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Infinite Quantum Group Symmetry of Fields

arXiv:hep-th/9108007v1 20 Aug 1991CLNS 91-1056Infinite Quantum Group Symmetry of Fieldsin Massive 2D Quantum Field TheoryAndr´e LeClairNewman LaboratoryCornell UniversityIthaca, NY 14853andF. A. SmirnovLeningrad Branch of the Steklov Mathematical InstituteFontanka 27, Leningrad 191011, USSRStarting from a given S-matrix of an integrable quantum field theory in 1 + 1 dimen-sions, and knowledge of its on-shell quantum group symmetries, we describe how to extendthe symmetry to the space of fields.

This is accomplished by introducing an adjoint actionof the symmetry generators on fields, and specifying the form factors of descendents. Thebraiding relations of quantum field multiplets is shown to be given by the universal R-matrix.

We develop in some detail the case of infinite dimensional Yangian symmetry. Weshow that the quantum double of the Yangian is a Hopf algebra deformation of a level zeroKac-Moody algebra that preserves its finite dimensional Lie subalgebra.

The fields forminfinite dimensional Verma-module representations; in particular the energy-momentumtensor and isotopic current are in the same multiplet.3/91

1. IntroductionThe study of quantum field theory in 1+1 spacetime dimensions can provide importantinsight into the structure of quantum field theory in general.

Of particular interest arethe exactly solvable massive models, which are integrable Hamiltonian systems. In manycases, the exact on-shell S-matrices are known.

(For some well-known examples see [1] . )The exact form factors for some fields in many of the models have also been computed [2] .Despite the integrability of the models, many of their important properties, most notablytheir correlation functions, remain unknown.More recently it has been recognized that the massive integrable quantum field theoriesare to a large extent characterized by quantum group symmetries, which can serve todefine the theory non-perturbatively [3][4] [5] .

The quantum group symmetry algebrasof interest are infinite dimensional, and non-abelian. Explicit currents that generate aq-deformation of affine Kac-Moody algebras [6] [7] were constructed in the sine-Gordontheory and its generalization to imaginary coupling affine Toda theory [5], and shown tocompletely characterize the S-matrices.

At a specific value of the coupling constant wherethe theory is invariant with respect to a finite dimensional Lie algebra G, these theoriesare the Gross-Neveu models, and there the infinite dimensional symmetry becomes theYangian of Drinfel’d, which contains G as a subalgebra. The non-local conserved currentsthat generate this symmetry were actually constructed some time ago by L¨uscher[8] .However the connection of these currents to Yangians was only recently recognized byBernard [4].The existence of these infinite dimensional symmetries leads to the idea that thesemodels may be solvable in the spirit that the conformal field theories are solved [9] .

Inthis methodology the emphasis is on how the fields realize infinite dimensional Verma-module representations of the symmetry algebra.In this work, assuming the exact S-matrix and its on-shell symmetry for a generalmodel are given, we describe how the symmetry can be extended off-shell to the space offields. We begin by defining an adjoint action of the symmetry algebra on fields, whichgeneralizes the notion of commutator one encounters for ordinary symmetries, to an ar-bitrary quantum group symmetry.

Form factors of ‘descendent’ fields are shown to beexplicitly computable from the form factors of ‘ancestor’ fields. The braiding of fields inthe multiplets obtained by adjoint action is given by the universal R-matrix of the quan-tum group.

A result of this kind was also found for the finite dimensional quantum groupsymmetry of conformal field theory by Gomez and Sierra[10].1

The example we develop in some detail, which displays all of the interesting issuesinvolved, is characterized by sl(2) Yangian symmetry. As mentioned above, these modelscan be formulated as the G-invariant Gross-Neveu model [11] , where G is here taken tobe sl(2).

This model can also be formulated in a way more suitable for our discussion asa massive current-current perturbation of the G-invariant Wess-Zumino-Witten-Novikov(WZWN) model at level k = 1, with the actionS = Slevel 1G+ gZd2x Ja(x)Ja(x),(1.1)where Ja, Ja are the level 1 Kac-Moody currents of the WZWN conformal field theory SG.For this model, we are able to construct infinite dimensional multiplets through adjointaction. The derivative of the sl(2) current is actually a descendent of the energy-momentumtensor.

We construct the appropriate braiding matrix by studying the quantum double forthe Yangian. These braiding matrices are infinite dimensional.It is interesting to compare some of the features of the massive verses the conformaltheory.

The conformal model described by SG can be completely solved using its Kac-Moody symmetry [12] [13], where the Kac-Moody generators are modes of the currents,Jan =Hdz2πi zn Ja(z), n ∈Z. These generators satisfy the algebraJan, Jbm= f abc Jcn+m + 12k n δab δn,−m,(1.2)The full symmetry of the conformal model also includes and additional antiholomorphicKac-Moody symmetry generated by Jan =Hdz2πi zn Ja(z).

As we will describe, the Yangianalgebra of the massive theory is a deformation of a Kac-Moody algebra that preserves itsfinite dimensional Lie subalgebra G. However the Yangian symmetry does not appear tobe related in any simple way to the Kac-Moody symmetry of the conformal model. In fact,setting the deformation parameter of the Yangian to zero one recovers a level 0 Kac-Moodyalgebra.

This zero value for the level is a general feature of the infinite dimensional algebrasrelevant for the massive theories. Indeed the theories with q-deformed affine Kac-Moodysymmetry, which the Yangian invariant theories are a limiting case of, also have zero level[5].This zero value of the level has a physical explanation.

Level zero Kac-Moody algebras(sometimes called loop algebras) possess finite dimensional representations. In conformalfield theory, since there is a one-to-one correspondence between fields and states, a non-zerolevel is required for the infinite dimensional representations corresponding to the fields.

In2

contrast, for massive models, if the symmetry algebra commutes with the particle numberoperator, then the symmetry algebra must possess finite dimensional representations onthe space of particle states, since the number of states at fixed particle number is finite.Thus, assuming that at least some of the representations of the Kac-Moody algebra canbe deformed into representations of the Yangian, the level must be zero.However, indistinction with conformal field theory, since there is not this correspondence betweenstates and fields in massive theory, the fields of the theory may still form distinct infinitedimensional representations.This paper is organized as follows. In the first section we review some aspects of thetheory of quantum groups that we will need.

The general, model independent formalismis presented in section 3. The example of Yangian symmetry is treated in section 4.2.

Quantum GroupsIn this section we review some general aspects of the theory of quantum groups, fol-lowing mainly the presentation of Drinfel’d [14] and Faddeev, Reshetikhin, and Takhtajan[15] .Quantum groups are non-trivial examples of Hopf algebras. Let A denote such a Hopfalgebra.

