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Quantum Symmetries in 2D Massive Field Theories.

arXiv:hep-th/9109058v1 30 Sep 1991SPhT-91-124Quantum Symmetries in 2D Massive Field Theories.Cargese-91

SPhT-91-124QUANTUM SYMMETRIESIN 2D MASSIVE FIELD THEORIES..Denis BERNARDService de Physique Th´eorique de Saclay *F-91191 Gif-sur-Yvette, France.We review various aspects of (infinite) quantum group symmetries in 2D massive quantumfield theories. We discuss how these symmetries can be used to exactly solve the integrablemodels.

A possible way for generalizing to three dimensions is shortly described.. Series of lectures given at the 91 Cargese school, New symmetry principles in quantum fieldtheory. * Laboratoire de la Direction des sciences de la mati`ere du Commissariat `a l’´energie atomique.1

Content:1 - Quantum Symmetries in 2D Lattice Field Theories.1a- Quantum symmetries and conserved currents.1b- Fields multiplets.1c- Examples.2 - Classical Origin of Quantum Symmetries: Dressing Transformations.2a- What are the dressing transformations.2b- Few of their properties.3 - An Example of Dressing Transformations: Current Algebras.3a- The equations of motions.3b- Dressing transformations and the Riemann-Hilbert problem.3c- Non-local conserved currents.4 - Quantification: Yangians in Massive Current Algebras.4a- OPE’s in massive quantum current algebras.4b- The quantum non-local currents.4c- The non-local conserved charges and their algebra.4d- Action on the asymptotic states and the S-matrices.4e- Action on the fields and the field multiplets.5 - Conclusions.3D generalizations ? ?2

INTRODUCTIONSymmetries of the S-matrices of 4D quantum field theories are subject to the severeconstraints of the Coleman-Mandula theorem [1]. In general, these possible symmetries donot allow for non-perturbative solutions of the theories.

In two dimensions, this theorembreaks down and there is room for richer symmetries. The aim of the lecture is to describenew symmetries of 2D massive QFT, known as quantum group symmetries.

This paper ismainly a review of published papers completed by few remarks and comments.There are at least two motivations for studying quantum symmetries in 2D quantumfield theories:(i) The study of the possible symmetries of 2D QFT. The quantum group symme-tries we will analyse in these lectures are characterized by the fact that, unlike standardLie algebra symmetries, they do not act additively, and by the fact that they are gener-ated by non-local currents having in general non-integer Lorentz spins.

They thus providenon-Abelian extension of the Lorentz group. (ii) An algebraic formulation of 2D QFT and exact solutions.

Our (desesperate?) goal is to formulate an algebraic approach to 2D QFT, based on their symmetries (localand non-local), which could offer a way to solve the integrable two-dimensional quantumfield theories from symmetry data, in analogy with the approach used in conformal fieldtheories [2].It is of some interest to compare the approach which as been used recently in CFT andin 2D integrable models.

a) Rational conformal field theories are invariant under chiralvertex operator algebras, which could be local, e.g. the Virasoro or affine algebras, ornon-local, e.g.

the parafermionic algebras. The conformal field theories are reformulatedas representation theories of the chiral algebras: Hilbert spaces of the CFT’s are directsums of representations of the chiral algebras; conformal primary fields are intertwiners forthe chiral algebras, etc...

The chiral algebras are non-abelian and this is, for a large part,at the origin of the exact solvability of the CFT’s. Being completely integrable, the CFT’salso possess infinitely many local integrals of motion in involution.

However, almost noneof these integrals of motion are actually used to solve the conformal models. (It is evendifficult to express them in terms the generators of the chiral algebras.) b) In contrast, away of studying integrable models [3] consists in extracting local integrals of motion whichare in involution.

These integrals of motion thus form an abelian algebra. There existenceensures the integrability of the theory and the factorization of the S-matrix.

In general,3

they do not provide enough informations for solving the theory, e.g. for determining theS-matrix.Thus, almost none of the techniques used in one of these fields is used in the other.However, besides these local integrals of motion, the 2D integrable models and the confor-mal models also possess non-local integrals of motion.

These non-local conserved chargesare the generators of non-abelian algebras known as the quantum symmetry algebras of themodels. It is hoped that these new symmetry algebras will allow us to define a frameworkwhich could be apply simultaneously to the conformal field theories and to the massiveintegrable models.To characterize the massive quantum field theories uniquely by their symmetry alge-bras requires :(I) that the asymptotic particles form multiplets of the symmetry algebras and that theinvariance for the S-matrix determine it uniquely.

As we will see in the course of thislecture, because quantum group symmetries in 2D do not commute with the Lorentzgroup, they provide algebraic relations on the S-matrices. (II) that all the fields of the QFT can be gathered into field multiplets transformingcovariantly under the symmetry algebras and that the intertwining properties for the fieldmultiplets determine them uniquely (in analogy with the minimal assumption in rationalconformal field theories).

This amounts to demand that the Ward identities have uniquesolutions. As we will describe, the components of the field multiplets, the descendents andthe highest vector fields, are related through the Ward identities, once more in completeanalogy with conformal field theories.The symmetry algebras having these requirements could be called complete symmetryalgebras.

For a given model there could be more than one complete symmetry algebra.The problem of solving the integrable massive models reduce to the problem of finding acomplete symmetry algebra. As we already said, the local integrals of motion which areinvolution do not form (in general) a complete symmetry algebra.

The quantum symmetrywe are going to describe provide in general more informations than these abelian integralsof motion. To our knownledge, it is not known if they form or not a complete symmetryalgebra.

However, it is tempting to conjecture that the algebra generated by the localintegrals of motion together with the generators of the quantum symmetry form a completesymmetry algebra.4

An example:dslq(2) -symmetry in the Sine-Gordon models. The quantum sine-Gordon theory is described by the Euclidean actionS =14πZd2zh∂zΦ∂zΦ + λ : cosbβΦ:i.

(1)The parameter bβ is a coupling constant; it is related to the conventially normalized cou-pling by bβ = β/√4π. For bβ ≤√2 the action can be renormalized by normal-orderingthe cosbβΦinteraction and absorbing the infinities into λ; the coupling constant bβ isthereby unrenormalized [4].

The sine-Gordon theory has a well known topological current:Jµ(x, t) = bβ2π ǫµν∂νΦ(x, t) where ǫµν = −ǫνµ. The topological charge is:T=bβ2πZ +∞−∞dx ∂xΦ =bβ2πΦ(x = ∞) −Φ(x = −∞).

(2)The topological solitons that correspond to single particles in the quantum theory aredescribed classically by field configurations with T = ±1. These solitons are kinks thatconnect two neighboring vacua in the cos(bβΦ) potential.The sine-Gordon model possesses infinitely many local integrals of motion with oddLorentz spins, we denote them by Jn.

Besides those, the sine-Gordon model also admitsfour non-local conserved currents [5]:∂µ J±µ (x, t) = ∂µJ±µ (x, t) = 0. (3)The Lorentz spin s of the currents J±µJ±µare s =2bβ2−2bβ2.

From these conservedcurrents we define four conserved charges, Q± and Q±, respectively associated to thecurrents J±µ (x, t) and J±µ (x, t). The Lorentz spins of the conserved charges are :spin (Q±) = −spinQ±= 2 −bβ2bβ2= 8π −β2β2.

(4)The conserved currents whose exact expressions are given in ref. [5] are Mandelstam likevertex operators [6] and are thus non-local.The algebra of the non-local charges is :Q± Q± −q2 Q± Q± = 0(5a)Q± Q∓−q−2 Q∓Q± = a1 −q±2T (5b)hT , Q±i= ±2 Q±(5c)hT , Q±i= ±2 Q±,(5d)5

where q = exp(−2πi/bβ2).and a some constant.The algebra (5) is a known infinitedimensional algebra, namely the q-deformation of the sl(2) affine Kac-Moody algebra,denoteddslq(2), with zero center [7] [8]. Only the Serre relations fordslq(2) are missing in(5).The non-local charges (3) provide relevent information; for example, the S-matrix ofthe Sine-Gordon solitons [9] can be deduced from this dslq(2) symmetry plus its unitary andcrossing symmetry property.

However, they probably do not form a complete symmetryalgebra because the local integrals of motion do not seem to be generated by them. To provethe conjecture that the local conserved charges Jn and these non-local charges generate acomplete symmetry algebra for the sine-Gordon models will be very illuminating.1.QUANTUM SYMMETRIES IN 2D LATTICE FIELD THEORY.We consider vertex models, i.e.

models of two-dimensional statistical mechanics inwhich the discrete spin variables live on the midpoints of the links of a square lattice,and the Boltzmann weights are associated to the vertices of the lattice [10]. The Boltz-mann weight of a given vertex depends on the spin variables σ1, .

. ., σ4 at the four sitessurrounding the vertex, and is denoted by Rσ3σ4σ1σ2.

It is useful to view R as an operatorV ⊗V →V ⊗V , where V is the vector space spanned by a set of basis vectors eσ labeledby the possible values of the spin variable:Reσ1 ⊗eσ2 = Rσ3σ4σ1σ2 eσ3 ⊗eσ4. (1.1)Partition function and correlation functions are defined as follows.

