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ON THE SYMMETRIES OF INTEGRABILITY

arXiv:hep-th/9112067v1 19 Dec 1991ON THE SYMMETRIES OF INTEGRABILITYM. Bellon, J-M. Maillard, C. VialletAbstract.We show that the Yang-Baxter equations for two dimensional models admit as a groupof symmetry the infinite discrete group A(1)2 .

The existence of this symmetry explains thepresence of a spectral parameter in the solutions of the equations. We show that similarly, forthree-dimensional vertex models and the associated tetrahedron equations, there also existsan infinite discrete group of symmetry.

Although generalizing naturally the previous one,it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help toresolve the Yang-Baxter equations and their higher-dimensional generalizations and initiatethe study of three-dimensional vertex models.

These symmetries are naturally representedas birational projective transformations. They may preserve non trivial algebraic varieties,and lead to proper parametrizations of the models, be they integrable or not.

We mentionthe relation existing between spin models and the Bose-Messner algebras of algebraic combi-natorics. Our results also yield the generalization of the condition qn = 1 so often mentionedin the theory of quantum groups, when no q parameter is available.Talk given at the COMO meeting “Advanced quantum field theory and Critical Phenomena” (1991)and the LUMINY “Colloque Verdier sur les syst`emes int´egrables” (1991)work supported by CNRSPostal address: Laboratoire de Physique Th´eorique et des Hautes EnergiesUniversit´e Paris VI, Tour 16, 1er ´etage, boˆıte 126.4 Place Jussieu/ F–75252 PARIS Cedex 05

1IntroductionThe results presented here appear in a series of papers by M. Bellon, J-M. Maillard andC-M. Viallet [1, 2, 3, 4, 5, 6].The Yang-Baxter equations, which appeared twenty years ago1, have acquired a predom-inant role in the theory of integrable two-dimensional models in statistical mechanics [12, 13]and field theory (quantum or classical). They have actually outpassed the borders of physicsand have become fashionable in some parts of the mathematics literature.

They in particularsupport the construction of quantum groups [14, 15].The Yang-Baxter equations [13] and their higher dimensional generalizations are nowconsidered as the defining relations of integrability. They are the “Deus ex machina” ina number of domains of Mathematics and Physics (Knot Theory [16], Quantum InverseScattering [17], S-Matrix Factorization, Exactly Solvable Models in Statistical Mechanics,Bethe Ansatz [18], Quantum Groups [19, 15], Chromatic Polynomials [20] and more awaiteddeformation theories).

The appeal of these equations comes from their ability to give globalresults from local ones. For instance, they are a sufficient and, to some extent, necessary [21]condition for the commutation of families of transfer matrices of arbitrary size and even ofcorner transfer matrices.

From the point of view of topology, one may understand theserelations by considering them as the generators of a large set of discrete deformations of thelattice. This point of view underlies most studies in knot theory [16] and statistical mechanics(Z-invariance [22, 23]).We want to analyze the Yang-Baxter equations and their higher dimensional general-izations [24, 25, 26, 27] without prejudice about what should be a solution, that is to sayproceed by necessary conditions, and have as an input the form of the matrix of Boltzmannweights.We will exhibit an infinite discrete group of transformations acting on the Yang-Baxterequations or their higher dimensional generalizations (tetrahedron, hyper-simplicial equa-tions).These transformations act as an automorphy group of various quantities of interest inStatistical Mechanics (partition function,.

. .

), and are of great help for calculations, evenoutside the domain of integrability (critical manifolds, phase diagram,. .

. ) [28].We show here is that they form a group of symmetries of the equations defining inte-grability.

They consequently appear as a group of automorphisms of the Yang-Baxter ortetrahedron equations. We will denote this group Aut.The existence of Aut drastically constrains the varieties where solutions may be found.

Inthe general case, it has infinite orbits and gives severe constraints on the algebraic varietieswhich parametrize the possible solutions (genus zero or one curves, algebraic varieties whichare not of the general type [29]). In the non-generic case, when Aut has finite order orbits,the algebraic varieties can be of general type, but the very finiteness condition allows fortheir determination [1].In the framework of infinite group representations, it is crucial to recognize the essentialdifference between what these symmetry groups are for the Yang-Baxter equations andwhat they are for the higher dimensional tetrahedron and hyper-simplicial relations: thenumber of involutions generating our groups increases from 2 to 2d−1 when passing from1In fact, fifty years ago, Lars Onsager was totally aware of the key role played by the star-triangle relationin solving the two-dimensional Ising model, but he preferred to give an algebraic solution emphasizing Cliffordalgebras [7, 8, 9, 10, 11].1

two-dimensional to d-dimensional models and the group jumps from the semi-direct productZ × Z2 to a much larger group, i.e., a group with an exponential growth with the length ofthe word.The existence of Aut as a symmetry of the Yang-Baxter equations has the followingconsequence: we may say that solving the Yang-Baxter equation is equivalent to solvingall its images by Aut. These images generically tend to proliferate, simply because Aut isinfinite.

