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DILOGARITHM IDENTITIES, PARTITIONS AND

arXiv:hep-th/9212150v1 24 Dec 1992Preliminary versionDILOGARITHM IDENTITIES, PARTITIONS ANDSPECTRA IN CONFORMAL FIELD THEORY, IANATOL N. KIRILLOV*Isaac Newton Institute for Mathematical Sciences,20 Clarkson Road, Cambridge, CB3 OEH, U.K.andSteklov Mathematical Institute,Fontanka 27, St.Petersburg, 191011, RussiaABSTRACTWe prove new identities between the values of Rogers dilogarithm function and de-scribe a connection between these identities and spectra in conformal field theory (part I).We also describe the connection between asymptotical behaviour of partitions of someclass and the identities for Rogers dilogarithm function (part II).Introduction.The dilogarithm function Li2(x) defined for 0 ≤x ≤1 byLi2(x) :=∞Xn=1xnn2 = −Z x0log(1 −t)tdt,is one of the lesser transcendental functions. Nonetheless, it has many intriguingproperties and has appeared in various branches of mathematics and physics suchas number theory (the study of asymptotic behaviour of partitions, e.g.

[RS], [AL];the values of ζ-functions in some special points [Za]) and algebraic K-theory (theBloch group and a torsion in K3(R) - A.Beilinson, S.Bloch [Bl], A.Goncharov),geometry of hyperbolic three-manifolds [Th], [NZ], [Mi], [Y], representation theoryof Virasoro and Kac-Moody algebras [Ka], [KW], [FS] and conformal field theory(CFT).In physics, the dilogarithm appears at first from a calculation of magnetic sus-ceptibility in the XXZ model at small magnetic field [KR1], [KR2], [Ki1], [BR].More recently [Z], the dilogarithm identities (through the Thermodynamic BetheAnsatz (TBA)) appear in the context of investigation of UV limit or the criticalbehaviour of integrable 2-dimensional quantum field theories and lattice models [Z],* This work is supported by SERC Grant GR59981

2Anatol N. Kirillov[DR], [K-M], [KBP], [Kl], [KP],...). Evenmore, it was shown (e.g.

[NRT]) usinga method of Richmond and Szekeres [RS], that the dilogarithm identities may bederived from an investigation of the asymptotic behaviour of some characters of2d CFT. Thus, it seems a very interesting problem to lift the dilogarithm identityin question to some identity between the characters of certain conformal field the-ory.

A partial solution of this problem (without any proofs!) contained in [Te] and[KKMM].One aim of this paper (part I) is to prove some new identities between the valuesof Rogers dilogarithm function using the analytical methods (see [Le], [Ki1]).

Basedon such identities we give an answer on one question of W.Nahm [Na]: how big maybe the following abelian groupW := XiniL(αi)L(1)| ni ∈Z,αi ∈Q ∩R for all i∩Q ?Theorem ♣. The abelian group W coincides with Q, i.e.

any rational numbermay be obtained as the value of some dilogarithm sum.A proof of Theorem ♣follows from Proposition 4.5. We also give a proof (andthe different generalizations) of an identity (3.1) from [NRT] (see our Theorem 3.1and Proposition 5.4).

Note the following “reciprosity law” for dilogarithm sums(see Theorem 3.2)s(j, n, r) + s(j, r, n) = nr −1.This is a consequence of the corresponding reciprosity law for the Dedekind sums[Ra]. Note also that the author don’t know any CFT interpretation for the diloga-rithm identities from Propositions 4.4 and 5.4.Part II will be deal with the partition identities which”materialise” the diloga-rithm ones.§1.

Definition and the basic properties of Rogers dilogarithm.Let us remind the definition of the Rogers dilogarithm function L(x) forx ∈(0, 1)L(x) = −12Z x0log(1 −x)x+ log x1 −xdx =∞Xn=1xnn2 + 12 log x · log(1 −x). (1.1)The following two classical results (see e.g.

[Le2], [GM], [Ki1]) contain the basicproperties of the function L(x).Theorem A. The function L(x) ∈C∞((0, 1)) and satisfies the following func-tional equations1.

L(x) + L(1 −x) = π26 ,0 < x < 1,(1.2)2. L(x) + L(y) = L(xy) + Lx(1 −y)1 −xy+ Ly(1 −x)1 −xy,(1.3)where 0 < x, y < 1.

Dilogarithm identities, partitions and spectra in conformal field theory3Theorem B. Letf(x) be a function of class C3((0, 1)) and satisfies the relations(1.2) and (1.3). Then we havef(x) = const · L(x)We continue the function L(x) on all real axis R = R1 ∪{±∞} by the followingrulesL(x) = π23 −L(x−1),if x > 1,(1.4)L(x) = L11 −x−π26 ,if x < 0,(1.5)L(0) = 0,L(1) = π26 ,L(+∞) = π23 ,L(−∞) = −π26 .The present work will concern with relations between the values of the Rogersdilogarithm function at certain algebraic numbers.

More exactly, let us consider anabelian subgroup W in the field of rational numbers QW =(XiniL(αi)L(1)| ni ∈Z,αi ∈Q ∩R for all i)∩Q. (1.6)According to a conjecture of W. Nahm [Na] the abelian group W “coincides”with the spectra in rational conformal field theory.

Thus it seems very interestingtask to obtain more explicit description of the group W (e.g. to find a system ofgenerators for W) and also to connect already known results about the spectra inconformal field theory (see e.g.