Consider the sys-tem in a finite square box of size N × N. Then the square lattice Λ = ZZ2N 2ZZ2 containspoints with integer coordinates (x, t) called space and time. The spin variables live on thelattice Λ′ of points of the form (x + 12, t) and (x, t + 12) with x, t integers modulo N. Foreach i ∈Λ′ introduce a copy Vi of the space V .

For (x, t) ∈Λ define R(x, t) be the matrixR mapping V(x−12 ,t) ⊗V(x,t−12 ) to V(x+ 12 ,t) ⊗V(x,t+ 12 ). The matrix of Boltzmann weights,B =O(x,t)∈ΛR(x, t),(1.2)is an operator from Ni∈Λ′ Vi to Nj∈Λ′ Vj.The partition function is ZN = tr B.6

For any operator O ∈EndV define the insertion O(j) of O at the point j ∈Λ′ to bethe operator 1 ⊗· · ·⊗1 ⊗O ⊗1 ⊗· · ·⊗1 acting on Vj in the tensor product Ni∈Λ′ Vi. Thecorrelation functions of operator insertions are defined as⟨O1(j1) · · · On(jn)⟩N =1ZNtrnYk=1Ok(jk) B.

(1.3)Classical examples are Oeσ = σeσ and Oeσ = δσσeσ for some σ. Their correlation functionsare the usual spin correlation functions and the joint probabilities that the spin σjk assumegiven values.An alternative formalism is the transfer matrix formulation.

In this formalism, oneassigns to each x = 1, · · ·, N a copy Vx of V . The transfer matrix T is:T = tr0R0N · · ·R02 R01(1.4)with Rnm the matrix R acting on Vn and Vm in V0 ⊗V1 ⊗· · · ⊗VN, and the trace is overV0.For O ∈EndV and x ∈{1, · · ·, N} define O(x) as O acting on Vx in V1 ⊗· · · ⊗VNand O(x −12) as O(x −12) = T −1tr0R0N · · · R0x O0 R0,x−1 · · · R01.

Heisenberg fieldsO(j), j ∈Λ′ are defined as:O(x −12, t) = T −t O(x −12) T tO(x, t −12) = T −t O(x) T t(1.5)The partition function in the transfer matrix formalism is Z = trT N, and if the timecoordinates of j1, · · ·, jn are ordered (with smaller times on the right of larger times) thecorrelation function ⟨O1(j1) · · ·On(jn)⟩N defined above coincides with⟨O1(j1) · · · On(jn)⟩N =1ZNtrO1(j1) · · ·On(jn) T N(1.6)We have defined point-like operator insertions in two different formalisms. It is some-times useful to define operator insertions associated to some finite set of neighboring pointsin Λ′ as linear combination of products of point-like insertions at the points of the set.

Thisis the lattice analogue of the operator product expansion of field theory.1a) Quantum symmetries and conserved currents.7

(i) Lie algebra symmetry.Suppose that the Boltzmann weights are invariant undersome Lie algebra G. This means that V carries a representation of G and for each generatorTa of G in that representation we haveR(Ta ⊗1 + 1 ⊗Ta) = (Ta ⊗1 + 1 ⊗Ta)R(1.7)i.e.R is an intertwiner.It is useful to represent this equation graphically.If Ta isrepresented by a little cross, we have−−−−+−−−−=−−−−+−−−−(1.8)Introduce now a local current Jµ(x, t; X), linear in X ∈G, for each vertex (x, t) of thelattice. The components Jt(x, t; X), Jx(x, t; X) are defined by the insertion of the matrixX = Pa XaTa at the site (x, t −12) or the site (x −12, t), respectively.

Graphically,Jt(x, t; X) =−−−−(x,t)(1.9a)Jx(x, t; X) =−−−−(x,t)(1.9b)Then, in any correlation function (with no insertion of other fields at the sites surrounding(x, t)), (1.7) readsJt(x, t + 1; X) −Jt(x, t; X) + Jx(x + 1, t; X) −Jx(x, t; X) = 0(1.10)which is the lattice version of the continuity equation ∂µJµ = 0. As in the continuum, thisequation implies the conservation of the charge Q(X) = Px Jt(x, t; X).As it is obvious from the pictures (1.9), the operators Jµ(x; X) are local operators:they satisfy equal-time commutation relations:Jµ(x; X) Jν(y; Y ) = Jν(y; Y ) Jµ(x; X);∀x ̸= y(1.11)for all X, Y ∈G.8

(ii) Quantum invariance.We now generalize [11] the preceding construction to aninvariance under a Hopf algebra. Recall that a Hopf algebra A is an algebra with unit 1and associative product m : A⊗A →A, equipped with a coproduct ∆: A →A⊗A, a counitǫ : A →C, and an antipode S : A →A so that: (i) ∆, ǫ are algebra homomorphisms, Sis an algebra antihomomorphism; (ii) (1 ⊗∆)∆(X) = (∆⊗1)∆(X); (iii) (1 ⊗ǫ)∆(X) =(ǫ⊗1)∆(X) = X; (iv) m(1⊗S)∆(X) = m(S ⊗1)∆(X) = ǫ(X)1, for all X ∈A.

The Hopfalgebra A we will consider are those generated by elements Ta, Θba, bΘba with, among therelations, Θca bΘbc = bΘcaΘbc = δba. We also assume that the comultiplication in A is definedby:∆(Ta) = Ta ⊗1 + Θba ⊗Tb∆(Θba) = Θca ⊗Θbc∆(bΘab) = bΘcb ⊗bΘac(1.12)The definition of the counit and the antipode in A are found from the Hopf algebra axioms.The motivation for introducing these algebras will be given latter.Lie superalgebrasprovide an example of such algebras.The correct generalization of the invariance eq.

(1.7) is R ∆(X) = σ ◦∆(X) R, withσX ⊗Y = Y ⊗X. Explicitly,R (Ta ⊗1 + Θba ⊗Tb) = (1 ⊗Ta + Tb ⊗Θba) R(1.13a)R Θca ⊗Θbc = Θbc ⊗Θca R(1.13b)These equations have a graphical interpretation.

The generators Ta, Θba, bΘba are conve-niently represented in terms of crosses and oriented wavy lines:Ta = a ×;Θba = ab;bΘba = baThe graphical representation of (1.13a) is then−−−−+−−−−=−−−−+−−−−with the convention that where pieces of wavy lines join an implicit contraction of indicesis understood.The currents Jµ(x, t; X), X = Pa XaTa, are then constructed as for9

parafermionic currents, namely with a disorder line (the wavy line) attached:Jta(x, t) = Jt(x, t; Ta) =a−−−−−−· · ·· · ·· · ·−−−(x,t)−−−(1.14a)Jxa (x, t) = Jx(x, t; Ta) =a−−−−−−· · ·· · ·· · ·−−−(x,t)−−−(1.14b)The disorder line ends at some specified point on the boundary of the lattice. The identity(1.13b) implies that the disorder line may be deformed (away from insertions of observables)without changing the value of correlation functions.

It behaves as the holonomy of a flatconnection, just as for ordinary disorder fields [12]. Equation (1.13a) implies the continuityequation (1.10) for non-local currents.In the operator formalism, the time component of the current is an operator (in theSchr¨odinger picture) acting on the finite volume Hilbert space V ⊗· · · ⊗V (N factors) asJta(x) = Jt(x; Ta) = Θa1a ⊗Θa2a1 ⊗· · · ⊗Θax−1ax−2 ⊗Tax−1 ⊗1 ⊗· · · ⊗1.

(1.15)The space component has a more cumbersome operator representation which we omit. (iii) The braiding relations.

By construction the currents (1.14) are non-local. Theysatisfy braided equal-time commutation relations.

These braiding relations arise due tothe topological obstructions that one encounters when trying to move the wavy stringattached to the currents through a point on which a field is located. In order to writesimple closed formula for the braiding relations we now assume that we have completedthe set of generators, Ta, Θba, bΘba such that they close under the adjoint action.

Thisimplies that there exists a c-number matrix Rbdac such that:Θna Tc bΘbn = Rbdac TdRabnm Θnc Θmd= Θbm Θan Rnmcd(1.16)Then, a simple computation shows that:Jµa (x) Jνb (y) = Rcdab Jνd (y)Jµc (x),for x > y(1.17)10

(iii) Global symmetry algebra. The algebra A acts on the Hilbert space V ⊗N by thecoproduct ∆N, defined recursively by ∆2 = ∆, ∆n+1 = ∆n(1 ⊗∆).

For generators, wehave the formulae∆N(Θba) = Θa1a ⊗Θa2a1 ⊗· · · ⊗ΘbaN−1∆N(bΘba) = bΘba1 ⊗bΘa1a2 ⊗· · · ⊗bΘaN−1a∆N(Ta) =NXx=1Θa1a ⊗Θa2a1 ⊗· · · ⊗Θax−1ax−2 ⊗Tax−1 ⊗1 ⊗· · · ⊗1. (1.18)Comparing with (1.15), we see that ∆N(Ta) is the charge corresponding to the currentJµa (x):∆N(Ta) =NXx=1Jta(x).

