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Graph IRF Models and Fusion Rings

arXiv:hep-th/9306143v2 6 Jul 1993June, 1993Graph IRF Models and Fusion RingsDoron Gepner⋆Division of Physics, Mathematics and AstronomyMail Code 452–48California Institute of TechnologyPasadena, CA 91125ABSTRACTRecently, a class of interaction round the face (IRF) solvable lattice models wereintroduced, based on any rational conformal field theory (RCFT). We investigatehere the connection between the general solvable IRF models and the fusion ones.To this end, we introduce an associative algebra associated to any graph, as thealgebra of products of the eigenvalues of the incidence matrix.

If a model is basedon an RCFT, its associated graph algebra is the fusion ring of the RCFT. A numberof examples are studied.

The Gordon–generalized IRF models are studied, and areshown to come from RCFT, by the graph algebra construction. The IRF modelsbased on the Dynkin diagrams of A-D-E are studied.While the A case stemsfrom an RCFT, it is shown that the D −E cases do not.

The graph algebrasare constructed, and it is speculated that a natural isomorphism relating these toRCFT exists. The question whether all solvable IRF models stems from an RCFTremains open, though the D −E cases shows that a mixing of the primary fieldsis needed.⋆On leave of absence from the Weizmann Institute.

Incumbent of the Soretta and HenryShapiro Chair.

Recently, this author has put forward four categorical isomorphisms amongimportant problems that arise in two dimensional physics [1]. These are integrableN = 2 supersymmetric models, rational conformal field theories (RCFT), fusioninteraction round the face models (IRF), and integrable soliton systems.

It hasbeen shown that the latter three categories are equivalent, and evidence was givenfor the equivalence with the first category.The purpose of this note is to further explore these equivalences. In particular,many solvable IRF models are known to be connected to certain (very special)graphs.

The graphs are the admissibility conditions for the state variables thatare allowed to be on the same link on the lattice. One could try to enlarge theisomorphisms mentioned above to all solvable IRF lattice models.

This would meanthat the graph must be obtained from the fusion ring of some rational conformalfield theory.Specifically, this raises the question: is an IRF model solvable ifand only if its admissibility graph arises from the fusion ring of some RCFT? Inthis note we wish investigate this question.

In particular, for each graph we willassociate a commutative algebra, which is essentially the multiplication table forthe eigenvalues of the incidence matrix. If our conjecture holds, this very same ringmust be the fusion ring of some rational conformal field theory.

We shall explorea variety of examples.Let the pair (S, K) where K ∈S ×S be a graph, where S is the set of points ofthe graph and K is the incidence matrix. More generally, we shall allow orientedgraphs with multiple links, more conveniently described by the incidence matrixMi,j, whose non–negative integer entries describe the number of links from thepoint i to j where i, j ∈S.

For such a graph we can associate (ambiguously)an interaction round the face (IRF) lattice model. The partition function of themodel, which is defined on a square lattice, is given byZ =XstatesYfacesw abcd!,(1)where a, b, c, d are the state variables, and a and b are allowed to be on the same2

link iffa and b are connected by the graph, denoted by a ∼b. w abcd!is someBoltzmann weight, which is still undefined, and can be considered as the parametersof the model.

For multiple links, the Boltzmann weight depends on the particularlink, in an obvious generalization. For some graphs and some choices of Boltzmannweights the models are solvable, in the sense that two other Boltzmann weightsw′ and w′′ can be found, obeying the same admissibility condition, such that thefollowing relation holds,Xcw bdac!w′ acgf!w′′ ccfe!=(2)Xcw′′ abgc!w′ bdce!w cegf!This relation is called the star–triangle equation (STE).

It is a very powerful toolin the calculation of the partition function eq. (1), and forms the basis for thesolvability of the model.This raises the important question of which graphs and which choices of Boltz-mann weights lead to solvable IRF models.

