JHEP11(2006)021 . hep112006021.pdf October 17, 2006 October 12, 2006 November 9, 2006 Received: Accepted: iscuss magnon kinematics Published: wed to fluctuate. We will ict the allowed magnon irreps = 4 Yang-Mills with the existence N = 4 supersymmetric Yang-Mills is the N . We also consider general features of the http://jhep.sissa.it/archive/papers/jhep112006021 /j Z b as central Hopf subalgebra, and discuss the giant magnon [email protected] symmetry with non-trivial co-multiplication and antipode , Z Published by Institute of Physics Publishing for SISSA Z and Rafael Hern´andez ab AdS-CFT Correspondence, Gauge-gravity correspondence. The planar dilatation operator of [email protected] Instituto UAM/CSIC de Te´orica F´ısica Universidad de Aut´onoma Madrid Cantoblanco, 28049 Madrid, Spain Theory Group, Physics Department, CERN CH-1211 Geneva 23, Switzerland E-mail: SISSA 2006 b a c ´srG´omez C´esar The magnon kinematics ofcorrespondence the AdS/CFT hamiltonian of anidentify integrable the spin dynamics of chain lengthof fluctuations whose of an length planar abelian is HopfThe allo algebra intertwiner conditions forto this those Hopf leading algebra toand will the crossing restr magnon symmetry dispersion on relation. the spectrum We of will d Abstract: ° underlying Hopf algebra with semiclassical regime. Keywords: JHEP11(2006)021 4 5 8 1 4 11 11 -matrix S the spin chain at one-loop for the pported on the two- asymptotic ang-Baxter triangular completely the scattering y side integrability of the ma model [14]. The Bethe d on the classical equivalence AdS/CFT correspondence has y, need to be imposed [27]. tors [4]-[6], to the hamiltonian lity. The dilatation operator of rity, bootstrap in the case of a rong ’t Hooft coupling of this by the spectral density suggested – 1 – [7] allowed a resolution of the theory in terms of 5 S -matrix on both sides of the correspondence. The string S × = 4 Yang-Mills was recently derived in [16]. Integrability 5 and the sine-Gordon integrable model [25, 26]. On the gauge N 2 AdS S -matrix and crossing transformations 9 × S R 2) | -matrix of geometry and the dispersion relation 6 -matrix describing the scattering of magnon fluctuations of S S Z = 4 supersymmetric Yang-Mills has been shown to correspond, -matrix describes the scattering of some classical lumps su S N 2.1 Dynamics and2.2 representations Dynamical co-multiplication2.3 rules Spec 2.5 The SU(1 2.4 The kinetic plane 3.1 Cyclic two-dimensional irreps -matrix has been proposed in [17] (see also [18]-[24]), base spectral curves [8]-[13]. Thesoon integral after equations a satisfied discreteequations Bethe of ansatz the for gauge the quantum theory string were sig then shown to arise from an in [15], and the 1. Introduction The appearance of integrable structuresplayed on a both central sides role of in the ourplanar current understanding of the dua complete theory [1]-[3], andof at several a loops one-dimensional inclassical some integrable subsec sigma system. model On on the string theor Contents 1. Introduction 2. The central Hopf subalgebra theory side the 3. The Hopf algebra symmetry 4. Magnon kinematics and the sine-Gordon model 13 is thus encoded in a factorizable theory can be constrained byequation the [16]. symmetries However of thesematrix, the symmetries and system are additional and not physical enough bynon-trivial requirements to such spectrum the as fix of Y unita bound states, and crossing symmetr dimensional worldsheet. AS semiclassical description at st of strings moving on JHEP11(2006)021 (1.4) (1.3) (1.5) (1.6) (1.2) (1.1) plications, = 4 Yang-Mills, ate at weak but N trivial central Hopf .4) with a non-symmetric ing process correspond to a Hopf algebra of symme- -matrices can then always . with the leading quantum S ) ssing symmetry in [28]. The nd a solution to the crossing , ntertwiner condition (1.4) the k ents, to live on certain Fermat 1 1 π ,p estricts the allowed irreps, that anar limit of π ion law that defines the action of cular, not all magnon irreps are j 2 in the Hopf algebra. The physical V p π ( rizable a ]. The quantum dressing factor [31] R ⊗ ) . 