PoS(Solvay)002 http://pos.sissa.it/ superalgebra. ) 2 | 2 ( su ∗ [email protected] Speaker. We review the algebraic construction of thealgebra S-matrix which of turns out AdS/CFT. We to also be present a its Yangian of symmetry the centrally extended ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Max-Planck-Institut für Gravitationsphysik Albert-Einstein-Institut Am Mühlenberg 1, 14476 Potsdam, Germany E-mail: Bethe Ansatz: 75 years later October 19-21 2006, Brussels, Belgium Niklas Beisert The S-Matrix of AdS/CFT and Yangian Symmetry PoS(Solvay)002 ) 3 × 2 | 5 2 (2.1) (2.2) (2.3) ( AdS ]. It is su 14 . In Sec. 2 Niklas Beisert . , and the central δ d c γ , see [ β ) a , S Q ] which interpolate 2 ] and in planar IIB β | α α S β β γ 6 2 10 δ δ , ( L – , b 1 2 2 1 α 9 4 δ α + δ − psu Q d β − c ] better. Since the discovery α or , δ 3 S Q α C ) α γ δ β 2 . α of the Lie superalgebra L δ | δ δ 3 γ δ β 2 h c ( b δ − δ ) n R su 2 + ) ] = ] = ] = + 2 b d δ δ | c γ c γ 2 n R ( L R Q S ) , , , α δ 2 β , the supercharges β β | δ psu α , β α α . 2 α ( L + ] in the excitation picture above a ferromagnetic L L P K [ = [ [ 4 supersymmetric gauge theory is given by two L δ 2 2 bd 11 βδ , su α generators take the standard form , ε , = ε b ( ) L δ d a c c γ αγ ac 2 b U n R ( R ε δ ε , N ) S Q b a a 2 b c b su = = = | 4 supersymmetric gauge theory [ δ δ ]; today it provides exact solutions for the spectra of certain 2 R } } } 2 1 2 1 2 ( a = ] d d δ δ γ c c δ − + su 13 b δ Q S S − N , γ , , , a = : d b b β generators a Q S 12 ] the tools for computing and comparing the spectra of both models a α α h c a b d ) 8 R S Q Q δ δ 2 , c b ( { { { 7 δ − [ su 5 S ] = ] = + ] = × d d δ ) × The symmetry in the excitation picture for light cone string theory on ] for solving a one-dimensional integrable model was and remains a powerful c γ c 2 5 1 ( R Q S , , , su b b b a AdS a a . The Lie brackets of the K R R R [ [ [ , P , C In this note we will focus on the S-matrix [ Bethe’s ansatz [ In this section the results on the S-matrix of AdS/CFT shall be reviewed from an algebraic and for single-trace local operators in 5 charges It is a centralgenerated extension by of the the standard Lie superalgebras The Lie brackets of two supercharges yield The remaining Lie brackets vanish. smoothly between the perturbative regimesavailable data. in gauge and string theory and which agree with all we subsequently show that this S-matrix has indeed a larger symmetry algebra: a2. Yangian. The Universal Enveloping Algebra gauge and string theories and thus helps us understand their duality [ have evolved rapidly. We now have complete asymptotic Bethe equations [ ground state. We start by reviewing the algebraic construction of the S-matrix in Sec. The S-Matrix of AdS/CFT and Yangian Symmetry 1. Introduction and Overview tool in contemporary theoretical physics: 75mechanics, years the ago Heisenberg spin it chain solved [ one of the first models of quantum of integrable structures in planar string theory on point of view. The applicable symmetry is a central extension which we consider first.enveloping algebra We and continue its by fundamentalits representation. presenting dressing the Finally, phase we factor. Hopf comment algebra on the structure S-matrix of and itsLie Superalgebra. universal S copies of the Lie superalgebra [ PoS(Solvay)002 (2.8) (2.5) (2.7) (2.6) (2.4) Niklas Beisert . 