In fact, study have shown, that onlyfor very special graphs such solvable Boltzmann weights exists at all, and then theyare more or less unique (for a review, see e.g., [2]). One might speculate that sucha solution exists if and only if the graph in question corresponds to the fusion ringof some RCFT, and then the Boltzmann weights are described uniquely by thebraiding matrices of the RCFT.

Actually, there are obvious counter examples tothis conjecture. However, these IRF models do not have second order phase tran-sition points and thus can be considered as ‘bad’ models in the aforementionedsense.

Precisely put: does all solvable IRF models with a second order fixed pointstem from an RCFT, in the above sense?Let us thus delve into the definition of fusion IRF models. Let O be a rationalconformal field theory, and let x be a field, typically primary, in the theory.

For3

an explanation of these notions see for example [1]. In such a theory, the fusion ofthe primary fields defines a commutative semi–simple ring,[p] × [q] =XrNrp,q[r],(3)where [p], [q], [r] denote the primary fields, and Nrp,q are the structure constants,which are non–negative integers.

Now, for any such ring we can associate a familyof graphs in the following fashion. We let the points of the graph be the primaryfields of the theory, and we identify the incidence matrix Mp,q with the structureconstants with respect to a fixed field in the theory [x], Mp,q = Nqx,p.

Now, givensuch a pair (O, x), we can define an IRF model based of the fusion graph of thefield x, i.e., p ∼q iffNqxp > 0. We denote the resulting lattice model by IRF(O, x).It was shown in ref.

[1] that indeed all such models, termed fusion IRF models aresolvable, and that Boltzmann weights satisfying the STE, eq. (2), can be found.The Boltzmann weights are extensions of the braiding matrices of the correspondingRCFT.

The questions is then, is the converse true and all such solvable models arefusion IRF?In any event, we can examine known solvable IRF models, to determine if theiradmissibility graph comes from an RCFT. If this is the case, such graphs has toobey some very special properties that are nearly enough to settle the question,case by case, as well as to determine the specific RCFT.It was shown in ref.

[3] that the fusion ring in an RCFT is connected to aunitary matrix which is the matrix of modular transformations. The importantthing about S is that it diagonalizes the eigenvalues of the fusion ring.

Namely, ifwe define{i} =XjSi,jSi,0[j],(4)then {i} obeys the fusion product{i} × {j} = δij{i}. (5)4

As the matrix S is unitary, this determines it uniquely from the fusion ring, up toa permutation of the rows, as it is simply the matrix that diagonalizes the fusionring. (More precisely, it is the point basis in the affine variety defined by the ring[4]).

However, not all rings lead to sensible S matrices, and those that do are veryspecial. The reason is that in RCFT, the S matrix needs to be symmetric, and notjust unitary, S = St.

This alone is a very strong constraint on the allowed fusionrings. A further restriction arises from the fact that every such ring must admit anon–degenerate symmetric bi–linear form (a, b), where a and b are primary fields,defined by (a, b) = 1 iffN1ab = (a, b), where 1 stands for the unit in the ring (whichis a primary field).

Further (a, b) must be either 0 or 1 for all the primary fields aand b, and for each primary a, (a, b) is zero, for all b except for a unique choice.Thus the bi-linear form defines a unique conjugate for each field, ¯a which is theunique field for which (a, ¯a) = 1.Thus, the question whether a given graph stems from an RCFT can be ex-amined on the basis of whether the above properties holds for the graph, and itsassociated fusion ring. Let (S, K) be an arbitrary graph then, with the incidencematrix Mi,j.

Denote the eigenvalues of M by vαj , i.e.,XjMi,jvαj =Xγλαvi. (6)We can normalize the eigenvalues to unity, Pj vαj vαj† = 1.

The eigenvalues arethus uniquely defined (up to a phase). We can now write down a commutativeassociative algebra associated to the eigenvalues.

We do so by specifying a uniquechoice for the ‘unit’ element, denoted by say 1. Further, we define the product ofthe elements α and β to be,[α] × [β] =XγNγα,β[γ],(7)5

where the structure constants Nγα,β are defined byvαjv1jvβjv1j=XγNγα,βvγjv1j,(8)for all j. Since the eigenvectors are linearly independent, N is so uniquely defined.The eigenvalues can be normalized, in which case the matrix vhj is unitary,Xjvhj (vpj )∗= δh,p,(9)where h and p are any two exponents.