(1.19)The global charges ∆N(Θba) can be interpreted as topological charges.The global charges ∆N(Ta) and ∆N(Θba), satisfy the same algebra as the originalgenerators Ta, Θba (because the comultiplication ∆is an homorphism from A to A ⊗A). Ifwe assume, as we did in the previous section, that the generators Ta, Θba are closed underthe adjoint action, then there also exist c-numbers f abc such that :TaTb −Rcdab TdTc = f cab Tc.

(1.20)These are generalized braided Lie commutation relations.Remark: Because the currents are non-local, the local conservation laws for the currentsdo not systematically imply those of the global charges. The conservation laws for thecharges can be broken by boundary terms which depends on the sector on which thecharges are acting.1b) Fields multiplets.The symmetry algebra A acts also on the field operators.

In the operator formalism,the fields are operators O ∈End(V ⊗N ). Any element X ∈A of an Hopf algebra A acts anoperator O byQX O =XiXi O S(Xi)(1.21)with ∆(X) = Pi Xi ⊗Xi and S the antipode in A.

In our case, for the generators Ta andΘba, this becomes:QaO = ∆N(Ta) O −(QbaO) ∆N(Tb)QbaO = ∆N(Θca) O ∆N(bΘbc)(1.22)11

The multiplets of fields are collections of fields transforming in a representation ofthe symmetry algebra A. More precisely, let VΛ be a representation space for A.

A fieldmultiplet at x is an operator ΦΛ(x; v) acting on V ⊗N depending linearly on a vector v inVΛ, with transformation propertyQXΦΛ(x; v) = ΦΛ(x; Xv). (1.23)It is clear that, in general, the fields ΦΛ(x; v) are necessarily non-local.

However, in a givenmultiplet there can be local fields.Field multiplets can be constructed from the following data: Representation spacesW, W ′ and operators φ and Ω, φ : VΛ ⊗W →W ′ and Ω: VΛ ⊗V →VΛ ⊗V , with (twisted)intertwining propertiesφ∆(X) = XφΩ∆(X) = σ ◦∆(X)Ω,(1.24)for all X ∈A. The spaces W, W ′ are in the simplest case equal to V , or may be tensorproducts V ⊗n, V ⊗n′.

In terms of a basis {ei} of VΛ, with φ(ei ⊗w) = φiw, Ω(ei ⊗v) =ej ⊗Ωjiv, Xei = ejXji , the field multiplets are operators acting on V ⊗N defined by,ΦΛ(x; ei) ≡ΦΛi (x) = Ωi1i ⊗Ωi2i1 ⊗· · · ⊗Ωix−1ix−2 ⊗φix−1 ⊗1 ⊗· · · ⊗1,(1.25)with φix acting on the tensor product of the xth to the (x −1 + n)th factor *. Graphicallythis is represented asΦΛi (x) ≡i========· · ·· · ·· · ·====⃝··where the circle represents an insertion of φ and each crossing of the double line with asingle line represents an Ω.The transformation property (1.23) follows from (1.24) and Hopf algebra properties.The action of the algebra of fields is by definition (1.22):QbaΦΛ(x; v) = Θa1a ⊗Θa2a1 ⊗· · · ⊗ΘcaN−1 ΦΛ(x; v) bΘbb1 ⊗bΘb1b2 ⊗· · · ⊗bΘbN−1c,QaΦΛ(x; v) =NXy=1Jta(y) ΦΛ(x; v) −QbaΦΛ(x; v) Jtb(y).

(1.26)* If n ̸= n′ the insertion of a field produces a deformation of the lattice. It is understood thatwe consider correlation functions where the total n is equal to the total n′ so that at infinity thelattice is the regular square lattice.12

Because the terms with y > x + n −1 cancel in the sum, the action of the charges Qa canwritten as a contour integral on the lattice. Indicating graphically the summation by anintegration contour on the dual lattice, we have:QaΦΛi (x) ≡ai========· · ·· · ·· · ·====⃝··(1.27)The integration contour is surrounding the fields.

The intertwining properties (1.24) ofthe microscopic data φ and Ωimply that the field multiplets defined in (1.25) transformcovariantly:Qba ΦΛi (x) = ΘbjaiΦΛj (x)Qa ΦΛi (x) = T jaiΦΛj (x)(1.28)Remark: Because the field multiplets are non-local they satisfy equal-time braiding re-lations. But the braiding relations between the field multiplets are constrained by thequantum invariance.

Let VΛ and VΛ′ be two representation spaces of A with basis ei ∈VΛand e′α ∈VΛ′. Let Φi(x) ≡Φ(x; ei) and Φ′α ≡Φ′(y; e′α) be two field multiplets.

Denote byR : VΛ′ ⊗VΛ →VΛ′ ⊗VΛ, with R(e′α ⊗ei) = Rβjαi e′β ⊗ej, the braiding matrix of thesefield multiplets:Φi(x)Φ′α(y) = Rβjαi Φ′β(y)Φj(x)for x > y(1.29)By quantum invariance,R ∆(X) = σ ◦∆(X) R,∀X ∈A(1.30)Thus, the braiding matrices intertwines the quantum algebra. Examples will be given inthe following sections.1c) Examples.

(i) Yangian invariance. The Yangians are deformations of loop algebras which havebeen introduced by Drinfel’d [7].

They are related to rational solutions of the quantumYang-Baxter equation.Let us first recall what are the Yangians. Let G be a simple Lie algebra with structureconstants fabc in an orthonormalized basis.

The Yangian, denoted Y (G), is the associative13

algebra with unity generated by the elements ta and Ta, a = 1, · · ·, dimG, satisfying therelations:hta , tbi= fabc tchta , Tbi= fabc TchTa,hTb, tcii−hta,hTb, Tcii= Almnabcntl, tm, tno(1.31)with Adefabc =124fadkfbelfcfmf klm and {x1, x2, x3} = Pi̸=j̸=k xixjxk. In particular, theelements ta generate the Lie algebra G and the elements Ta are G - intertwiners takingvalues in the adjoint representation of G. (for G = SU(2) one must add another Serre-likerelation.) The Yangians Y (G) are Hopf algebras with comultiplication ∆, counit ǫ andantipode S defined by:∆ta = ta ⊗1 + 1 ⊗ta ;(1.32c)ǫ(ta) = 0;S(ta) = −ta∆Ta = Ta ⊗1 + 1 ⊗Ta −12fabc tb ⊗tc ;(1.32b)ǫ(Ta) = 0;S(Ta) = −Ta −CAd4 tawith CAd the Casimir in the adjoint representation of G: fabcfbcd = CAd δad.The Yangian invariant R - matrices are those which satisfy the intertwining relationR ∆(Y )= σ ◦∆(Y ) R for all Y ∈Y (G).

We suppose that the vertex models we areconsidering in this section are defined from Y (G) - invariant Boltzmann weights. The non-local conserved currents we will describe in this section are the lattice analogues of thosehidden in 2D massive current algebras [13] [14], see section 4.From the defining relations of Y (G), it is obvious that the Yangians Y (G) possessthe properties we need in order to apply our formalism.

We therefore can define Yangiancurrents. For simplicity, we just define the currents associated to the generators ta and Ta;we denote them Jµa (x, t) and J µa (x, t), respectively.

By applying the general construction,we deduce that the conserved currents Jµa (x, t) are local (as it should be); they are definedby local insertions of the matrices ta representing the Lie algebra G. The conserved currentsJ µa (x, t) are non-local; in the operator formalism we have:J µa (x, t) = T µa (x, t) + 12fabc Jµb (x, t) φc(x, t)(1.33)14

withφc(x, t) =Xy y(1.35)(ii) Quantum universal enveloping algebras.

For any (affine) Kac-Moody algebraG with Cartan matrix aij, 0 ≤i, j ≤r and any complex number q ̸= 0, Drinfel’d [7]and Jimbo [8]define an universal quantum enveloping algebra (QUEA) Uq(G). Let di bepositive integers such that the matrix diaij is symmetric, and let qi = qdi.

The algebraA = Uq(G) has generators E+i , E−i , K2i , K−2i, 0 ≤i ≤r, and relationsK2i K±2j= K±2jK2i ,K2i K−2i= K−2iK2i = 1,K2i E±j = q±aijiE±j K2i ,E+i E−j −q−aijiE−j E+i = δij(K4i −1),(1.36)plus Chevalley-Serre relations to be written below. The Hopf algebra structure is definedby the coproduct∆(K±2i) = K±2i⊗K±2i,∆(E±i ) = E±i ⊗1 + K2i ⊗E±i ,(1.37)counit ǫ(E±i ) = 0, ǫ(K±2i) = 1, and antipode S(E±i ) = −K−2iE±i , S(K±2i) = K∓2i.

Theadjoint representation is then defined as usual, eq. (1.21), and the Chevalley-Serre relationsareAd1−aijE±iE±j = 0.

(1.38)15

We see that this is a very simple example of the Hopf algebras described in the introduction:the generators Ta are E±i and Θba is diagonal with entries K2i .Statistical models with QUEA symmetry are defined by trigonometric solutions of theYang-Baxter equation, the simplest case being the six-vertex model [10].The simple currents are defined by insertions of E±iwith disorder lines given byinsertions of K2i . In the operator formalism, the time components of the currents areJt±i(x) = K2i ⊗· · · ⊗K2i ⊗E±i ⊗1 ⊗· · · ⊗1.