It is equipped with a multiplication map m : A ⊗A →A, a comultiplication ∆:A →A ⊗A, antipode s : A →A, and counit ε : A →C, where C is the complex numbers.We suppose that A contains the unit element 1, with ∆(1) = 1⊗1, s(1) = ε(1) = 1. Theseoperations have the following properties:m(a ⊗1) = m(1 ⊗a) = a(2.1a)m(m ⊗id) = m(id ⊗m)(2.1b)(∆⊗id)∆= (id ⊗∆)∆(2.1c)∆(a)∆(b) = ∆(ab)(2.1d)m(s ⊗id)∆(a) = m(id ⊗s)∆(a) = ε(a) · 1(2.1e)s(ab) = s(b)s(a)(2.1f)∆s = (s ⊗s)P∆(2.1g)(ε ⊗id)∆= (id ⊗ε)∆= id(2.1h)ε(ab) = ε(a)ε(b),(2.1i)3

for a, b ∈A, and P is the permutation operator P(a ⊗b) = b ⊗a. Eq.

(2.1a)is thedefinition of the unit element, (2.1b, c) are the associativity and coassociativity ofA ,(2.1d) defines ∆to be a homomorphism of A to A ⊗A, and (2.1e −i) are the definingproperties of the counit and antipode.Let {ea} denote a linear basis for A . The above operations can be formulated bymeans of structure constants:ea eb = mcab ec(2.2a)∆(ea) = µbca eb ⊗ec(2.2b)s(ea) = sba eb(2.2c)ε(ea) = εa,(2.2d)where mcab, µbca , sba, and εa are constants in C. We also define constants εa such that1 = εaea.

(2.3)The properties (2.1) are easily expressed as consistency conditions on the structureconstants:mcab εa = mcba εa = δcb(2.4a)mcab mbde = mbad mcbe(2.4b)µbca µdeb = µdba µecb(2.4c)µija µklb mcik mdjl = miab µcdi(2.4d)mbij sjk µika = mbji sjk µkia = εaεb(2.4e)miab sci = sjb ska mcjk(2.4f)µabi sic = saj sbk µkjc(2.4g)µbca εc = µcba εc = δba(2.4h)mcab εc = εaεb(2.4i)For the applications we are interested in, we need to introduce the concept of thequantum double. Let A∗denote an algebra dual to A in the quantum double sense, with4

basis {ea}. The elements of the dual basis are defined to satisfy the following relations:eaeb = µabc ec(2.5a)∆(ea) = mabc ec ⊗eb(2.5b)s(ea) = (s−1)ab eb(2.5c)ε(ea) = εa(2.5d)εaea = 1.

(2.5e)Due to the properties of the structure constants (2.4), one has a Hopf algebra structureon A∗.The quantum double is a Hopf algebra structure on the space A ⊗A∗. The relationsbetween the elements ea and the dual elements ea are defined as follows.Define thepermuted comultiplication ∆′:∆′(ea) = µbca ec ⊗eb,(2.6)and skew antipode s′:m(s′ ⊗1)∆′(a) = εa · 1.

(2.7)The universal R-matrix is an element of A ⊗A∗defined byR =Xaea ⊗ea,(2.8)satisfyingR ∆(ea) = ∆′(ea)R,(2.9)for all ea. The equation (2.9) implies the following relation between elements of A andA∗:µbca mdib ei ec = µcba mdbi ec ei.

(2.10)Using the property (2.7) of s′, and (2.4b, c), the above equation can be written asea eb = µkclambidk s′il ed ec(2.11a)eb ea = µlckambkdi s′il ec ed(2.11b)whereµlcka= µlia µcki ;mbkdm = mikd mbim. (2.12)5

The universal R-matrix constructed in this fashion satisfies the Yang-Baxter equationR12R13R23 = R23R13R12. (2.13)One also hasR−1 =Xas(ea) ⊗ea.

(2.14)When A has an invariant bilinear form, A∗can often be identified with A , and Rbecomes an element of A⊗A. However in some important examples, such as the Yangian,this is not the case.3.

General TheoryLet us suppose we are given the exact factorizable S-matrix of an integrable quantumfield theory. As usual, the energy-momentum of an on-shell particle is parameterized bythe rapidity βp0(β) = m cosh(β),p1(β) = m sinh(β).

(3.1)In general, the one-particle states have isotopic degrees of freedom taking values in somevector space V . A complete infinite dimensional set of states is provided by the multipar-ticle basis|B⟩= |βn, βn−1, ...., β1⟩.

(3.2)The above state is a vector in V ⊗n; the isotopic indices are not explicitly displayed. In-states have βn > βn−1 > · · · > β1, whereas out-states have β1 > β2 > · · · > βn.The two-particle to two-particle S-matrix is an operatorS12(β1 −β2) :V ⊗V →V ⊗V.

(3.3)The S-matrix S12(β1 −β2) satisfies the usual constraints of the Yang-Baxter equations,unitarity and crossing symmetry [1]:S12(β1 −β2)S13(β1 −β3)S23(β2 −β3) = S23(β2 −β3)S13(β1 −β3)S12(β1 −β2)(3.4a)S12(β)S21(−β) = 1(3.4b)S12(iπ −β) = c2S21(β)c2,(3.4c)6

where c is the matrix of charge conjugation, and the subscripts refer to the space wherethe operator acts. The multiparticle S-matrix is factorized into 2-particle S-matrices:Si,i+1(βi −βi+1) |βn, βn−1, .., βi+1, βi, .., β1⟩= |βn, βn−1, .., βi, βi+1, .., β1⟩.

(3.5)We further suppose that the S-matrix is invariant under some symmetry algebra A .The algebra A is assumed to be a quantum group with the structures outlined in section2. The space of multiparticle states H forms an infinite dimensional representation ρHof the algebraA .

It is assumed that the generators ofA commute with the particlenumber operator. This implies that the representation ρH is necessarily reducible intofinite dimensional representations of fixed particle number.

(More specifically we havefinite dimensional representations at fixed rapidities, i.e. we do not count the additionalmultiplicity involved in varying the rapidity.

)Given the representation ρH of A on one-particle states, which is in general rapidity dependent, the representation on multiparticlestates is provided by the comultiplication ∆. The on-shell symmetry of the S-matrix is thestatementS12(β1 −β2) ρV ⊗ρV (∆(ea)) = ρV ⊗ρV (∆′(ea)) S12(β1 −β2).

(3.6)For the models of interest, (3.6) completely characterizes the S-matrix. Comparingeq.

(3.6) with the defining equation (2.9) for the universal R-matrix it is evident that theS-matrix is a specialization of the R-matrix to the finite dimensional rapidity dependentrepresentation ρV of A (up to overall scalar factors s0(β) required for unitarity and crossingsymmetry):S12(β) = s0(β) ρV ⊗ρV (R) . (3.7)It is of primary interest to extend the above on-shell symmetries to the space of fields.This problem is more complex than for the conformal field theories since there is no one-to-one correspondence between fields and states here.

The fields in the theory are completelydefined by specifying their matrix elements in the space of states H, i.e. their form factors.Let us begin by reviewing some basic properties of form factors in integrable quantumfield theory [2].