2 . would anticommute − ) h . The construction ( U h ] = K [ of ) . to manually impose the correct A h , ( J and therefore the coproduct 2 ] U A h [ − ) with graded commutators. For ] = + , U , . b β 0 P 2.4 − for the generators. The Lie brackets α [ a ] A , A Q S , , J β J ) = b , 16 , ) = a α ⊗ A ⊗ P , 1 A K ⊗ . 1 J 1 in the generator ] L J R C − ( ⊗ A 15 + ( ⊗ − , [ J ε F 2 ⊗ S 2 ⊗ ] of the algebra C ) + U U 1 1 − AB U 1 ] = C f ⊗ 14 − , , + + + U ( + + S U 1 1 . 1 [ 1 1 1 3 1 1 − + + + ] = U ⊗ ⊗ ⊗ 1 ⊗ ⊗ B 1 , 1 ) = U b β b β ⊗ J 1 A ⊗ a a α ⊗ , ⊗ U α J U A ( R L Q S U C P K ) = J ε = [ ======A ] = + U ( for the individual generators. The above grading is derived b b J β β C K Q ) = a P S a α [ ∆ U α ∆ 1 1 ∆ automorphism [ ∆ ( R ∆ L S Q ) (a priori unrelated to the algebra) and the grading ε ∆ , , ∆ ∆ ∆ 2 1 ( 0 U sl ] = ) = 1 C ( The coproduct of the braided universal enveloping algebra U S ] = [ L generator 1 Next we consider the universal enveloping algebra U ] = [ Table 1: R [ ]. The antipode is an anti-homomorphism which acts on the generators as 16 , We should define the remaining structures of the Hopf algebra: the antipode S and the counit Curiously, we can include the supersymmetric grading Where appropriate, we shall use the collective symbol 1 15 [ ε the coproduct one can introduce a non-trivial braiding [ with some abelian is compatible with the algebra relations. The counit acts non-trivially only on 1 and from the Cartan charge of the statistics. This is helpful for an implementation within a computer algebra system. In this case The S-Matrix of AdS/CFT and Yangian Symmetry with fermionic generators. The coproduct is spelt out in Tab. then take the standard form For simplicity of notation, wefermionic generators shall by pretend insertion that of all suitable signs generators and are graded bosonic; commutatorsHopf is the Algebra. straightforward. generalisation to of the product is standard, and one identifies the Lie brackets ( PoS(Solvay)002 ] 16 (2.9) (2.13) (2.12) (2.15) (2.16) (2.14) (2.10) (2.11) as ] which external α ) , 13 2 g . ( Niklas Beisert . K which read sl . and two fermi- as follows [ + − ⊗  U i x x
a − + U 2 φ x x s − | and − = U K covariant way 1 , U ) − P  2 , . 1 (
+ γ C , i 2 x and the constants su − =  g γ ×
− + , √ U 2 ) x x ± . − 2 − x = i ( , − γ 1 i . d , U 1 . , c su i ψ 0 i 1 | i g  φ β − b . Note that the corresponding quadratic | − α β , = 1 4 1 a φ , ψ b g . α δ | α | = i i δ with the braiding factor + 2 1 K g a γ α = x 2 1 ab i = − ⊗ αβ α 1 K ε φ α x ψ = ε generators has to be canonical | g , | K K i − β b g ) α αβ PK a i − . To make it quasi-cocommutative we have to ab K − P α − 2 a δ δ ε 1 ε √ ( has a four-dimensional representation [ − − 4 ψ , φ a b d c P | | x 2 h 1 c γ = su b β  , = = = = C − − δ δ c , + − i i i i P
α x x b b + β β = = 1 2 g using the parameters x in Tab. ⊗ φ φ , is singled out by being invariant under the + i i are represented by the values ψ ψ | | − d
| | c γ − a α  + , K 2 a 1 α φ U α a c ψ | + U , α a − + | +  , PK b x x x Q S − β b PK a P Q S ) label the representation and they must obey the constraint α U , α − 1 The algebra γ g − R a and 2 L − 1 C α ] that the coproduct is quasi-triangular, at least at the level of central = 1 K  g , P 17 γ P α = = , g