2 a π ) gauge ( , R . -matrix is, we should require, for any 1 V a 2 S π ( 12 π R ) 2 2) algebra [16]. k π 21 | → R ⊗ ,p 2 j 0 12 1 π p ( S π V – 2 – iθ )=∆ )=∆ = = e ⊗ -matrix a a 2 1 ( ( R 12 π π -matrix such that 2 1 12 S π V )= π R -matrix through a dressing phase factor [14], 1 k : ∆ π S R 2 ,p ∆ π j 2 1 p π π ( 1 -matrix for the Hopf algebra of symmetries, and the factor π R R R string S -matrix follows as a consequence of non-symmetric co-multi R is the intertwiner -matrix for the quantum string Bethe ansatz should extrapol in the central Hopf subalgebra, that of the Hopf algebra, and the S ) the co-multiplication for an arbitrary element i a 12 a π R V -matrix is derived by imposing the intertwiner condition (1 In many occasions, underlying an integrable system there is In case the underlying Hopf symmetry algebra contains a non- The is the dressing phase. The magnons entering into the scatter S 0 12 with ∆( is determined by the intertwiner condition meaning of the co-multiplication is tosymmetry provide transformations the on composit multimagnon states. Fromnon-triviality the of i the In fact, independently of what the intertwiner element i.e. “non-classical” composition laws. In the case of the pl This condition, with aare non-symmetric now co-multiplication, parameterized by r the eigenvalues of the central elem the irreps S where tries (see for instance [35] and references therein). Facto Constraints on this phase factordressing have factor been can obtained fromcorrection also to cro be the energy constrained ofwas semiclassical through in strings comparison fact [29]-[32 shownrelation in has [33] recently to been satisfy proposed the in crossing [34]. equations, a subalgebra [36], new interestingallowed, features and appear. there is In not parti a universal co-multiplication for the generators of the SU(2 be written in the form finite coupling to the gauge theory JHEP11(2006)021 3) K | (1.7) , with Z to define cross- irreps, independently econstruct the magnon \ SL(2)) is the affine Hopf ( mation. It was originally q orphic to the central Hopf onal central generator slate the dynamic SU(2 ls coupling enters through U ra of symmetries, ersymmetry. Multimagnon ral parameter, the “crossing the generators of the central algebra. This, together with es that solve condition (1.6). e under the Drinfeld quantum -matrix. The idea is simply to ion for magnons is generated d be defined by promoting the etry. This program was devel- In this note we will identify a e application of Schur’s lemma his lifted action of the antipode central Hopf subalgebra leading ctrum of the central subalgebra can be derived from the central R determined by the sine-Gordon gebra, as for instance the chiral 1 ing transformations can be given q he allowed magnon irreps to those ications for the existence of a universal , = 4 Yang-Mills, with three generators 1 − N R = = 4, as well as for other integrable models R N – 3 – 11) = 4 Yang-Mills, is dynamically generated. In this ⊗ for planar N γ ( a root of unity a central Hopf subalgebra, isomorphic Z q -matrix a root of unity. It is known that the quantum affine Hopf algebra of the sine-Gordon model, R q A as defined in (1.5) should in principle exist for any couple of 2 π 1 π \ SL(2)) with ( ) [39], with R q ∗ U A , = 4 Yang-Mills. However, for A ( the dual algebra. In [28] this approach was suggested as a way N D ∗ A The article is organized as follows. In section 2 we will tran Finally, let us just briefly comment on the crossing transfor These constraints on the allowed irreps have important impl 1 on the rapidity plane. -matrix. In fact, leads to a purely algebraicoped implementation for of the crossing sine-Gordon symm modeldouble in [37] by imposing invarianc ing for and enjoying invariance under a non-trivial central Hopf subal Potts model, an intrinsicallyindependently algebraic of definition the of assumption of cross existence of a universal realize that the rapidity planedefined is by the the submanifold in intertwiner conditions. thewe spe Therefore, can as lift to a the simpl subalgebra, rapidity and plane thus the define action crossing of transformationsγ to the be antipode t on chain into the existence of an abelian central Hopf subalgeb symmetry algebra of the sine-Gordon model [37], with coupling. At the special value of to the one that we have identified for case a new dispersion relationelements for in the a sine-Gordon way solitons analogousdispersion to relation. the one we have used in order to r suggested in [38] that crossingaction for an of affine the Hopf antipode algebra intotransformation”, coul a that certain becomes change an in innerthe the automorphism property affine of of spect the the universal R curves in the spectrum of the central Hopf subalgebra. central Hopf symmetry subalgebra of whether they satisfy condition (1.6). with the BMN-like dispersionphysical states relation are then found defined in asto those [16] invariant the under from the total sup zerothe momentum undetermined Virasoro constants condition. parameterizing TheThe the Yang-Mil central Fermat curv Hopf subalgebraby governing the the two dispersion centralrelated relat to elements the introduced magnon momentum. insubalgebra This of [16], Hopf subalgebra and is isom an additi and non-symmetric co-multiplications that will