The form factors f(β1, ...., βn) of a field φ(x) are defined as the particularmatrix elementsf(β1, ..., βn) = ⟨0| φ(0) |βn, βn−1, ..., β1⟩. (3.8)The form factor in (3.8) is a map from V ⊗n to a function of n rapidity variables; it is thusa vector in the dual space (V ⊗n)∗.7

The form factors satisfy the following axiomatic properties:Si,i+1(βi −βi+1)f(β1, ..., βi, βi+1, ..., βn) = f(β1, ..., βi+1, βi, ..., βn),(3.9)and for local fieldsf(β1, ..., βn−1, βn + 2πi) = f(βn, β1, β2, ..., βn−1). (3.10)The form factors also satisfy a third axiom that relates residues of n-particle form factorsto n −1-particle form factors, but we will not need it here.

The general matrix elementsfollow from the functions (3.8) by crossing symmetry⟨α1, ..., αm| φ(0) |βn, ..., β1⟩= c1 · ·cm f(αm −iπ, ..., α1 −iπ, β1, ..., βn). (3.11)(The equation (3.11) is correct as it stands for the set of rapidities {α1, ..., αm} disconnectedfrom the set {β1, ..., βn}.

When some of the rapidities α coincide with β, there are someadditional δ-function terms in (3.11). ) Finally the x dependence of the matrix elementscan be trivially restored due to translational invariance⟨α1, ..., αm| φ(x) |βn, ..., β1⟩= exp ixµ mXi=1pµ(αi) −nXi=1pµ(βi)!

!· ⟨α1, ..., αm| φ(0) |βn, ..., β1⟩. (3.12)For the more familiar symmetries of quantum field theory, such as global Lie-algebrainvariance, the symmetry is realized on the space of fields through the commutator withthe global conserved charges.

The Jacobi identity ensures that fields related by this adjointaction fall into finite dimensional representations of the symmetry algebra. It is this featurethat we now generalize to an arbitrary quantum group A .

We define two adjoint actionson the space of fields:1adea (φ(x)) ≡µija s(ei) φ(x) ej(3.13a)ad′ea (φ(x)) ≡µija s′(ej) φ(x) ei . (3.13b)1 For arbitrary Hopf algebras there exists a standard definition of the adjoint action of thealgebra on itself which is analagous to the adjoint action on fields defined below [16].8

The significance of the two different adjoint actions will be explained shortly. The adjointactions as we have defined them are primarily characterized as being homomorphismsad, ad′ : Ahomo.−→A φ(x) A.

(3.14)More precisely,adea (adeb (φ(x))) = mcab adec (φ(x)) . (3.15)This follows from the fact that the comultiplication ∆is a homomorphism fromA toA ⊗A (2.1d) , and from (2.1f) .

For elements a ∈A with the trivial comultiplication∆(a) = a ⊗1 + 1 ⊗a, and counit ε(a) = 0, ada is just the usual commutator.Repeated adjoint action on a given field φ(x) generates a tower of its ‘descendents’:φ(x)ad−→{φa1,a2,...(x)} ≡Φ(x)(3.16a)φ(x)ad′−→{φ′a1,a2,...(x)} ≡Φ′(x). (3.16b)The form factors of any descendent of a field φ(x) are explicitly computable from knowledgeof the form factors for φ(x) and the given action ρH of A on the space of multiparticlestates.This is evident from the definition (3.13), where when computing the matrixelements of adea (φ(x)) the elements of A to the right or left of the field give a knowntransformation on states.By virtue of the symmetry of the S-matrix (3.6), the descendent form factors satisfythe axiom (3.9).However they do not generally satisfy (3.10), hence descendents aregenerally not local fields.

This is due to the non-locality of the conserved currents thatgenerate A . Indeed, as explained in [4][5], and similarly for conformal field theories in[10], the non-trivial comultiplication of A is due to the non-locality of these currents.A highest weight field is defined as a field which cannot be reached by adjoint actionon another field.

A precise definition of highest weight field requires a definition of raisingand lowering operators for the algebra A , or at least for its quantum double. The towersof descendents (3.16) of a highest weight field by construction fill out a representationρΛofA , denoted ΦΛ(x).

This is due to the property (3.15) of the adjoint action. Morespecifically,adea (ΦΛ(x)) = ρΛ(ea)ΦΛ(x).

(3.17)9

The fields are thus intertwiners for A .Specific models are characterized by which representations ρΛ span its field content.For algebras A generated by a finite number of elements these representations are expectedto be finite dimensional. For the case where A is simply a finite dimensional Lie algebra,this corresponds to the fact that fields are irreducible tensors.

For infinite dimensionalalgebrasA the possibility arises that the representations ρΛ are infinite dimensional,Verma-module representations. For the example considered in the next section, this is thecase.The fields ΦΛ(x) are in part characterized by their braiding relations.

We will provethe fundamental braiding relationΦΛ2(y, t) ΦΛ1(x, t) = RρΛ1 ,ρΛ2 ΦΛ1(x) ΦΛ2(y)x < y,(3.18)where RρΛ1 ,ρΛ2 is the universal R matrix specialized to the representations ρΛ1,2 of thefields ΦΛ1,2:RρΛ1 ,ρΛ2 = ρΛ1 ⊗ρΛ2 (R) . (3.19)One way to prove (3.18) is to use a generalization of the locality theorem used in [3].One first uses this locality theorem to show thatΦΛ1(x, t) Φ′Λ2(y, t) = Φ′Λ2(y, t) ΦΛ1(x, t)x < y.

(3.20)The proof of (3.20) is model dependent; however it is ultimately due to the crossingsymmetry of the S-matrix.The relation (3.20) will be established for the example ofthe next section.The fields ΦΛ(x) and Φ′Λ(x) are related by the following expression:Φ′Λ(x) = RρH,ρΛ ΦΛ(x),(3.21)where RρH,ρΛ is again the universal R-matrix evaluated in the representations indicated.The formula (3.21) is established by showing that RρH,ρΛ must satisfy its defining relation(2.9). To prove the last statement, note thatρH ⊗ρΛ (∆(ea)) ΦΛ(x) = µija ei adej (ΦΛ(x))= µija µklj ei s(ek) ΦΛ(x) el= ΦΛ(x) ea,(3.22)10

where in the last step we used the coassociativity (2.4c) , the defining properties of theantipode (2.4e) , and (2.4h) . Similarly,ρH ⊗ρΛ (∆′(ea)) Φ′Λ(x) = Φ′Λ(x) ea.

(3.23)Therefore Φ′Λ(x)ea = RρH,ρΛΦΛ(x)ea impliesρH ⊗ρΛ (∆′(ea)) RρH,ρΛΦΛ(x) = RρH,ρΛ (ρH ⊗ρΛ) (∆(ea)) ΦΛ(x),(3.24)which establishes the relation (2.9) for RρH,ρΛ.The relation (3.21) can be inserted into (3.20) to derive the braiding relation (3.18)for the fields. We haveΦΛ2(y) ΦΛ1(x) = R−1ρH,ρΛ2 ΦΛ1(x)RρH,ρΛ2 ΦΛ2(y)= s(ei)ΦΛ1(x)ej ρΛ2(ei)ρΛ2(ej) ΦΛ2(y)= (ρΛ1(ea)ΦΛ1(x)) (ρΛ2(ea)ΦΛ2(y)) ,(3.25)where we have used (2.8), and the homomorphism property (3.15) for the relation (2.5a)in the dual space.