, C ] P 2 √ − − + 15 x x = This coproduct is in general not quasi-cocommutative as can easily be seen + + U ]. x x b (together with / / − 18 1 1 ± 1 , . The action of the two sets of x + − γ i g α 1 1 ⊗ ψ √ 1 2 P | = = a C We can write the four parameters combination of central charges The supersymmetry generators must also act in a manifestly The three central charges They furthermore obey the quadratic relation automorphism. we will call fundamental. The correspondingonic states multiplet has two bosonic states The parameters charges, see also [ Fundamental Representation. The S-Matrix of AdS/CFT and Yangian Symmetry Cocommutativity. satisfy the constraints [ They are solved by identifying the central charges This leads to the following quadratic constraint by considering the central charges It was furthermore shown in [ PoS(Solvay)002 : ] ) ]. h 30 33 2.17 (2.18) (2.19) (2.17) ]. h 17 ]). Niklas Beisert ]. The latter 24 10 ∞ n ] of the one-dimensional Hub- − = x ] which will thus satisfy the standard 21 n x 17 b 1 ∞ = ∑ n ] establish a link between an important ] for the phase factor into the crossing ∞ ] Bethe ansatz. 17 = 32 → 0 and at as analytic functions of the index, we can x ]. Note that we have to fix the parameters . ] contain two copies of the Lieb-Wu equa- 20 n ) = . 0 x b 17 n 13 ( x − f b ] = ]. The problem of an algebraically undetermined and − 5 S n 29 , a = A – , n J n a x ∆ 26 ]. n [ a 20 0 , ∞ = ∑ ] an S-matrix acting on the tensor product of two fundamental n 19 , 0 ] and the algebraic [ 17 . The phase factor was computed to some approximation from → = 13 , x ]. So far this step has been performed in two different ways: by h 13 ]. These observations of [ 9 ) 13 x ( 23 f In [ 1. When we consider − ]. By substituting a suitable ansatz [ in order to make the action of the generators compatible with the coproduct = 31 n − . It has the following expansions at ] and from string theory [ The remaining overall phase factor of the S-matrix clearly cannot be determined b x ) ]. Furthermore the Bethe equations [ / x 25 + 22 − x 1 1 and p ( / = 1 = n 2 a U This identification removes all braiding factors from the S-matrix in [ A corresponding all-orders expansion at weak coupling was presented in [ This S-matrix has several interesting properties. Firstly, it is not of difference form; it cannot Intriguingly this S-matrix is equivalent to Shastry’s R-matrix [ Finally, let us note that one can derive (asymptotic) Bethe equations from the S-matrix and ). 2 ) = ]. This unusual fact is related to an unusual feature of representation theory of the algebra x = ( 13 2.5 f Yang-Baxter (matrix) equation, see also [ Phase Factor. by demanding invariance under gauge theory [ as was shown in [ phase factor is in fact generic.a Usually constraint one on imposes it. a Indeed further the crossing known symmetry string relation phase to factor obtain is consistent with crossing symmetry [ ( ξ [ The tensor product of two fundamental representations is irreducible in almost all cases [ tions for the Hubbard model [ model of condensed matter physics and string theory (complementary to the one in [ means of the nested coordinate [ symmetry relation a conjecture for the all-orders phase factor at strong couplingconjecture was made was in obtained [ by aLet sort us of illustrate analytic this continuation principle by in means the of perturbative a order very of simple the example: series. Consider the rational function with be written as a function of thebe difference determined of uniquely some up spectral to parameters. one Secondly, the overall S-matrix function could merely by imposing a Lie-type symmetry ( make the observation (“reciprocity”) thus confirm the conjecture in [ The S-Matrix of AdS/CFT and Yangian Symmetry Fundamental S-Matrix. representations was derived. It was constructed by imposing invariance under the algebra bard model [ We will not reproduce the result here, it is given in [ PoS(Solvay)002 ]. is a 40 (3.2) (3.1) ] that , ¯ at the h . Their 36 39 [] A J b ] for more 52 Niklas Beisert and , 51 {} ] and on taking a ] and non-critical string 47 34 , 46 It was proved in [ . with two indices lowered , } 1 could be related, but the 3 F − 36 J AD C , f E J 44 . The brackets D { 1 and BE J ] for the above S-matrix. We will g + 50 and the Yangian generators GHK , f and , , A 49 C C J CK F J J b AD f g AB AB C C is a deformation of the Serre relation for affine f f BH E f 4 6 ]. The dressing phase can also be obtained as an ] = ] = AG D f B B 38 2 ], analytic methods [ J J b , -generators ¯ , , h , , g A A 4 1 45 0 0 37 J J , [ [ = = = is a deformation of the universal enveloping algebra of half 44    ] ] ] ) ] ] ] ]) from the scattering mediated by a non-trivial vacuum state g 2 C C C ]. J J J b 40 represents the structure constants , , , 48 B B B n R J J J b . The third relation Typically the symmetries of rational S-matrices are of Yangian type. DE [ [ [ ) C f , , , 2 AB | C A A A [ [ [ f 2 BE J J J b ( g    . g AD su ] (see also [ ( g of a Lie algebra 39 Y = ) it has to be replaced by a quartic relation. g ( ) 2 ABC ( f ], numerical evaluation [ ) appears to work for a surprisingly large class of functions! su ]. While this is certainly encouraging in general, it is at the same time strange from the 43 = ] where this reciprocity can be applied. Furthermore, summation by the Euler-MacLaurin formula (also known 42 g 2.19 , 35 Among other physical examples, we have identified circular Maldacena-Wilson loops [ For In the section we investigate Yangian symmetry [ More plainly, it is generated by the Several tests of the phase have recently appeared, they are based on four-loop unitary scattering 24 3 4 , 41 scale parameter whose valueon plays the no structure physical constants role. The first two relations lead to a constraint extensions of Lie algebras. level of indices imply total symmetrisation and anti-symmetrisation, respectively. Finally, as zeta-function regularisation) is consistentdiscussions with of it. this principle. I thank Curt Callan, Marcos Mariño and Tristan McLoughlin for methods [ it does apply for thefor conjectured the expansion phase of have the recently phase appeared factor. in [ Very useful integral expressions This calls for further investigations. certain highly non-trivial limit [ 3. The Yangian extensive reviews), and then we apply the framework to theYangians S-matrix and discussed S-Matrices. above. The Yangian Y the affine extension of commutators take the generic form The symbol by means of the inverse of the Cartan-Killing forms theory [ Hopf algebra point of view to use an S-matrix which does not obey the crossing relation [ effective quantity [ [ The S-Matrix of AdS/CFT and Yangian Symmetry Of course there are variouschoice ways ( in which the two functions start with a very brief review of Yangian symmetry for generic S-matrices (see [ and they should obey the Jacobi and Serre relations PoS(Solvay)002 i 2 u , (3.7) (3.8) (3.5) (3.6) (3.3) (3.4) 1 (3.10) u | Niklas Beisert of spectral parameters . This Yangian repre- 2 u u . , drops out from the second − C D 2 1 J J u u . ⊗ C BC D J B f J . ⊗ 0 A BC A BC B . 