[BPZ], [FF], [GKO], [FQS], [Bi], [Ka], [KP]) withsuitable elements in W. One of our main results of the present paper allows todescribe some part of a system of generators for abelian group W. As a corollary,we will show that the spectra of unitary minimal models [BPZ], [GKO] and someothers are really contained in W.But at first we remind some already knownrelations between the values of the Rogers dilogarithm function.§2. Some dilogarithm relations.It is easy to see from (1.2) and (1.3) thatL(12) = π212 ,L(12(√5 −1)) = π210 ,L(12(3 −√5)) = π215 .

(2.1)Proof.Let us put α :=12(√5 −1).It is clear that α2 + α = 1.So we haveL(α2) = L(1 −α) = L(1) −L(α). Now we use the Abel’s formulaL(x2) = 2L(x) −2Lx1 + x,(2.2)

4Anatol N. Kirillovwhich may be obtained from (1.3) in the case x = y. From (2.2) we find thatL(α2) = 2L(α) −2L(α1 + α) = 2L(α) −2L(α2).So, 3L(α2) = 2L(α).

But as we already saw,L(1) = L(α) + L(α2) = 53L(α).This proves (2.1).Apparantly, there are no other algebraic point from the interval (0, 1) at whichthere is such an elementary evaluation of Rogers dilogarithm function. However,there are many identities relating the values of dilogarithm function at variouspowers of algebraic numbers.

It is interesting to write some of this identities inorder to compare the elements of the abelian group W obtained by such mannerwith the spectra of known conformal models. As concerning identities between thevalues of dilogarithm function, we follow [Le1], [Le2], [Lo] and [RS].6L(13) −L(19) = π23 ,(2.3)nXk=2L 1k2+ 2L1n + 1= π26 ,(2.4)consequently,∞Xk=2L 1k2= π26 .This identities may be easily deduced from (1.2) and (2.2).

Again, if α =√2−1,we have the relations4L(α) + L(1 −α2) = 5π212 ,4L(α) + 4L(α2) + L(1 −α4) = 7π212(2.5)Proof. We use Abel’s formula (2.2) and relation (1.3) with y = 12L(12) + L(x) = Lx2+ Lx2 −x+ L1 −x2 −x.

(2.6)We haveL(α2) = 2L(α) −2Lα1 + α= 2L(α) −2L 2 −√22!,L(12) = 2L(√22 ) −2L(α) = π23 −2L(2 −√22) −2L(α),L(α4) = 2L(α2) −2Lα21 + α2= 2L(α2) −2L 2 −√24!,L 2 −√22!+ L(12) = L 2 −√24!+ L(α) + L(α2).

Dilogarithm identities, partitions and spectra in conformal field theory5Excluding successively L( 2−√22) and L( 2−√24) from these relations we obtain (2.5).Watson [Wa] found three relations involving the roots of the cubic x3+2x2−x−1.Namely, if we take α = 12 sec 2π7 ,β = 12 sec π7 ,γ = 2cos 3π7 , then α, −β and −1γare the roots of this cubic andL(α) + L(1 −α2) = 4π221 ,2L(β) + L(β2) = 5π221 ,(2.7)2L(γ) + L(γ2) = 4π221 .Lewin [Le1] and Loxton [Lo] found three relations involving the roots of thecubic x3 + 3x2 −1. Namely, if we take δ = 12 sec π9 , ǫ = 12 sec 2π9 , ζ = 2cos 4π9 , thenδ,−ǫ and −1ζ are the roots of this cubic and3L(δ) + 3L(δ2) + L(1 −δ3) = 17π218 ,6L(ǫ) + 9L(1 −ǫ2) + 2L(1 −ǫ3) + L(ǫ6) = 31π218 ,(2.8)6L(ζ) + 9L(1 −ζ2) + 2L(1 −ζ3) + L(ζ6) = 35π218 .§3.

Basic identities and conformal weights.In this section we present our main results dealing with a computation of thefollowing dilogarithm sumn−1Xk=1rXm=1Lsin kϕ · sin(n −k)ϕsin(m + k)ϕ · sin(m + n −k)ϕ:= π26 s(j, n, r),(3.1)where ϕ = (j + 1)πn + r ,0 ≤j ≤n + r −2.It is clear that s(j, n, r) = s(n + r −2 −j, n, r), so we will assume in sequel that0 ≤2j ≤n + r −2.The dilogarithm sum (3.1) corresponds to the Lie algebra of type An−1. Thecase j = 0 was considered in our previous paper [Ki1], where it was proved thats(0, n, r) = (n2 −1)rn + r. It was stated in [Ki1] that this number coincides with thecentral charge of the SU(n) level r WZNW model.

Before formulating our result

6Anatol N. Kirillovabout computation of the sum s(j, n, r) let us remind the definition of Bernoullipolynomials. They are defined by the generating functiontextet −1 =∞Xn=0Bn(x)tnn!,|t| < 2π.We use also modified Bernoulli polynomialsBn(x) = Bn({x}),where {x} = x −[x]be a fractional part of x ∈R.

It is well-known thatB2n(x) = (−1)n 2(2n)! (2π)2n∞Xk=1cos 2kπxk2n,(3.2)B2n+1(x) = (−1)n−1 2(2n + 1)!

(2π)2n+1∞Xk=1sin 2kπxk2n+1 .Theorem 3.1. We haves(j, n, r) = 6(r+n)[ n−12]Xk=016 −B2(n −1 −2k)θ−142n2+1+3(−1)n, (3.3)where θ = j + 1r + n and g.c.d.

(j + 1, r + n) = 1.Theorem 3.2. (level-rank duality [SA], [KN1])s(j, n, r) + s(j, r, n) = nr −1.

(3.4)Corollary 3.3. We haves(j, n, r) = c(n)r−24h(r,n)j+ 6 · Z+,(3.5)wherec(n)r= (n2 −1)rn + r,h(r,n)j= n(n2 −1)24· j(j + 2)r + n ,0 ≤j ≤r + n −2,(3.6)are the central charge and conformal dimensions of the SU(n) level r WZNW pri-mary fields, respectively.Proof.

