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DILOGARITHM IDENTITIES, q-DIFFERENCE EQUATIONS

arXiv:hep-th/9212094v1 16 Dec 1992DILOGARITHM IDENTITIES, q-DIFFERENCE EQUATIONSAND THE VIRASORO ALGEBRAEDWARD FRENKEL AND ANDR´AS SZENESDecember, 19921. IntroductionIn this paper we give a new proof of the identityn−1Xj=1L(δj) = π262n −22n + 1,(1)where L(z) is the Rogers dilogarithm function:L(z) = −Z z0 log(1 −w) d log w + 12 log z log(1 −z),0 ≤z ≤1,and δj = sin2(π2n+1)/ sin2(π j+12n+1).This equality was proved by Richmond and Szekeres from the asymptotic analysisof the Gordon identities [1] and also by Kirillov using analytic methods [2].

There is awide class of identities of this type, which emerged in recent works on two-dimensionalquantum field theories and statistical mechanics. They appear in the calculation ofthe critical behavior of integrable models, using the so-called Thermodynamic BetheAnsatz equation ([3, 4, 5, 6, 7, 8]).

This is an elegant, but mathematically non-rigorous method.In this paper, we present a new approach to the identities, which relies on theconcepts of quantum field theory, and is rigorous at the same time. We hope thiswill lead to a better understanding of the mathematical structures behind generalintegrable two-dimensional quantum field theories.The idea goes back to Feigin [9]: There is a certain increasing filtration on eachirreducible minimal representation of the Virasoro algebra (or another conformal alge-bra) by finite-dimensional subspaces, whose characters are connected by a q-differenceequation.

This equation can be used to obtain an expression for the asymptotics ofthe character of this representation in terms of values of the dilogarithm. On theResearch of the first author was supported by a Junior Fellowship from the Harvard Society ofFellows and by NSF grant DMS-9205303E-mail addresses: frenkel@math.harvard.edu, szenes@math.mit.edu1

2EDWARD FRENKEL AND ANDR´AS SZENESother hand, it is known that this asymptotics is determined by the (effective) centralcharge ceff(cf. [10]), and this gives us an identity.To illustrate that the values of the dilogarithm should appear in the asymptoticsof the solutions to q-difference equations, consider the equationf(qx) = (1 −aq)f(x),0 < q < 1.Thenlimn→∞f(qn) = f(1)∞Yn=1(1 −aqn).

(2)Let us calculate the asymptotics of this solution as q →1. We havelog q log f(qn) = log q(log f(1) +∞Xn=1log(1 −aqn)).This expression can be interpreted as a Riemann sum.

Thus in the limit q →1 itgives−Z 10 log(1 −ax) d log x = L(a) −12 log a log(1 −a).We will show that this approach can be carried out for the irreducible repre-sentations of the (2, 2n + 1) models of the Virasoro algebra.In this case ceff=(2n −2)/(2n + 1), and the corresponding identity is (1).Let us remark that according to Goncharov [11], the dilogarithm identities of thistype define elements of torsion in K3(R).2. (2, 2n + 1) minimal models of the Virasoro algebraThe Virasoro algebra is the Lie algebra generated by elements {Li, i ∈Z} and C,with the relations[Li, Lj] = (i −j)Li+j + 112(i3 −i)δi,−jCand[Li, C] = 0.Fix an integer n > 1.

Let 1n be the one-dimensional representation of the subalge-bra generated by {C, Li, i ≥−1}, on which the elements Li act by 0, and C acts bymultiplication by 1 −3(2n −1)2/(2n + 1). Denote by ˜Vn the induced module of theVirasoro algebra.

It is Z-graded with respect to the action of L0, deg(L−i) = i. Bythe Poincar´e-Birkhoff-Witt theorem this module has a Z-graded basis consisting ofthe monomials {L−m1 . .

. L−mpv| m1 ≥m2 ≥· · · ≥mp > 1}, where v is the generatorof 1n.

The module ˜Vn has a unique vector u of degree 2n + 1, such that Liu = 0 fori > 0. The quotient Vn of the module ˜Vn by the submodule generated by u inheritsthe Z-grading, and is irreducible [12, 13].

It is called the vacuum representation ofthe (2, 2n + 1) minimal model of the Virasoro algebra [14].Let Cn = {(m1, . .

., mp)| m1 ≥· · · ≥mp > 1, mi ≥mi+n−1 + 2}.

DILOGARITHM IDENTITIES3Proposition 1 ([15]). The image of the set{L−m1 .

. .

L−mpv| (m1, . .

. , mp) ∈Cn}under the homomorphism ˜Vn →Vn gives a linear basis of Vn.Remark 1.