0 (3.9) J ¯ h f 1 ¯ h f 1 4 ) = . A is proportional to the Lie generators V BC A i + . ] = 1 A + J b i ¯ ⊗ h f A ( u J b A u 1 2 2 , J S ε | J b 2 , A ∆ V u + A 2 , − J J b u A A ) = → ¯ ∆ J J b hu A 7 + 2 ) = J i ∼ | ⊗ ⊗ 1 ( 2 = A V ε u 1 1 J b i ] = [ ⊗ , ( ⊗ u 1 | + + A S 1 S u , A J 1 1 | , V 1 ) J b A : 2 , ⊗ ⊗ S -representation is. One merely has to ensure that the Serre J u A A A g ∆ J ) = S [ J J b − 1 − 1 ( = = u ε ( A A ) = J J b ' ) ensures that the term proportional to A ∆ ∆ J A ( 3.9 J b S ∆ . The existence of such an S-matrix is equivalent to quasi-cocommutativity A J b . The antipode S is defined by , A AD C J f takes the standard form ε BD g ) is satisfied. This is indeed not the case for all representations of all Lie algebras. The = is some state of the evaluation module with spectral parameter 3.2 A BC . Note that only the difference of spectral parameters appears in the invariance condition: i f ) u | g Let us finally consider the connection to the S-matrix. The S-matrix is a permutation operator; The Yangian is a Hopf algebra and the coproduct takes the standard form For the study of integrable systems, the evaluation representations of the Yangian are of special ( In particular, for the tensor product of two evaluation modules one has equation. Therefore the S-matrix typically dependsonly. on the difference Invariance of the S-matrix under the Yangian means it acts by interchanging two modules of the algebra Here sentation is finite-dimensional if the relation ( power of the Yangian symmetryare lies in typically the irreducible fact (except that forsimple tensor proofs special products (e.g. values of for of the evaluation Yang-Baxter their representations relation) spectral by representation parameters). theory arguments. This allows for interest. For these the action of the Yangian generators The S-Matrix of AdS/CFT and Yangian Symmetry where and the counit for all generators of Y We can write the action of the coproduct of Yangian generators on the evaluation module Here the first equation in ( as PoS(Solvay)002 ], A and 53 6 h (3.11) (3.12) (3.13) (3.14) ] Yangian obeying the 56 Niklas Beisert – A J 54 ] and it is likely that 60 – for each . 57 A C , J b J 25 ⊗ . . ] 0 )) C [ is unaffected by the redefinition. 4 | , b 2 U P . ) = 0 . , , , B 0 ¯ K h A
b 2 , P K 1 2 J b K 1 J b (
( α − A , − ⊗ and the counit BC ⊗ ε α 0 g α 2 b K 2 g 5 psu b C ¯ g h f + ( 2 − . For that purpose it is favourable to choose − P 2 ⊗ 1 2 b K + − U P U 1 − , A , 1 − A + P J b b − b C + K P + 8 J b 1 C , ] α 1 1 ⊗ C A C g b ] C [ = ⊗ ⊗ . We have already studied the universal enveloping ⊗ A 0 0 + [ − 0 ) − 0 + b . Our Yangian should arise as a subalgebra of the full P b h b b C P K b ) b b C P K ( U U K h 2 1 = = = ( − = = = + 0 0 0 − 0 0 0 1 0 b b C K b b b C P K b b P K ) = ∆ ∆ ⊗ ∆ ]. Effectively, lowering an index leads to an interchange of the A P A 2 1 J b J b 17 ( − S 0 = b C , where we also fix the value of A ) turn out to be very important for the Yangian. It can easily be seen that without them C 2 J b Yangian symmetries for planar AdS/CFT have been investigated in [ and the Yangian would be Y ∆ 3.11 when acting on asymptotic excitation states. ) 4 | )) 0 here, so there is no contribution from the Lie generators. An important question is if this coproduct can be quasi-cocommutative. 2 4 , | = 2 2 Let us now consider Y ( , BC ), and define its coproduct, antipode as well as counit. D 2 f ( the Cartan-Killing form is degenerate and we need to extend the algebra by the psu A 3.2 BC . All we still need to do is to introduce one generator , f h ) psu the coproduct almost trivialises ) we have defined a braided coproduct for the universal enveloping algebra. For consis- h ( 3.1 ( 7 2.5 outer automorphism, see [ Note that The braiding factors in ( Note that the scalar product Here we consider a different picture of well-separated excitations