We thus find from eq. (8), the followingform for the structure constants,Nrp,q =Xjvpjvqj(vrj)∗v1j.

(10)If vhj is a modular matrix of an RCFT [5], eq. (10) gives the fusion coefficients [5],according to the formula of ref.

[3].For a non RCFT, since the matrix of eigenvalues, vpj , is inherently non–symmetric,we can define a transposed algebra, based on the nodes of the diagram, instead, ina similar fashion. The structure constants of the algebra are then given byMkij =Xhvhi vhj (vhk)∗vh1,(11)where the structure constants, Mkij describe the product of the nodes of the graph,vhivh1vhjvh1=XkMkijvhkv1k.

(12)For an RCFT the two algebras are, of course, the same. The algebra so definedsuffers from a number of ambiguities.First, the phase of the eigenvectors was6

arbitrary. However, this is simply a redefinition of the basis elements.

More im-portantly the choice for the unit field ‘1’ was arbitrary, and for each such choice adifferent algebra is found. To summarize, for a pair of any graph and a point in itwe defined uniquely an algebra, denoted by A(G, p), where G is the graph and pis the point.

Further, the algebra has a unique preferred basis, up to a phase.Now, if the graph in question stems from a fusion ring, then the graph algebra,so defined, is identical with the fusion ring of the theory, provided that we take forthe preferred point the unit field of the fusion ring. Further, up to a phase, thepreferred basis of the graph algebra is the primary field basis of the fusion ring, upto the phase ambiguity mentioned above.It follows that the question whether an IRF model stems from an RCFT boilsdown to the question of whether its associated graph algebra is a fusion ring.

Inlight, of the many properties of such fusion rings, only very special graphs canbe candidates for fusion rings. Further, the RCFT may be constructed from thefusion ring itself.

Thus by studying the graph algebra, the question raised in theintroduction can be settled.Let us illustrate this construction by an example.A class of solvable IRFmodels called the Gordon–Generalized (GG) hierarchy has been found [6]. Thestate variables in these models take the values a = 0, 1, .

. .

, k −1, where a are thestate variables, and k is any integer. The admissibility condition for the graph isa ∼biffa + b ≤k −1.

(13)It was shown in ref. [6] that the models so obtained are solvable, and Boltzmannweights satisfying the STE were found.

The case of k = 2 is the well known hardhexagon model [7]. Now, let us construct the graph algebra associated to thisgraph, for any k. For the unit field we choose the element [k −1].

It is a straightforward calculation that the algebra so obtained assumes the form,[i] × [j] =2k−1−i−jXm=|i−j|m−i−j=0 mod 2[m],(14)7

where we identified [2k −1 −i] ≡[i]. It can be checked that this algebra hasall the properties of a fusion ring.

In fact, this is the known fusion ring of theRCFT SU(2)2k−1/SU(2)1/(2k−1), described in ref. [1].

The graph itself is obtainedfrom the field x = [k −1]. We conclude that the GG hierarchy is the fusion IRFmodel IRF(SU(2)2k−1/SU(2)1/(2k−1), [k −1]).

It can be further checked that theBoltzmann weights described in ref. [6] are indeed the extensions of the braidingmatrices of this RCFT.It is quite straight forward to see directly that the admissibility condition forthe GG hierarchy models, eq.

(13), is indeed precisely what is obtained by fusionwith respect to the field [k −1] in the theory G = SU(2)2k−1/SU(2)1/(2k−1). Weidentify the state (σ) of the GG hierarchy model with the primary field [k −1 −σ]in the theory G. We can now compute the fusion with respect to the field [k −1].We find, according to eq.

(14),[k −1] × [k −1 −σ] = [σ] + [σ + 2] + . .