From the expression (2.8) for R, one has thus established (3.18).The above proof of the braiding relation (3.18) relied on (3.20), which is established ina specific model using form factors. Eq.

(3.18) can be understood in an alternative modelindependent way. We first show that the adjoint action on a product of fields is given bythe comultiplication, namely:adea (ΦΛ2(y) ΦΛ1(x)) = µbca s(eb) ΦΛ2(y) ΦΛ1(x) ec= µbca adeb (ΦΛ2(y)) adec (ΦΛ1(x))= ρΛ2 ⊗ρΛ1 (∆(ea)) ΦΛ2(y) ΦΛ1(x).

(3.26)Above we have used the coassociativity (2.4c) and (2.4h) . Applying adea to both sidesof (3.18), and using (3.26), one finds that R must satisfy the defining property of theuniversal R-matrix (2.9).Let us suppose that we can associate a conserved current Jµa (x), satisfying ∂µJµa (x) =0, to an element ea of A , such that ea =RdxJta(x).

The braiding of these currents withother fields can be expressed in a simple way. Note first that due to the (co)associativity11

(2.4b, c) of the algebra A , there exists an adjoint representation ρadj of A , A∗whosematrix elements are given in terms of the structure constants:⟨b|ρadj(ea)|c⟩= mbac(3.27a)⟨c|ρadj(ea)|b⟩= µabc(3.27b)This fact is analogous to the fact that the structure constants of a Lie algebra form theadjoint representation of the algebra due to the Jacobi identity. The currents Jµa (x) areassociated to the fields in Φadj(x).

The braiding relation (3.18) for ρΛ2 = ρadj reads:Jµa (y) ΦΛ(x) = ρΛRbaΦΛ(x) Jµb (y)x < y,(3.28)whereRba ≡µcba ec. (3.29)The above equations imply that the braiding with the currents can be simply determinedfrom the comultiplication ∆.4.

The example of Yangian SymmetryIn this section we apply the general theory of the last section to models with Yangiansymmetry. As explained in the introduction these models can be described by the action(1.1).

For simplicity we consider only the case of sl(2).4.1Symmetries of the S-matrixThe spectrum of particles in this model is known to consist only of kinks |β; ±⟩trans-forming under the 2-dimensional spin-1/2 representation |±⟩of sl(2). The S-matrix hasthe following simple formS12(β) = s0(β) (β −iπP12) ,(4.1)where P12 is the permutation operator:P12 = 12 3Xa=1σa ⊗σa + 1!,(4.2)(σa are the Pauli spin matrices), ands0(β) = Γ(1/2 + β/2πi) Γ(−β/2πi)Γ(1/2 −β/2πi) Γ(β/2πi) .

(4.3)12

We now describe the symmetries of the S-matrix (4.1). Let Qa0, a = 1, 2, 3 denote theglobal sl(2) charges satisfyingQa0, Qb0= f abcQc0(4.4)f abc = iǫabc.Define an additional charge Qa1, satisfyingQa0, Qb1= f abcQc1.

(4.5)These charges have the following action on 1-particle states:Qa0 |β⟩= ta |β⟩,Qa1 |β⟩= taβ |β⟩,(4.6)where ta = σa/2, and |±⟩are eigenstates of t3, t3|±⟩= ±|±⟩/2. The action on multiparticlestates is given by the comultiplication∆(Qa0) = Qa0 ⊗1 + 1 ⊗Qa0(4.7a)∆(Qa1) = Qa1 ⊗1 + 1 ⊗Qa1 + α f abc Qb0 ⊗Qc0,(4.7b)whereα = −2πicA,(4.8)and cA = 2 is the quadratic casimir in the adjoint representationf abcf bcd = −cAδad.

(4.9)The comultiplication (4.7) impliesQa0 |βn, βn−1, ..., β1⟩=nXi=1tai |βn, βn−1, ..., β1⟩(4.10)Qa1 |βn, ..., β1⟩=nXi=1tai βi + αf abc Xi>jtbi tcj|βn, ..., β1⟩≡T a(β1, . .

. , βn) |βn, .

. ., β1⟩.

(4.11)That these are symmetries of the S-matrix, i.e. that (3.6) is satisfied, is easily checked byexplicit computation.

The value (4.8) for α is a consequence of the crossing symmetry ofS12.13

The equations (4.4), (4.5), and (4.7), are part of the defining relations of the YangianY, as formulated by Drinfel’d [14]. The complete set of relations for the Yangian includethe additional ‘terrific’ 2 relationsQa1,Qb1, Qc0−Qa0,Qb1, Qc1= α2Aabcdef{Qd0, Qe0, Qf0}(4.12a)Qa1, Qb1,Qc0, Qd1+Qc1, Qd1,Qa0, Qb1= α2 Aabklmnf cdk + Acdklmnf abk{Ql0, Qm0 , Qn1},(4.12b)where Aabcdef = f adkf belf cfmf klm, and {, } denotes symmetrization{x1, x2, ..., xn} = 1n!Xi1̸=i2···̸=inxi1xi2 · · · xin.

(4.13)These last relations (4.12) were constructed to be consistent with the comultiplication.For sl(2), eq. (4.12a) is superfluous, i.e.

it follows from (4.5) with the right hand side of(4.12a) equal to zero. For other groups, (4.12b) follows from (4.5) and (4.12a).The remaining Hopf algebra properties of the Yangian areε(Qa0) = ε(Qa1) = 0,(4.14)s(Qa0) = −Qa0;s(Qa1) = −Qa1 + αf abcQb0Qc0.

(4.15)The antipodes (4.15) are derived from the definition (2.1e) .The Yangian is a Hopf algebra for any constant value of the parameter α. Settingα = 0 one recovers half of the Kac-Moody algebra. By ‘half’ we mean the generators witheither positive or negative eigenvalue of the derivation element, or positive or negativefrequency modes in the sense of current algebra.

The Yangian is the unique deformationof the Kac-Moody algebra that preserves the finite dimensional Lie subalgebra (4.4). Ithas a natural grading where Qa0 has degree zero, and Qa1 and α have degree +1.

Elementsof the Yangian corresponding to the other half of the Kac-Moody algebra are missingin Drinfeld’s work. For our purposes, more specifically for constructing the universal R-matrix that characterizes the braiding of fields, it is necessary to introduce the quantumdouble of the Yangian D(Y).

The deformation of the other half of the Kac-Moody algebrais located in D(Y) as we will show in the next section.2We are here quoting Drinfel’d.14

3.2 The Yangian Double and Deformations of the Full Kac-Moody AlgebraAs explained in the last section, for our goals an extension of the Yangian Y is neededwhich we are now going to describe. For the sake of simplicity we consider only the sl(2)case 3.