restrict t JHEP11(2006)021 as 3) ral | act Z (2.1) 2) factors | as the enlarged Z e special features of the is defined in terms of the non-trivial center for the algebra 2) spin chain, let us first ter, but the quark constituents al states and therefore are | h is part of the symmetry ing Hopf algebra with f the chain two extra central nteracting hamiltonian. The correspond to some quantum urely algebraic grounds. But ameterized by the eigenvalues and the elliptic curve on the Hopf algebra symmetry of the G , l momentum. However they act 2) in order to induce the SU(2 milar to the one played by the center 2 1 | om intertwiner conditions. These . l show how the generators of this 3) dynamic chain, and a convenient − 3) sector we have 12 supercharges, | | ]= ] = 0 , satisfy H, G [ δH,G δH [ – 4 – 2) chain. In particular, we will find the origin | , and Q , 2 1 2 2, three complex scalar fields and two spinors. The H , 3) chain arises because we can find states with the ] = 0 | ]= = 1 α -matrix. H,Q δH,Q [ [ S 4) algebra splits into two equal pieces. Both SU(2 . | 2 N = 4 supersymmetric Yang-Mills can be constructed using the f , 3 and Z , (2 2 N 2) algebra need to be introduced [16]. These additional cent , | = 1 P SU i 3) integrable system [5]. In the SU(2 | , with ) in QCD. Physical states are singlets with respect to the cen i α N G -matrix of planar S Before constructing suitable representations for the SU(2 and From a physical point of view the role of this center is very si of SU( 2 i α N recall the SU(2 Q share a central chargealgebra. that But in behaves order aselements to the deal to hamiltonian, with the the whic dynamical SU(2 properties o elements act trivially onin physical a states non-trivial of vanishing wayrelevant tota on to the fix the magnon scattering constituents of the physic dynamics. We will then describeof how the the generators possible of irreps, theconditions central par algebra, determine are constrained the fr dispersionrapidity plane. relation In for section magnons, 3 we will wonder about the underly algebra is enlarged with afull U(1) and subalgebra the generated interacting by hamiltonians, the i of the Virasoro constraints and the dispersion relation on p central subalgebra. We willHopf provide affine evidence algebra that at it acenter should root at of such unity, with amagnon the root kinematics central Hopf of with sub unity. theunderlying quantum conditions In affine for symmetry section of the 4 the existence we sine-Gordon of model. will a identify2. th The central Hopf subalgebra In this section we willcentral describe extensions the geometry of underlying the the integrable SU(2 before doing that we will reviewchoice in of some detail representations. the SU(2 2.1 Dynamics and representations The that the complete Z non-trivial co-multiplication rules and antipodes.algebra turn We out wil to be the central elements added to SU(2 transform non-trivially under same quantum numbers and energy, but different length, which The dynamical nature of the SU(2 JHEP11(2006)021 , , , i i 3 i i 1 to ] 2] φ 2] 1 2), . ere − | i and ψ Z φ 2 1 ψ (2.5) (2.3) (2.4) (2.2) 2 1 Z 2] [1 [1 3 1 G 2 acting φ φ φ ψ | ψ , where Q ψ | [1 1 2 1 2 [1 a α φ φ Q Q | | G and 1 1 1 1 in SU(2 and with central 1 1 Q Q as i ' | B , 3] 2 β G a remove one i and φ φ R 2 | G 2 a α 1 1 ψ φ | , and we can now Q G 3 1 [1 i with and φ and Q 2] 1 b from the first and last φ α B ψ [1 1 Q φ upercharges φ | ly on the dynamics of the r two supercharges , respectively. We will now action 2 c . 2) is the existence of central harges ps. | ψ α β make the length of the chain a s, between δ , can be employed to define the b . a , entral term, and also of the central δ . Z and i leads to i 1 i 1 + Z i − 1 Z ψ . Using this we get a B , Ψ b j Z i | ψ are indeed the same state. However, as ab β φ 1 R 1] | ab ² i . We can act on β ψ ² ψ α − ij | a α | Z into δ ² αβ 2 1 Z ab αβ ² G Ψ 3 2 ² ² | + G αβ φ ψ 1 1 ² = β αβ α [1 – 5 – G ² } . After this we remove L i' b ψ and , and transform then the resulting state β 3 b | a i' i Ψ φ i δ | 3 β 3) dynamics into a deformed co-multiplication for a i' | ,Q b β φ Ψ b ψ a | = | 3 α and 2) sector, with only 8 supercharges φ Q i α | | φ } a α Q i | a α b β or, in general [16], Q { . Thus, 2] Q supercharges, is asymmetric and non-trivial. We will use 2 G i 2) symmetry algebra. It is generated by two bosonic gen- φ | φ , G a α Z [1 α a 2 G φ | φ Q { and . They transform 3 2. In this case we only have two scalar fields, and therefore th 1 . This action defines in a natural way a central term φ , . Using a fluctuation we transform now this state into i φ i ' | i Ψ 2 2] . Through the same argument as above we can consider | ψ 1 = 1 , together with the supersymmetry generators ψ | φ β 1 1 is playing the role of α [1 α Q L ψ on a formal state | Z through a fluctuation. After these formal manipulations, we 3 2 i Q and 3] 2 and 3 1 associated to the φ b , a 2 Q R R , φ c = 1 [1 3) chain to construct irreps. Consider for instance the two s φ | a | A direct consequence of these dynamic irreps for SU(2 Now, we will move to the SU(2 In order to construct the irreps, let us for instance conside , acting on the field 3 2 Following [16], we will extend the algebra with two central c or, in general [16], charge terms. In fact, we find that for any generic state because with respect to this algebra this observation to translate thecentral SU(2 Hopf subalgebra. 