While the argument in [15] relied on the Gordon identities, this statementcan be proved directly, using only the representation theory of the Virasoro algebra(see [16]).For every integer N > 0, introduce the subspaces W rN, 0 ≤r < n, of the moduleVn, which are linearly spanned by the vectors{L−m1 . .

. L−mpv| (m1, .

. ., mp) ∈Cn, m1 ≤N, mr+1 ≤N −1}.For any Z-graded vector space V = ⊕∞m=0V (m) with dim V (m) < ∞, let ch V =P∞m=0 dim V (m)qm, which is called the character of V .

Denote by wrN the characterof W rN, and introduce wN = (w0N, . .

. , wn−1N) and w0 = (0, .

. .

, 1).Lemma 2. We have the following recursion relation:wN = Mn(qN)wN−1,whereMn(x) =00.

. .0100. .

.x1. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

.0xn−2. .

.x1xn−1xn−2. .

.x1.This gives us the following formula for the character of Vn:ch Vn = wt0∞YN=1Mn(qN)w0 = wt0 . .

. Mn(qN) .

. .Mn(q2)Mn(q)w0.

(3)Thus ch Vn is equal to wt0 limn→∞f(qn), where f(x) is the solution to the q-differenceequation f(qx) = Mn(x)f(x) with the initial condition f(1) = w0.The character of Vn is a function in q for 0 < q < 1. We will study its asymptoticsas q →1.The following result was proved by Kac and Wakimoto, and follows from the mod-ular properties of ch Vn.Proposition 3 ([10]).−limq→1 log q log ch Vn = π262n −22n + 1.Remark 2.

The number 2n−22n+1 is called the effective central charge of the (2, 2n + 1)minimal model.In the next section, we will use (3) to derive another expression for this asymptotics.

4EDWARD FRENKEL AND ANDR´AS SZENES3. Asymptotics of the infinite productThe following lemma relates the asymptotics of (3) to the asymptotics of the infiniteproduct of the highest eigenvalues of Mn(qN).Lemma 4.

Denote by dn(x) the eigenvalue of Mn(x) of maximal absolute value. (a) For any x, 0 < x ≤1, the matrix Mn(x) has n different real eigenvalues anddn(x) > 0.

(b) The asymptotic behavior of ch Vn is the same as that of the infinite product ofthe dn(qN), more precisely−limq→1 log q log ch Vn = −limq→1 log q log∞YN=1dn(qN). (4)Similarly to the calculation of (2), the right hand side of (4) can be written as thefollowing integral:Z 10 log dn(x) d log x.

(5)The rest of this section is devoted to the calculation of this integral.Denote by Qn(λ, x) the characteristic polynomial of Mn(x). Our computation isbased on an explicit rational parametrization of the curve Qn(λ, x) = 0.Remark 3.

On this curve, λ and x define two algebraic functions. Our integral (5) isthe integral of log λ d log x along a path on the curve.

We would like to stress thatit is the rationality of this curve, that allows us to express this integral in terms ofvalues of the dilogarithm function.Introduce the n × n matrixLn(x) =1 + x−10. .

.000−x1 + x−1. .

.0000−x1 + x. . .000. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .000. .

.1 + x−10000. .

.−x1 + x−1000. .

.0−xx,and observe that Mn(x)−2 = x−nLn(x). Then the roots of the characteristic polyno-mial Pn(µ, x) = det(Ln(x)−µ) are connected to the roots of Qn(λ, x) via the formulaλ2 = xn/µ.

On the other hand, we have the following recursion for Pn(x):Pn(µ, x) = (1 + x −µ)Pn−1(µ, x) −xPn−2(µ, x),with initial conditions P0(µ, x) = 1 and P1(µ, x) = x −µ.

DILOGARITHM IDENTITIES5Introduce t = x/(x −µ + 1)2 and u = 1/(x −µ + 1). In these new variables ourrecursion can be written asu−1Pn−1Pn=11 −tu−1 Pn−2Pn−1.This can be solved by the continuous fractionu−1Pn−1Pn=11 −t1 −t.

. .. .

.1 −t1 −u.As a consequence the equation Pn = 0 is equivalent setting the denominator of thiscontinuous fraction to 0. This leads to the following parametrization for u:u = An(t),whereAn(t) = 1 −t1 −t.

. .. .

.1 −t1 −t.Naturally, An(t) satisfies the recursion An(t) = 1 −t/An−1(t) with the initial con-dition A1(t) = 1. A quick calculation shows that as functions of the parameter t ouroriginal variables λ and x take the formxn(t) =tA2n(t)andλn(t) = A1(t)A2(t) .

. .

An−1(t)An−1n(t).Lemma 5. For any α, 0 < α ≤1, there are n real positive solutions t1(α) < · · ·

The numbers λn(t1(α)), . .