(1.39)The corresponding charges are the generators ∆N(E±i ) acting on the whole space V ⊗N.The braiding relations are, for x > y:Jµ±i(x)Jν±j(y) = q±aijiJν±j(y)Jµ±i(x),Jµ±i(x)Jν∓j(y) = q∓aijiJν∓j(y)Jµ±i(x). (1.40)These relations are the same as the braiding relations of chiral vertex operators of a freemassless field φ taking value in the Cartan subalgebra of G with canonical momentum π.This suggests the continuum limit identificationJt±j∼expiβαj±φ(x) +Z x−∞π(y)dy,(1.41)with q = eiβ2, αj the simple roots, with inner product αiαj = diaij.

The space componentin the continuum limit is Jx±j= ∓iJt±j .2.CLASSICAL ORIGIN OF QUANTUM SYMMETRIES: DRESSINGTRANSFORMATIONS.The dressing transformations form the (hidden) symmetry groups of solitons equa-tions. Dressing transformations were first introduced by V. Zakharav and A. Shabat [15]and futher developped by the Kyoto group in their Tau-function appraoch to soliton equa-tions [16].

Their Poisson structure was disantangled by M. Semenov-Tian-Shansky [17].The author’s understanding of these transformations emerged from a joint work with O.Babelon [18].2a) What are the dressing transformations?16

(i) Equations of motion and Lax connexions. Suppose that the equations of motionof a set of fields φ are described by a set of non-linear differential equations.

Supposemoreover that these equations admit a Lax representation. This means that there exists afield dependent connexion, called the Lax connexion, Dµ,Dµ = ∂µ −Aµ[φ],such that the equations of motion are equivalent to the zero curvature condition for Dµ,hDµ , Dνi= 0(2.1)The Lax connexion takes value in some Lie algebra G with Lie group G.Notice that, thanks to the zero-curvature condition, the Lax connexion is a pure gauge;i.e.

there exists a G-valued function Ψ(x, t) such that:∂µ −AµΨ = 0orAµ =∂µΨΨ−1(2.2)The function Ψ(x, t) is defined up to a right multiplication by a space-time independentgroup element. This freedom is fixed by imposing a normalization condition on Ψ; e.g.Ψ(x0) = 1 for some point x0.

(ii) Construction of the dressing transformations. The dressing transformationsare non-local gauge transformations acting on the Lax connexion Aµ →Agµ and leaving itsform invariant.

They thus induce a transformation of the field variables φ →φg mappinga solution of the equations of motion into another.They are constructed as follows. First let us study the set of gauge transformationsmapping the Lax connexion Aµ on a given connexion Agµ.

Suppose that there exist two Gvalued functions, Θg+ and Θg−, such that:Agµ =∂µΘg±Θg±−1 + Θg± Aµ Θg±−1(2.3)Since Aµ is a pure gauge, Aµ=(∂µΨ) Ψ−1, Agµ is also a pure gauge, Agµ=∂µΘg±Ψ Θg±Ψ−1. This implies thatΘg+ΨandΘg−Ψdiffer by a right multiplicationby a space-time independent group element which we denote by g. Equivalently:Θg−−1 Θg+ = Ψ g Ψ−1(2.4)The main idea underlaying the dressing tranformations is to consider eq.

(2.4) as a fac-torization problem; i.e. we look for two subgroups B± ⊂G such that any element h ∈G17

admits a unique decomposition h = h−1−h+ with h± ∈B±. The requirement that Θ±belongs to B± then specify them uniquely from eq.

(2.4). The subgroups B± are foundby demanding that the transformations (2.3) preserve the form of the Lax connexion.The factorization problem in G,g = g−1−g+withg± ∈B±(2.5)is a called an algebraic Riemann Hilbert problem (by analogy with the classical RiemannHilbert problem).

For the dressing transformations to be well-defined this decompositionas to be unique.The gauge transformation (2.3) induces a transformation of the group valued functionΨ: Ψ →Ψg. Decompose the group element g ∈G as g = g−1−g+ with g± ∈B±, in theway specified by the algebraic factorization problem discussed above, then,Ψg =ΨgΨ−1+ Ψ g−1+=ΨgΨ−1−Ψ g−1−(2.6)The transformation (2.6) is well defined on the phase space because it preserves the nor-malization condition Ψ(x0) = 1.

(iii) The composition law for the dressing transformations. It is not the compos-tion law in G [17] [19].

Let g, h ∈G with decomposition, g = g−1−g+ and h = h−1−h+,their composition law in the dressing group is:(h+, h−) • (g+, g−) = (h+g+, h−g−)(2.7)In particular, the plus and minus components commute. We denote by GR the new groupequipped with this multiplication law.

The group law (2.7) can be derived by using thedressing of Ψ. First, we dress Ψ →Ψg by the g = g−1−g+ according to eq.

(2.6). Then, wedress Ψg →(Ψg)h by h = h−1−h+:(Ψg)h = Θh± Ψg h−1±withΘh± =Ψg h Ψg −1±(2.8)Using the definition (2.6) of Ψg, the factorization of ΨghΨg −1 can be written as follows:Θh−−1Θh+ = ΨghΨg −1 = Θg−Ψ(h−g−)−1(h+g+)Ψ−1Θg+This implies that (Θh−Θg−)−1(Θh+Θg+) = Ψ(h−g−)−1(h+g+)Ψ−1, or equivalently:Ψ(h−g−)−1(h+g+)Ψ−1± =ΨghΨg −1±ΨgΨ−1± = Θh± Θg±18

This proves eq. (2.7).For infinitesimal transformations, g ≃1 + X, X ∈G, and g± ≃1 + X± with X =X+ −X−, the dressing transformations are:δX Ψ = Y± Ψ −ΨX±withY± =ΨXΨ−1± .

(2.9)The composition law is, for X, Z ∈G :hX , ZiR =hX+ , Z+i−hX−, Z−i(2.10)This defines a new Lie algebra GR which is the Lie algebra of GR.2b) Few of their properties. (i) They are non-local.

This is obvious from their definition as Ψ is non-local. The dressingtransformations can be used to construction solutions of the soliton equations having non-trivial topological numbers from solutions with trivial topological numbers:φ(x) →φg(x);∀g ∈GR;(2.11)In particular, by dressing local conserved currents, the dressing transformations provide away to construct non-local conserved currents :Jµ(x, t) →Jgµ(x, t);∀g ∈GR.

(2.12)In the quantum theories, these non-local currents are turned into the generators of thequantum group symmetries. (ii) They induce Lie Poisson actions.

The dressing transformations induce an action of thegroup GR on the space of solutions of the classical equations of motion, i.e. on the phasespace.

This action is (in general) compatible with the Poisson structure; more precisely,it is a Lie Poisson action. It means that the Poisson brackets transform covariantly if thegroup GR is equipped with a non-trivial Poisson structure.

This Poisson structure, which,by construction, is compatible with the multiplication in GR, turns the group GR into aLie Poisson group.Let us be a more precise. Denote by P the phase space and by {, }P the Poissonbracket on it.

The dressing transformations define an action of GR on the function overthe phase space: f g(x) = f(g−1 · x) for f ∈Funct(P) and x ∈P. Suppose that the groupGR of is equipped with a Poisson bracket which we denote by {, }GR.

The statement that19

the dressing transformation are Lie Poisson action is equivalent to the covariance of thePoisson brackets:{f1, f2}gP = {f g1 , f g2 }P×GR∀f1, f2 ∈Funct(P) ; ∀g ∈GR(2.13)The Poisson bracket on P × GR is the product Poisson structure. (iii) A standard example.

As is well known, a Lie group G can be equipped with thefollowing Poisson bracket (Sklyanin’s Poisson bracket) [20]:nΨ(x) ⊗, Ψ(x)o=hrǫ , Ψ(x) ⊗Ψ(x)i(2.14)with matrices rǫ, ǫ = ±, solutions of the classical Yang-Baxter equation. By G-invariance,in eq.

(2.14) we can choose any of two solutions r+ or r−of the classical Yang-Baxterequation provided that their difference is the tensor Casimir C = r+ −r−.A directcomputation [17] [21] shows that the Sklyanin’s Poisson brackets are covariant under thetransformation (2.6), Ψ →Ψg, only if there are non-trivial Poisson brackets among the g’sbut vanishing Poisson brackets between the g’s and the fields Ψ. The Poisson brackets inGR are:ng+ ⊗, g+o=hr± , g+ ⊗g+i(2.15a)ng−⊗, g−o=hr± , g−⊗g−i(2.15b)ng−⊗, g+o=hr−, g−⊗g+i(2.15c)ng+ ⊗, g−o=hr+ , g+ ⊗g−i(2.15d)For g = g−1−g+, the Poisson brackets are the Semenov-Tian-Shansky brackets:ng ⊗, go= (g ⊗1)r+(1 ⊗g) + (1 ⊗g)r−(g ⊗1)−(g ⊗g)r± −r∓(g ⊗g)(2.16)It is easy to check that the multiplication in GR (not in G) is a Poisson mapping for thePoisson structure defined in eq.