Let us remind thatB2(x) = x2 −x + 16 and B2(x) = B2({x}).

Dilogarithm identities, partitions and spectra in conformal field theory7Thus,6(r + n)[ n−12]Xk=016 −B2(n −1 −2k)θ== 6(r + n)[ n−12]Xk=0(n −1 −2k)θ −6(r + n)[ n−12]Xk=0(n −1 −2k)2θ2++ 6(r + n)[ n−12]Xk=0(n −1 −2k)θ(n −1 −2k)θ −1 +(n −1 −2k)θ:== 6Σ1 −6Σ2 + 6Σ3.Now if we take θ = j + 1r + n, then it is clear that Σ3 ∈Z+. In order to compute Σ1and Σ2 we use the following summation formulae[ n−12]Xk=0(n −1 −2k) = 2n2 −1 + (−1)n8=hn2i·n + 12,[ n−12]Xk=0(n −1 −2k)2 = n(n2 −1)6.Consequentlys(j, n, r) == 3(2n2 −1 + (−1)n)(j + 1)4−n(n2 −1)(j + 1)2r + n−2n2 + 1 + 3(−1)n4+ 6Σ3 == (n2 −1)rr + n−n(n2 −1)j(j + 2)r + n+ 6jhn2i·n + 12+ 6Σ3.For small values of j we may compute the sum in (3.3) and thus to find corre-sponding positive integer in (3.5).Corollary 3.4.i) if j ≤r, thens(j, 2, r) = c(2)r−24h(r,2)j+ 6j =3rr + 2 + 6 j(r −j)r + 2 .

(3.7)ii) if 2j ≤r + 1,thens(j, 3, r) =8rr + 3 −24 j(j + 2)r + 3+ 12j. (3.8)iii) if (n −1)j < r + 1, thens(j, n, r) = c(n)r−24h(r,n)j+ 6j ·hn2i·n + 12.

(3.9)

8Anatol N. KirillovProof.An assumption(n −1)j < r + 1is equivalent to a condition(n −1)(j + 1)n + r< 1.So the term Σ3 (see a proof of Corollary 3.3) is equal tozero.It seems interesting to find a meaning of the positive integer in (3.5). Now wewant to find a “dilogarithm interpretation” of the central charges and conformaldimensions for some well-known conformal models.Corollary 3.5.

We haves(j1, 2, k)+s(0, 2, 1)−s(j2, 2, k+1) = ck−24hj1+1,j2+1+6(j1−j2)(j1−j2+1), (3.10)whereck = 1 −6(k + 2)(k + 3),h(k)r,s = [(k + 3)r −(k + 2)s]2 −14(k + 2)(k + 3)(3.11)are the central charge and conformal dimensions of the primary fields for unitaryminimal conformal models [BPZ], [Ka], [GKO].Corollary 3.6.s(j1, n, k) + s(0, n, 1) −s(j2, n, k + 1) = ck,n −24h(n)j1+1,j2+1 + 12Z+,(3.12)whereck,n = (n −1)1 −n(n + 1)(k + n)(k + n + 1),(3.13)h(n)r,s (k) = n(n2 −1)24· [(k + n + 1)r −(k + n)s]2 −1(k + n)(k + n + 1)are the central charge and conformal dimensions of the primary fields for Wn models[Bi], [CR].Corollary 3.7.s(j1, 2, k) + s12(1 −(−1)j1−j2), 2, 2−s(j2, 2, k + 2) =(3.14)= c(k) −24ehj1+1,j2+1 + 12j1 −j2 + 12·j1 −j2 + 22,wherec(k) = 321 −8(k + 2)(k + 4),(3.15)ehr,s := ehr,s(k) = [(k + 4)r −(k + 2)s]2 −48(k + 2)(k + 4)+ 1 −(−1)r−s32,are the central charge and conformal dimensions of the primary fields for unitaryminimal N = 1 superconformal models [GKO], [MSW].We give a generalization of Corollaries (3.5)-(3.7) to the case of non-unitaryminimal models.

Dilogarithm identities, partitions and spectra in conformal field theory9Corollary 3.8. If p ≥q ≥2, then(p −q)s(j1, 2, q −2) + s(0, 2, 1) −(p −q)s(j2, 2, p −2) =(3.16)= c −24hj1+1,j2+1 + 6(j1 −j2)(p −q + j1 −j2),wherec = 1 −6(p −q)2pq,(3.17)hr,s := hr,s(c) = (pr −qs)2 −(p −q)24pq,r < q, s < p,are the central charge and conformal dimensions of the primary fields for non-unitary (if p −q ≥2) Virasoro minimal models [FQS].

Note that “remainder term”in (3.16)6(j1 −j2)(p −q + j1 −j2)appears to be positive for all 0 ≤j1 < q,0 ≤j2 < p iffp −q = 0 (trivial case) orp −q = 1 (unitary case).Corollary 3.9. Let p ≥q ≥2 and p −q ≡0 (mod 2).