.λn(tn(α)) are the eigen-values of the matrix Mn(α), anddn(α) = λn(t1(α)).The lemma implies that our integral (5) can be written asZ t=t1(1)t=0log λn(t) d log xn(t). (6)Introduce the rational functions fi(t) = 1−Ai+1(t)/Ai(t).

The following key formulacan be proved by induction:fi(t) =ti(1 −f1(t))2(i−1) · · · (1 −fi−1(t))2. (7)

6EDWARD FRENKEL AND ANDR´AS SZENESIn terms of the these functions our variables can be expressed as follows:xn =t(1 −f1)2(1 −f2)2 . .

. (1 −fn−1)2λn =1(1 −f1)(1 −f2)2 .

. .

(1 −fn−1)n−1Now we can calculate the integral (6). We begin with a more general integralZ γ0 log λn(t) d log xn(t),where γ > 0 is such that λn(t) and xn(t) have no poles on the interval [0, γ].

Substi-tuting the formulas above, we obtain−Z γ0n−1Xj=1j log(1 −fj(t)) d log t −n−1Xi=12 log(1 −fi(t))!=−Z γ0n−1Xj=1log(1 −fj(t)) d log tjQj−1i=1(1 −fi(t))2(j−i)!+Z γ0n−1Xi,j=12 min(i, j) log(1 −fj(t)) d log(1 −fi(t)).Applying formula (7) to the first summand, and partial integration to the second,our integral takes the form−Z γ0n−1Xj=1log(1 −fj(t)) d log fj(t) +n−1Xi,j=1min(i, j) log(1 −fj(γ)) log(1 −fi(γ)).Now by the definition of the Rogers dilogarithm, this can be written asn−1Xj=1L(fj(γ)) −12n−1Xj=1log fj(γ) log(1 −fj(γ))+(8)n−1Xi,jmin(i, j) log(1 −fj(γ)) log(1 −fi(γ)).To complete our proof, we need to describe some properties of the numbers δi of (1).Lemma 6. Fix an integer n > 1.

The numbers δi, 1 ≤i ≤n−1, satisfy the followingproperties:(a) fi(δ1) = δi,(b) δi =Qn−1i=1 (1 −δj)2 min(i,j),(c) δ1 = An(δ1)2, thus xn(δ1) = 1. Moreover, t1(1) = δ1.

DILOGARITHM IDENTITIES7Combining the calculation above with property (c), we see that the integral (6)equals to the expression (8), with γ = δ1. Using property (a) this is simplyn−1Xj=1L(δj) −12n−1Xj=1log δj log(1 −δj) +n−1Xi,jmin(i, j) log(1 −δi) log(1 −δj).Finally, according to property (b), the last two terms cancel.

Hence we obtain thefollowingProposition 7.Z 10 log dn(x) d log x =n−1Xj=1L(δj).This result, together with Lemma 4 and Proposition 3, proves identity (1).Remark 4. We are certain that this method is applicable to other models of conformalfield theory.Acknowledgements.

We are grateful to B. Feigin for sharing his ideas with us. Wewould like to thank A. Goncharov, V. Kac, A. Levin and T. Nakanishi for valuablediscussions.References1.

B.Richmond, G.Szekeres, Some formulas related to dilogarithms, the zeta function and theAndrews-Gordon identities, J. Australian Math. Soc.

(Series A) 31 (1981) 362-3732. A.N.Kirillov, On identities for the Rogers dilogarithm function related to simple Lie algebras,Zap.

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LOMI, 164 (1987) 121-1333. A.N.Kirillov, N.Reshetikhin, Exact solution of the XXZ Heisenberg model of spin S, Zap.

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V.Bazhanov, N.Reshetikhin, Restricted solid-on-solid models connected with simply laced alge-bras and conformal field theory, J. Phys. A 23 (1990) 1477-14925.

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W.Nahm, A.Recknagel, M.Terhoeven, Dilogarithm identities in conformal field theory, PreprintBONN-HE-92-35, November 19927. Al.Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models: scaling 3-state Pottsand Lee-Yang models, Nucl.

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8EDWARD FRENKEL AND ANDR´AS SZENES13. B.Feigin, D.Fuchs, Representations of the Virasoro algebra, in “Representations of infinite-dimensional Lie groups and Lie algebras”, eds.

A.M.Vershik and D.P.Zhelobenko, Gordon andBreach, 1990, 465-55414. A.Belavin, A.Polyakov, A.Zamolodchikov, Infinite conformal symmetry in two-dimensionalquantum field theory, Nucl.

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B.Feigin, E.Frenkel, in preparation.Department of Mathematics, Harvard University, Cambridge, MA 02138Department of Mathematics, Massachusetts Institute of Technology, Cambridge,MA 02139

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