Considering that the equations form an overdetermined set, it is easy to believethat the total set of equations is “less overdetermined” when the orbits of Aut are of finiteorder. One can therefore imagine that the best candidates for the integrability varieties areprecisely the ones where the symmetry group possesses finite orbits: the solutions of Au-Yanget al.

[30, 31, 32] seem to confirm this point of view [33, 34].A contrario, if one gets hold of an apparently isolated solution, the action of Aut willmultiply it until building up, in experimentally not so rare cases, a continuous family ofsolutions from the original one. This is the solution to the so-called baxterization problem [4].We first show that the simplest example of Yang-Baxter relation which is the star-trianglerelation [13] has an infinite discrete group of symmetries generated by three involutions.These involutions are deeply linked with the so-called inversion relations [35, 36, 37, 38].which two-dimensional model.This analysis can be extended to the “generalized star-triangle relation” for InteractionaRound the Face models without any major difficulties [12, 39].2The star-triangle relations2.1The settingWe consider a spin model with nearest neighbour interactions on square lattice.

The spins σican take q values. The Boltzmann weight for an oriented bond ⟨ij⟩will be denoted hereafterby w(σi, σj).

The weights w(σi, σj) can be seen as the entries of a q × q matrix. In thefollowing we will introduce a pictorial representation of the star-triangle relation.

An arrowis associated to the oriented bond ⟨ij⟩. The arrow from i to j indicates that the argumentof the Boltzmann weight w is (σi, σj) rather than (σj, σi).

This arrow is relevant only forthe so-called chiral models [30], that is to say that the q × q matrix describing w is notsymmetric. An interesting class of q × q matrices has been extensively investigated in thelast few years [30, 32, 31]: the general cyclic matrices.

It is important to note that we donot restrict ourselves to this particular class of matrices.The finite algebras we are lead to consider contain in particular Bose-Messner algebras,i.e. stuctures occurring in graph theory, and more precisely algebraic combinatorics [40].The detailed relationship will not be explained here.

The reader could compare paragraph5 of [1] and the definition of Bose-Messner algebra of association schemes in [41] ,page 44,and [42].Let us give the following non cyclic nor symmetric 6 × 6 matrix as another illustrative2

example:xyzyzzzxyzyzyzxzzyyzzxzyzyzyxzzzyzyx(1)2.2The relationsWe introduce the star-triangle equations both analytically2 and pictorially:Xσw1(σ1, σ) · w2(σ, σ2) · w3(σ, σ3) = λ w1(σ2, σ3) · w2(σ1, σ3) · w3(σ1, σ2). (2)✚✚✚✚✚❃❩❩❩❩❩⑦✁✁✁✁✕❆❆❆❆❯❄✲¯1¯2¯3(3)(2)(1)(1)(2)(3)123=(st1.1)One should note that satisfying equation (2) together with the relation (st1.2) obtained byreversing all arrows, is a sufficient condition for the commutation of the diagonal transfermatrices of arbitrary size M with periodic boundary conditions TM(w2, w2) and TM(w3, w3):✲✛❏❏❏❏❏❫✻❏❏❏❏❏❫❏❏❏❏❏❫✻✻❏❏❏❏❏❫❏❏❏❏❏❫✻✻❏❏❏❏❏❫❏❏❏❏❏❫✻✻❏❏❏❏❏❫✻Mσσ¯2¯2¯2¯22¯333¯322¯333¯32Note that for cyclic matrices ([30, 32, 31]) the star-triangle relations (st1.1) and (st1.2)give the same equations since one exchanges (st1.2) and (st1.1) by spin reversal.One could obviously imagine many other choices for the arrows on the six bonds, howeveronly three of them lead to the commutation of diagonal transfer matrices.