Thenp −q2s(j1, 2, q −2) + s12(1 −(−1)j1−j2), 2, 2−p −q2s(j2, 2, p −2) =(3.18)= ec −24ehj1+1,j2+1 + 6(j1 −j2)(j1 −j2 + p −q)2+ 1 −(−1)j1−j24,whereec = 321 −2(p −q)2pq,(3.19)ehr,s = (pr −qs)2 −(p −q)28pq+ 1 −(−1)r−s32,r < q, s < p,are the central charge and conformal dimensions of primary fields for non-unitary(if p −q > 2) Neveu-Schwarz (if r −s even) or Ramond (if r −s odd) minimalmodels [FQS].Corollary 3.10.Let p ≥q ≥n, then(p −q)s(j1, n, q −n) + s(0, n, 1) −(p −q)s(j2, n, p −n) =(3.20)= c −24hj1+1,j2+1(c) + 6Z,

10Anatol N. Kirillovwherec = (n −1)1 −n(n + 1)(p −q)2pq,(3.21)hrs(c) = n(n2 −1)24(pr −qs)2 −(p −q)2pq,r < q, s < p,are the central charge and conformal dimensions of primary fields for non-unitary(if p −q ≥2) Wn minimal models [Bi].Finally we give a “dilogarithm interpretation” for the central charges and con-formal weights of restricted solid-on-solid (RSOS) lattice models and their fusionhierarchies [KP].Corollary 3.11. We haves(l, 2, N) + s(N −1, 2, N −2) −s(m −1, 2, N −2) =(3.22)= c + 1 −24∆+ 6(l −|m|),m ∈Z,0 ≤l ≤N,wherec = 2(N −1)N + 2and ∆= l(l + 2)4(N + 2) −m24N(3.23)are the central charge and conformal weights of ZN parafermion theories [FZ].

Themembers of (3.23) may be also realized as the central charge and conformal weightsof fusion N + 1-state RSOS(p, p) lattice models [DJKMO], [BR] on the regime I/IIcritical line. Note that physical constraints|m| ≤l,m ≡l (mod 2)for value of m in (3.23) are equivalent to a condition that “remainder term” in(3.22), namely, 6(l −|m|), must be belongs to 12Z+.Corollary 3.12.Let us fix the positive integers k,p = 1, 2, .

. .

(the fusionlevel), j1 and j2 such that 0 ≤j1 ≤k,0 ≤j2 ≤k + l.Let r0 = pj1 −j2pbe the unique interger determined by0 ≤r0 ≤p,r0 ≡±(j1 −j2)mod 2p. (3.24)Then we haves(j1, 2, k) + s(r0, 2, p) −s(j2, 2, k + p) =(3.25)= c −24∆+ 12 (j1 −j2)(p + j1 −j2) + r0(p −r0)2p,

Dilogarithm identities, partitions and spectra in conformal field theory11wherec =3pp + 21 −2(p + 2)(k + 2)(k + p + 2),(3.26)∆= [(k + p + 2)(j1 + 1) −(k + 2)(j2 + 1)]2 −p24p(k + 2)(k + p + 2)+ r0(p −r0)2p(p + 2)are the central charge and conformal weights of the fusion (k+p+1)-state RSOS(p, p)latice models [KP] on the regime III/IV critical line. It is easy to see that “remainderterm” in (3.25) belongs to 12Z+.Note also that the fusion RSOS(p, q) latticemodels, obtained by fusing p×q blocks of face weights together, are related to cosetconformal fields theories obtained by the Goddard-Kent-Olive (GKO) construction[GKO].

Namely, c and ∆in (3.26) are the central charge and conformal dimensionsof conformal field theory, which corresponds to the coset pair [GKO]A1⊕A1⊃A1levelskpk + pThus the members of (3.26) are reduced to those of (3.11) if p = 1 and of (3.15) ifp = 2.§4. A1-type dilogarithm identities.As is well-known [Le2], the Rogers dilogarithm function L(x) admits a continu-ation on all complex plane C. Follow [Le], [KR] we define a functionL(x, θ) : = −12Z x0log(1 −2x cosθ + x2)xdx + 14 log |x| · log(1 −2x cosθ + x2) == ReL(xeiθ),x, θ ∈R(4.1)Our proof of Theorem 3.1 is based on a study of properties of the function L(x, θ).Proposition 4.1.For all real ϕ, θ we haveL sin θsin ϕ2= π2B2θ + ϕπ−B2ϕπ−B2 θπ+ 16+(4.2)+ 2L−sin(ϕ −θ)sin θ, ϕ−2L−sin ϕsin θ , ϕ + θ.Before proving a Proposition 4.1 let us give the others useful properties of function(4.1) (compare with [Le2]).

12Anatol N. KirillovLemma 4.2. (i)L(x, 0) = L(x),L(−x, ϕ) = L(−x, π −ϕ)(4.3)(ii)L(x, ϕ) = L(x, 2πk ± ϕ),k ∈Z(4.4)(iii) L(−1, ϕ) = π2B2 ϕ2π + 12,L(1, ϕ) = π2B2 ϕ2π(4.5)(iv)L(x, ϕ) + L(x−1, ϕ) = 2π2B2 ϕ2π,x > 0L(−x, ϕ) + L(−x−1, ϕ) = 2π2B2 ϕ2π + 12,x < 0(4.6)(v)L(0, ϕ) = 0,L(+∞, ϕ) = 2π2B2 ϕ2π,L(−∞, ϕ) = 2π2B2ϕ + π2π(4.7)(vi)L(2 cosϕ, ϕ) = π2B2ϕπ+ 112(4.8)(vii) L(xn, nϕ) = nn−1Xk=0Lx, ϕ + 2kπn,x ∈R+,L(xn) = nn−1Xk=0Lx · exp 2kπin,x ∈(0, 1).

(4.9)More generally (Rogers’ identity [Ro])L(1 −yn) =nXk=1nXl=1L(λk/λl) −L(xkλl),where {xk}nk=1 are the roots of the equation1 −yn =nYk=1(1 −λkx).Proof. At first let us remind some properties of modified Bernoulli polynomials.dBn(x)dx= nBn−1(x),Bn(x) = Bn(x + 1),Bn(−x) = (−1)nBn(x),(4.10)Bp(nx) = nnXk=1Bpx + kn.Note that identities (4.5) follows from the Fourier expansion for B2(x) (see(3.2)).