2.2 Dynamical co-multiplication rules Let us now introduceerators, the SU(2 define a fluctuation relating we will discuss in section 3,element the co-multiplication of this c produce, respectively, states to obtain Similar formal manipulations, with the spectator field irreps for the remaining set of supercharges, where the field which leads to on the scalar field now are no allowed fluctuations.SU(2 However we can still rely formal field from the original and final states, and read from this the into act with Q play a fundamental role in the definition of the different irre number of constituents. Fluctuationsdynamical between these variable. states In particular, the allowed fluctuation JHEP11(2006)021 w al- (2.6) (2.7) (2.9) (2.8) (2.11) (2.10) on with a given 3 , commutes with all , nvariance under the K i K Ψ as a first step to relate | ip and Z ∓ . e R ymmetric co-multiplication. eds to be introduced in the bra governing the integrable i , = . B tral elements is that they define i ] pf subalgebra must be equipped c r is part of its center. We can now ][ b [ ts on the allowed intertwiners. We R , Z 1) , ...Z... on defined by , B , ⊗ i − R . n 1 , 1 ...Z... )( ⊗ Ψ i − − | Ψ αβ 1 1 bd ² Ψ ± K ± ab ⊗ n Z z + 11 BK | ² Ψ ac − = K . = = K 11+ on the excited state will correspond to i – 6 – i } ...Z... ⊗ ⊗ ⊗ β Ψ | )=( Ψ b ) = | Z | d 1 K R B = 4 supersymmetric Yang-Mills with the existence B ± K ⊗ ipn , G ( ...Z... e = = c = α γ | a K N )( G b n R B K ipn { X e ⊗ ∆ . ∆ ∆ . It is immediate to check now that a = Z ( n ip i X − define with the co-multiplication (2.11) an abelian Hopf sub Ψ e | = K i ≡ 2), and thus belongs to the center of the algebra. With this ne | Ψ z ± and Z | R , = 4 Yang-Mills is, that the subalgebra B will be geometry and the dispersion relation N p denotes a generic magnon state. Following [16], an excitati Z i Ψ the eigenvalue | z In all our equations we implicitly use a graded multiplicati 3 The operators The first thing to be noticedan concerning action these additional on cen multiple magnonIn states order with to a non-trivial exhibitalgebra and through this as the co-multiplication, relation a new element ne where gebra, that we will denote by We will now useintegrability invariance in under the the planar central Hopf limit subalgebra of 2.3 Spec of an underlying Hopf algebra symmetry. Let us first show how i central subalgebra already implies non-trivialwill constrain assume, independently ofstructure what of the underlying Hopf alge momentum read from the co-multiplications (2.11)with that an the antipode central Ho operator we easily find the following co-multiplications fo the generators of SU(2 with and thus we find Therefore, inserting or removing a field JHEP11(2006)021 i− Ψ | (2.16) (2.15) (2.12) (2.14) (2.13) K , in the ( thus the α . Notice Z = cd i (2.15). , Ψ = | xist for irreps characterizing Z B y β defined by (2.13), ill characterize each Z and and In fact, this condition α ab 4 h the constraint imposed ress that in the previous = z . ling formula [40] the choice [36], and use Schur’s lemma ve a meaning of their own, . tes. x : they determine the explicit nto Spec tants ) multi-magnon physical states, Z at type in Spec e 1 )= rivial co-multiplication implied . to the central algebra . The dependence on the Yang- eads to relations between both ). In fact irreps for which an irectly from the structure ofgin the of (2.15) is thus independent − z K ] on the eigenvalues of the central eps. . Now, given two different irreps N z ( ( β , Z c Z − )). The plus sign will correspond to ∈Z = z 1), respectively. These relations arise z as gauge transformations, a ( 1 c − ∀ − . ± − ip y 1 (2 1 − y, π z, , − e , − KR , ) ( z αβ a Z − K β ( )= ), existence of an intertwiner requires – 7 – 21 2 R = as the spectrum of ( ) = ) = α, 1 + 4 , z cd 2 Z R K = p ( ( ,y 1 2 γ )=∆ γ 2 1 a x ± ( x = 1, to solve for the central extension, − x, π 12 1) and z bc )= ∆ − z − )= ( ip ) and ( c e B 1 ( ad ( ) is only valid once we have imposed the Virasoro constraints i