We therefore havethree systems of equations to study. For example, if the Boltzmann weights are given bythe 6 × 6 matrix (1), these three systems of equations are respectively made of 20 differentequations or 35 or 36.3The Yang-Baxter relation for vertex models.We shall not get here into the arcanes of this relation, which appears in the theory ofintegrable models [15], the theory of factorizable S-matrix in two-dimensional field theory,2Since the wi and wi are homogeneous variables, there will always be a global multiplicative factor λfloating around in the star-triangle equations.3

the quantum inverse scattering method [17], knot theory and has been given a canonicalmeaning in terms of Hopf algebras [43] (quantum groups [14, 15, 44, 45, 46]) and the list isfar from exhaustive. We just want to fix some notations for later use.We consider a vertex model on a two-dimensional square lattice.

To each bond is associ-ated a variable with q possible states and a Boltzmann weight w(i, j, k, l) is assigned to eachvertexkjliIn order to write the Yang-Baxter relation, the q4 homogeneous weights w(i, j, k, l) arefirst arranged in a q2 × q2 matrix R:Rijkl = w(i, j, k, l). (3)The Yang-Baxter relation is a trilinear relation between three matrices R(1, 2), R(2, 3) andR(1, 3):Xα1,α2,α3Ri1i2α1α2(1, 2)Rα1i3j1α3(1, 3)Rα2α3j2j3 (2, 3) =Xβ1,β2,β3Ri2i3β2β3(2, 3)Ri1β3β1j3(1, 3)Rβ1β2j1j2 (1, 2).

(4)The assignation (3) is arbitrary and we may specify it by complementing the vertex with anarrow and attributing numbers to the lines✒jgidjdigR= Rigidjgjd(g, d).With these rules relation (4) has the following graphical representation◗◗◗◗◗◗◗◗◗◗◗✑✑✑✑✑✑✑✑✑✑✑✲✑✑✑✸✁✁✁✁✕✁✁✁✁✕✑✑✑✸✲◗◗◗◗◗◗◗◗◗◗◗✑✑✑✑✑✑✑✑✑✑✑β2β1β3i2i1i3j3j2j1j1j2j3i3α3α2α1i2i1=123321(5)The lines carry indices 1,2,3.Some especially interesting solutions depend on a continuous parameter called the “spec-tral parameter”. The presence of this parameter is fundamental for many applications inphysics, as for example the Bethe Ansatz method [47, 11, 17, 18].

One of the main issues in4

the full resolution of (4) is precisely to describe what is this parameter and the algebraic va-riety on which it lives, although its presence may obscure the algebraic structures underlyingthe Yang-Baxter equation (the discovery of quantum groups was allowed by forgetting thisparameter [46, 14, 48, 15]). The problem of building up continuous families of solutions froman isolated one, known as the baxterization [16], is made straightforward by our study.

In-deed our results explain the presence of the spectral parameter in the solution of the equation(see also [1]).4The symmetry group of the star-triangle relation4.1The inversion relationTwo distinct inverses act on the matrix of nearest neighbour spin interactions: the matrixinverse I and the dyadic (element by element) inverse J.We write down the inversionrelations both analytically and pictorially:Xσw(σi, σ) · I(w)(σ, σj)=µ δσiσj,(6)w(σi, σj) · J(w)(σi, σj)=1. (7)where δσiσj denotes the usual Kronecker delta.ssss✲✲σi = σjσjσiσI(w)w=andssss✛✚✘✙✲✲σjσiσiσj=wJ(w)The two involutions I and J generate an infinite discrete group Γ (Coxeter group) iso-morphic to the infinite dihedral group Z2 × Z.

The Z part of Γ is generated by IJ. In theparameter space of the model, that is to say some projective space CPn−1 (n homogeneousparameters), I and J are birational involutions.

They give a non-linear representation of thisCoxeter group by an infinite set of birational transformations [1]. It may happen that theaction of Γ on specific subvarieties yields a finite orbit.

This means that the representationof Γ identifies with the p-dihedral group Z2 × Zp.4.2The symmetries of the star-triangle relationsThe two inversions I and J act on the star-triangle relation. Let us give a pictorial repre-sentation of this action, starting from (st1.1) as an example:5

✛✙✘✛✙✘✲✲❄✚✚✚✚✚✚❃❩❩❩❩❩❩⑦✲✁✁✁✁✁✕❆❆❆❆❆❯✚✚✚✚✚✚❃❩❩❩❩❩❩⑦❄✁✁✁✁✁✕❆❆❆❆❆❯✲I(1)I(1)J(¯1)J(¯1)==¯3¯2¯1321J(¯1)32I(1)¯3¯2(tst)The transformed equation reads:λXσ1I(w1)(τ, σ1) · w2(σ1, σ3) · w3(σ1, σ2) = w2(τ, σ2) · w3(τ, σ3) · J(w1)(σ2, σ3). (8)We get an action on the space of solutions of the star-triangle relation.If (w1, w2, w3, w1, w2, w3) is a solution of eq(2) (see picture (st1.1) for the specific arrange-ment of arrows), then (I(w1), w3, w2, J(w1), w3, w2) is also a solution of eq(2), at the price ofa permitted redefinition of λ.