Dilogarithm identities, partitions and spectra in conformal field theory13In order to prove the identity (4.2) let us differentiate LHS and RHS of the lastone with respect to ϕ using the following formuladL(x, ϕ) =−14log(1 −2x cos ϕ + x2)x+ 12 log |x|x −cos ϕ1 −2x cos ϕ + x2dx(4.11)+−tan−1x sin ϕ1 −x cos ϕ+ 12 log |x|x sin ϕ1 −2x cos ϕ + x2dϕ.Acting in such a manner we findddϕL−sin ϕsin θ , ϕ + θ== ϕ + 12cot(ϕ + θ) logsin ϕsin θ−12cotϕ logsin(ϕ + θ)sin θ,(4.12)ddϕL−sin θsin ϕ, ϕ + θ== θ −12cot(ϕ + θ) logsin ϕsin θ−12cotϕ logsin(ϕ + θ)sin θ,(4.13)if 0 < ϕ,θ < π,ϕ + θ < π;(4.13a)ddϕL sin θsin ϕ · sin(θ + ψ)sin(ϕ + ψ)!== 12cotϕ + cot(ϕ + ψ)logsin(ϕ −θ) sin(ϕ + θ + ψ)sin θ sin(θ + ψ)−−12cot(ϕ −θ) + cot(ϕ + θ + ψ)logsin ϕ sin(ϕ + ψ)sin θ sin(θ + ψ),(4.14)if 0 < θ < ϕ < π,0 < ψ < π,ϕ + θ + ψ < π. (4.14a)Further, let us use the reduction rules (4.3), (4.4) and (4.6) (compare with (1.4)and (1.5)) if the angles ϕ, θ (or ϕ, θ, ψ) do not satisfy the condition (4.13a) (or(4.14a)).

As a result one can obtain that a derivative of difference between LHSand RHS of (4.2) with respect to ϕ is equal to2πB1ϕ + θπ−B1ϕπ.Integrating (see (4.10)), then taking ϕ = 0 and using (4.5) to determine the inte-gration constant, we obtain the equality (4.2).It is easy to see that (4.8) follows from (4.2) when ϕ + θ = π.In order toprove (4.9) let us differentiate LHS and RHS of (4.9) with respect to x and use asummation formulan−1Xk=0exp i(ϕ + 2πkn )1 −x exp i(ϕ + 2πkn ) = nxn−1 exp(inϕ)1 −xn exp(inϕ).

14Anatol N. KirillovThus, the proofs of Proposition 4.1 and Lemma 4.2 are finished.Proof of the Theorem 3.1 for the case n = 2.If we substitute θ = mϕ in (4.2)then obtainL sin mϕsin ϕ2 != π2B2(m + 1)ϕπ−B2mϕπ−B2ϕπ+ 16++ 2L−sin((m −1)ϕ)sin ϕ, mϕ−2L−sin mϕsin ϕ , (m + 1)ϕ. (4.15)Futher let us introduce notationfm(ϕ) := 1 −Qm−1(ϕ)Qm+1(ϕ)Q2m(ϕ)=1Q2m(ϕ).Then using (4.15) we findrXm=1L(fm(ϕ)) = −2L−Qr(ϕ), (r + 2)ϕ−π26+(4.16)+ π2B2(r + 2)ϕπ−16+ (r + 2)π216 −B2ϕπ−π22 .Now let us put ϕ = (j + 1)πr + 2,0 ≤j ≤r + 1.

Then Qr(ϕ) = (−1)j and it is clearfrom (4.3) and (4.4) thatL(−1)j+1, (j + 1)π= L(1) = π26 .Note that polynomials Qm := Qm(ϕ) satisfy the following recurrence relationQ2m = Qm−1Qm+1 + 1,Q0 ≡1,m ≥1,where as the polynomials ym := ym(ϕ) = Qm−1(ϕ) · Qm+1(ϕ) satisfy the followingoney2m = (1 + ym−1)(1 + ym+1),y0 ≡0,m ≥1.Follow [Le2] we define a function W(x, ϕ) byW(x, ϕ) := W(x, ϕ, θ) = Lsin2 θsin ϕ(sin ϕ + x sin(ϕ + θ))+(4.17)+ L−x2 sin ϕ + x sin(ϕ −θ)x sin(ϕ + θ) + sin ϕ−Lx sin(ϕ + θ)x sin(ϕ + θ) + sin ϕ.

Dilogarithm identities, partitions and spectra in conformal field theory15Note the following particular casesW(0, ϕ) = L sin θsin ϕ2!,W(−2 cosθ, ϕ) = L−sin2 θsin ϕ sin(ϕ + 2θ),(4.18)W(1, ϕ) = Lsin θ sin 12θsin ϕ sin(ϕ + 12θ)+ Lsin( 12θ −ϕ)sin( 12θ + ϕ)−Lsin(ϕ + θ)2 sin(ϕ + 12θ) cos 12θ,W(−1, ϕ, θ) = W(1, ϕ, π + θ).Proposition 4.3. We haveW(x, ϕ) = 2L(−x, θ) + 2L(−x1, ϕ) −2L(−x2, ϕ + θ) ++ π2B2ϕ + θπ−B2ϕπ−B2 θπ+ 16,(4.19)wherex1 =x sin ϕ + sin(ϕ −θ)sin θand x2 = x sin(ϕ + θ) + sin ϕsin θ.A proof of Proposition 4.3 may be obtained by the same manner as that of Propo-sition 4.1.Note that x2 is obtained from x1 by replacing ϕ by ϕ+θ.