on a ferromagnetic ground For the sake of completeness we state the antipode In ( ) 5 6 7 2 ( Hopf Algebra. Note that lowering anthe index case requires of to usesl the inverse Cartan-Killing form of the algebra. In the coproduct cannot be quasi-cocommutative. Thisbraided is as in well contradistinction as to the the unbraided universal enveloping coproduct algebra are where quasi-cocommutative. the Yangian Y algebra U state and of their scattering matrix. In this picture the Lie symmetry reduces to two copies of both for classical string theory and for gauge theoryit at also exists leading at order, finite see coupling.on also This an [ Yangian infinite can world be sheet understood orthis as as picture a an is symmetry expansion of of the the Hamiltonian full monodromy matrix. The Lie symmetry in the corresponding Yangian would be Y tency with the Serre relations, wecoproduct also have to apply an analogous braiding to the standard Yangian automorphism generators with the central charges.form We refrain or from the spelling out modified theYangian structure Cartan-Killing generators in constants. Tab. Instead we present the complete set of coproducts of The S-Matrix of AdS/CFT and Yangian Symmetry Yangians in AdS/CFT. symmetry also persists to higher perturbative orders in both models [ relations ( Cocommutativity. suitable combinations first step is to consider the central generators for whom PoS(Solvay)002 (3.15) cocom- 0 b K , 0 Niklas Beisert b P , , c γ δ , d , . Q ) P K S h ( ⊗ ⊗ ⊗ ⊗ 1 2 2 1 + also become cocommutative + − − b c K U U U U γ γ δ α , c d a c d δ γ b Q P S S S K Q , Q ⊗ b c K α γ β a ⊗ b bd C β c b a 1 βδ δ ε α γ 1 δ ε + 1 4 R R igu 1 4 − L L αγ ac C ⊗ U C ⊗ ε − γ c = ⊗ ε ⊗ + U , c 1 2 β a 0 2 1 δ α β b b γ d ⊗ c ⊗ α c c γ γ P a γ b K L R b − β + S Q S S Q Q S α a Q ⊗ γ c 1 2 ⊗ ⊗ 1 2 1 2 1 2 1 2 1 2 , b d 2 β δ . ⊗ Q S b β 9 ⊗ α a C + c + 1 2 1 2 S 1 γ + − C − + − 1 Q , b b c Q S β − b ⊗ b L γ R β + β U ⊗ c ⊗ + b − γ γ ) a c ⊗ α 2 1 P 2 1 b a α ⊗ 1 ⊗ P b K b P U 1 K β S K U P Q Q S Q S 1 c 1 − b b a α L + γ − + R ⊗ − δ + + ⊗ − d ⊗ − b ⊗ ⊗ . With this choice, 2.10 ⊗ ⊗ ⊗ β Q S 2 ⊗ , U igu c 2 ⊗ 1 b γ C U K Q 1 1 K 1 1 1 S P K + U U 1 − 1 − ⊗ ⊗ u P + + L K − + − R a α = b β ⊗ ⊗ 2.9 1 1 ⊗ ⊗ + 0 + + U δ δ + bd 2 U 1 ⊗ U ⊗ only by central elements. βδ + − 2 2 U U U U U 1 4 1 4 ε 1 1 1 c b − 1 ε γ c c γ γ P 0 b + + − β and + + a c a a α γ α α b U U K ⊗ αγ ac + − ⊗ ⊗ 1 ⊗ 1 1 U , U U P R C ε R L C ε S L Q L R b The coproduct of the Yangian generators in Y β b β 0 ⊗ 1 1 1 1 2 2 2 2 2 2 2 2 1 2 2 2 1 1 1 2 1 1 1 1 u P C C a a α ⊗ ⊗ α b P b b b b b b b P L C K R Q S + − + − + − − − − + − + + + + , 0 b C ======b b β β b b C K b a P Table 2: a α α ∆ ∆ ∆ b is already cocommutative, and in order to make the generators b R b b L S Q 0 ∆ ∆ ∆ ∆ b C because they differ from The S-Matrix of AdS/CFT and Yangian Symmetry The combination mutative we have to set as above in ( with two universal constants PoS(Solvay)002 ) 9 . 2 | K 2 u ( (3.16) (3.19) (3.18) (3.20) (3.17) = su P u = Niklas Beisert 0 u ). . ]. ) − 61 x . ). 3.11 ) + 0 -representation parame- x 3.2 u h / ). Superficially it is very . 1 + K + u 0 2.15 ( 1 , presumably to zero. ig igu )( 0 4 1 and the u − = u x . = ]. 