In this transformation, the weights w1 and w1 play a specialrole.At this point, it is better to formalize this action by introducing some notations. Wemay choose as a reference star-triangle relation ST , the symmetric configuration:✚✚✚✚✚❃❩❩❩❩❩⑥✁✁✁✁☛❆❆❆❆❑❄✲t1t2t3(3)(2)(1)(1)(2)(3)s1s2s3=(ST )Any configuration may be obtained by reversing some arrows and permuting some bonds.With evident notations, we will denote by Rs1, Rs2, Rs3, Rt1, Rt2, Rt3 the reversals of arrows,and by Psi,sj, Psi,tj, Pti,tj the permutations of bonds.Moreover I and J act on the bondsas Is1, Is2, .

. .

The action of I and J described above (where 1 was playing a special role)identifies with the action ofK1 = Rs2Rt3Is1Jt1Ps2,t3Ps3,t2. (9)It is easy to check that K1 is an involution.We may construct two similar involutions K2 and K3, obtained by cyclic permutation ofthe indices 1, 2, 3.

The involutions Ki(i = 1, 2, 3) verify the defining relations of the Weyl6

group of an affine algebra of type A(1)2[49]:(K1K2)3 = (K2K3)3 = (K3K1)3 = 1. (10)We denote Aut the group generated by the three involutions Ki (i = 1, 2, 3).5The symmetry group of the Yang-Baxter equation.5.1The inversion relations.The R-matrix appears naturally as a representation of an element of the tensor productA ⊗A of some algebra A with itself.

This algebra is a nice Hopf algebra in the context ofquantum groups. We shall not dwell on this here but recall some simple operations on R.In A ⊗A we have a product inherited from the product in A:(a ⊗b)(c ⊗d) = ac ⊗bd.

(11)R is an invertible element of A ⊗A for this product and we shall denote by I(R) the inversefor this product:R · I(R) = I(R) · R = 1 ⊗1. (12)In terms of the representative matrix this reads:Xα,βRijαβ I(R)αβuv = δiu δjv =Xα,βI(R)ijαβ Rαβuv .

(13)This is nothing else but the so-called inversion relation for vertex models [35, 36, 39, 50, 29].On A ⊗A we have a permutation operator σ:σ(a ⊗b)=b ⊗a,(14)(σR)ijuv=Rjivu,for the matrix R.(15)Note that the representation of σ is just the conjugation by the permutation matrix P:P ijkl=δilδjk,(16)σR=PRP. (17)In the language of matrices we have a notion of transposition.Let us define partialtranspositions tg and td by:(tgR)ijuv=Rujiv ,(18)(tdR)ijuv=Rivuj,(19)and the full transpositiont = tgtd = tdtg.

(20)We shall in the sequel use another inversion J defined by:J = tgItd = tdItg,(21)7

or equivalently:Xα,βRαuvβ J(R)αijβ = δiu δjv =Xα,βJ(R)iβαj Ruβαv(22)These operators verify straightforwardly:I2=J2 = 1,It = tI,Jt = tJ,σ2=t2 = 1,σI = Iσ,σJ = Jσ,(σtg)2=(σtd)2 = t,σtgσtd = 1. (23)Each of these operations has a graphical representation.

For the inversion I or more preciselyfor σI it is:✒✒vuij=ivσI(R)Rjuαβ(24)the inversion J reads:❙❙♦❙❙♦=uijvviJ(R)σRujαβ(25)and the transposition reads:❅❅❅❅■✒=AσtdAkjliiljkNote that the two inversions I and J do not commute. They generate an infinite discretegroup Γ, the infinite dihedral group, isomorphic to the semi-direct product Z × Z2.