The angular parameterin (4.19) also increases in this way and the terms L(−x, ϕ) and L(−x2, ϕ + θ) haveopposite signs. So if we substitute successively the angles ϕ, ϕ+θ, .

. .

, ϕ+rθ insteadof ϕ in (4.19) and after this will add all results together, we will obtain the followinggeneralisation of (4.16):rXk=0W(x, ϕ + kθ) = 2(r + 1)L(−x, θ) + 2L(−x1, ϕ) −2L(−xr+2, ϕ + (r + 1)θ)++ π2B2ϕ + (r + 1)θπ−B2ϕπ+ (r + 1)π216 −B2 θπ,(4.20)wherexn+1 := x sin(ϕ + nθ) + sin(ϕ + (n −1)θ)sin θ,0 ≤n ≤r + 2.Now let us take ϕ = θ in (4.20). Then we find x1 = x, so that (4.20) becomesr+1Xk=1W(x, kθ) = 2(r + 2)L(−x, θ) + (r + 2)π216 −B2 θπ−π23 +(4.21)+ π2B2(r + 2)θπ−16−2L(−xr+2, (r + 2)θ) −π26,wherexn+1 := xn+1(x, θ) = x sin(n + 1)θ + sin nθsin θ.

16Anatol N. KirillovNote the following particular cases (0 ≤n ≤r + 2)xn+1(0, θ) = sin nθsin θ ,xn+1(−2 cosθ, θ) = −sin(n + 2)θsin θ,xn+1(1, θ) = sin 12(2n + 1)θsin 12θ,xn+1(−1, θ) = xn+1(1, θ + π).One can show that an identity (4.21) is reduced to (4.2) if x = −2 cosθ (or x = 0).Now assumex ̸= −2 cosθand takeθ = (j + 1)πr + 2in (4.21).Then we findxr+2 = (−1)j, so that (4.21) becomesr+1Xk=1W(x, kθ) = 2(r + 2)L(−x, θ) + π26 (1 + s(j, 2, r)),(4.22)wheres(j, 2, r) =3rr + 2 + 6 j(r −j)r + 2(see (3.7)).Finally let us take x = ±1 in (4.22). After some manipulations we obtain thefollowing result.Proposition 4.4.Let functionsW(±1, θ)are defined by (4.18) andθ = (j + 1)πr + 2.

Then we haver+1Xk=1W(−1, kθ) = π262r + 2 −3(j + 1)2r + 2,r+1Xk=1W(+1, kθ) = π262 −r −3(j + 1)2r + 2+ 6j. (4.23)Now we propose a generalisation of (4.16).Given a rational number p anddecomposition of p into the continued fractionp = [br, br−1, .

. .

, b1, b0] = br +1br−1 +1· · · +1b1 + 1b0. (4.24)We will assume that bi > 0 if 0 ≤i < r and br ∈Z.

Using the decomposition (4.24)we define the set of integers yi and mi:y−1 = 0,y0 = 1,y1 = b0, . .

. , yi+1 = yi−1 + biyi,0 ≤i ≤r,(4.25)m0 = 0,m1 = b0,mi+1 = |bi| + mi,0 ≤i ≤r.

Dilogarithm identities, partitions and spectra in conformal field theory17It is clear that p = yr+1yrandyi+1yi= pi := bi +1bi−1 +1· · · +1b1 + 1b0,0 ≤i ≤r.The following sequences of integers were first introduced by Takahashi andSuzuki [TS]r(j) = i,if mi ≤j < mi+1,0 ≤i ≤r,nj = yi−1 + (j −mi)yi,if mi ≤j < mi+1 + δi,r,0 ≤i ≤r.Finally we define a dilogarithm sum of “fractional level p”:mr+1Xj=1(−1)r(j)L sin yr(j)θsin(nj + yr(j))θ2 ! := (−1)r π26 s(k, 2, p),(4.26)where θ =(k + 1)πyr+1 + 2yr.The dilogarithm sum (4.26) (in the case k = 0) was considered at first in [KR],where its interpretation as a low-temperature asymptotic of the entropy for theXXZ Heisenberg model was given.Proposition 4.5.

We have(i)s(0, 2, p) =3pp + 2,(4.27)(ii)s(k, 2, p) =3pp + 2 −6k(k + 2)p + 2+ 6Z.Proof. We start withLemma 4.6.Given an integer σ such that mi < σ ≤mi+1.

Then for anyθ ∈R we haveσ−1Xj=miL sin yr(j)θsin(nj + yr(j))θ2 ! := (σ −mi)π216 −B2yiθπ++ π2B2(nσ−1 + 2yi)θπ−B2(yi−1 + yi)θπ+(4.28)+ 2L−sin yi−1θsin yiθ , (yi + yi−1)θ−2L−sin(nσ−1 + yi)θsin yiθ, (nσ−1 + 2yi)θ.A proof of Lemma 4.6 follows from Proposition 4.1.From (4.28) one can easily deduce the following generalisation of (4.16)

18Anatol N. KirillovCorollary 4.7. Given an integer σ,σ < mi ≤mi+1.

Thenσ−1Xj=0L sin yr(j)θsin(nj + yr(j))θ2 ! := π2i−1Xj=0(−1)jbj16 −B2yjθπ++ (−1)i(σ −mi)π216 −B2yiθπ+(−1)iπ2B2(nσ−1 + 2yi)θπ−−2π2i−1Xj=0(−1)jB2(yj−1 + yj)θπ−π2B2 θπ+(4.29)+ 2i−1Xj=0(−1)j+1L−sin yj−1θsin yjθ , (yj + yj−1)θ+ L−sin yjθsin yj−1θ, (yj + yj−1)θ++ (−1)i+12L−sin(nσ−1 + yi)θsin yiθ, (nσ−1 + 2yi)θ.In order to go further we must compute the last sum in (4.29).