For this case, the second relation in (4.12) can be rewritten as followsf acd[[Qc1, Qd1], Qb1] + f bcd[[Qc1, Qd1], Qa1] = −2α2 f acd{Qb0, Qc0, Qd1} + f bcd{Qa0, Qc0, Qd1}(4.16)(by multiplying (4.12b) by f abif cdj), while the first relation is satisfied trivially. As men-tioned above, the Yangian is a deformation of the universal enveloping algebra of thesubalgebra bsl(2)+ of the Kac-Moody algebra bsl(2).

In this work, for reasons that will soonbe apparent, we need only consider the bsl(2) loop algebra, which is obtained from theKac-Moody algebra by setting the level to zero. We begin by presenting several simplefacts concerning the loop algebra itself.The loop algebra bsl(2) has the set of generators Jan, n ∈Z satisfying the relations[Jam, Jbn] = f abcJcm+n(4.17)It contains two subalgebras bsl(2)+ and bsl(2)−generated by {Jan, n ≥0} and {Jan, n < 0}respectively.

One can define the set of generators in a more economical way; for example,Ja1 , Ja0 , Ja−1 are sufficient to generate bsl(2), because the relations (4.17) allow one to con-struct the remaining generators. However, one must impose additional relations due tothe fact that the free Lie algebra generated by Ja1 , Ja0 , Ja−1 is larger than the loop algebra.Using the relations (4.17) one has many ways of defining Jam, |m| ≥3 in terms of Ja0 , Ja±1.

(Ja±2 are defined uniquely for bsl(2).) Thus, it is necessary to require that these differentdefinitions define the same algebra.

It can be shown that for the bsl(2) case it is sufficientto require that two different definitions of Ja3 , Ja−3 coincide:f acd[[Jcm, Jdm], Jbm] + f bcd[[Jcm, Jdm], Jam] = 0m = ±1. (4.18)Provided (4.18) is satisfied, all possible definitions of Jam, |m| ≥3 coincide automatically.The relations (4.18) play the same role as the Serre relations [17] .

Thus the loop algebrabsl(2) is generated by Ja±1, Ja0 satisfying (4.17), (4.18), with the trivial comultiplication∆(Jam) = Jam ⊗1 + 1 ⊗Jamm = 0, ±1. (4.19)3 Below we will use the special properties of the sl(2) structure constants: f abef cde = δadδbc −δacδbd.15

The loop algebra can be equipped with the inner product⟨Jam, Jbn⟩= δabδm+n+1,0. (4.20)The subalgebras bsl(2)+, bsl(2)−are dual with respect to this inner product.

So, the struc-ture of the classical double can be introduced [18]and the classical r-matrix can bedefined:r =Xn≥0Jan ⊗(Jan)∗=Xn≥0Jan ⊗Ja−n−1. (4.21)For the tensor product of finite dimensional representations depending on spectral param-eters λ, µ, where Jam →λmta, µmta, the r-matrix is equal to1λ −µta ⊗ta(4.22)and coincides with the celebrated classical r-matrix found by Sklyanin [19].Let us now return to the Yangian.

We want to demonstrate first that Y is a defor-mation of the universal enveloping algebra U(bsl(2)+) of bsl(2)+. The basis of U(bsl(2)+)consists of the following elements{Ja1k1 · · · Jankn } ≡JAK(4.23)where ki ≥0, { } means symmetrization, and A, K are multi-indices.

We definedeg(JAK) =Xk∈Kk. (4.24)Define the following elements of Y:Qak =1(−2)k−1 f aa1a2f a2a3a4 · · · f a2k−4a2k−3a2k−2 ×[Qa11 [Qa31 · · · [Qa2k−31, Qa2k−21] · · ·] (4.25)and consider the vector space whose basis consists of the elements{Qa1k1 · · · Qankn} ≡QAK.

(4.26)Using induction in the degree of the elements one can show that the vector space definedin this way coincides with Y, just as the vector space generated by (4.23) coincides withU(bsl(2)+). An important point is that the relations (4.16) allow one to express the dif-ference between two possible definitions of Qa3 in terms of the elements of lower degree.16

Thus Y is indeed a deformation of U(bsl(2)+) where Qak corresponds to Jak . We present thefollowing examples of the multiplication map in Y:Qa0Qb0 = Qa,b0,0 + 12f abcQc0(4.27)Qa,b0,0Qc0 = Qa,b,c0,0,0 + 12f acdQb,d0,0 + 12f bcdQa,d0,0−112(δacδbd + δbcδad −2δabδcd)Qd0⇒mQa0Qb0Qc0 = 12f abc,mQa0Qde00Qc0 = 112(2δdeδac −δdcδae −δecδad).

(4.28)For what follows we need the comultiplication of QAK, which can be shown to be ofthe form∆(QAK) =Xdeg∗QAK=deg∗QBL +deg∗QCMµ QBL ,QCMQAKQBL ⊗QCM(4.29)wheredeg∗QAK= −Xk∈K(k + 1). (4.30)For example,∆(Qa1) = Qa1 ⊗1 + 1 ⊗Qa1 + αf abcQb0 ⊗Qc0(4.31)∆(Qa2) = Qa2 ⊗1 + 1 ⊗Qa2 + αf abc(Qb0 ⊗Qc1 −Qc1 ⊗Qb0)−α2(Qb0 ⊗Qa,b0,0 + Qa,b0,0 ⊗Qb0).We wish to extend Y in order to obtain a deformation of the complete loop algebra.This can be accomplished by following the general construction of the quantum doublediscussed in Section 2.

Namely, let us construct the Hopf algebra Y∗dual to Y. Theelement dual to Qa0 is denoted by 2αQa−1. It can be shown that:1.

The algebra Y∗is generated by Qa−1.2. The generators Qa−1 satisfy the relation:f acd[[Qc−1, Qd−1], Qb−1] + f bcd[[Qc−1, Qd−1], Qa−1] = 0(4.32)which is the undeformed form of (4.18) for m = −1.

The relation (4.32) will be provenbelow.The algebra Y∗is equivalent as an algebra (but not as coalgebra!) to U(bsl(2)−).

So,its basis consists ofQA−K ≡{Qa1−k1 · · · Qan−kn},ki > 0(4.33)17

whereQa−k =1(−2)k−1 f aa1a2f a2a3a4 · · · f a2k−1a2k−3a2k−2 × [Qa1−1[Qa3−1 · · · [Qa2k−3−1, Qa2k−2−1] · · ·]. (4.34)The degree of QA±K is defined asdegQA±K=Xk∈K±k.

(4.35)The multiplication map m∗in Y∗is described by the equation:QA−K QB−L =Xdeg QA−K+deg QB−L=deg QC−Mm∗QC−MQA−K,QB−L QC−M. (4.36)From the general theory of section 2, namely (2.5a), the constants m∗are computablefrom the comultiplication map in Y.To exhibit the properties of the double D(Y), knowledge of the basis elements isneeded.

