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On Solvable Lattice Models and Knot Invariants

arXiv:hep-th/9305182v1 31 May 1993May, 1993On Solvable Lattice Models and Knot InvariantsDoron GepnerDivision of Physics, Mathematics and AstronomyMail Code 452–48California Institute of TechnologyPasadena, CA 91125ABSTRACTRecently, a class of solvable interaction round the face lattice models (IRF)were constructed for an arbitrary rational conformal field theory (RCFT) and anarbitrary field in it. The Boltzmann weights of the lattice models are related in theextreme ultra violet limit to the braiding matrices of the rational conformal fieldtheory.

In this note we use these new lattice models to construct a link invariantfor any such pair of an RCFT and a field in it. Using the properties of RCFT andthe IRF lattice models, we prove that the invariants so constructed always obey theMarkov properties, and thus are true link invariants.

Further, all the known linkinvariants, such as the Jones, HOMFLY and Kauffman polynomials arise in thisway, along with giving a host of new invariants, and thus also a unified approach tolink polynomials. It is speculated that all link invariants arise from some RCFT,and thus the problem of classifying link and knot invariants is equivalent to thatof classifying two dimensional conformal field theory.⋆On leave from the Weizmann Institute, Israel.

Incumbent of the Sorrela and Henry ShapiroChair.

The intriguing interplay between knot theory and two dimensional physics hasbenefited considerably both fields (for a review, see, e.g., [1]).The purpose ofthis note is to put forwards a general framework for link invariants stemming fromsolvable lattice models.It was recently shown that solvable fusion interactionround the face (IRF) lattice models are in a one–to–one correspondence with apair of a rational conformal field theory and a field in it [2]. It follows as we shallsee that for each such pair one can form a link invariant, and that this class of linkinvariants is in a one to one correspondence with such pairs.Let us review the construction of the Boltzmann weights described in ref.

[2].Consider a rational conformal field theory (RCFT) O, and a field in it, x, whichfor simplicity we shall assume to be a primary field. We than construct a solvableIRF model, denoted by IRF(O, x) following [2], whose admissibility conditions aregiven by fusion with respect for x and whose Boltzmann weights reduce in theextreme ultra violet limit to a specialization of the braiding matrix of the RCFT(see [2] for more detail).

We put on the vertices of the lattice, which is a square twodimensional one, state variables which are the primary fields of O and are labeledby a, b, c, . .

.. The pair a and b is allowed to be on the same link, a ∼b, if and onlyif, the fusion coefficient Nbax > 0.

The partition function of the model isZ =XconfigurationsYfacesw abcdu!,(1)where a, b, c and d are the four states (primary fields) on the vertices of the face,w abcdu!is the Boltzmann weight associated to the face, and u is a spectralparameter which labels a family of models. The Boltzmann weights obey the startriangle equation (STE), from which it follows that the transfer matrices for dif-ferent values of the spectral parameter u commute, and thus the model is solvable.The Boltzmann weights of the model IRF(O, x) were given in ref.

[2], and areconveniently described in an operator form. To do so define the operator Xs(u),2

the face transfer matrix, byXs(u)m1,m2,...,mnl1,l2,...,ln=Yi̸=sδ(li, mi)w li−1milili+1u!,(2)where li and mi are the states on two adjacent diagonals of the lattice. The facetransfer matrix of the model IRF(O, x) is [2]Xs(u) =NXa=1P as fa(u),(3)where a = 1, 2, .

. .

, N labels the fields appearing in the operator product x·x, P a isa projection operator of the braiding matrix on the a field in the operator productdefined byP a =NYj=1j̸=aBs −λjλa −λj,(4)and where Bs is the braiding matrix of the RCFT at the face s, and λj are itseigenvalues, which are given byλj = eiπ(2∆x−∆j),(5)and ∆x and ∆j are the conformal dimensions of the field x and the j field in theoperator product x · x, respectively.The functions fa(u) are defined by,fa(u) =a−1Yj=1sin(ζj + u)N−1Yj=asin(ζi −u),(6)whereζi = π(∆i+1 −∆i)/2,(7)3

and λ = ζ1 is the crossing parameter of the model. The projection operators obey,P as P bs = δabP as ,1 =NXa=1Pa,Bs =NXa=1P as λa,(8)from which it follows that the face transfer matrix obeys the unitarity condition,Xs(u)Xs(−u) = ρ(u)ρ(−u),(9)where the unitarity factor isρ(u) = fN(u) =N−1Yi=1sin(ζi + u).