Thisgroup is represented on the matrix elements by birational transformations [1, 51, 52] actingon the projective space of the entries of the matrix R. Remark that for the vertex models,the birational transformations associated to the two involutions I and J are naturally relatedby collineations (see (21): this should be compared with the situation for nearest neighbourinteraction spin models [1, 53].5.2The symmetries of the Yang-Baxter equations.At the price of the redefinitions:A=tR(2, 3),(26)B=σtdR(1, 3),(27)C=R(1, 2),(28)8

we may picture the Yang-Baxter relation in a more symmetric way:✁✁✁✁☛✁✁✁✁☛❆❆❆❆❑❆❆❆❆❑◗◗◗◗◗◗◗◗◗◗◗✑✑✑✑✑✑✑✑✑✑✑✲✲◗◗◗◗◗◗◗◗◗◗◗✑✑✑✑✑✑✑✑✑✑✑CBBAAC=123321(29)We may bracket (29) with◗◗◗◗◗◗✑✑✑✑✑✑✲˜C12, where ˜C = σI(C). We get✲✲❏❏❏❏❏❏✡✡✡✡✡✡✡✡✡✡✡✡❏❏❏❏❏❏✲✲❏❏❏❏❏❏✡✡✡✡✡✡❙❙♦✠❅❅❘✓✓✼❏❏❏❏❏❏✡✡✡✡✡✡✲✡✡✡✡✡✡❏❏❏❏❏❏❏❏❏❏❏❏✡✡✡✡✡✡✲=˜CC˜CABBA˜CC˜C(30)that is to say✁✁✁✁☛✁✁✁✁☛❆❆❆❆❑❆❆❆❆❑◗◗◗◗◗◗◗◗◗◗◗✑✑✑✑✑✑✑✑✑✑✑✲✲◗◗◗◗◗◗◗◗◗◗◗✑✑✑✑✑✑✑✑✑✑✑tgAtdBtI(C)tI(C)tdBtgA=123321(31)This relation is nothing but (29) after the redefinitionsA→tgA,B→tdB,C→tI C.(32)We may denote by K3 the operation (32).

We have two other similar operations K1 and K2K1 :A →tI AB →tgBC →tdC,K2 :A →tdAB →tI BC →tgC.The discrete group Aut generated by the Ki’s (i = 1, 2, 3) is a symmetry group of the Yang-Baxter equations. These generators Ki (i = 1, 2, 3) are involutions.

The Ki’s satisfy the9

relation (K1K2K3)2 = 1. Actually, the operation K1K2K3 is just the inversion I on R.Among the elements of the discrete group generated by the Ki’s we have in particular:(K1K2)2 :A→ItgItgA = tIJA,(33)B→tdItdIB = tJIB,(34)C→C.

(35)Since IJ is of infinite order, we have generated an infinite discrete group of symmetries. Thisis exactly the phenomenon that we described in section 4.2 for the star-triangle equations.Under this form it is not so evident to find the actual structure of the group.

Let usintroduce KA, KB and KC, which are simply related to the Ki’s by the transposition of twovertices:KA :A →σtIAB →tgσCC →σtgB,KB :A →σtgCB →σtIBC →tgσA,KC :A →tgσBB →σtgAC →σtIC.It is easily verified that:K2A = K2B = K2C = 1,(36)and(KAKB)3 = (KBKC)3 = (KCKA)3 = 1,(37)with no other relations. We recover the affine Coxeter group A(1)2we already encountered insection 4.2.We have here a very powerful instrument: it defines adequate patterns for the matrixR [54].

It permits the so-called baxterization of an isolated solution just acting with tIJ.Indeed if a set of relations among the entries of R are preserved by IJ (or at least by tIJ),they will stay for every transforms of the initial Yang-Baxter relation. We shall illustrate insection 7.1.1 the baxterization on the Baxter eight-vertex model [55, 22] and show in section7.1.2 how to introduce a spectral parameter for the solutions of the Yang-Baxter equationsassociated to sl(n) algebras.6The tetrahedron equations and their symmetries.This equation is a generalization of the Yang-Baxter equation to three dimensional vertexmodels [25, 24, 27].

We give a pictorial representation of the three-dimensional vertex bymjlkniRThe Boltzmann weights of the vertex are denoted w(i, j, k, l, m, n) and may be arranged ina matrix of entriesRijklmn = w(i, j, k, l, m, n). (38)10

The tetrahedron equation has a pictorial representation:✒❅❅❅❅❅❅❅❅❅❅❘❄✟✟✟✟✟✟✟✟✟✟✟✯❏❏❏❏❏❏❏❏❏❏❏❏❏❫❳❳❳❳❳❳❳③❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳③❏❏❏❏❏❏❏❏❏❏❏❏❏❫✟✟✟✟✟✟✟✟✟✟✟✯❄❅❅❅❅❅❅❅❅❅❅❘✒541236362415=The algebraic form isR123R543R516R426 = R426R516R543R123. (39)We may here again introduce an inverse IXαg,αm,αd(IR)igimidαgαmαd · Rαgαmαdjgjmjd= δigjgδimjmδidjd.