. + [2k −2 −σ] =k−1−ρXρ=0[k −1 −ρ], (15)where we used the identification of fields, [σ] = [2k −1 −σ], which holds in thetheory G. As the state (σ) is identified with the primary field [k −1 −σ], eq.

(15) implies precisely the GG admissibility condition, eq. (13).

This concludes theproof that the GG models are fusion IRF.The theory G may be constructed explicitly, ref. [1], as a sub–sector of thetheory SU(2)2k−1 × (E7)1, with an extended algebra (for an example).

The caseof k = 1 corresponds to (G2)1 current algebra.Consider now, as another example, the KAW hierarchy of models defined inref. [8].

The state variables of the models along with their admissibility conditionare given by,σi = 0, 1, 2, . .

., k −1,k −2 ≤σi + σj ≤k,(16)where k is some integer, and σi and σj are any two neighboring states. Let G ≡SU(2)k be the current algebra theory associated to SU(2)k. Denote as before by8

[j] the field with the isospin j/2, and let p = [k −1] + [k] be a field which is amixture of two primary fields. We can compute the fusion with respect to the fieldp, according to the usual rules of SU(2)k, eq.

(24), and we find,p × [σi] = [σi] × [k] + σi × [k −1] =Xσjk−2≤σi+σj≤k[σj],(17)which are precisely the fusion admissibility conditions of the KAW hierarchy,eq. (16).Thus, we conclude that the GAW model is the fusion lattice modelIRF(SU(2)k, [k] + [k −1], [k] + [k −1]).

This is an example of a model based ona mixture of primary fields. Such models where considered in ref.

[1], and theirBoltzmann weights given as an extension of the conformal braiding matrices ofthe RCFT. It would be an interesting exercise to compare the Boltzmann weightsgiven in ref.

[1], based on the fusion properties, and those given in ref. [8], by adirect solution, and to show that they indeed coincide.Let us turn now to another example.

Consider the so called grand hierarchy ofsolvable IRF lattice models, discussed in refs. [8, 9].

These models are describedbyli = 0, 1, . .

. , k,li −lj = −N, −N +2, .

. .

, N,li +lj = N, N +2, . .

. , k −N,(18)where li are the state variables, li and lj are adjacent, and k and N are arbitraryintegers.

For each k and N, a solvable model was found [9], using compositionsof the eight vertex model.In fact, eq. (18), is exactly the well known fusionrules of SU(2)k. It is rather evident from these fusion rules, eq.

(24), that li isadmissible to lj, if and only if Nlilj,p ≥0, where p = [N] primary field, in theprevious notation. We conclude that the grand hierarchy is exactly the fusion IRFmodels IRF(SU(2)k, [N], [N]), for any k and any N. Again, it would be interestingto verify that the Boltzmann weights coming from RCFT [1], and those computeddirectly in ref.

[9], are identical. For N = 1 (the Andrews–Baxter–Forrester model9

[10]), this was done in ref. [1], and the results indeed agree.

For larger N thisverification is left to further work.Before proceeding, let us discuss one subtlety in the logic of identifying thegraph algebra with the fusion rules. Recall that the primary fields were identifiedwith the eigenvalues of the incidence matrix, and that this identification was un-ambiguous up to a phase.

However, in case the incidence matrix has degenerateeigenvalues, we can no longer distinguish which mixture of these eigenvectors arethe primary fields, and additional information may be needed.Let us now proceed to another interesting family of solvable IRF lattice models.These are the lattice models based on the simple Lie algebras which are An [10]and Dn, E6, E7 and E8 [11]. The states of the ADE IRF models are in one–to–one correspondence with the simple roots of the respective Lie algebra.

Similarly,the admissibility graph is the Dynkin diagram of the algebra. Thus, the incidencematrix of the model is given by Mab = 2δab −Cab where Cab is the Cartan matrixof the algebra.