(2.16), or in eq. (2.15).

Therefore GR is a Poisson Liegroup and the actions (2.6) are Lie Poisson actions.3. AN EXAMPLE OF DRESSING TRANSFORMATIONS: CURRENT AL-GEBRAS.We illustrate the general construction explained in the previous section on the exampleof the classical current algebras.

We essentially follow [22]. Dressing transformations inthe Toda theories were traited in [18].20

3a) The equations of motion.The field variables are one-forms, denoted by Jµ(x), valued in a semi-simple Lie algebraG: Jµ(x) = Pa Jaµ(x)ta where ta, a = 1, · · ·, dimG, form a basis of G 1. By definition theequations of motion impose to Jµ(x) to be a curl-free conserved current:∂µJaµ(x) = 0∂µJaν (x) −∂νJaµ(x) + f abcJbµ(x)Jcν(x) = 0(3.1)The equations of motion (3.1) admit a Lax representation: they are equivalent to the zerocurvature condition, [Dµ(λ), Dν(λ)] = 0, for the connexion Dµ(λ),Dµ(λ) = ∂µ +λ2λ2 −1 Jµ(x) +λλ2 −1 ǫµνJν(x)(3.2)The Lax connexion is an element of the loop algebra bG = G ⊗C[λ, λ−1].Remark 1: The linear problem (∂µ −Aµ(x)) Ψ(x) = 0 associated to the Lax representa-tion (3.2) is equivalent to the following (ǫµνǫνσ = δσµ):∂µ −λǫµν∂ν −λǫµνJνΨ(x) = 0(3.3)Remark 2: In the light-cone components of the Lax connexion are (ǫ±± = ±1):A± = −λλ ∓1 J±(3.4)The Lax connexion is therefore completely characterized by the following two conditions: i)A± have a simple pole at λ = ±1 (we then set Resλ=±1 A± = ∓J±) and ii) A±(λ = 0) = 0.Therefore, for a gauge transformation to be a symmetry it only has to preserve these twoconditions.Remark 3: The gauge condition A±(λ = 0) = 0 implies that Ψ(λ = 0) is space-timeindependent.

In the following we fix the gauge on Ψ by setting Ψ(λ = 0) = 1. Moreoverwe have:Jν(x) = ∂λǫνµ Aµ(x)λ=0 = ∂λǫνµ (∂µΨ) Ψ−1λ=0.

(3.5)3b) Dressing transformations and the Riemann Hilbert problem.1 We suppose the ta orthonormalized. We use the convention: ta, tb= f abctc where f abcdenote the structure constants of G.21

First, because the Lax connexion takes value in the loop algebra, we have to definethe factorization problem (2.5) in the loop group. It can be formulated as follows: Let Γbe a contour around the origin λ = 0.

Denote by Γ−(Γ+) the exterior (interior) domainof Γ. We choose Γ such that the points λ = ±1 belong to Γ−.

The factorization problemconsists in factorizing any regular element of the loop group, G(λ), λ ∈Γ, into the productof two λ-dependent group elements G±(λ) respectively analytic on Γ±:G(λ) = G−1−(λ) G+(λ);λ ∈Γ(3.6)This is the Riemann-Hilbert factorization problem. It is known that it admits a uniquesolution up to a left multiplication, G± →MG±, by a λ-independent group element M.The definition of the Riemann-Hilbert factorization is cooked up such that the transforma-tions we will now define are symmetries of the equations of motion of the classical currentalgebras.To dress the Lax connexion (3.2), we follow the general procedure:(i) We pick up an element G(λ) of the loop group.

We fix the gauge in the Riemann-Hilbertfactorization by imposing G+(λ = 0) = 1. (ii) We define ΘG(λ) = Ψ G(λ)Ψ−1 and factorize it according to the Riemann-Hilbertproblem:ΘG(λ) = Ψ G(λ)Ψ−1 = ΘG−(λ)−1 ΘG+(λ).

(3.7)We impose the gauge condition ΘG+(λ = 0) = 1. The solution to eq.

(3.7) is then unique. (iii) We define the dressed Lax connexion by:AGµ =∂µΘG±ΘG±−1 + ΘG± Aµ ΘG±−1.

(3.8)Because we can implement the dressing either using Θ+ or using Θ−, it easy to check thatthe dressed connexion AGµ possesses the same poles with the same orders than the originalconnexion Aµ. The gauge conditions we choose for the Riemann-Hilbert factorization alsoensure that AGµ satisfy the same gauge condition as Aµ.

Therefore, the dressing transfor-mation Aµ →AGµ , preserving the structure of the Lax connexion, induce a symmetry ofthe equations of motion. The dressed currents JGµ (x) are defined via the eq.

(3.5) withAGµ instead of Aµ:JGµ (x) = Jµ(x) + ǫµν∂ν∂λΘG+λ=0(3.9).22

(iv) For infinitesimal transformations, G(λ) = 1 + X(λ) + · · ·, where X(λ) ∈bG, X(λ) =X+(λ) −X−(λ) with X±(λ) analytic in Γ±, the dressings are:δX Aµ = ∂µ Y± +hY±, AµiδX Ψ = Y± Ψ −Ψ X±(3.10)with Y (λ) =ΨX(λ)Ψ−1(λ) = Y+(λ) −Y−(λ). In particular for the current:δX Jµ = ǫµν∂ν∂λY+λ=0(3.11)3c) Non-local conserved currents.The problem consists now in solving the Riemann-Hilbert factorization, eq.

(3.6). Themain point is that we will find differential equations which solve this problem recursively.In the following we restrict ourselves to the dressing of the current Jµ(x).

(i) Projection on bG+. Recall that by definition, eq.

(3.6), bG+ is the algebra of G-vectorfields regular at the origin λ = 0. Therefore, if Y (λ) is an element of the loop algebra bG,its projection Y+(λ) on bG+ is:Y+(λ) =IΓdz2iπY (z)z −λ;λ ∈Γ+(3.12)(ii) The dressing transformations act on Jµ by non-local gauge transformations.Thedressing of Jµ is defined in eq.

(3.9), or its infinitesimal form (3.11). To compute it we usethe explicit expression of Y (λ) and the projector (3.12):∂µ∂λY+λ=0 =Idz2iπz2h(∂µΨ(z)) Ψ−1(z) , Ψ(z)X(z)Ψ−1(z)i= ǫµν∂νZ+ + [Jν, Z+](3.13)with:Z+ =Idz2iπzΨ(z)X(z)Ψ−1(z).

(3.14)To derive the last equation we used the linear problem in the form (3.3). The variation ofJµ is therefore:δX Jµ = ∂µZ+ + [Jµ, Z+] .

(3.15)23

(iii) Recursion relation for Z+. The last step consists in solving for Z+ by recursion.

LetX ∈bG be Xn(λ) = vλ−n with n = 0, 1, · · · and v ∈G. Denote by Zn the correspondingsolution to eq.

(3.14):Zn =Idz2iπzΨ(z)vΨ−1(z)z−n. (3.16)Then using once more the differential equation (3.3), we have:∂µZn+1 = ǫµν∂νZn + [Jν, Zn].

(3.17)As advertised, this solves recursively the Riemann-Hilbert problem. The dressed currents,δnv Jµ, are recursively defined by eqs.

(3.15) and (3.17). This recursive construction isequivalent to the construction of ref.

[23]. The conservation law for the dressed currentsδnv Jµ can be checked directly.

(iv) The two first conserved currents. The first ones are the local currents Jaµ(x) since, forX = v ∈G,δ0v Jµ(x) =hJµ(x), vi.

(3.18)For X = vλ−1, v ∈G, we have ∂µZ1 = ǫµν [Jν, v], or equivalently,Z1(x) =hΦ(x), viwithΦ(x) =ZCx⋆J(3.19)where Cx is a curve ending at the point x. The dressing of Jµ is:δ1vJµ(x) = ǫµνhJν(x), vi+hJµ(x),hΦ(x), vii.

(3.20)In particular, projecting on the adjoint representation, we find the following non-localconserved currents:f abcδ1tb Jcµ(x) ∝J(1)aµ(x)J(1)aµ(x) = ǫµνJaν (x) + 12f abc Jbµ(x) Φc(x)(3.21)4. QUANTIFICATION: YANGIANS IN MASSIVE CURRENT ALGEBRAS.We use the example of the massive current algebras in order to describe how non-localconserved currents can be defined in a non-perturbative way and to illustrate few of their24

properties, (e.g. how they act on the states, on the fields, etc...).

But the approach is moregeneral, see e.g. ref.

[5].The currents J(1)aµ(x) are the currents we want to quantize. There are different waysto specify the quantum theory, e.g.

by defining it on the lattice, or as perturbation of itsU.V. fixed point, etc...

Here we use an alternative approach: we look for the conditionsthat we have to impose on the operator algebra in order to be able to define the quantumnon-local conserved currents. Therefore, we are interested in a quantum models satisfyingthe following hypothesis:(a) There exist quantum local conserved currents, Jaµ(x), taken values in the Lie algebraG :∂µ Jaµ(x) = 0.