(40)We also introduce the partial transpositions tg, tm and td with(tgR)igimidjgjmjd = Rjgimidigjmjd,(41)and similar definitions for tm and td.We redefineA = R123,B = tdR543,C = tgtmR516,D = tR426,(42)where t is the full transposition tgtmtd. Equation (39) then takes the more symmetric formXs1,...,s6Ai1i2i3s1s2s3Bi5i4j3s5s4s3Cj5j1i6s5s1s6Dj4j2j6s4s2s6 =Xr1,...,r6Dr4r2r6i4i2i6 Cr5r1r6i5i1j6 Br5r4r3j5j4i3 Ar1r2r3j1j2j3 .

(43)We may multiply the previous equation by (IA)u1u2u3i1i2i3and (tIA)v1v2v3j1j2j3 and sum over (i1, i2, i3)and (j1, j2, j3). This amounts to a bracketing of the tetrahedron equations by two times thesame vertex, in a procedure trivially generalizing the one for the Yang-Baxter equation (30).We recover (43) with A, B, C and D transformed byK1 : A→tIAB→tdBC→tmCD→tmD.

(44)We have in a similar way the operationsK2 :A →tdAB →tIBC →tgCD →tgD,K3 :A →tgAB →tgBC →tICD →tdD,K4 :A →tmAB →tmBC →tdCD →tID.11

Each of these four operations is an involution. They satisfy various relations, for instance(K1K2K3K4)2 = 1.The Ki’s generate a group Aut3 which is a symmetry group of thetetrahedron equations.

This group is “monstrous” since the number of elements of lengthsmaller than l is of exponential growth with respect to l, unlike the case of the affine Coxetergroups (as A(1)2for the Yang-Baxter equation) where this number is of polynomial growth.The operations playing a role similar to the one of I and J in the two-dimensional Yang-Baxter equations are the four involutionsI,J = tgItmtd,K = tmItdtg,L = tdItgtm. (45)We call Γ3 the group generated by these four involutions.

Γ3 is also a symmetry group forthe three dimensional vertex model even if [37] the model does not satisfy the tetrahedronequation.In order to precise the algebraic structure of the group Γ3 generated by I, J, K and L, itis simpler to consider as generators two of the partial transpositions tg and td, I and the fulltransposition t. The third partial transposition can be recovered as the product ttgtd and tcommutes with all other generators and so contributes a mere Z2 factor in the group. Weare thus considering the Coxeter group generated by three involutions tg, td and I, with twoof them commuting: this is represented by the following Dynkin diagramrrr∞Itdtg∞For this group again, the number of elements of length smaller than l is greater than 2l/2.This is in fact a hyperbolic Coxeter group [56].7Use of the symmetry group.7.1The baxterizationThe problem of the baxterization is to introduce a spectral parameter into an isolated solutionof the Yang-Baxter equations [16].

We have solutions of this problem by acting with thesymmetry group Γ.7.1.1Baxterization of the Baxter modelConsider the matrix of the symmetric eight vertex modelR =a00d0bc00cb0d00a. (46)Notice that this form is preserved by I and J and that tR = R. The action of I isa→aa2 −d2b →bb2 −c2(47)c→−cb2 −c2d →−da2 −d2(48)and the action of J is12

a→aa2 −c2b →bb2 −d2(49)c→−ca2 −c2d →−db2 −d2(50)We shall look at the solutions of the Yang-Baxter equations for matrices R of the form (46).The leading idea is that the parametrization of the solutions is just the parametrizationof the algebraic varieties preserved by tIJ in the projective space CP3 of the homogenousparameters (a, b, c, d). The remarkable fact is that not only these varieties exist but can becompletely described.

We use the visualization method we have already used [1, 2] for spinmodels, that is to say just draw the orbits obtained by numerical iteration and look.This is best illustrated by figure 1. This figure shows the orbit of point (*), which is amatrix of the form (46).

It is drawn by the iteration of IJ acting on the initial point (*). Theresulting points densify on the elliptic curve given by the intersection of the two quadrics∆1 = constant and ∆2 = constant (Clebsch’s biquadratic), with ∆1 and ∆2 the Γ invariants∆1=a2 + b2 −c2 −d2ab + cd,∆2=ab −cdab + cd.