The dual elements to QAK ∈Y should be expressed in terms of the basis elementsof Y∗. Let A, B, C ∈Y and X, Y, Z ∈Y∗.

The comultiplication coefficients µ of Y andthe multiplication map m∗in Y∗have the usual definitions: ∆(A) = µBCAB ⊗C andX · Y = m∗ZXY . Let us define a matrix of coefficients T , whereA∗=XX∈Y∗T AX XA ∈Y.

(4.37)Then by (2.5a), T must satisfyµABCT CZ = T AX T BY m∗ZXY . (4.38)We were unable to deduce an general formula for T .

However a study of particular examples(up to degree-4 in Y ) convinced us that the defining relations (4.38) allow a solution.Furthermore the matrix T is triangular:T AX = 0ifdeg∗(A) ̸= deg(X)ordeg(A) > deg∗(X),(4.39)wheredeg∗QA−K=Xk∈K(k −1). (4.40)18

Let us present some examples of dual elements. Consider (Qa1)∗.

From (2.5a) one has(Qa0)∗Qb0∗= (2α)2Qa,b−1,−1 + 12Qa−1, Qb−1=XD∈YµQa0,Qb0DD∗. (4.41)The only non-zero contributions to the above sum over D comes from µQa0,Qb0Qc1= αf abc andµQa0,Qb0Qab00= 1.

Thusαf abc (Qc1)∗= 2α2 Qa−1, Qb−1(4.42)Qab00∗= 4α2 Qa,b−1,−1.The above equation implies (Qa1)∗= 2αQa−2 and [Qa−1, Qb−1] = f abcQc−2.One can deduce (Qak)∗inductively as follows. Let us suppose (Qak−1)∗= 2αQa−k, and∆Qak−1= Qak−1 ⊗1 + 1 ⊗Qak−1 + αf abc Qb0 ⊗Qck−2 −Qck−2 ⊗Qb0+ · · ·(4.43)Note that ∆(Qa2) is of this form.

Representing Qak as −f abcQb1Qck−1, from (4.43) one showsthat ∆(Qak) is of the same form as (4.43) with k −1 →k. We have(Qa0)∗Qbk−1∗= (2α)2Qa−1Qb−k =XD∈YµQa0,Qbk−1DD∗.

(4.44)Now it is not hard to realize that the only contributions in the above sum over D comefrom D = Qak and Qab0,k−1. Thus(Qak)∗= 2α Qa−k−1(4.45)Qab0k∗= 4α2Qa,b−1,−k−1,which impliesQa−1, Qb−k= f abcQc−k−1.

(4.46)The ‘Serre’ relation (4.32) in Y∗is now seen to be a consequence of (4.46). The relation(4.46) also implies the following examples of the multiplication map in Y∗:Qa−1Qb−1 = Qa,b−1,−1 + 12f abcQc−2(4.47)Qa,b−1,−1Qc−1 = Qa,b,c−1,−1,−1 + 12f acdQb,d−1,−2 + 12f bcdQa,d−1,−2−112(δacδbd + δbcδad −2δabδcd)Qd−3.19

Dual elements generally have corrections to the classical formulas, unlike (4.45). Con-sider for exampleQabc000∗.

By examining (4.38) with A = Qab00 and B = Qc0, one findsQabc000∗= 8α3 Qa,b,c−1,−1,−1 + 43α3 δacδbd + δbcδad + δabδcdQd−3. (4.48)Now using the general formulas (2.11) one can deduce the commutation relations inthe Yangian double.

A straightforward computation gives[Qa0, B∗] = mB[D,Qa0]D∗(4.49)[Qa1, B∗] = mB[D,Qa1] D∗+ αf acd mBQc0 D + mBD Qc0D∗Qd0+ αf acdmBQc0,[D,Qd0] D∗where B, D ∈Y, mB[D,Qa0] = mBD Qa0 −mBQa0 D, and similarly for mBQc0,[D,Qd0]. From (4.49)with B = Qb0, and the above dual elements, one findsQa0, Qb−1= f abcQc−1.

(4.50)One can also compute the first few terms in the commutatorQa1, Qb−1= f abcQc0 −α2 f abcQc−1 −2α2Qa,b−1,−1 + 2α23 δab Qc,c−1,−1+ 2α23 f abdQc,c−1,−1Qd0 −2α23 f acdQc,b−1,−1Qd0 + . .

. (4.51)The comultiplication of Qa−1 can be found as follows.

Consider all possible productsof elements of Y which contain Qa0. Evidently the only elements of this type are Qa1···an0,···0 :Qb1···bk0···0 Qc1···cl0···0= mQa0Qb1···bk0···0,Qc1···cl0···0· Qa0 + · · ·(4.52)According to the general rules∆Qa−1= 12αXC,D∈YmQa0DC C∗⊗D∗.

(4.53)The first several terms in (4.53) are∆Qa−1= Qa−1 ⊗1 + 1 ⊗Qa−1 −αf abcQb−1 ⊗Qc−1 + 2α23δacδbd + δbaδdc −2δbcδad·Qd−1 ⊗Qb,c−1,−1 + Qb,c−1,−1 ⊗Qd−1+ . .

. (4.54)20

Finally the R-matrix of D(Y) is equal toR = I+2α∞Xk=0Qak ⊗Qa−k−1 + 4α2Qab0k ⊗Qab−1,−k−1+ 8α3Qabc000 ⊗Qa,b,c−1,−1,−1 + 4α3Qaab000 ⊗Qb−3 + · · ·(4.55)In the classical limitR = 1 + αr + o(α2)(4.56)r = Qa0 ⊗Qa−1 + Qa1 ⊗Qa−2 + Qa2 ⊗Qa−3 + · · ·which coincides with the classical r-matrix (4.21).We remark that one does not have the freedom to introduce a level k into the Yangiandouble, as in (1.2). Indeed, upon setting α to zero in (4.51), one recovers the loop algebra.Though it remains an interesting mathematical problem to determine whether a centralextension of the Yangian can be introduced by other means, this does not appear to berelevant for the physical applications we are considering.3.3 Adjoint Action on FieldsFrom the definition (3.13) we have the following adjoint actionsadQa0 (φ(x)) = ad′Qa0 (φ(x)) = [φ(x), Qa0](4.57a)adQa1 (φ(x)) = [φ(x), Qa1] −α f abc Qb0 [φ(x), Qc0](4.57b)ad′Qa1 (φ(x)) = adQa1 (φ(x)) + 2α f abc Qb0 [φ(x), Qc0] .

(4.57c)Highest weight fields φΛ(x) can be defined to satisfyadQ+0 (φΛ(x)) = adQa−1 (φΛ(x)) = 0,(4.58)where Q±0 =Q10 ± iQ20/√2. The vacuum is invariant under all the chargesQa0|0⟩= Qa1|0⟩= Qa−1|0⟩= 0.

(4.59)Descendents of φΛ(x), which comprise the multiplet ΦΛ(x), are denoted asφΛa1,···,am;N(x) = adQa11◦adQa21◦· · · ◦adQam1◦adNQ−0(φΛ(x)) . (4.60)21

Let f(β1, . .