α , z , π 1 } Ψ -plane defined by the function on which intertwiners for arbitrary pairs of points exist is = z ,y 1 Z i − | ab x Q, G , Ψ { | 1 R − some undetermined constants. Intertwiners will then only e K ( β and β needs to be done [16]. = B i 2 and g Ψ | α R We can now use the intertwiner condition (2.15) together wit = 2 Notice also that the interpretation of the central elements ), 4 i Ψ | Mills coupling constant appearsthe through intertwiner Fermat the curve. arbitrary Inαβ order cons to recover the BMN sca by the closure of simply means that physical states are singlets with respect same way as in QCD physical states are singlets under Next we will introduce a manifold Spec in [16] by imposing invariance under the central elements of with vanishing total momentum.independently of However, the these condition relationsform of ha vanishing of total the momentum derivation single we have magnon identified dispersion the origincentral relation. Hopf of algebra these and relations Let the d intertwinerof condition. us the The also condition ori of st vanishing total momentum on physical sta The region in Spec with satisfying (2.15). In the notation of reference [16], we hav branch cover of the that condition (2.15) are preciselyelements those introduced in [16 to construct a map from the space of irreps into Spec parameterized by ( intertwiner exist are characterized by the pair ( Notice that the antipode isby non-trivial the because dynamics of the ofirrep non-t the by the spin eigenvalues chain. of the Using generators now of Schur’s lemma we w When the co-multiplication isirreps. non-trivial Using this now the condition co-multiplication l (2.11) and the map i irreps for particles, and the minus sign to antiparticle irr condition (2.14) leads to the following set of curves of Ferm JHEP11(2006)021 goes (2.22) (2.20) (2.17) (2.18) (2.19) (2.21) ± ). They x z 1. When , ( rse irreps. 1, and the -plane. At c z eigenvalues, /z À ) are located 1 along a path → y z as a particular ( z ± are taken to be c → z ip α, β y and e z x , and ¶ 1 d weak coupling regimes. x . ) − 2 icles, respectively. Moving . . We get 1 β ent crossing on a generalized ¤ ± 2 o a change in − e defined by ) as coordinates, and we will 1 r x α z z ( , while αβ c z µ + 8 , the branch points 1 , ) , − 3 z r g ) . and αβ ( 1) 2 1 + 16 ¶ c 2 − z β 16 ) 2 − →∞ p 2 z define an elliptic curve. When written in 1) x z ( 1 − ± α 2 ( c ± ± − g αβ 1 1 2 − z ) define a double covering of the z g − − i z ( z ± αβ 2 + 16 ( = – 8 – z 3 )( 1 c x z ) with particle and antiparticle irreps. Let us first = ), or to zero for the negative branch. The magnon µ αβ αβ , depending on the choice of branch for z )= z 2 1 + 8 ( 1 ( ± z r = 4 £ y c c ( ). In particular, the branch points of − q 2 r z − 1 ) ( y 1 αβ x c (1 + 8 ig 8 − (1 + 16 2 r 1 1 = − 12 216 ( i ± z = = 2 3 )= g g z as the fundamental parameter, without assuming , subject to the constraint (2.15), to parameterize the dive ( = 1 with respect to the antipode transformation, r z z q z ), which together with z = 0, the branch points and ( z y − are large there are two possible irreps: those with generic , zx x close to 1, and those with generic values of β z = + and x α Let us now discuss the different magnon irreps in the strong an but with to infinity for the positive branch of charges are now given by representation. We first consider the strong coupling regim both And we will employ the self-dual point at We will use Weierstrass form, the elliptic invariants are curve degenerates to correspond to thefrom magnon one charges particle for irrep particles into and an antipart antiparticle irrep amounts t while which provides two different values, 2.4 The kinetic plane We will now parameterize different irreps using Together with going through the branch cuts of This is precisely therapidity curve plane. derived in In [28] the in strong order coupling regime to implem identify the two possible branches of translate these coordinates into the ones employed in [16], JHEP11(2006)021 2. 