(51)Similar calculations can be performed for a general 16-vertex model for which:R =a1a2b1b2a3a4b3b4c1c2d1d2c3c4d3d4(52)Amazingly the baxterization of the 16-vertex model leads to curves. These curves arealso intersection of quadrics (even in the general case for which their is no solution for theYang-Baxter equations), and lead to a remarkable elliptic parametrization of the model [6].7.1.2Baxterization of the R matrix of slq(n)Another example corresponds to the baxterization of solutions associated to sl(n) alge-bras [19].There are special solutions generally denoted R+ and R−.For the simplestfour-dimensional representation of the sl(2) case, we haveR+ =q00001q −q−100010000q.

(53)and a similar expression for R−[19].Looking for a family containing both R+ and R−our baxterization procedure leads to the well-known [11] six-vertex model R-matrix R =λR+ + 1/λR−.We let as an exercise for the reader to treat the sl(3) case. In a forthcoming publicationwe will show that these ideas can be generalized to all the universal R-matrices [15] forevery representation [57].

This group appears in field theory, in the analysis of classicalR-matrices [58].13

7.1.3q root of unityOne of the most studied cases of quantum group is obtained when the parameter q is a rootof unity. The structure of the representation theory is then extremely rich and differs fromthe generic one.Proposition: qn = 1 is equivalent to: the orbit of Γ is finite.This applies even if the parameter q is not defined (e.g.

in the elliptic case). See [57].7.2Three dimensional modelsOur strategy for finding solutions of the tetrahedron equations is to seek for patterns of theBoltzmann weights of the three dimensional vertex compatible with the symmetry group Γ3.By this we mean that its form should be preserved by Γ3.7.2.1A first modelWe will therefore consider a simple model where i, j, k, l, m and n take only two values +1and −1.

The matrix (38) is an 8 × 8 matrix. We will require that its pattern is invariantunder the inverse I [54] and the various partial transpositions tg, tm and td.

We aim at havinga generalization of the Baxter eight-vertex model and we impose the following restrictions:w(i, j, k, l, m, n)=w(−i, −j, −k, −l, −m, −n),(54)w(i, j, k, l, m, n)=0if ijklmn = −1. (55)These constraints amount to saying that the 8 × 8 matrix splits into two times the same4 × 4 matrix.

It is further possible to impose that this matrix is symmetric since, in thiscase, tgR (and any other partial transpose) is also symmetric. Let us introduce the followingnotations for the entries of the 4 × 4 block of the R matrixad1d2d3d1b1c3c2d2c3b2c1d3c2c1b3.

(56)The four rows and columns of this matrix correspond to the states (+, +, +), (+, −, −),(−, +, −) and (−, −, +) for the triplets (i, j, k) or (l, m, n). The R-matrix can be completedby spin reversal, according to the rule (54).

tg simply exchanges c2 with d2 and c3 with d3,tm and td can be similarly defined and I acts as the inversion of this 4 × 4 matrix.For this three dimensional model, the coefficients of the characteristic polynomial of the4 × 4 matrix (56) give a good hint for invariants under Γ3. They areσ(3d)1=a + b1 + b2 + b3,(57)σ(3d)2=a(b1 + b2 + b3) + b1b2 + b2b3 + b3b1 −(c21 + c22 + c23 + d21 + d22 + d23),(58).

. .

.Since σ(3d)2is invariant by tg, tm amd td and takes a simple factor (the inverse of the determi-nant) under the action of I, the variety σ(3d)2= 0 is invariant under Γ3. Given the hugenessof the group Γ3, it is already an astonishing fact to have such a covariant expression.

In fact14

we can exhibit five linearly independent polynomials with the same covariance, which givefour invariants, as follows:ab1 + b2b3 −c21 −d21,c2d2 −c3d3,(59)and the ones deduced by permutations of 1, 2 and 3. They form a five dimensional space ofpolynomials.

Any ratio of the five independant polynomials is invariant under all the fourgenerating involutions. In other words CP9 is foliated by five dimensional algebraic varietiesinvariant under Γ3.To have some flavour of the possible (integrable ?) algebraic varieties invariant under Γ3,we study its orbits [1, 2].

We start with the study of the subgroup generated by some infiniteorder element namely IJ. This element gives a special role to axis 1.

The transformation IJdoes preserve the symmetry under the exchange of 2 and 3. If the initial point is symmetricunder the exchange of 2 and 3, the orbit under IJ is thus a curve.

