math. Phys. 22 (1981) 1149.
 T. Bargheer, N. Beisert, and F. Loebbert, Exact Superconformal and Yangian
Symmetry of Scattering Amplitudes, arXiv:1104.0700.
 J. Bartels, L. Lipatov, and A. Prygarin, Integrable spin chains and scattering
 D. Fioravanti and M. Stanishkov, On the null-vectors in the spectra of the 2D
integrable hierarchies, Phys. Lett. B430 (1998) 109–119,
 D. Fioravanti and M. Stanishkov, Hidden local, quasi-local and non-local
symmetries in integrable systems, Nucl. Phys. B577 (2000) 500–528,
 D. Fioravanti and M. Rossi, On the commuting charges for the highest
dimension SU(2) operator in planar N = 4 SYM, JHEP 08 (2007) 089,
 J. M. Maldacena and I. Swanson, Connecting giant magnons to the pp-wave:
An interpolating limit of AdS5×S5, Phys. Rev. D76 (2007) 026002,
 T. Klose, T. McLoughlin, J. A. Minahan, and K. Zarembo, World-sheet
scattering in AdS(5) x S**5 at two loops, JHEP 08 (2007) 051,
 T. Klose, T. McLoughlin, J. Minahan, and A. Torrielli (unpublished).
 V. Regelskis, The secret symmetries of the AdS/CFT reflection matrices, to
 G. Delius, N. MacKay, and B. Short, Boundary remnant of Yangian symmetry
and the structure of rational reflection matrices, Phys.Lett. B522 (2001)
 N. MacKay and V. Regelskis, Yangian symmetry of the Y=0 maximal giant
graviton, JHEP 1012 (2010) 076, [arXiv:1010.3761].
 S. Ramgoolam, Schur-Weyl duality as an instrument of Gauge-String duality,
AIP Conf.Proc. 1031 (2008) 255–265, [arXiv:0804.2764].
 V. De Comarmond, R. de Mello Koch, and K. Jefferies, Surprisingly Simple
Spectra, JHEP 1102 (2011) 006, [arXiv:1012.3884].
 A. V. Belitsky, G. P. Korchemsky, and D. Mueller, Towards Baxter equation
in supersymmetric Yang-Mills theories, Nucl. Phys. B768 (2007) 116,