., βn) denote the form factor for a field φ(x), not necessarily highest weight,as in (3.8). We suppose that φ(x) transforms under some finite dimensional representationof sl(2):[Qa0, φ(x)] = taφ φ(x).

(4.61)The sl(2) invariance implies taφ +nXi=1tai!f(β1, . .

., βn) = 0. (4.62)Consider now the form factors of the descendents with respect to Qa1f a1,a2,...,am(β1, .

. ., βn) = ⟨0| φa1,a2,...,am;0(0) |βn, .

. ., β1⟩.

(4.63)From (4.57), (4.59), and (4.11), one obtains these form factors in terms of the form factorsof φ(x)f a1,a2,...,am(β1, . .

., βn) = T a1(β1, . .

., βn) · · ·T am(β1, . .

., βn) f(β1, . .

. , βn).

(4.64)A similar expression holds for descendents with respect to ad′Qa1, where α →−α in thedefinition for T a.We now show that the two sets of φ’s descendent fields Φ and Φ′ commute for x < y(3.20). The proof of (3.20) relies on a generalization of the locality theorem, which can bestated as follows [2].

Let the form factors of two fields O1(x) and O2(x) be given. Thematrix elements are defined asf1,2(αm, .

. ., α1|β1, .

. ., βn) = ⟨α1, .

. ., αm| O1,2(0) |βn, .

. ., β1⟩.

(4.65)These matrix elements can be defined in two different waysf1,2(αm, . .

., α1|β1, . .

., βn) = f1,2(αm −iπ, . .

., α1 −iπ, β1, . .

., βn)(4.66a)f ′1,2(αm, . .

., α1|β1, . .

., βn) = f1,2(β1, . .

., βn, αm + iπ, . .

., α1 + iπ)(4.66b)(For simplicity of notation we don’t display the charge conjugation matrices.) For localoperators, by (3.10), f = f ′.

The locality theorem states that all matrix elements satisfyO1(x) O2(y) = O2(y) O1(x)x < y. (4.67)22

The proof of (4.67) involves inserting a complete set of states between the operators,representing the matrix elements of O1 and O2 with f1 and f ′2 respectively, and usinganalytic properties of form factors [2].The details of the proof of the locality theorem admit the following generalization [3].Let us be given any two operators O(x) and O′(y), whose matrix elements are of the formf and f ′ respectively, as in (4.66). ThenO(x) O′(y) = O′(y) O(x)x < y.

(4.68)Let us apply the above result to the comparison of descendent form factors in Φ andΦ′. For simplicity consider the case where the ancestor field φ is local.

We adopt thenotation that ⟨A|, |B⟩represent the states ⟨α1, . .

., αm| and |βn, . .

., β1⟩, respectively. Theform factor f a(β1, .

. ., βn) has the following analytic propertyf a(β1, .

. ., βn + 2πi) = T a(β1, .

. ., βn) f(βn, β1, .

. ., βn−1) + 2πi tan f(βn, β1, .

. ., βn−1).

(4.69)The operator T a has the following propertyT a(β1, . .

., βn) = T a(βn, β1, . .

., βn−1) −2πi tan −2αf abc tbB tcn,(4.70)where tbB is the representation of sl(2) on the state |B⟩, tbB = Pni=1 tbi. Using (4.62), theequations (4.69) and (4.70) imply more generally thatf a(B1, B2 + 2πi) = f a(B2, B1) + 2αf abc tbφ tcB2 f(B2, B1).

(4.71)Consider now the two kinds of matrix elementsf a(A|B) = f a(A −iπ, B)(4.72a)f ′a(A|B) = f a(B, A + iπ). (4.72b)From (4.71)f ′a(A|B) = f a(A −iπ, B) + 2αf abc tbφ tcA f(A −iπ, B).

(4.73)Comparing (4.73) with (4.57), one sees that f a(A|B) and f ′a(A|B) represent the matrixelements of adQa1 (φ) and ad′Qa1 (φ) respectively. Therefore, from the generalized localitytheorem, these two fields commute for x < y.

Repeated adjoint action yields the result(3.20).23

Let Jaµ(x) and J aµ (x) denote the currents generating the conserved charges Qa0 andQa1 respectively. Form the general result (3.28) one has the following braiding relationsJaµ(y) ΦΛ(x) = ΦΛ(x) Jaµ(y)x < y(4.74)J aµ (y) ΦΛ(x) = ΦΛ(x) J aµ (y) −αf abc Qb0, ΦΛ(x)Jcµ(y).In applying (3.28) we have used the fact that the current generating the identity is zero.Non-local currents J aµ (x) generating the charges Qa1 were constructed by L¨uscher andBernard [8][4].

As was shown in these works, the current J aµ (x) can be constructed solelyin terms of the current Jaµ(x), through a non-local bilinear expression. Using this explicitconstruction of the current it was shown in [4] that it satisfies the braiding relation (4.74).Let us say a few words about what one expects for the field content of the specificmodel we are considering.

The conserved energy momentum tensor Tµν, being an sl(2)singlet, of course has no descendents with respect to Qa0. However repeated adQa1 actiongenerates an infinite multiplet of conserved currents, which we analyze in more detail inthe next section.

As we will show, the energy momentum tensor and the sl(2) current Jaµare in the same multiplet. There are also fields ψ±(x), ψ±(x) which are sl(2) doublets thatcreate the kink particles asymptotically.

The fields ψ+, ψ+ are highest weight fields forsome additional infinite dimensional Verma module representations.3.4Descendents of the Energy-Momentum TensorThe form factors of the energy-momentum tensor Tµν(x) and the sl(2) current Jaµ(x)are known [2]. They have the following presentation.

Define two operators OT and OaJ,related to Tµν and Jaµ byTµν(x) = 12 (ǫµµ′ǫνν′ + ǫνµ′ǫµν′) ∂µ′∂ν′ OT (x)Jaµ(x) = ǫµν∂νOaJ(x),(4.75)where ǫµν is the antisymmetric tensor. The energy-momentum tensor and sl(2) currentcan be represented this way since they are conserved.

The form factors for Tµν and Jaµfollow simply from those of OT , OaJ due to the relation⟨A| ∂µO(x) |B⟩= i mXi=1pµ(αi) −nXi=1pµ(βi)!⟨A| O(x) |B⟩. (4.76)24

For simplicity we discuss only the two-particle form factors. They take the followingform⟨0| OT (0) |β2, β1⟩=12πiζ(β1 −β2)β2 −β1 −iπ |singlet⟩21⟨0| OaJ(0) |β2, β1⟩= ζ(β1 −β2)2πi|triplet; a⟩21,(4.77)where ζ(β) is the functionζ(β) = sinh(β/2) expZ ∞0dx sin2 (x(β + iπ)/2π)x sinh(x) cosh(x/2) e−x/2.