Such computationis based on the following result.Lemma 4.8. Given the real numbers ϕ and θ, let us defineǫ(ϕ, θ) = { ϕ2π + 12} + { θ2π + 12} −{ ϕ2π} −{ θ2π}(mod 2),(4.30)where {x} = x −[x] be a fractional part of x ∈R.Then we haveL−sin ϕsin θ , ϕ + θ+ L−sin θsin ϕ, ϕ + θ= 2π2B2ϕ + θ + ǫ(ϕ, θ)π2π.

(4.31)A proof of Lemma 4.8 follows from (4.6).Now we are ready to finish a proof of Proposition 4.5. Namely from (4.29) and(4.31) it follows that s(k, 2, p) ∈Q,(yr+1 + 2yr) · s(k, 2, p) ∈Z, ands(0, 2, p) =3pp + 2.Finally, we observe that (4.27) follows from (4.29) and (4.31) if we replace all mod-ified Bernoulli polynomials by ordinary ones.Proposition 4.9.

For all positive p ∈Q the remainder term in (4.27) lies in6Z+. More exactly, given a positive p ∈Q we define a set of integers {sk},k =0, 1, 2, .

. .

such thatj + 1p + 2= k iffsk ≤j < sk+1, s0 := 0.

Dilogarithm identities, partitions and spectra in conformal field theory19Further, let us define a functiont(j) := t(j, p) = (2k + 1)j + k −2kXa=0sa iffsk ≤j < sk+1.Then we haves(j, 2, p) =3pp + 2 −6j(j + 2)p + 2+ 6t(j, p).It is clear that t(j, p) ∈Z+.Corollary 4.10. Let us fix the positive integers l = 1, 2, 3, .

. .

(the fusion level),p > q,j1 and j2. Thensj1, 2,qlp −q −2+ s(r0, 2, l) −sj2, 2,plp −q −2= c −24∆+ 6Z,(4.32)where r0 = l ·j1 −j2landc =3ll + 21 −2(l + 2)(p −q)2l2pq,(4.33)∆= [p(j1 + 1) −q(j2 + 1)]2 −(p −q)24lpq+ r0(l −r0)2l(l + 2)are the central charge and conformal dimensions of RCFT, which corresponds tothe coset pair [GKO]A1⊕A1⊃A1levelsqlp −q −2lplp −q −2§5.

Proof of Theorem 3.1.We start with a generalization of identity (4.2).Proposition 5.1. For all real ϕ, ψ and θ we haveL sin θsin ϕ · sin(θ + ψ)sin(ϕ + ψ)!=(5.1)= π2B2θ + ϕ + ψπ−B22θ + ψ + ǫ(θ, θ + ψ)π2π−

20Anatol N. Kirillov−B22ϕ + ψ + ǫ(ϕ, ϕ + ψ)π2π+ 16++ L−sin(ϕ −θ)sin θ, ϕ+ L−sin(ϕ −θ)sin(θ + ψ), ϕ + ψ−−L−sin ϕsin(θ + ψ), ϕ + θ + ψ−L−sin(ϕ + ψ)sin θ, ϕ + θ + ψ,where ǫ(ϕ, θ) = 1 −ǫ(ϕ, θ) and ǫ(ϕ, θ) is defined by (4.31).Proof. First of all we consider the case when 0 < ϕ + θ + ψ < π and ϕ, θ, ψ > 0.In this case ǫ(θ, θ +ψ) = ǫ(ϕ, ϕ+ψ) = 0 and one can use the identities (4.12)-(4.14)in order to show that a derivative of difference between LHS and RHS of (5.1) withrespect to ϕ is equal to2πB1θ + ϕ + ψπ−B12ϕ + ψ2π= 2θ + ψ.Integrating (see (4.10)) we find that the difference between LHS and RHS of (5.1)is a function c(θ, ψ) which does not depend on ϕ.

In order to find c(θ, ψ) let us takeϕ = θ in (5.1). After this substitution we obtain an equalityπ26 =12(2θ + ψ)2 + c(θ, ψ)−−L−sin θsin(θ + ψ), 2θ + ψ−L−sin(θ + ψ)sin θ, 2θ + ψ.Comparing the last equality with (4.31) (in our case ǫ(θ, θ+ψ) = 1) we find c(θ, ψ) =0.

In general case we use the reduction rules (4.3), (4.4) and (4.6) and the followingproperties of function ǫ(ϕ, θ):ǫ(θ, θ) = 1,ǫ(θ, ϕ) = ǫ(ϕ, θ),ǫ(θ, π + θ) = 0,ǫ(ϕ, −θ) = ǫ(ϕ, π + θ),ǫ(θ, π −θ) = 1,ǫ(−ϕ, −θ) = ǫ(ϕ, θ).Corollary 5.2. We haveLsin(ϕ + θ)sin θ, ϕ+ Lsin(ϕ + θ)sin ϕ, θ= 2π2B2ϕ + θ + ǫ(ϕ, θ)π2π+ 112.(5.2)Proof.