(For a review on simple Lie algebras see, e.g., [12]. )The Boltzmann weights of the ADE models [11] have the relatively simplegraph state form (see, e.g., [1] and ref.

therein),w abcdu!= sin(λ −u)δbc + ψbψcψaψd 12sin u,(19)where u labels the different Boltzmann weights satisfying the STE, eq. (2), whichare given by the values u, u + v and v, for w, w′ and w′′, respectively, for anycomplex u and v. ψa is the eigenvector with the largest eigenvalue of the incidencematrix (the so called Perron–Frobenius vector),XbMabψb = 2 cos λψa,(20)where β = 2 cos λ is the maximal eigenvalue of the incidence matrix, given byλ = πg ,(21)10

Table 1.AlgebraCoxeter NumberExponentsAnn + 11, 2, . .

. , nDn2n −21, 3, 5, .

. .

, 2n −3, n −1E6121, 4, 5, 7, 8, 11E7181, 5, 7, 9, 11, 13, 17E8301, 7, 11, 13, 17, 19, 23, 29where g is the Coxeter number of the algebra. The entire set of eigenvalues of theincidence matrix is given byλh = 2 cos(πh/g),(22)where h is any of the exponents of the Lie algebra (which can be degenerate).

Theexponents, along with the Coxeter number are described in table (1).Now, in light of the conjecture raised in the introduction, we would like toexamine if the ADE IRF models are fusion IRF models, i.e. if they arise from aconformal field theory.

To this end, let us proceed with constructing the graphalgebras associated with the Dynkin diagrams of simple Lie algebras. If our con-jecture is correct, this should be the fusion ring of some RCFT.

To do so, we firstneed to calculate the eigenvectors of the Cartan matrix of each Lie algebra, andthen insert these into eq. (10).We shall skip the An cases (ABF models), as these have already been demon-strated to be the fusion IRF models associated with the RCFT SU(2)n−1 [1].

Forcompleteness sake, the eigenvectors for the An graph are given byvij =2n + 1 12sin( πijn + 1),(23)where i labels the simple roots and j labels the exponents, and i, j = 1, 2, . .

. , n.This is non–else but the toroidal modular matrix of the RCFT SU(2)k [5], showing11

that the graph algebra is identical to the fusion ring of the model, which has theproduct rule [5],[i] × [j] =min(2k−i−j,i+j)Xl=|i−j|l−i−j=0 mod 2[l],(24)where [l] labels the lth primary field. The Boltzmann weight, eq.

(10), can beseen to give at the limit u →i∞the braiding matrix of the respective RCFT [1],concluding the proof that the ABF model is a fusion IRF.Let us turn now to the Dn algebras.From table (1) the exponents of thealgebra are, 1, 3, 5, . .

., 2n −3, n −1, and thus are all different for odd n, and havea twofold degeneracy at n −1 for even n. The eigenvalues of the incidence matrixare given by eq. (22), λh = 2 cos(πh/(2n −2)), where h is any of the exponents.The eigenvectors of the incidence matrix are readily computed and are found tobe,vhj =q2n−1 sin( πhj2n−2)for j ≤n −2(−1)j√2(n−1)for j = n −1, n −2,(25)for the odd exponents h = 1, 3, 5, .

. .

, 2n −3. For the exceptional exponent, h =n −1, we find,vn−1j=( 0forj ≤n −2,(−1)j√2for j ≥n −1.

(26)We have normalized the eigenvectors to have the absolute value one, and thus vhjis the unitary matrix which diagonalizes the incidence matrix.We next proceed to calculate the graph algebra, using eq. (10).

Denote by [h],h = 1, 3, . .

. 2n−3, the elements of the algebra associated to the regular exponents,and by z = [n −1] the element associated to the exceptional one.

Then the graph12

algebra can be computed from eq. (10), and we find,[p] × [q] =min(p+q−1,4n−5−p−q)Xr=|p−q|+1r=1 mod 2[r],z × [p] = (−1)(p−1)/2z,z × z =12(n −1)2n−3Xr=1r=1 mod 2(−1)(r−1)/2[r].

(27)It is striking that up to a trivial rescaling of z, z →zp2(n −1), all the structureconstants are integers. This implies that the graph algebra of Dn, any n, is actuallya commutative ring with a unit.