2) | / 1 both . All e I is (2.24) (2.23) ± β y -matrix -matrix. S S )= and z and . Thus, the ( α c Z x , type I irreps ip e ion of the SU(1 nergy will diverge, as are close to zero. In z ). β + z − and to understand the strong type II, respectively. The z 1, will start competing with α so that in this approach the )( rreps will diverge, something h large values of ' − s from the physical spectrum. coupling limit of the z z mat curves in Spec rong coupling regime the genuine − we now read z ( close to one, but with 2.15) are those with generic values of z + , and z xy , y = . However, those irreps with arbitrary z=z 2 xy . y y 1+ √ and p x and 1 2 '± – 9 – ± x ) − z ( -matrix can be directly derived from the )= II S z=z z c ( I 2) c | The algebraic curve close to zero. In the case of irreps of type II the momentum p 1 have finite energies. Thus the contribution of irreps of typ ' , but large values of z close to zero. For all of these irreps the energy is fixed at -matrix and crossing transformations z Figure 1: S y 2) | and 2) sector. In this section we will briefly discuss the derivat | x will be decoupled in the strong coupling limit, because its e SU(1 z We will now analyze the weak coupling regime, where both can reach generic values, but the magnon energy is , but with for the SU(1 As it was shown in [16], the SU(2 equal or close to zero. 2.5 The are those with momentum p large. We willmagnon refer energy to for these irreps representations of as type I of is type I and The existence ofcoupling these regime two of semiclassical kinds strings. ofBMN As we irreps, irreps reach with the generic is st and crucial finite values in of order which is finite for both particle and antiparticle irreps. If those with generic Therefore, the magnon chargesthat (2.18) we can for interpret theIn in antiparticle the terms weak i of coupling decoupling region of we also antiparticle find irreps with values of while the ones with the most natural onecoupling to constant the is strong a coupling free parameter region. labeling Notice different al Fer condition of strong coupling simply fixes a certain curve wit points on this curve are natural contributions to the strong this region, irreps satisfying the intertwinerz condition ( JHEP11(2006)021 1 π 2) er- | V irreps (2.27) (2.28) (2.29) (2.26) (2.25) i -matrix -matrix π S S V 2) , is the triv- | a ⊗ 2). Denoting by | ver the 2). The solution , e SU(2 the magnon rapidi- In this reference the (1 2) we will impose the , 11+ 11 iscussed in the previous 2) group was considered. (1 P R ertwiner condition (2.26) | ⊗ i dity. The main difference S a the different magnon irreps , et us consider two irreps algebra. This new condition nts. In this case, we can lift Z 2) 2) = ) = twiner conditions in the case of , ra , . a , ˜ (1 G , (1 2 , S i S 1 ˜ ) ⊗ Q , ∈Z P a ( a ⊗ x , 2) 1 to the space of irreps. We thus define ∀ 1 . , π + 11 2) that are not in SU(1 2 | − 2) in (2.25) by imposing the intertwiner 1 1 π (1 , Z z zy , i + 11 , − − − − S (1 z ¢ i K K ) 2)11. Since we are interested in the SU(2 S = a i = -matrix as = , ( ⊗ ⊗ )=∆ X – 10 – ¯ ¯ ¯ R y x z γ (1 a ˜ ˜ ¡ G Q ( 0 2 π S → → → π 1 2) = z y x π , ) = ) = = )= ˜ ˜ (1 G Q a . In order to fix the functions 2 , ( 2 i 2)∆ R 1 ¯ π π , ∆( ∆( P 2) algebra, and where V | i (1 ⊗ S 2) with a deformed co-multiplication, and an additional int P | 1 π 2) V is the one introduced in (2.7). , (1 -matrix through the co-multiplication (2.27) leading to an K 0 -matrix for a quantum deformation of the SU(1 R S R 2), and let us define the | 2) = , are the projector intertwiners, and with the sum extending o the additional supersymmetry generators, we can recover th (1 2 , of SU(1 i 1 R ˜ G 2 P -matrix was constrained by two conditions, which are the int π 2) are not associated with any form of affinization. Moreover, | R V It is worth to compare this construction with the one in [41]. Let us now consider the issue of crossing. As we have already d and -matrix, we can try to fix the functions -matrix, and the meaning of the crossing transformations. L ˜ Q condition (2.26), but for those elements on SU(2 affinization of the where the generator The for the generators oftwiner SU(1 condition involving thedepends extra on generators of the the spectralwith affine parameter, the which previous turns constructionSU(2 is to that be the the additional rapi ties inter enter into the This map leads to by imposing (2.26) with ial one S depending on the two magnon rapidities. subsections, due to the existence ofcan a be central Hopf characterized subalgeb bythe the action eigenvalues of of the the antipode central on eleme the generators of to (2.26) if we consider symmetric co-multiplications, ∆( intertwiner condition for any element in the SU(1 in the decomposition of where and S JHEP11(2006)021 , i 2 ss K (3.1) nR (2.30) and 2) | i F , in such a wing cycle are part of . , Z i l i that encodes A E K Z -matrix. We can R = 1, such that the and l fold given by Schur’s l i q we have employed the n on the origin of the F algebra Z , . In this section we will l i e of K E n parameter at a particular it an enlarged central Hopf al momentum through the definition of crossing on the efined through (2.11) in the ith generators nt to stress that these central ... , lgebra f a universal which is completely determined on. i α Q −→ . In this sense, this implementation with central extension i i Z A φ Z Z cross   y j β 2) algebra. Q | −→ | Spec Spec i = 1. In this case α with the rapidity plane allows a kinematical def- l ψ – 11 – π π q Z , but rather arise as a consequence of the quantum i α −−−−→ −−−−→ Q A −→ | , which cannot a priori be identified with SU(2   y Z Z i i γ A φ , let us first consider the kind of periodic irreps that we , together with the generator A 2 j β and “cross” is the crossing transformation. Notice that cro Q −→ | nR Z are in a one-to-one correspondence with irreps of being a root of unity. With this observation in mind, what we . . . 