Other starting pointslead to orbits lying on a two dimensional variety given by the intersection of seven quadrics(see figure 2,3,4). However, what we are interested in are the orbits of the whole Γ3 group.The size of this group prevent us from studying exhaustively the full set of group elementsof a given length even for quite small values of this length.

We have nevertheless exploredthe group by a random construction of typical elements of increasingly large length [4]. Thisconfirms that we generically only have the four invariants described previously.15

7.2.2A second modelWe also consider a simple model where i, j, k, l, m and n take only two values +1 and −1and which is also a generalization of the Baxter eight vertex model. The Boltzmann weightsw(i, j, k, l, m, n) are given by:w(i, j, k, l, m, n) = f(i, j, k) δil δjm δkn + g(i, j, k) δi−l δj−m δk−n(60)f(i, j, k) = f(−i, −j, −k) and g(i, j, k) = g(−i, −j, −k)(61)Equations (61) are symmetry conditions reducing the numbers of homogeneous parametersfrom 16 to 8.As for the previous model, there exists an invariant of the action of the whole group Γ3:f(+, +, +)f(+, −, −)f(−, +, −)f(−, −, +)g(+, +, +)g(+, −, −)g(−, +, −)g(−, −, +)(62)Considering the subgroup of Γ3 generated by the infinite order element IJ, one can easilyfind other invariants, namelyf(+, +, +)f(+, −, −)g(+, +, +)g(+, −, −)(63)andf(+, +, +)2 + f(+, −, −)2 −g(+, +, +)2 −g(+, −, −)2g(+, +, +)g(+, −, −)(64)For this model [5], the trajectories under IJ are curves in CP7.8ConclusionAn important problem in statistical mechanics and field theory, is the understanding ofthe role of the dimension of the lattice on both the algebraic aspects and the topologicalaspects.All this touches various fields of mathematics and physics: algebraic geometry,algebraic topology, quantum algebra.

Indeed the Coxeter groups we use are at the sametime groups of automorphisms of algebraic varieties, symmetries of quantum Yang-Baxterequations (and their higher dimensional avatars). They also provide an extension to severalcomplex variables functions of the notion of the fundamental group Π1 of a Riemann surface,with of course a much more involved covering structure [37, 2].We believe moreover that the space of parameters seen as a projective space is the appro-priate place to look at, if one wants to substantiate the deep topological notion embodied inthe notion of Z-invariance [22] and free the models from the details of the lattice shape.Actually, we have exhibited an infinite discrete symmetry group for the Yang-Baxterequations and their higher dimensional generalization acting on this parameter space.

Thisgroup is the Coxeter group A(1)2(semi-direct product of Z × Z by some finite group). Wehave shown that this symmetry is responsible for the presence of the spectral parameter.

Inother words, the discrete symmetry gives rise to a continuous one (see [3]). A similar studyfor the generalized star-triangle relation of the Interaction aRound a Face model, sketchedin [39], can be performed rigorously along the same lines, leading to the same result.

Alsonote that the same groups generated by involutions appear in the study of semi-classical19

r-matrices [58]. An interesting point will be to exhibit the action of our symmetry group onthe underlying quantum group for the Yang-Baxter equations [57].Our symmetry group is a group of automorphisms of the integrability varieties.

Thisshould give precious informations on these varieties. In particular one should decide if, up toLie groups factors (which cannot be excluded because of the existence of “gauge” symmetries,weak graph duality [59], .

. .

), these varieties can be anything else than abelian varieties,or even product of curves: can they be for example K3 surfaces, are they homologicalobstructions to the occurence of anything but curves ?For three-dimensional vertex models, the symmetry group, though generalizing very nat-urally the previous group (generated by four involutions with similar relations) is drasticallydifferent: it is so “large”3 that the chances are quite small that it leaves enough room forany invariant integrability varieties. It is not useless to recall the unique non-trivial knownsolution of the tetrahedron equations (Zamolodchikov’s solution) [25, 24, 27].

For this modelthe three axes are not on the same footing, so that we do not have a “true” three dimensionalsymmetry for the model (two-dimensional checkerboard models coupled together). Is therestill any hope for a three-dimensional exactly solvable model with genuine three-dimensionalsymmetry?

We think that the group of symmetries we have described gives the best line ofattack to this problem. We will show that Γ3 and even more Aut3 are generically too “large”to allow any non-trivial solution of the tetrahedron equations with genuine three dimensionalsymmetry [61].References[1] M.P.

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