As no two eigenvectors are the same, the ring isa semi–simple one, which is a finite dimensional algebra, with vanishing nil andJacobson radicals. It is straight forwards to present this ring in terms of generatorsand relations.

Let Tn(x) be the Chebishev polynomial of the second kind, definedby Tn(2 cos φ) = sin[(n+1)φ]sin φ. Then the generators of the ring may be taken to be zand x = [3], along with the relations p1(x) = p2(z, x) = p3(z, x) = 0, wherep1(x) = T2n−4(√1 + x) + T2n−2(√1 + x),p2(x, z) = z2 −12(n −1)2n−3Xh=1h=1 mod 2(−1)(h−1)/2[h],p3(z, x) = zx + x,[h] = Th−1(√1 + x),(28)where [h] expresses the basis element [h] as a polynomial in x, and is used toexpress p2 as a polynomial in x and z.

In other words,R ≈P[x, z](p1, p2, p3),(29)where R denotes the graph algebra, P[x, z] is the algebra of polynomials in x andz, and (p1, p2, p3) is the ideal in it generated by the three polynomials p1, p2 andp3.13

It remains to be seen, now, if this graph algebra satisfies any of the proper-ties of a fusion ring of an RCFT, in accordance with the conjecture raised in theintroduction. Quite evidently the answer is no!

There are a number of problems.1) Some of the structure constants are negative integers. 2) There is no appropri-ate symmetric bilinear form.

3) In a related way, the S matrix cannot be madesymmetric. To be more precise let pjh be the alleged symmetric bi–linear form,where since the matrix p has a unique 1 in each row and column, it is actually apermutation, expressing a map between exponents h, denoted by k(j), where j isa Dynkin node, and k(j) is an exponent, and the nodes of the Dynkin diagram.From the properties of RCFT, the modular matrix must be symmetric, when p isused to lower the index.

Thus, RCFT requires that,vk(l)j= vk(j)l.(30)More generally, we could allow for a change of normalizations of the eigenvectors,which are defined only up to a phase, in which case, eq. (30) assumes the form,Xhvhj phl =Xhvhl phj.

(31)It is readily seen that eq. (31), has no solutions for phj, when vhj is taken to be theeigenmatrix of Dn, eqs.

(25-26). Thus, this graph algebra is not the fusion ring ofany RCFT.This appears to be quite a catastrophe for the original conjecture we raised,and in fact, a counter example for it.

The question is if there is a way in which theconjecture we raised could be relaxed, and that this graph algebra can be relatedto an RCFT? More precisely, we assumed in our entire discussion, that the nodesof the graph are the primary fields of the RCFT.

There is actually no reason forthis assumption, as we can well build a fusion IRF model based on non–primaryfields [1]. Clearly, this entails a change of basis for the graph algebra.

Thus, thequestion becomes whether the graph algebra is isomorphic to a fusion ring of an14

RCFT. Unfortunately, this is a rather meaningless question, since it is well knownthat two finite dimensional algebras are isomorphic if they are both semi–simple,and have the same dimension.

Thus, any graph algebra is isomorphic to any fusionalgebra, provided they have the same number of nodes, respectively, primary fields.Clearly, this is too weak a criteria to be of much use.We conclude the discussion of the Dn cases by noting one particular basis inwhich things look rather close to an RCFT. We form the basis,α+h = 12([h] + [2n −2 −h]),α−h = (−1)(h−1)/212([h] −[2n −2 −h]),z,(32)where h = 1, 3, .

. .

, n −3. In this basis, the algebra decouples to a direct sum oftwo subalgebras.

These are the subalgebras generated by α+h (along with z, foreven n), and α−h (along with z, for odd n). Denote these two subalgebras by A andB.

Then AB = 0, and G ≈A ⊕B. The question now is any of A and B are thefusion rings of an RCFT?