2) | Z q 2). We can construct two-dimensional subspaces by the follo | is the antipode in is the map from the central subalgebra into the rapidity mani γ π The most natural candidate to = 4 Yang-Mills case. This would provide a natural explanatio corresponding central subalgebra isN isomorphic to theadditional one central d elements extending the SU(2 3.1 Cyclic two-dimensional irreps In order to uncover the algebra can define in SU(2 of transformations, the generators of theelements central are Hopf not subalgebra. added to It the is algebra importa deformation parameter are searching for must be a Hopf algebra and a particular valu in the Cartan-Chevalley basis, and with Notice now that the identification of Spec by the definition of the antipode for the generators of inition of crossing as transformation on the rapidity plane of crossing only dependspace on of the irreps central is Hopf completely subalgebra. independent This of the existence o is a quantum group Hopf algebra,root with of the unity, quantum because deformatio subalgebra. quantum groups Let at us roots consider of to unity clarify exhib ideas a Hopf algebra w represent the previous implementation of crossing through where lemma, preserves the curves obtained from the intertwiner conditi 3. The Hopf algebra symmetry In the previous section we have identified an abelian Hopf sub pose the question of the existence of a Hopf algebra way that points in Spec the information onintertwiner the conditions. multi-magnon In states order tocentral of construct elements vanishing the of Hopf tot SU(2 suba JHEP11(2006)021 , 2 E / the 1 ∼ (3.6) (3.5) (3.4) (3.3) (3.2) Z K β 12 21 ψ ˜ Q \ SL(2)) at \ SL(2)) at operators. ( ∼ ( q q stency with i ˜ U , with G U φ i β K Q = , so that there are 2 is the central Hopf = 1, where K ji βα e affine Z 4 ˜ Q q or the supersymmetry challenge of uncovering t part of the underlying = ntify them with the two- ij αβ ...... ˜ ... , Q suitable formal way to respect onal vector spaces, to interpret end the Hopf algebra structure i α eps of the type 12 21 12 21 G ˜ ˜ G −→ Q −→ −→ . Dynamics was introduced in rep- i \ i G , SL(2)) with i ( 2 α 1 , and such that Z q , , ⊗ Q , ψ φ ψ A U R . B , j β j β 1 Q , − 1 G Q j β 12 21 12 21 αβ αβ ˜ ˜ − ² ² G K K⊗ G Q −→ | −→ | −→ | + + ij ij = 1. In fact, we can define operators i i i ² ² i α . The same holds true for the i α i 1 2 4 −K φ q 11+ G Q 11+ φ 12 12 ψ have non-trivial co-multiplications and, as con- = = whose central subalgebra ˜ – 12 – Q ⊗ ⊗ 2 2 ≡ ≡ i α R 12 21 ) ) ) = 12 21 A G ˜ ˜ G Q G −→ | Q −→ | −→ | Q = 4 Yang-Mills. ij αβ ij αβ ij αβ ij αβ ( i and i i ˜ ˜ ˜ ˜ = = and γ G Q 2 G Q α 1 N ( ( ψ φ ψ 12 21 B G Q ˜ being parts of the central Hopf subalgebra of Q ∆ ∆ j β 12 21 12 21 \ SL(2)) with R ˜ ˜ ( G G Q −→ | −→ | −→ | q U ...... and B operators, . In a similar way we can define β . The advantage of these definitions of irreps is formal consi , consistent with the one defining ˜ Q 2 G / 6= = 4 Yang-Mills. From (3.4) it follows that 1 α Z . The corresponding antipodes are i N and φ Z , and with Q F and ∼ would be part of the Cartan subalgebra of j β ∼ ψ 6= K i α 12 21 i G Q As a step further, we will also suggest a co-multiplication f = 1, which therefore appears as a natural candidate to at leas = 1. This should be just considered as a formal hint toward the 4 4 generator in where the fact that the centralsequence, elements non-trivial antipodes. Thusby we requiring will formally ext and q Hopf algebra of the planar limit of generators resentation theory through fluctuations ofthese the fluctuation form equivalences (2.4). would A be the definition of irr q subalgebra of the underlying Hopf symmetry algebra only two different Thus, we find the adequate number of operators for the map to th and In this way we can think of (3.5) as a cyclic irrep of Then, the cyclic representation can be chosen as satisfying It is of course very tempting,them once we as have these some two-dimensi dimensional sort cyclic of irreps cyclic of irreps. In particular, we will ide with JHEP11(2006)021 5 θ is θ (4.4) (4.2) (4.3) (4.5) (4.1) (3.7) en in = 2. cosh / β cosh 1 2) as the / 1 p/ ' ( ) 2 ' θ ( sin E E 2 λ/π 2) with the momentum the dispersion relation, 1+ p/ n system we have xcitations. don solitons, with the only ( p soliton energy. Introducing ns on semiclassical strings. 