(10)Also, this implies the regularity condition,Xs(0) = ρ(0) · 1. (11)An important, and highly non trivial, property of the Boltzmann weights is thecrossing symmetry,w abcdλ −u!= ψbψcψaψd 12w cadbu!,(12)where the crossing multiplier ψa is given in terms of the torus modular functionSab,ψa = Sa,0S0,0,(13)where ‘0’ denotes the unit field.

Repeating the crossing transformation twice im-4

plies the charge conjugation symmetry:w abcdu!= w dcbau!. (14)It is convenient to define the two braiding operators,G±i =limu→±∞Xi(u)/ρ(u),(15)where G+i (denoted also for simplicity by Gi) differs from the conformal braidingmatrix Bi by an irrelevant phase.

In terms of the Boltzmann weights, this isσ abdc±!=limu→±∞w abcdu!/ρ(u),(16)from which it follows that G+i = (G−i )†, i.e., they are complex conjugates of eachother, and that G±i obey the Braid group relationships which areGiGi+1Gi = Gi+1GiGi+1,GiGj = GjGifor |i −j| > 1,(17)which is the relation obeyed by the generators of the braiding group, i.e., Gi canbe considered as the generator of the braiding of the i and i + 1 strands in a braid.By Artin theorem these are the generating relations for the braid group.A link is formed by connecting the end points of a braid. Labeling the endpoints l1, l2, .

. .

, ln and m1, m2, . .

. , mn, as before, we connect with a strand the liand mi end points, for all i.

This procedure is ambiguous as different braids maygive the same (topologically) link. We call such braids equivalent.

It was shownby Markov [3], that two braids are equivalent if an only if they can be related bythe sequence of moves of the two types,(I)AB →BAfor A, B ∈Bn,(18)(II)A →AG±1nfor A ∈Bn,(19)where Bn denotes the braid group on n elements, defined by the relations eq. (17).5

In order to classify links we wish to form a functional α which assigns a complexnumber for each link, in such a way that topologically equivalent links will havethe same value of α, α(A) = α(B) if A and B are equivalent topologically. Todo so, it is thus sufficient to demand that α is invariant under the Markov moves.We define a Markov trace on a braid, φ(A), for A ∈Bn, to be a complex numberobeying the properties,((I)φ(AB) = φ(BA),A, B ∈Bn,(II)φ(AGn) = τφ(A),φ(AG−1n ) = ¯τφ(A),A ∈Bn,(20)and where the parameters τ and ¯τ areτ = φ(Gi),¯τ = φ(G−1i ).

(21)The link invariant α(A) is formed in terms of the Markov trace φ(A), byα(A) = (τ ¯τ)−(n−1)/2(τ/¯τ)e(A)/2φ(A),(22)where e(A) is the exponent sum of the braid, i.e.,e(nYi=1Gaii ) =nXi=1ai,(23)which is evidently a well defined grading, since it is preserved by the braid grouprelationships, eqs. (17).We next proceed to describe a Markov trace based on the lattice model IRF(O, x).Note that any element of the braid group, A ∈Bn is represented by some diagonalto diagonal transfer matrix, Am1,m2,...,mnl1,l2,...,ln, where the generators are represented bythe conformal braiding matrix Gi.

Now, define the diagonal matrix,(Hn)m1,m2,...,mnl1,l2,...ln=nYi=1δ(li, mi)Sln,0Sl1,0,(24)where S is, as before, the torus modular matrix, which gives the crossing multiplier.6

Define also a constrained trace by,ˆTr(A) =Xl2,l3,...,lnAl1,l2,...,lnl1,l2,...,ln. (25)Then the Markov trace is defined byφ(A) =ˆTr(HnA)ˆTr(Hn),(26)for any element of the braid group A.

It remains to show that the Markov trace sodefined, φ(A) obeys the properties (I) and (II), eqs. (18–19).

Property (I) followstrivially from the definition, while property (II) follows from a straight forwardscalculation, provided that the Boltzmann weights obey the Markov property,Xb∼aw baacu!Sb,0Sa,0= H(u)ρ(u),(27)where H(u) is some function independent of a and c. The parameters τ and ¯τ aregiven byτ, ¯τ =limu→±∞H(u)/H(0),(28)where τ (¯τ) corresponds to the plus (minus) sign in the limit.Using the crossing property, eq. (12), it is straight forwards to show that theextended Markov property holds provided that the following relation is valid,Xi(λ)Xi(u) = β(u)Xi(λ),(29)where β(λ−u) = H(u)ρ(u).

We shall now show that for the models IRF(O, x) theproperty eq. (29) holds and that thus φ is always a good Markov trace.

This is7

a simple calculation using eqs. (8).