(4.78)The states in (4.77) are|singlet⟩21 = (|+⟩⊗|−⟩−|−⟩⊗|+⟩)(4.79)|triplet; 1, 2, 3⟩21 = (|−⟩⊗|−⟩−|+⟩⊗|+⟩) /2,i (|−⟩⊗|−⟩+ |+⟩⊗|+⟩) /2,(|+⟩⊗|−⟩+ |−⟩⊗|+⟩) /2.Consider now the first descendent of Tµν whose form factors are given by the generalformula (4.63). Since Tµν is an sl(2) singlet, its first descendent is a local operator, asindicated by (4.71).

One can show by a simple computation thatT a(β1, β2) |singlet⟩21 = (β2 −β1 −iπ) |triplet; a⟩21. (4.80)Therefore, comparing the 2-particle form factors of this first descendent of Tµν with theform factors of Jaµ, one derives[Qa1, Tµν(x)] = −12ǫµα∂αJaν (x) + ǫνα∂αJaµ(x).

(4.81)Introduce the generator L of Lorentz boostsL = −Zdx xT00(t = 0). (4.82)Then the global version of (4.81) is[L, Qa1] = −Qa0.

(4.83)The generator L thus has degree −1. The equation (4.83) was expected from the fact thaton-shell, one has (4.6) and L = −∂∂β .

Thus one concludes that the Yangian must actually25

be extended to include the Poincar´e algebra with generators L, Pµ in order to realize itsfull implications. Though we have presented the derivation of (4.81) using only the twoparticle form factors, the same relation is implied by the general multiparticle form factors.We consider next the first descendents of Jaµ.

One finds12f abc adQa1Jbµ(x)= J cµ(x) + α2 Jcµ(x),(4.84)up to possible additional total derivatives. This follows by comparing the braiding of thecurrent J a on the right hand side of (4.84) computed from the generalized locality theoremwith the braiding of the more abstractly defined current in (4.74), and also by consideringthe global relation (4.5).As a final example consider the conserved current and chargeNµ(x) = adQa1Jaµ(x);N =Zdx Nt(x).

(4.85)In order to characterize this current, we determine the matrix elements of the charge N.Matrix elements of conserved charges can be generally computed as follows. Let Jµ(x) bea general current for a conserved charge Q, and define Jµ(x) = ǫµν∂νO(x).

The operatorO is non-local. The form factors of O(x) correspond to the partially integrated currentZ x−∞dy ⟨A| Jt(y) |B⟩= ⟨A| O(x) |B⟩(4.86a)Z ∞xdy ⟨A| Jt(y) |B⟩= −⟨A| O(x) |B⟩.

(4.86b)The matrix elements on the RHS’s of (4.86a, b) should be defined by the continuations in(4.66a, b) respectively, since these are consistent with the range of integration. Therefore⟨A| Q |B⟩= limǫ→0f(A + iǫ|B) −f(A −iǫ|B),(4.87)where f are the form factors of O.

The RHS of (4.87) is non-zero due to the fact that Ois non-local.Returning to the charge N, let Nµ(x) = ǫµν∂νON (x). The form factors of ON arecomputed to be⟨0| ON (0) |β2, β1⟩= −c1/22πi ζ(β1 −β2) (β1 −β2 −iπ) |singlet⟩,(4.88)26

where c1/2 = 3/4 is the casimir in the spin 1/2 representation. Using the fact that ζ(β1−β2)has a simple pole at β2 = β1 + iπ, and the identity1β −iǫ −1β + iǫ = 2πi δ(β),(4.89)we obtain⟨β1, ∓| N |β2, ±⟩= 3πi2 δ(β1 −β2).

(4.90)Thus on one-particle states, N is just proportional to the particle number operator. Asimilar computation for multiparticle states verifies this identification:N |βn, .

. ., β1⟩= 3πi2n |βn, .

. ., β1⟩.(4.91)5.

Concluding RemarksThe relation (4.81) expressing the derivative of the sl(2) current as a descendent of theenergy-momentum tensor is remarkable. It implies the space-time properties of the theoryare closely related to the sl(2) properties.

Since the non-local current generating Qa1 can beexpressed in terms of the sl(2) current Jaµ(x), (4.81) implies the energy-momentum tensorcan be expressed exactly in terms of the sl(2) current also, thereby indicating an exactSugawara construction in the massive theory. The details of such a construction deservesfurther study.

This idea taken together with the fact that the action (1.1) is expressedonly in terms of the current Jaµ implies that the theory is fully characterized by the currentalgebra and the Yangian symmetry.We have shown that the level k of the Kac-Moody symmetry in the conformal modeldoes not enter into the Yangian algebra. This is somewhat paradoxical, since the spectrumof fields in the WZWN model is governed by k. Furthermore, the level k is still defined inthe massive theory as appearing in the current commutatorJat (x), Jbx(0)= f abcJcx(0) δ(x) −kiπ δab δ′(x).

(5.1)The resolution of this puzzle can be found by considering the exact spectrum and S-matrices of the model (1.1) for arbitrary level k, which were proposed in [20] . SimilarS-matrices were found for spin-chain realizations of (1.1) in [21] .

Here it was found thatthe spectrum of massive particles K±ab, a, b ∈{0, 1/2, 1, . .

., k/2} is still governed by aninteger k, and reflects the spectrum of primary fields of the conformal model SG it is a27

perturbation of. However the S-matrix factorizes into two pieces, one which is the Yangianinvariant factor (4.1) for the indices ± of K±ab, the other is characterized by an additionalfractional supersymmetry of order k which acts on the indices a, b.

Thus the Yangiansymmetry is unaffected by varying the level k. Rather, increasing k enlarges the Yangiansymmetry to include an independent fractional supersymmetry.The precise mathematical nature of the field representations of the Yangian should beexplored. The mathematics literature has so far focused only on finite dimensional rapiditydependent representations, which are only applicable to on-shell objects.

Indeed this partlyexplains why the precise connection of the Yangian with a deformation of the full Kac-Moody algebra, which arises is the study of its quantum double, was never precisely madebefore.The correlation functions are also constrained by the quantum symmetry, and weconclude with a few remarks on this problem. It is evident that the Ward identities aregiven by∆(n) (ea) ⟨0| φ1(x1) · · · φn(xn) |0⟩= 0.

(5.2)This is derived by inserting the generator ea to the left of the fields in the correlationfunction and using ⟨0| ea = 0, with (3.26). This equation can be used to relate correlationfunctions of various fields related by adjoint action.

However by itself, it cannot determinecorrelation functions. What is missing are the analogs of ‘null-fields’ that are available inthe conformal theory.

The possibility of such null-fields is a very interesting issue.AcknowledgementsWe gratefully thank D. Bernard, I. Frenkel, N. Reshetikhin, and M. Semenov-Tian-Shansky for valuable discussions. We wish to thank J. Cardy for the opportunity to visitthe Institute in Santa Barbara, where this work was begun.

F.S. thanks the Cornell groupfor hospitality and the MSI at Cornell for financial support.

This work was supportedin part by the US National Science Foundation under grants no. PHY-8715272 , PHY-8904035, and by the US Army.28

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