(4.1)Furtheremore, because the currents Jaµ have to be one-forms, we impose that they havescaling dimensions one. (b) The currents Jaµ(x) satisfy the quantum version of the equations of motion (3.1); i.e.the quantum currents are curl-free:∂µJaν (x) −∂νJaµ(x) + f abc : Jbµ(x) Jcν(x) : = 0(4.2)where the double dots denote an appropriate regularization of f abcJbµ(x)Jcν(x), e.g.

by apoint splitting. This hypothesis imposes constraints on the operator product expansion(OPE) of the currents.

(c) The only fields taking values in the adjoint representation of G and having scalingdimensions zero, one or two are either Jaµ or ∂νJaµ. This fixes the OPE f abcJbµ(x)Jcν(0) upto the order O(|x|1−0):f abc Jbµ(x)Jcν(0) = Cρµν(x) Jaρ (0) + Dσρµν(x)∂σJaρ (0)+ O(|x|1−0)(4.3)The quantum currents Jaµ(x) satisfying these three hypthesis generate what could becalled a massive current algebra.4a) OPE’s in massive quantum current algebras.We now show that these hypothesis ensure that the currents satisfy the commutationrelations of a Kac-Moody algebra but also that they satisfy the following OPE’s 2:2 We used the following space-time conventions xν ≡(x0 = t, x1 = x); x± = x ± t; andds2 = ηµνdxµdxν = dt2 −dx2.25

Jb±(x)Jc±(0) = −kδab8iπ1(x±)2 −f abc2iπJc±(0)x±+ O(|x|−0)(4.4a)12f abc Jb+(x)Jc−(0) −Jb−(x)Jc+(0)(4.4b)= CAdj8iπ logM 2x+x−∂+Ja−(0) −∂−Ja+(0)+ O(|x|1−0)CAdj is the Casimir of G in the adjoint representation and M is the mass scale.Theproduct Ja±(x)Jc∓(0) is logarithmically divergent.We solve for the OPE (4.3) using our hypothesis. The proof goes in few steps:(i) First, locality, PT-invariance and Lorentz covariance determine the general tensor formof Cρµν(x) and Dσρµν(x).

Notice also that the conservation law for Jaµ allows us to chooseDσρµν(x) to be traceless: ησρDσρµν(x) = 0. Therefore, under the conditions (a) to (c), thegenerators Jaµ(x) of a massive current algebra satisfy the following OPE’s [13] :f abc Jbµ(x)Jcν(0) =C1 x2ηµνxρ + C2 x2 xµδρν + xνδρµ+ C3 xµxνxρ Jaρ (0) + 12xσ∂σJaρ (0)+D1 xρ xµδσν −xνδσµ+ D2 xσ xµδρν −xνδρµ∂σJaρ (0)+ O|x|1−0(4.5)The coefficients Ci and Di only depend on x2.

Furthermore, the conservation law for thecurrents implies the following differential equations for the functions Ci and Di [13] :x2 ddx2 C2 = −12 (C1 + 5C2)(4.6a)x2 ddx2 (C1 + C2 + C3) = −(C1 + C2 + 2C3)(4.6b)andx2 ddx2 D1 = −D1 −x24 C1(4.7a)x2 ddx2 D2 = −D2 −x24 C2(4.7b)x2 ddx2 (D1 + D2) = x24 C3(4.7c)26

(ii) The differential equations (4.6) and (4.7) do not specify uniquely the unknown coeffi-cients Ci(x2) and Di(x2). But we can use the hypothesis on the scaling dimension of thecurrents to fixe the leading behaviour of the functions Ci(x2):Ci(x2) =αi(x2)2 + O(|x|−3−0);i = 1, 2, 3,(4.8)with αi some constants.

We assume that there is no leading logarithmic corrections.Solving the differential equations (4.6) and (4.7), we find:C1(x2) = −α(x2)2 + O(|x|−3−0)(4.9a)C2(x2) =α(x2)2 + O(|x|−3−0)(4.9b)C3(x2) = −γα(x2)2 + O(|x|−3−0)(4.9c)Dk(x2) = −αk4x2 log−M 2k x2+ O(|x|−1−0);k = 1, 2(4.9d)The constants Mk are related to the mass scale and γ = 2 log(M2/M1). The constantα depends on the normalization of the currents: we fixe the normalization such thatα = −Cadj2iπ(iii) We finally impose the zero curvature condition.

From eqs. (4.5) and (4.9), we have:ǫµνhf abcJbµ(x)Jcν(0) + Z(−x2)∂µJaν (0) −∂νJaµ(0)i= −αγ4x2xµ2 (xρǫµσ + xσǫµρ)∂σJaρ (0) + ∂ρJaσ(0)(4.10)with Z(−x2) = α4 log(−M1M2x2).The curl-free equation (4.2) is then an immediat consequence of (4.10) if γ vanishes.The normal order in (4.2) is defined in such a way to cancel the logarithmic divergence inf abcJbµJcν.

Therefore, the curl-free equation fixes the two mass scale to be equalγ = 2 log(M2/M1) = 0The same conclusion could have been reached by imposing the chiral splitting of theleading terms of the OPE (4.5)3; the approach based on the chiral splitting assumptionwas described in ref. [14].3 The case γ ̸= 0 is also quite interesting; it probably corresponds to the 2D O(n) models.In particular, in this case the leading terms of the OPE of the currents do not satisfy the chiralsplitting.

In other words the chiral components of the currents, J a−and J a+, are mixed in theleading terms of the OPE.27

(iv) Commutation relations of the currents.Finally, other current OPE’s can bededuced using the same techniques. In particular we have:Jaµ(x)Jcν(0) = −kδab2iπ1(x2)2xµxν −12x2ηµν−f abc2iπ1x2xµδρν + xνδρµ −x2ηµνxρJaρ (0) + O(|x|−0).

(4.11)These OPE’s reduce to eq. (4.4a).

The products of the quantum operators are defined by:Jaµ(x, t)Jbν(y, t) = limǫ→0+ Jaµ(x, t + iǫ)Jbν(y, t). (4.12)Therefore, using limǫ→0+iǫx2+ǫ2 = iπδ(x), the OPE’s (4.11) implies:hJat (x) , Jbx(0)i= f abcJcx(0)δ(x) −k2δabδ′(x)hJat (x) , Jbt (0)i= f abcJct (0)δ(x)hJax(x) , Jbx(0)i= f abcJct (0)δ(x)(4.13)They are the commutation relations of a current algebra: the light cone component Ja±satisfy the commutation relations of the affine Kac-Moody algebra G(1).Remark 1: Two hidden consequences of the definition of the massive current algebraswe choose are: i) their ultra-violet limits are WZW models with G(1) ⊗G(1) symmetry;and ii) they describe perturbations of affine Kac-Moody algebras by the perturbing fieldsΦpert.

(x) = Pa Jaµ(x) Jaµ(x).Remark 2: The massive current algebras are characterized by the level K of the affineKac-Moody algebras. However the OPE’s (4.4) and the curl-free equation (4.2) are modelindependent in the sense that they do not depend on the level.Remark 3: Because the WZW models are the U.V.

fixed point of the massive chiralalgebras, they are also Y (G) invariant.Actually, The WZW models are Y (G) ⊗Y (G)invariant (at least classically [24]). They are also Uq(G)×Uq(G) invariant.

The perturbingfield JaµJaµ breaks these symmetries down to the diagonal Y (G) symmetry times a fractionalsupersymmetry. It could be interesting to solve the WZW models from their non-localsymmetries.

This will provide a test of the idea we are trying to develop for the massiveintegrable models.4b) The quantum non-local conserved currents.28

(i) Their definition. Having proved that the quantum conserved currents satisfy thequantum form (4.2) of the equations of motion (3.1), it is now easy to defined the quantumconserved currents J(1)(x, t).

We define them by a point splitting regularization (δ > 0):J(1)aµ(x, t) =limδ→0+ J(1)aµ(x, t|δ)J(1)aµ(x, t|δ) = Z(δ)ǫµνJaν (x, t) + 12f abc Jbµ(x, t)φc(x −δ, t)(4.14)where φc(x, t), which satisfies dφc = ⋆Jc, is defined by: φc(x, t) =RCx ⋆Jc The contourof integration Cx is a curve from −∞to x.The renormalization constant Z(δ) is fixed by requiring that J(1)aµ(x, t) are finiteand conserved. First it is easily seen from eq.

(4.4b) that J(1)aµ(x, t) is finite wheneverZ(δ) = α2 log(δ) + constant. The constant is fixed by demanding the conservation law forJ(1)aµ.

(The other subleading terms in Z(δ) are meaningless.) Using eq.

(2.4) we deduce,∂µJ(1)aµ(x, t|δ) = 12ǫµνhZ(δ)∂µJaν −∂νJaµ(x, t) + f abc Jbµ(x, t)Jcν(x −δ, t)i(4.15)Therefore, from eq. (4.4b) or (4.10), we learn that ∂µJ(1)aµ(x, t|δ) vanishes when δ →0 ifZ(δ) = α2 log(Mδ) + O(δ1−0).

(ii) Non-locality: the braiding relations. The non-local character of the currentsJ(1)(x, t) is encoded in their braiding relations, the equal time commutation relations.The latter are described as follows: Let Φ(y, t) be a quantum field local with respect tothe currents Jaµ(x, t).

