The hexagon equations for dilogarithms and the Riemann-Hilbert problem
aa r X i v : . [ m a t h . C A ] M a r The hexagon equations for dilogarithms andthe Riemann-Hilbert problem
Shu Oi ∗ Kimio Ueno † Abstract
In this article we present the hexagon equations for dilogarithms whichcomes from the analytic continuation of the dilogarithm Li ( z ) to P \{ , , ∞} . The hexagon equations are equivalent to the coboundary re-lations for a certain 1-cocycle of holomorphic functions on P , and aresolved by the Riemann-Hilbert problem of additive type. They uniquelycharacterize the dilogarithm under the normalization condition. Let D α ( α = 0 , , ∞ ) be domains in the Riemann sphere P defined by D = C \ { z = x | ≤ x < + ∞} , D = C \ { z = x | − ∞ < x ≤ } , D ∞ = P \ { z = x | ≤ x ≤ } , which are open neighborhoods of the points 0 , , ∞ , respectively. They give anopen covering of the Riemann sphere P = D ∪ D ∪ D ∞ , furthermore, satisfy D ∩ D ∩ D ∞ = H + ∪ H − where H + ( H − ) is the upper (lower) half plane.According to Theorem 2 in [4], the dilogarithms Li ( z ) and Li (1 − z ) arecharacterized as the solutions f ( z ) and f ( z ) holomorphic in D and D , re-spectively, to the following functional equation f ( z ) + log z log(1 − z ) + f ( z ) = ζ (2) ( z ∈ D ∩ D ) (1.1)under the asymptotic condition f ′ α ( z ) → z → ∞ , z ∈ D α , α = 0 ,
1) and thenormalization condition f (0) = 0. Mathematics Subject Classification.
Primary 34M50,11G55; Secondary 30E25,11M06,32G34; ∗ Department of Mathematics, College of Science, Rikkyo university.3-34-1, Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan.e-mail: [email protected] † Department of Mathematics, School of Fundamental Sciences and Engineering,Faculty of Science and Engineering, Waseda University.3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan.e-mail: [email protected] − log z log(1 − z ) + ζ (2), which isholomorphic in D ∩ D , decomposes to the sum of f ( z ) and f ( z ) holomor-phic in D and D . That is to say, (1.1) is a Riemann-Hilbert problem (or,Plemelj-Birkhoff decomposition) of additive type [1, 3, 8]. However the previ-ous asymptotic condition does not naturally come from the Riemann-Hilbertproblem, is a rather technical one. So we would like to avoid that, and givemore natural formulation of the Riemann-Hilbert problem. Let log(1 − z ) be the principal value of the logarithm in D , namely, log 1 = 0.Define a branch of the dilogarithm Li ( z ) in D byLi ( z ) = − Z z log(1 − t ) t dt. (2.1)Then the Taylor expansion at z = 0 isLi ( z ) = ∞ X n =1 z n n ( | z | <
1) (2.2)and by Abel’s continuous theorem, we havelim z → ,z ∈ D Li ( z ) = ζ (2) . (2.3)It is well known that the dilogarithm satisfies the following functional rela-tions [2];Li ( z ) + log z log(1 − z ) + Li (1 − z ) = ζ (2) ( z ∈ D ∩ D ) , (2.4)Li (cid:0) zz − (cid:1) = − Li ( z ) −
12 log (1 − z ) ( z ∈ D ) . (2.5)The first formula describes the direct analytic continuation of Li ( z ) along thepath 01, and the second one is regarded as the half-monodromy continuation ofLi ( z ) at z = 0. We call them Euler’s inversion formula and Landen’s inversionformula, respectively.To consider the global analytic continuation of the dilogarithm, let us intro-duce automorphisms t i ( z ) of P as follows: (cid:26) t ( z ) = z = z, t ( z ) = z = 1 − z, t ( z ) = z = z − z ,t ( z ) = z = z , t ( z ) = z = − z , t ( z ) = z = zz − . (2.6)They induce permutations σ t i of the points { , , ∞} , σ t = (cid:18) ∞ ∞ (cid:19) , σ t = (cid:18) ∞ ∞ (cid:19) , σ t = (cid:18) ∞∞ (cid:19) ,σ t = (cid:18) ∞∞ (cid:19) , σ t = (cid:18) ∞ ∞ (cid:19) , σ t = (cid:18) ∞ ∞ (cid:19) , t i : D α −→ D t i ( α ) . Furthermore, we shouldnote that z j − = z j z j − , z j +1 = 1 − z j , ( j = 0 , , t j ( j = 0 , , Proposition 2.1.