0 i 0 b + K 13 b P we have confirmed that the S-matrix is + X K x | 2 , ( 1 2 A P 1 2 . J 0 has to be related to the parameters of the ) − 0 0 0 = u u b igu K g i ] = P + 2 = 1 2 u 0 10 S ( For the fundamental evaluation representation we + b , P − A 0 ig − 1 J b b x C , = ∆ [ C + i C 0 = − x X b ] to Shastry’s R-matrix, our Yangian is presumably an exten- | P igu A = K 17 J b + 1 2 g i 2 C ) we can infer that − ) have to both coincide with the universal constant 4 igu b − K P = Using the coproducts in Tab. 3.15 + 3.15 1 , 1 2 0 x b Yangian symmetry of the Hubbard model found in [ C − in ( + ) b 2 3.13 C + K ( x u C su = × u and 8 ) ). The latter is again related to the braiding in the coproduct ( 2 P ( u To show invariance requires heavy use of the identity ( su 3.17 10 in ( ± We have also confirmed the invariance of the singlet state found in [ It is conceivable that a further consistency requirement fixes the value of As our S-matrix is equivalent [ In this note we have reviewed the construction of the S-matrix with centrally extended We believe, but we have not verified that this is compatible with the Serre relations ( It is interesting to see that the S-matrix is based on standard evaluation representations of the The eigenvalues of the redefined central elements of the Yangian within the evaluation repre- x 9 8 10 sion of the 4. Conclusions and Outlook symmetry that appears in the context ofnal the planar Hubbard AdS/CFT correspondence model. and the one-dimensio- symmetry We whose have Hopf furthermore algebra structure shownYangian, we but that have its the presented. coproduct S-matrix This needs Yangian has to is an be not quite braided additional a in Yangian standard order to be quasi-cocommutative. This fact is Yangian. Nevertheless, it is notproperty a traces function back to of the the link difference between of the spectral spectral parameter parameters. This unusual By comparison with ( Fundamental S-Matrix. also invariant under all of the Yangian generators surprising to find allshould these be that additional the symmetries coproduct is ofwhen quasi-cocommutative. acting the We have on S-matrix. thus fundamental proved representations. quasi-cocommutativity The deeper reason however fundamental representation by sentation read The S-Matrix of AdS/CFT and Yangian Symmetry Fundamental Evaluation Representation. make the ansatz ters As an aside we also state the eigenvalue of the quadratic combination We have used a computer algebraS-matrix. system to evaluate the action of the Yangian generators and the Furthermore PoS(Solvay)002 . . Niklas Beisert . lift to evaluation . . h . . . . . hep-th/0206103 . . , hep-th/0303060 . . , . hep-th/0610251 , hep-th/0305116 hep-th/0406256 , , . hep-th/0608038 ] exist and what is its structure? 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London A276, 238 (1963) Phys. Rept. 405, 1 (2004) hep-th/0511082 J. Stat. Mech. 07, P01017 (2007) Nucl. Phys. B135, 149 (1978) Z. Phys. 71, 205 (1931) automorphism of the algebra be included at the Yangian level such that the coproduct is ) 2 ( In connection to the Yangian there are many points left to be clarified. Most importantly Then it would be highly desirable to construct a universal R-matrix for this Yangian and show Some further interesting questions include: Is this Yangian the unique quasi-cocommutative sl [4] J. A. Minahan[5] and K. Zarembo, N. Beisert, C.[6] Kristjansen and M. N. Staudacher, Beisert and[7] M. Staudacher, G. Mandal, N.[8] V. Suryanarayana and I. S. Bena, R. J. Wadia, [9] Polchinski and R. N. Roiban, Beisert and M. Staudacher, [3] J. M. Maldacena, [1] H. Bethe, [2] W. Heisenberg, [26] G. Arutyunov, S. Frolov and M. Staudacher, [28] R. Hernández and E. López, [11] M. Staudacher, [17] N. Beisert, [18] N. Beisert, P. Koroteev[19] and F. 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