Then it satisfies the following equal-time braiding relations [14]:J(1)aµ(x, t)Φ(y, t) = Φ(y, t)J(1)aµ(x, t);for x < y(4.16a)J(1)aµ(x, t)Φ(y, t) = Φ(y, t)J(1)aµ(x, t) −12f abc Qb0Φ(y, t)Jcµ(x, t);for x > y(4.16b)where Qb0 are the global charges associated with the local conserved current Jbµ. They arethe same as on the lattice, eq.

(1.35).The proof of the braiding relations (4.16 ) is the same as the proof of the braidingrelations for disorder fields. It only relies on the way to deform the contour Cx entering inthe definition of the currents J(1)(x, t) The relative positions of the contours Cx depend ifJ(1)(x, t) acts first or second: if J(1)(x, t) acts first (second) the contour is slightly under(above) the equal-time slice t = cst, we denote them C−x (C+x ).

(Remember that product ofoperators are defined by time ordering.) The relation (4.16a) follows because, in this case,29

there is no topological obstruction for moving the contour from the configuration C+x tothe configuration C−x . In the case of the relation (4.16b), these is an obstruction for movingthe contour C+x onto the contour C−x .

This implies non-trivial braiding relations. All thenon-locality of the currents J(1)aµ(x, t) is concentrated in the fields φc(x, t), eq.

(2.4). Forx > y the exchange relation between φc(x, t) and Φ(y, t) is:φc(x, t)Φ(y, t) =Zz∈C+x⋆Jc(z)Φ(y, t)=Zz∈γ(y)⋆Jc(z)Φ(y, t) +Zz∈C−x⋆Jc(z)Φ(y, t)= Qc0Φ(y, t)+ Φ(y, t) φc(x, t)(4.17)The contour γ(y) is a small contour surrounding the point y.

Plugging back eq. (4.17) intothe definition of the non-local current J(1)aµ proves the braiding relations (4.16b).4c) The non-local conserved charges and their algebra.Given conserved currents the associated charges are defined by integrating their dualforms along some curves.

The charges depend weakly on the contours of integration becausethe dual forms are closed. The global conserved charges acting on the states of the physicalHilbert space are defined by choosing the domain of integration to be an equal-time slice.Namely for a current Jµ(x, t):Q =Zt=cstdx Jt(x, t)(4.18)We denote by Qa0 and Qa1 the global charges associated to the currents Jaµ(x) and J(1)aµ(x).The (non-local) conserved charges generate a non-abelian extension of the two-dimensional Lorentz algebra.

In two dimensions the Poincar´e algebra which is generatedby the momentum operators Pµ and the Lorentz boosts L is abelian. The momentumoperators Pµ are the global charges associated with the conserved stress-tensor Tµν(x):∂µTµν(x) = 0 The Lorentz boost L is the global charge associated with the conservedboost current:Lµ(x) = 12ǫρσxρTµσ(x) −xσTµρ(x)(4.19)The (non-local) charges satisfy the following algebraic relations:hQa0 , Qb0i= f abcQc0;hQa0 , Qb1i= f abcQc1hL , Qa0i= 0;hL , Qa1i= −CAdj4iπ Qa0(4.20)30

The relations (4.20) are part of the defining relations of the semi-direct product of theYangians Y (G) by the Poincar´e algebra. Only the Serre relations are missing.

(They aremore difficult to prove because they involve commutation relations between the non-localcharges.) Moreover, as we will soon show, the comultiplications are those in Y (G).The three first relations are easily proved.

The last relation is more interesting andcan be proved in geometrical way. It consists in imposing a Lorentz boost R2π of angle(i2π) to the non-local currents J(1)aµ(x, t).

It is a rotation of (2π) in the Euclidian plane.Because the currents J(1)aµ(x, t) are non-local this transformation does not act trivially onthem: the string Cx winds around the point x. By decomposing this winding contour intothe sum of a contour from −∞to x plus a small contour surrounding x we obtain:R2π J(1)aµ(x, t) R−12π= J(1)aµ(x, t) −12f abc Qc0Jbµ(x, t)(4.21)Integrating the time-component of eq.

(4.21) over an equal-time slice givesR2π Qa1 R−12π= Qa1 −12 CAdj Qa0(4.22)in agreement with the relations (4.20) because R2π = exp(i2πL).4d) Action on the asymptotic states and the S-matrices. Non-perturbative resultson the S-matrices can be deduced by looking at the action of the quantum charges on theasymptotic states.

The constraints on the S-matrices we obtain arise by requiring thatthey commute with the non-local charges. These commutation relations imply algebraicequations which are nothing but the exchange relations for the quantum symmetry algebra(the Yangians Y (G) in the case of massive current algebras).

In general, these algebraicequations implies non-trivial constraints on the S-matrices which are sometimes enoughto determine them.Example: the SO(N) Gross-Neveu models. The SO(N) Gross-Neveu models are equivalentto the SO(N) massive current algebras at level K = 1.

In the SO(N) Gross-Neveu modelsthe fundamental asymptotic particles are Majorana fermions taking values in the vectorrepresentation of SO(N). In the SO(N) Gross-Neveu models, the Y (SO(N)) charges actingon the asymptotic fermions are given by:Qkl0= T kl(4.23a)Qkl1= −θ (N −2)iπT kl(4.23b)∆Qkl1 = Qkl1 ⊗1 + 1 ⊗Qkl1 −XnT kn ⊗T nl −T ln ⊗T nk(4.23c)31

where the T kl’s form the vector representation ⊔of SO(N):T klmn = δkmδln −δlmδkn.The charges Qa0 and Qa1 defined in eq. (4.23 ) satisfy the algebra (4.20); on-shell the boostoperator L acts as∂∂θ.

They define an irreducible representation of the SO(N)- Yangiansin the vector representation of SO(N). Eq.

(4.23c) is the comultiplication in Y ((SO(N)).Denote by S(θ12), θ12 = θ1 −θ2, the S-matrix of the two-fermion scattering. S(θ) actsfrom ⊔⊗⊔into itself.

As an SO(N) representation the tensor product ⊔⊗⊔decomposesinto ⊓⊔+⊥+•. We denote by P−, P+ and P0 the respective projectors.

By SO(N)-invariance, S(θ) decomposes on these projectors:S(θ) = σ+(θ)P+ + σ−(θ)P−+ σ0(θ)P0(4.24)where σn(θ) are scattering amplitudes.The non-local charges Qkl1are conserved andtherefore they commute with the S-matrix. For the two-fermion scattering, the Y (SO(N))exchange relations imply the following algebraic relations bewteen the scattering ampli-tudes:σ−(θ)σ+(θ) = θ(N −2) + i2πθ(N −2) −i2π;σ0(θ)σ−(θ) = θ + iπθ −iπ(4.25)Eq.

(4.25) determine S(θ) up to an overall function which could be fixed by closing thebootstrap program [9].4e) Action on the fields and the field multiplets. (i) The definition of the action.

The definitions of charges acting on the states and onthe fields differ by the choice of the contour along which the conserved current is integrated.The charges acting on a field Φ(y) located at a point y are defined by choosing the contourof integration γ(y) from −∞to −∞but surrounding the point y:QakΦ(y)=Zz∈γ(y)dzµǫνµ J(k)aν(z)Φ(y)(4.26)Compare with the lattice definition (1.27).For the currents Jaµ(x) and the charges Qa0 deforming the contour γ(y) proves thatQa0Φ(y)= Qa0 Φ(y) −Φ(y) Qa0(4.27)When the currents and the field Φ(y) are not respectively local the situation is moresubtle. The contour γ(y) can no more be closed and the action of the charges on the fieldis no more a pure commutator.

For the non-local conserved currents J(1)(x) the relation32

between the global charges (4.18) acting on the states and the charges (4.26) acting onthe fields is the following:Qa1Φ(y)= Qa1 Φ(y) −Φ(y) Qa1 + 12f abc Qb0Φ(y)Qc0(4.28)The proof of eq. (4.28) consists in decomposing the contour of integration γ(y) into thedifference of two contours γ+ and γ−which are respectively above and under the point y,and in using the braiding relation (4.16b) when the current Jb(x) is on γ−.

(ii) The comultiplications. We now derive the comutiplication from the braiding re-lations.

The comultiplications just encode how the charges act on a product of fields,say Φ1(y1)Φ2(y2) · · ·. We denote them by ∆.

In the case of the charges Qa0 and for fieldsΦn(yn) which are local with respect to the currents Jaµ(x) all the contours can be deformedwithout troubles and we have:Qa0Φ1(y1)Φ2(y2)= Qa0Φ1(y1)Φ2(y2) + Φ1(y1)Qa0Φ2(y2)∆Qa0 = Qa0 ⊗1 + 1 ⊗Qa0(4.29)It is the standard Lie algebra comultiplication as it should be.In the case of the non-local charges Qa1 the standard comultiplication is deformed dueto the non-trivial braiding relations between the non-local currents and the fields. LetΦn(yn) be quantum fields local with respect to the currents Jaµ(x).