The fusion rules in this basis become,α±r α±t =r+t−1Xs=|r−t|+1s=1 mod 2α±s ,zα+h = 0odd n,zeven n,zα−h = zodd n,0even n,z2 =n−2Xh=1h=1 mod 2α+hodd n,n−2Xh=1h=1 mod 2α−heven n.(33)It can now be seen that the subalgebra A (generated by α+h , for odd n, and by α−h ,for even n), gives rise to a symmetric S matrix. Namely, the eigenvectors matrix15

of this algebra, may be written as,Sh,t = C sin πht2n −2,(34)for h, t = 1, 3, . .

. n −2, and where C is some constant.

Clearly S is a symmetricunitary matrix, and the question remains whether it corresponds to an RCFT.Note, that if we take n to be half integral, this is exactly the S matrix of the RCFTSU(2)2n−4/SU(2)1/(2n−4). Unfortunately, for an integral n, it can be seen, exceptfor the trivial, n = 4, not to correspond to an RCFT, as the equation (ST)3 = 1has no solutions with a diagonal matrix T. As this is a necessary condition for S tobe a modular matrix, we conclude that the above S is not the modular matrix ofany RCFT.

For n = 5, 7, the entire eigenvalue matrix is symmetric, but still doesnot appear to stem from an RCFT. Other basis in which the S matrix is symmetriccan be found.

The significance of these observations remains to be studied. At thispoint we conclude that the relation of the Dn models with RCFT is moot, thoughshort of a counter example to our conjecture.Let us turn now to the case of the exceptional algebras En, for n = 6, 7, 8.The exponents of the respective algebras are listed in Table.

1. The eigenvaluesare thus, λh = 2 cos(πh/g), where h is any of the exponents.

The eigenvectors arefound to be,vj = sin(πjhg )for j ≤n −3,vn =sin( 3πhg )2 cos( πhg ),vn−2 = sin(π(n−2)hg) −sin( π(n−3)hg)2 cos( πhg ) ,vn−1 =vn−22 cos( πhg ),(35)The eigenvectors are further normalized to have absolute value one, vhj →vhj /qPj(vhj )2.Consider now the case of E6. It is more convenient to define here the transposedgraph algebra associated to the nodes, eq.

(11). We find that all the structure con-stants are positive integers.

Denoting by [j] the element of the algebra associated16

to the jth node, we find that the transpose ring is given by R ≡P[x,y](y3−2y,x2−xy−1),where the Dynkin nodes basis elements of the algebra are given by,[1] = 1,, [2] = x,, [3] = xy,, [4] = x(y2 −1),[5] = y2 −1,[6] = y. (36)All the products may be computed from the two relations, y3−2y = x2−xy−1 = 0.The first question now is whether the above graph algebra is a fusion ring.

It iseasy to see that this is not the case, and that the matrix of eigenvectors vhj cannotbe symmetrized. Thus, although close to the notion of a fusion IRF model, the E6IRF model does not stem from an RCFT.As in some of the Dn cases we can form combinations of the Dynkin nodesthat give rise to a symmetric matrix of eigenvalues.

These are the combinations[1] ± [5], [2] ± [4], [3] and [6] (up to normalizations). The combination [1] −[5],[2] −[4], generates a subalgebra which decouples from the rest of the algebra, i.e.,the fusion algebra is the direct sum of the two algebras, A generated by [1] −[5]and [2]−[4], and the algebra B generated by the rest, AB = 0.

The correspondingmatrix of eigenvalues is symmetric, and thus a candidate for a modular matrix ofan RCFT. The fusion ring A is seen to be isomorphic to that of SU(2)1 and thusis coming from an RCFT, R ≡P[x]/(x2 −1).

However, the structure constants ofB cannot be made integral, and thus it cannot be the fusion ring of any RCFT.Again, the significance of this observation is unclear.In the case of E7 we find the transpose graph algebra which is,R ≡P[x](x7 −6x5 + 9x3 −3x),(37)where the elements associated to the Dynkin nodes are given by,17

[1] = 1,[2] = x,[3] = x2 −1,[4] = x3 −2x,[5] = x6 −5x4 + 5x2,[6] = x5 −5x3 + 5x,[7] = −x6 + 6x4 −8x2 + 1. (38)Again, all the structure constants are non–negative integers, in terms of the Dynkinbasis elements, Mkij ≥0.In the case of E8, we find the the transposed graph ring again has all thestructure constants as positive integers.