The dilogarithm Li ( z ) satisfies Li ( z j ) + log z j log z j +1 + Li ( z j +1 ) = ζ (2) ( z ∈ D t − j (0) ∩ D t − j (1) ) , (2.7)Li ( z j − ) = − Li ( z j ) −
12 log z j +1 ( z ∈ D t − j (0) ) , (2.8) where j = 0 , , , and all the indices are defined modulo 6. We refer to them as the hexagon relations for dilogarithms, which describethe analytic continuation of Li ( z ) along the paths through the neighborhoodof 0 → → ∞ → H ± (see the figures below).0 1 ∞ ∞ j denotes 0 , ,
4, and all the indices are defined modulo6. Let f j ( z ) , f j − ( z ) be functions holomorphic in D t − j (0) , and consider thefollowing functional equations for them: f j ( z ) + log z j log z j +1 + f j +1 ( z ) = ζ (2) in D t − j (0) ∩ D t − j (1) , (3.1) f j − ( z ) = − f j ( z ) −
12 log z j +1 in D t − j (0) . (3.2)We call them the hexagon equations for dilogarithms. We show that they arerelevant to the cohomology theory on P :3 roposition 3.1. To the covering { D , D , D ∞ } of P , we attach a 0-cochain f = { f D α } α =0 , , ∞ and a 1-cochain F = { F D α D β } α,β =0 , , ∞ where f D α is aholomorphic function on D α and F D α D β is a holomorphic function on D α ∩ D β defined by f D t − j (0) ( z ) = f j ( z ) , F D t − j (0) , D t − j (1) ( z ) = 12 log z j − log z j log z j +1 + ζ (2) , (3.3) and F D β D α = − F D α D β . Then we have the following:1. F is a 1-cocycle. That is to say, F satisfies the cocycle condition F D D ∞ ( z ) − F D D ∞ ( z ) + F D D ( z ) = 0 ( z ∈ D ∩ D ∩ D ∞ ) . (3.4)
2. The condition (3.4) is equivalent to the Euler formula ζ (2) = π .3. The hexagon equations (3.1) and (3.2) are equivalent to (3.2) and thecoboundary relations F D t − j (0) , D t − j (1) ( z ) = f D t − j (0) ( z ) − f D t − j (1) ( z ) ( z ∈ D t − j (0) ∩ D t − j (1) ) . (3.5) Proof.
From (3.3), it follows that F D D ∞ ( z ) − F D D ∞ ( z )+ F D D ( z ) = 3 ζ (2)+ 12 (cid:18) log z + log (cid:0) z − z (cid:1) + log (cid:0) − z (cid:1)(cid:19) . As the logarithms satisfy the half-monodromy relationslog z + log (cid:0) z − z (cid:1) + log (cid:0) − z (cid:1) = ± πi ( z ∈ H ± ) , we have F D D ∞ ( z ) − F D D ∞ ( z ) + F D D ( z ) = 3 ζ (2) − π . Thus the claims (i)and (ii) are proved.The claim (iii) immediately follows from t − j +1 (0) = t − j (1). From Proposition 3.1 and the results on the cohomology theory that H ( P , O ) =0 , H ( P , O ) = C , where O denotes the sheaf of holomorphic functions on P ,one can deduce that the solutions to the hexagon equations exist uniquely un-der the normalization condition f (0) = 1. However one cannot know how thesolutions are relevant to dilogarithms. So let us solve the hexagon equation bythe Riemann-Hilbert problem of additive type.4 heorem 4.1. The hexagon equations for dilogarithms have a unique solutionunder the normalization condition f (0) = 0 , and the solutions are expressed as f i ( z ) = Li ( z i ) ( i = 0 , , . . . , . (4.1) Proof.
Differentiating the equations (3.1) and (3.2) with respect to the variable z = z , we have the following: f ′ j ( z ) + log z j +1 (log z j ) ′ = − f ′ j +1 ( z ) − log z j (log z j +1 ) ′ , (4.2) f ′ j +1 ( z ) + log z j (log z j +1 ) ′ = − f ′ j +2 ( z ) − log z j +3 (log z j +2 ) ′ (4.3)where j = 0 , ,
4. Put f ( z ) = f ′ ( z ) + log z (log z ) ′ = f ′ ( z ) + log(1 − z ) z . This is a function holomorphic in D . From (4.2), it is analytically continuedto g ( z ) and h ( z ) g ( z ) = − f ′ ( z ) − log z (log z ) ′ = − f ′ ( z ) + log z − z ,h ( z ) = − f ′ ( z ) − log z (log z ) ′ = − f ′ ( z ) + log (cid:0) z − z (cid:1) z , which are holomorphic in D and D ∞ , respectively. Hence f ( z ) is holomorphicon P so that f ( z ) = g ( z ) = h ( z ) = A where A is a constant. Substitute thisinto (4.2) and (4.3), and integrate them by usinglog z j +1 (log z j ) ′ = − (cid:0) Li ( z j ) (cid:1) ′ , log z j (log z j +1 ) ′ = − (cid:0) Li ( z j +1 ) (cid:1) ′ . Then we obtain f i ( z ) = Li ( z i ) + ( − i Az + b i ( i = 0 , , . . . , . Here b i are integral constants. From the normalization condition f (0) = 0, itis clear that b = 0. Hence from (3.1), we haveLi ( z ) + log z log(1 − z ) + Li (1 − z ) + b = ζ (2) . Taking the limit as z → z ∈ D ∩ D ) in this equation, we have b = 0 and f ( z ) = Li (1 − z ) − Az . In a similar way, we show that all the constants b i are0. Furthermore, since f ( z ), which is expressed as f ( z ) = Li (cid:0) z (cid:1) + Az, should be holomorphic at z = ∞ , we have A = 0. Thus the proof is completed.5 Discussions
In [5], the Riemann-Hilbert approach generalizes to the case of the fundamentalsolution normalized at the origin of the KZ equation of one variable. So it willbe worth trying to generalize the hexagon equations approach to the case of thefundamental solutions of the KZ equation of one variable (cf. [7]), and furtherto the case of the KZ equation of two variables (cf. [6]).
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