Take ψ = −θ −ϕ in (5.1).Let us continue a proof of Theorem 3.1 and take a specialisation θ →kϕ,ϕ →(m + k)ϕ and ψ →(n −2k)ϕ in (5.1). Then we obtainL sin kϕsin(m + k)ϕ ·sin(n −k)ϕsin(m + n −k)ϕ!=

Dilogarithm identities, partitions and spectra in conformal field theory21= π2B2(m + n)ϕπ−B2nϕ + ǫ(kϕ, (n −k)ϕ)π2π−−B2(n + 2m)ϕ + ǫ((m + k)ϕ, (n −k)ϕ)π2π+ 16++ L−sin mϕsin kϕ , (m + k)ϕ+ L−sin mϕsin(n −k)ϕ, (m + n −k)ϕ−−L−sin(m + k)ϕsin(n −k)ϕ , (m + n)ϕ−L−sin(m + n −k)ϕsin kϕ, (m + n)ϕ.Consequently, after summation we obtainn−1Xk=1rXm=1L sin kϕsin(m + k)ϕ ·sin(n −k)ϕsin(m + n −k)ϕ!= 2n−1Xk=1rXm=1L−sin mϕsin kϕ , (m + k)ϕ−−2n−1Xk=1rXm=1L−sin(m + n −k)ϕsin kϕ, (m + n)ϕ+ π2Σ3 == 2n−1Xk=1n−kXm=1L−sin mϕsin kϕ , (m + k)ϕ−2n−1Xk=1n−kXm=1L−sin(r + m)ϕsin kϕ, (m + r + k)ϕ++ π2Σ3 = 2Σ1 −2Σ2 + π2Σ3,whereΣ3 :=n−1Xk=1rXm=1B2(m + n)ϕπ−B2nϕ + ǫ(kϕ, (n −k)ϕ)π2π−(5.3)−B2(n + 2m)ϕ + ǫ((m + k)ϕ, (n −k)ϕ)π2π+ 16.At first, let us consider the sum2Σ1 := 2n−1Xk=1n−kXm=1L−sin mϕsin kϕ , (m + k)ϕ= 2[ n2 ]Xp=1L(−1, 2pϕ)+(5.4)+ 2nXp=3[p−12 ]Xk=1L−sin(p −k)ϕsin kϕ, pϕ+ 2L−sin kϕsin(p −k)ϕ, pϕ== 2nXp=3[ p−12 ]Xk=12π2B2pϕ + ǫ((p −k)ϕ, kϕ)π2π+ 2[ n2 ]Xp=1π2B2pϕπ + 12.Secondly, in order to compute the sum 2Σ2, let us remind that ϕ = (j + 1)πn + r .Hence sin(m + r)ϕ = sin (m + r)(j + 1)πn + r= (−1)j sin(n −m)ϕ and consequently

22Anatol N. Kirillov(see (4.3) and (4.4))L−sin(r + m)ϕsin kϕ, (m + r + k)ϕ== L(−1)j+1 sin(n −m)ϕsin kϕ, (j + 1)π −(n −m −k)ϕ== L−sin(m −n)ϕsin kϕ, (m + k −n)ϕ.So we have2Σ2 := 2n−1Xk=1n−kXm=1L−sin(r + m)ϕsin kϕ, (m + r + k)ϕ=(5.5)=n−1Xk=1n−kXm=1L−sin(m −n)ϕsin kϕ, (m + k −n)ϕ= 2[ n−12 ]Xp=1L(2 cospϕ, pϕ) + (n −1)π23++ 2n−1Xp=3[p−12 ]Xk=1Lsin pϕsin kϕ, (p −k)ϕ+ Lsin pϕsin(p −k)ϕ, kϕ= (n −1)π23++ 2n−1Xp=3[ p−12 ]Xk=12π2B2pϕ + ǫ(kϕ, (p −k)ϕ)π2π+ 112+ 2[ n2 ]Xp=1π2B2pϕπ+ 112.Let us sum up our computations. First of all we proved that s(j, n, k) ∈Q.

Sec-ondly, in order to compute the dilogarith sum s(j, n, k) moduloZ we may replaceall modified Bernoulli polinomials appearing in (5.3)-(5.5) by ordinary ones. Aftersome bulky calculations we obtain (3.5), except positivity of a remainder term in(3.5).

Finally, in order to obtain the exact formulae (3.3) and (3.4) we are basedon the properties of Dedekind sums [Ra]. Details will appear elsewhere.Now we propose a generalisation of (4.27).

For this goal let us define the follow-ing functionLk(θ, ϕ) := 2Lsin θ · sin kθsin ϕ · sin(ϕ + (k −1)θ)−k−1Xj=0L sin θsin(ϕ + jθ)2!. (5.6)Lemma 5.3.

We haveLk(θ, ϕ) = 2L−sin(ϕ −θ)sin kθ, ϕ + (k −1)θ−2L−sin ϕsin kθ, ϕ + kθ+ π2Q.Now let p ∈Q and consider a decomposition of p/k into continued fractionpk = br +1br−1 +1· · · +1b1 + 1b0,(5.7)

Dilogarithm identities, partitions and spectra in conformal field theory23where bi ∈N,0 ≤i ≤r −1 and br ∈Z. Using the decomposition (5.7) we define(compare with (4.25)):y−1 = 0,y0 = 1,y1 = b0, .

. .

, yi+1 = yi−1 + biyi,m0 = 0,m1 = b0,mi+1 = |bi| + mi,r(j) := rk(j) = i,if kmi ≤j < kmi+1 + δi,r,nj := nk(j) = kyi−1 + (j −kmi)yi,if kmi ≤j < kmi+1 + δi,r,where 0 ≤i ≤r.Finally, we consider the following dilogarithm sumkmr+1Xj=1(−1)r(j)Lkyr(j)θ, (nj + yr(j))θ= (−1)r π26 s(l, k + 1, p),(5.8)where θ =(l + 1)πkyr+1 + (k + 1)yr.Proposition 5.4. We have(i)s(0, k + 1, p) := ck = 3(p + 1 −k)p + k + 1 ,k ≥1,(5.8)(ii) s(l, k + 1, p) = ck −6k l(l + 2)p + k + 1 + 6Z,(5.9)(iii) if k = 1 or 2, then the remainder term in (5.9) lies in 6Z+.Riferences.

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