Then we have thefollowing comultiplication for the non-local conserved charges Qa1:Qa1Φ1(y1)Φ2(y2)= Qa1Φ1(y1)Φ2(y2) + Φ1(y1) Qa1Φ2(y2)−12f abc Qb0Φ1(y1)Qc0Φ2(y2)(4.30a)∆Qa1 = Qa1 ⊗1 + 1 ⊗Qa1 −12f abc Qb0 ⊗Qc0(4.30b)Eqs. (4.29) and (4.30) are the comultiplication in Y (G).

Equation (4.30a) can be proved bydecomposing the contour γ12 used in defining the action of Qa1 on the product Φ1(y1)Φ2(y2).The contour γ12 is surrounding the two points y1 and y2. It decomposes into the sum oftwo contours γ1 and γ2 surrounding y1 and y2 respectively.

But on the contour γ2 we haveto use the braiding relations (4.16 ) in order to pass the string Cz through the point y1.Eq. (4.30 ) can also be proved starting from the graded commutators (4.28).

(iii) The field multiplets. To any (local) field Φ(x, t) is associated a multiplet which isconstructed by acting on the field with as many charges as possible:QA1 · · · QAP Φ(x, t)(4.31)33

with, in the case Y (G) symmetry, QA = Qa0, Qa1 or any element of the algebra generatedby them. By construction, the fields (4.31) form a field multiplet in the sense of eq.

(1.22).In general the field multiplets are infinite dimensional.The main property of the field multiplets resides, (assuming the knowledge of theaction on the asymptotic states), in the fact that if the field Φ(x, t) is known, then all itsdescendents, QA1 · · · QAP Φ(x, t) are also known. In other words, the descendents arecompletely determined by the data of the fields Φ(x, t) and of the values of the charges onthe asymptotic states.This property follows from the Ward identities expressing the quantum invariance:∆(M) QA⟨Φ1(x1) · · ·ΦM(xM)⟩= 0(4.32)where ∆(M) the M th comultiplication with ∆QA = QA ⊗1 + ΘAB ⊗QB.

Here we haveassumed that the vaccuum is quantum group invariant: QA|0⟩= 0, ⟨0|QA = 0.Theidentity (4.32) can be formulated on the form factors The form factors are the matrixelements of the fields between asymptotic states.By crossing symmetry, only matrixelements between the vacuum and the asymptotic particles are relevent. Let us denoteby Zα(θ) the asymptotic particles with rapidity θ; they form a representation W of thequantum symmetry algebra.

The form factors of the fields Φi(x, t) are defined by:Fα1,···αMi(θ1, · · · , θM) = ⟨0|ΦΛi (0)|Zα1(θ1) · · ·ZαM (θM)⟩. (4.33)On the form factors, the Ward identities (4.32) become:⟨0|QAΦi(x)|Zα1(θ1), · · ·, ZαM (θM)⟩= −⟨0|ΘABΦi(x)∆(M)QB|Zα1(θ1) · · ·ZαM (θM)⟩= ⟨0|ΦΛ(x)∆(M)s(QA)|Zα1(θ1) · · ·ZαM (θM)⟩(4.34)with s the antipode.

Eqs. (4.34) give the form factors of the field QA(Φi(x, t)) in termsof those of the fields ΘABΦi(x, t) and of the action of the charges QA on the asymptoticparticles Zα(θ).Example: action on the stress-tensor.

In massive current algebras, the stress-tensor andthe current are in the same Y (G) - multiplets. We have:Qa1 (Tµν) ∝ǫµρ∂ρJaν + ǫνρ∂ρJaµ.

(4.35)34

This relation was proved in [25] using form factor technique, it can aslo be deduced fromthe hypothesis we made for defining the massive current algebras.Remark 1: The Ward identity (4.34) can written for any element in the envelopingalgebra. Choosing a particular element associated to the square of the antipode leading tothe so-called deformed KZ equations for the form factors [26].Remark 2: Assuming, as in conformal field theory, that the (complete) symmetry al-gebra possess free field vertex representations, the form factors will also admit free fieldrepresentations.

The Zamoldchikov creation operators Zα(θ) as well as the field operatorsΦΛ(x) will be represented as quantum vertex operators in analogy with the vertex operatorrepresentations of the quantum affine algebras. This is suggested by the explicit formulafor the form factors found by Smirnov [27].

Their generic forms are as follows:F(θ1, · · ·, θM) =Z Ykdµ(αk) P(α1, · · · , αk|θ1, · · ·, θM)×Yk

Thekernel between these operators can be deduced from the formula (4.36).Remark 3: The braiding relations between the quantum field multiplets are determinedby the quantum symmetries: the braiding matrices intertwine the quantum symmetries, cfe.g. eqs.

(1.29) and (1.30). Moreover the braiding relations are scale invariant; i.e.

they arerenormalization group invariant. This is obvious from their definitions but this also followsfrom the topological origin of the braiding relations.

The braiding relations just reflectthe monodromy of the field multiplet correlation functions. Therefore, the renormalizationgroup induces isomonodromy deformations [28].

The connection between isomonodromydeformations and quantum group symmetries could provide another starting point fordetermining the correlation functions in massive two dimensional quantum field theories.35

5. CONCLUSIONS.Few open problems: Quantum symmetries have been used with some success to studyintegrable perturbations of conformal field theories.

Some examples are [29]: the Φ(1,3)and the Φ(1,2), Φ(2,1) perturbations of the minimal conformal models, the GK ⊗GL/GK+Lcosets models, the ZN parafermionic models, and the fractional supersymmetric models,etc... Most of the results obtained in these papers concern the S-matrices of these massivemodels.

The main open problem is the derivation of the off-shell properties of the models,(the correlation functions and the form factors), from their quantum symmetries. As wementioned in the introduction, this requires checking if the quantum symmetries form acomplete symmetry algebra or not.

Other few technical problems have been formulatedin the previous sections, most of them as remarks. In particular, the connection betweenisomonodromy deformations and quantum group symmetries could open a new way ofsolving for the correlation functions.3D generalizations?

Let us discuss how these constructions could possibly be generalizedto three dimensions.In two dimensions, quantum group symmetries require non-localcurrents: the non-locality, which is reflected in the equal-time commutation relations,imply the non-trivial comultiplications. The 2D non-local currents are fields localized onpoints but with a “string” attached to them.The currents are generically express asproducts of disorder fields (the “wavy string” in the lattice description) by spin fields(which are local fields).

This is analogue to the definition of the 2D parafermions.In any dimensions, to have more general symmetry than supersymmetry we need fieldswith non-trivial equal-time commutation relations. In three dimensions, this requires toconsider fields localized on curves (with a sheet attached to them).

Once again, examplesare provided by disorder and parafermionic fields. The latters can be described as follows:consider a group G invariant spin lattice model in three dimensions.

The disorder fieldsµg(C), g ∈G, are defined by splitting all the spin variables σ which leave on a surfaceΣC bounded by C: σ →gσ. By G-invariance, µg(C) depend weakly on ΣC.

The 3Dparafermions Ψg(C; x) are defined as product of disorder fields µg(C) by spin fields σ(x):Ψg(C; x) = µg(C) σ(x);x ∈C. (5.1)They satisfy non-trivial commutation relations analogous to the two-dimensional case.

An-ions are particular examples of this construction, with the curve C a small two-dimensional36

cone extending to the spacial infinity [30]. Generalizing eq.

(5.1) by considering product ofspin fields all along the curve C gives the parafermionic string which has been consideredin the 3D Ising model [31].Thus, if quantum group symmetry exists in three dimensions, it is a theory of quantumfields localized on curves, i.e. a theory of quantum loops.

A (formal) example is given byPolyakov’s string representation of gauge theories in three dimensions [32]. Let W(C) bethe Wilson loops:W(C) = P expICA(5.2)where A is the Yang-Mills connection.

Define the functional current Jµ(C; x) by:Jµ(C; x) = W(C)−1δδxµW(C)(5.3)This functional current is conserved and curl-free:δδxµJµ(C; x) = 0δδxµJν(C; x′) −δδx′νJµ(C; x)+hJµ(C; x) , Jν(C; x′)i= 0tµJµ(C; x) = 0(5.4)with tµ the vector tangent to the curve C at the point x. Formally the following non-localcurrent, J (1)µ (C; x), localized on the curve C is functionally conserved:J (1)µ (C; x) = ǫµνρtνJρ(C; x) + 12hJµ(C; x) , P(C; x)i(5.5)with δP(C; x)/δxµ = ǫµνρtνJρ(C; x).

The analogy with the 2D current algebras is ap-pealing, cf. eq.

(3.21). It seems to indicate the possibility of having generalized non-localsymmetry in 3D gauge theories.

Unfortunately, the equations of motion (5.4) are not veryrigorous; only the discretized lattice version has been proved and, up to our knowledge, noconcrete result on the quantum continuous case has never been proved. However, to con-struct generalized quantum group symmetry in three dimensions remains a very attractivechallenge.Acknowledgements: It is a pleasure to thank my collaborators, Olivier Babelon, Gio-vanni Felder and Andr´e Leclair.

I thank the organizers of the 91 Cargese school gor givingme the opportunity to present this lecture in a very pleasant atmosphere.37

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