2 anti-ferromagnetic isotropic . ) case we have the coupling constant, sin h the co-multiplication and the )= βφ 2 p . , on the Dirac sea of Bethe strings, ( . ) ) cos( E θ φ λ/π ´ z 2). Let us recall that the low lying . 2 p 2 d θ p/ ³ Z = 1). sin(2 2 θ . (sinh 1 1 2 G . λ π 1 2 sin( − sin cosh m + − 2 ' −K λ φ = π tan ¯ = z ) (i.e. = cosh ´ p – 13 – φ − φ∂ ( = 1 8 p 2 ¤ z ) = sG p π 2 E σ are equivalent to the sine-Gordon integrable model ³ = G θ z∂ E ∂ ( 2 2 λ ' 2 d γ S sin . Notice also that the magnon energy − ) θ 2 Z τ , we should identify sG × ( sinh ∂ 2 π p 1 £ p R 4 /E = 2 and + = β 2 S , the soliton energy is given by m θ = [42]. The formal relation with the sine-Gordon model is hidd 2 1 2 E 2), goes like 1 p/ , as well as with relations (3.4). Z sin( λ/π √ This giant magnons of [17] correspond precisely to sine-Gor If we move now into weaker coupling and interpret We are normalizing the sine-Gordon model as 5 ' which leads to the dispersion relation and and they have spin excitations for the anti-ferromagnetic chain are the holes difference that the magnonthe energy sine-Gordon goes rapidity like the inverse of the Through the map of the string sigma model into the sine-Gordo strictly the same asHeisenberg the chain in one the of thermodynamic elementary limit. excitations In of fact in the this the special form of the rapidity dependence of the momentum e Thus, the magnonE energy, given by the large coupling limit of relativistic relation [25, 26]. The Virasoro constraints lead to square. Thus, a natural definition of rapidity is [17] This sine-Gordon model corresponds to a particular value of Thus, in the string case we have Notice that now (3.6) andantipode (3.7) of are perfectly consistent wit 4. Magnon kinematics and the sine-GordonIn model this section we will explore the kinematics of giant magno Semiclassical strings moving in JHEP11(2006)021 and and (4.9) (4.8) (4.7) (4.10) π ± olitons \ defined 2 SL(2)), Q ( / ) we get, p q 2 − 1) U ¯ Q − 1 − , with − . This provides θ 2 − z ip ( Q − λ , if we introduce an y side is isomorphic e )( dity in terms of the √ K 2 + = , ¯ = sinh Q π tum algebra wn result that the non- z = 1 is isomorphic to the algebra of symmetries of p 2 − and / n solitons in non-classical lassical irreps we also have 4 tum through [43] 2 + q F 1) . We can now represent the . In this case we can use the analysis. t magnons, up to the change e I we have a generic value of Q , y the same central subalgebra. 2 ± E . − 2) for vial way. The kinematic arena θ ¯ lation E F Q z − = ( , p/ ( ]. e ( 2 λ 1 2 1) + 1 (4.6) p and − and √ 2 ± − 2 ¯ K sin Q xy ± F , 1 2 E 4 θ (SL(2)), = , Q − = e q z p 2 − U ¯ πi λ/π P β 2 = ¯ ± Q 1)( − − = given by e , − xy , 2 2 – 14 – 2 ± = 1 we also have classical irreps, with = z p . If we write √ K ( K Q q 2 ¯ 4 2 P q F λ ∓ = π θ = E, Q − − θ = = e e P K P = 4 Yang-Mills. Thus, the constituents on both sides of p 2 θ = = p = e N + + and p ¯ Q Q 2 \ SL(2)), together with for ( F q Z , U 2 = 2. This affine quantum algebra at by the Serre relations. These are the standard sine-Gordon s E β ± ¯ Q \ SL(2)) in terms of the ones of ( and q . p U ± θ . Nicely enough, this central subalgebra on the string theor Q = 1 when 6 4 in the center of q = 0, 2 θ K Let us now try to understand the physical meaning of this rapi being non-vanishing elements in the center. For these non-c A similar phenomena takes place in the chiral Potts model [44 \ SL(2)) = 2 supersymmetry algebra, with generators = 6 ( ± q , respectively. This is just the relation ¯ when with the physical momentum elements in the center to construct a candidate for the momen K Q in terms of that are directly connectedin at the strong energy coupling with relations. the gian However for For regular solitonic irreps we get from here the standard re affine parameter, that plays the role of a rapidity, generators of with N the correspondence share the same kinematic arena, defined b to the central subalgebra The study of these common features clearly deserves further further evidence that magnonsirreps where must the be central relatedfor subalgebra these to is magnons realized sine-Gordo is inU determined a non-tri by the Spec of the central sub for the eigenvalues of quantum symmetries of thelocal sine-Gordon charges model. in It the is sine-Gordon a model well generate kno an affine quan so that From this identification we get Notice that in the strongthe coupling rapidity limit and for irreps of typ z JHEP11(2006)021 , , B 5 Phys. 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