The ring is given byR ≡P[x](1 −8x2 + 14x4 −7x6 + x8). (39)The Dynkin elements are now given by,[1] = 1,[2] = x,[3] = −1 + x2,[4] = −2x + x3,[5] = 1 −3x2 + x4,[6] = −2x + 9x3 −6x5 + x7,[7] = −2 + 9x2 −6x4 + x6,[8] = 5x −13x3 + 7x5 −x7,(40)The polynomials we find for En, n = 6, 7, 8, may be considered as the generaliza-tions of the Chebishev polynomials, which arise for An, to the exceptional algebras.18

A very interesting property of all the En graph algebras we find is that theproduct with x = [2] gives back the incidence matrix of the graph,x × [n] =XmMnm[m],(41)where Mnm is the incidence matrix of the graph, which is the Dynkin diagramof the respective algebra. Further, this rule alone, determines uniquely the entirealgebra.

Thus, the rings we found are exactly those giving the Dynkin graph as anadmissibility condition. Namely, the En models can be thought of as the modelsIRF(R, x), where R stands for the graph ring, and x = [2] is the element used in theadmissibility condition.

The fact that we managed to lift the admissibility relationto a full ‘fusion ring’ with positive integer structure constants is non–trivial, andmay be connected with the solvability of the model.In all the En cases, the matrix of eigenvalues is non–symmetric, and thus doesnot correspond directly to an actual RCFT. As noted earlier, they are certainlyisomorphic to fusion rings of RCFT, but this is a somewhat meaningless fact.

Asin the Dn case, we can also form the graph algebra, based on the exponents, eq.(10). Forming as in the Dn case, the combinations, A−= 12([h] −[g −h]) andA+ = 12([h] + [g −h]), where h is any of the exponents, we find that the twosubalgebras decouple, A−A+ = 0.

For E7 this leads also to a symmetric eigenvaluematrix, and fusion rules which are integers, and thus are full candidates for anRCFT. It remains to explore further whether an RCFT based on this can be built.For E6 and E8 we find non–symmetric eigenvalue matrix, indicating that the twosubalgebras do not represent an RCFT.In conclusion, we studied here a variety of examples of solvable IRF models,to judge if they stem from an RCFT.

We formed graph algebras based on theeigenvalues of the admissibility conditions. If a theory stems from an RCFT itsgraph algebra must be the fusion rules of the model.

This also gives immediatelythe solutions to the STE, eq. (2), by an extension of the braiding matrices ofthe theory, as described in ref.

[1]. We studied here two families of examples,19

the Gordon generalized (GG), KAW and grand hierarchies, and the Pasquier D–Emodels. Remarkably, it was shown that all the hierarchy models models stem fromsome RCFT related to SU(2) current algebra.

On the other hand, the D −Emodels were seen not to correspond directly to an RCFT. It remains to be studies,whether these models stem from a mixture of primary fields in an RCFT.

While,not ruling the possibility out, we found that, in general, there does not seem to bea natural way to relate these models to an RCFT. The most likely conclusion tobe drawn is that while most solvable IRF models studied to date come from someRCFT, other solutions to the STE exist, which do not appear to be related to anRCFT in the manner described in ref.

[1], with the D and E models as examples.The question certainly requires further study.We hope that we have further illuminated here the connection between RCFTand solvable lattice models. A great host of models, erratically constructed pre-viously, all stem from the unified construction described in ref.

[1]. While fewexceptions were found, it remains to study how these can be fitted into the generalframework.ACKNOWLEDGEMENTSI thank P. Di–Francesco, W. lerche and J.B. Zuber for helpful comments.

Whilewriting this work, I received [13], which somewhat relates to the present work.20

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