We note that Xi(λ) = P Ni fN(λ), since fa(λ)vanishes for a ̸= N. Thus Xi(λ) is indeed a projection operator and soXi(λ)Xi(u) = fN(λ)NXa=1P Ni P ai fa(u) = β(u)Xi(λ),(30)where we used eqs. (3,8), andβ(u) = fN(u) =NYa=1sin(ζi + u).

(31)It follows that the parameters areH(u) =NYi=1sin(λ + ζi −u)sin(ζi + u),(32)andτ = eiNλNYi=1sin(ζi)sin(λ + ζi),(33)and ¯τ = τ†. It follows that the invariant we defined, eq.

(22), indeed assumesthe same values for topologically equivalent links, and thus can be used to classifyknots and links.For a number of examples of IRF models, the link invariants we defined herewere previously calculated (for a review, see [1], and references therein). For exam-ple, the unrestricted Lie algebra model Am−1 give rise to the HOMFLY polynomial[4] (as a polynomial in m and the crossing parameter), which is a two variable gen-eralization of the original Alexander polynomial [5] (at the limit m →0) and themore recent Jones polynomial [6] (m = 2 case).

The unrestricted Bm, Cm andDm IRF models give the Kauffman polynomial [7]. These models correspond tothe current algebra RCFT based on the Lie algebras A, B, C, D, with the fieldwhich is the fundamental representation for An, and the vector representation forthe other algebras.8

It is noteworthy that the construction presented here, while encompassing allthe known link invariants, provides for a very far reaching generalization of these,along with a unified framework for their construction. Such new invariants areindeed needed in the problem of classifying links as it is well known that twotopological distinct links may certainly have identical classifying polynomials (seefor example Birman’s example [8] of two different knots that have the same Jonespolynomial).The link invariants we defined eq.

(22) may be calculated directly for each IRFmodel by substituting the Boltzmann weights and preforming the traces. This ishowever rather cumbersome for big links.

A considerable simplification is providedby the skein relations which relate the invariants of different links [5, 9]. To deriveskein relations for the invariants described here, first note that the Braiding matrixGi obeys a fixed Nth order polynomial equation,NXm=0amGmi =NYm=1(Gmi −λm) = 0,(34)where we used eq.

(5). Define the link Lm to be the link obtained with the insertionof the braid element Gmi , i.e., if L described by the braid A, then Lm is describedby the braid AGmi .

Using the polynomial relation, eq. (34), we find immediatelythe relation for the Markov trace,N−kXm=−kamφ(Lm) = 0,(35)for any k. Substituting this into the definition of the invariant, eq.

(22), we findthe skein relation,N−kX−kbmα(Lm) = 0,(36)wherebm = am(τ/¯τ)−m/2. (37)The skein relation, eq.

(36), is a very effective tool for the calculation of link9

invariants.We thus describe in this note a whole wealth of link invariants which are ina one to one correspondence with a pair of a rational conformal field theory anda field in this theory. The RCFT and the field chosen are arbitrary, and everyRCFT gives rise to different invariants.

It is tantalizing to speculate on this in anumber of directions. First, since all known link polynomials arise in this fashion,one might conjecture that the category of link invariants and the category of pairsof conformal field theory and a field in it are in fact equivalent ones, and thatthe problem of classifying link invariants is thus the same as that of classifyingconformal field theory.

Second, one might ponder the generalization of these ideasto all conformal field theories, not necessarily rational. There does not seem tobe any obstacle in doing so, and the entire construction might be carried, mutatismutandis.

This will also open up an entire different type of invariants, so calledirrational, which, in particular, obey an infinite order skein relations, i.e., a Laurentseries type rather than polynomial. Such invariants appear not to have been studiedbefore.Finally, it is hoped that the results described here will be of help in the furtherunderstanding of both knot theory and two dimensional physics, along with thefascinating interrelationship between them.10

REFERENCES1. M. Wadati, T. Deguchi and Y. Akutsu, Phys.

Rep. 180 (1989) 2472. D. Gepner, Foundations of rational conformal field theory, I, Caltechpreprint, CALT 68–1825, November (1992)3.

A.A. Markov, Recueil Math. Moscou 1 (1935) 734.

P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millet and A. Ocneanu,Bull.

Am. Math.

Soc. 12 (1985) 2395.

J.W. Alexander, Trans.

Am. Math.

Soc. 30 (1928) 2756.

V.F.R. Jones, Bull.

Am. Math.

Soc. 12 (1985) 1037.

L.H. Kauffman, Trans.

Am. Math.

Soc., to be published.8. J.S.

Birman, Invent. Math.

81 (1985) 2879. J.H.

Conway, in Computational Problems in Abstract Algebra, ed. J. Leech(Pargamon, New–York, 1970) p. 32911

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