The Dilogarithm Function

  • Don Zagier


The dilogarithm function, defined in the first sentence of Chapter I, is a function which has been known for more than 250 years, but which for a long time was familiar only to a few enthusiasts. In recent years it has become much better known, due to its appearance in hyperbolic geometry and in algebraic K-theory on the one hand and in mathematical physics (in particular, in conformal field theory) on the other. I was therefore asked to give two lectures at the Les Houches meeting introducing this function and explaining some of its most important properties and applications, and to write up these lectures for the Proceedings.


Functional Equation Modular Form Modular Function Theta Series Mock Theta Function 
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  1. 1.
    Beauville, A.; Blanc, J.: On Cremona transformations of prime order. C. R. Math. Acad. Sci. Paris 339 (2004) 257–259.zbMATHGoogle Scholar
  2. 2.
    Beilinson, A. and Deligne, P.: Motivic polylogarithm and Zagier conjecture. Preprint (1992).Google Scholar
  3. 3.
    Belabas, K. and Gangl, H.: Generators and relations for K2OF. K-Theory 31 (2004) 195–231.zbMATHCrossRefGoogle Scholar
  4. 4.
    Besser, A.: Finite and p-adic polylogarithms. Compositio Math. 130 (2002) 215–223.zbMATHCrossRefGoogle Scholar
  5. 5.
    Borel, A.: Cohomologie de SLn et valeurs de fonctions zêta aux points entiers. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977) 613–636.zbMATHGoogle Scholar
  6. 6.
    Bailey, D. and Broadhurst, D.: A seventeenth-order polylogarithm ladder. arxiv:math.CA/9906134.Google Scholar
  7. 7.
    Browkin, J.: Computing the tame kernel of quadratic imaginary fields. With an appendix by Karim Belabas and Herbert Gangl. Math. Comp. 69 (2000) 1667–1683.zbMATHCrossRefGoogle Scholar
  8. 8.
    Cathelineau, J.-L.: Remarques sur les di.érentielles des polylogarithmes uniformes. Ann. Inst. Fourier Grenoble 46 (1996), 1327–1347.zbMATHGoogle Scholar
  9. 9.
    Chern, S.-S. and Grifiths, P.: Abel's theorem and webs. Jahresber. Deutsch. Math.-Verein. 80 (1978) 13–110.zbMATHGoogle Scholar
  10. 10.
    Cohen, H., Lewin L. and Zagier, D.: A sixteenth-order polylogarithm ladder. Experimental Math. 1 (1992) 25–34.zbMATHGoogle Scholar
  11. 11.
    Coleman, R.: Dilogarithms, regulators and p-adic L-functions. Invent Math. 69 (1982), 171–208.zbMATHCrossRefGoogle Scholar
  12. 12.
    de Jeu, R.: Zagier's conjecture and wedge complexes in algebraic Ktheory. Compositio Math. 96 (1995) 197–247.zbMATHGoogle Scholar
  13. 13.
    Deninger, C.: Higher regulators and Hecke L-series of imaginary quadratic fields. I. Invent. Math. 96 (1989) 1–69.zbMATHCrossRefGoogle Scholar
  14. 14.
    Elbaz-Vincent, Ph. and Gangl, H.: On Poly(ana)logs I, Compositio Math. 130 (2002) 161–210.zbMATHCrossRefGoogle Scholar
  15. 15.
    Faddeev, L.D. and Kashaev, R.M.: Quantum dilogarithm. Modern Phys. Lett. A 9 (1994) 427–434.zbMATHCrossRefGoogle Scholar
  16. 16.
    Gangl, H.: Functional equations for higher logarithms. Selecta Math. 9 (2003) 361–377.zbMATHCrossRefGoogle Scholar
  17. 17.
    Gliozzi, F.; Tateo, R.: ADE functional dilogarithm identities and integrable models. Phys. Lett. B 348 (1995) 84–88.CrossRefGoogle Scholar
  18. 18.
    Gliozzi, F.; Tateo, R.: Thermodynamic Bethe ansatz and three-fold triangulations. Internat. J. Modern Phys. A 11 (1996) 4051–4064.zbMATHCrossRefGoogle Scholar
  19. 19.
    Goddard, P., Kent, A. and Olive, D.: Unitary representations of the Virasoro and supervirasoro algebras. Commun. Math. Phys. 103 (1986) 105–119.zbMATHCrossRefGoogle Scholar
  20. 20.
    Goncharov, A. B.: Geometry of configurations, polylogarithms, and motivic cohomology. Adv. Math. 114 (1995) 197–318.zbMATHCrossRefGoogle Scholar
  21. 21.
    Goncharov, A. B.: The double logarithm and Manin's complex for modular curves. Math. Res. Lett. 4 (1997) 617–636.zbMATHGoogle Scholar
  22. 22.
    Kirillov, A.N.: Dilogarithm identities. In Quantum field theory, integrable models and beyond, Kyoto 1994. Progress of Theoretical Physics Supplement 118 (1995) 61–142.Google Scholar
  23. 23.
    Kontsevich, M.: The 11 2 -logarithm. Compositio Math. 130 (2002) 211–214. (= Appendix to [14].)Google Scholar
  24. 24.
    Lawrence, R. and Zagier, D.: Modular forms and quantum invariants of 3-manifolds. Asian J. of Math. 3 (1999) 93–108.zbMATHGoogle Scholar
  25. 25.
    Levin, A.: Kronecker double series and the dilogarithm. In Number theory and algebraic geometry, London Math. Soc. Lecture Note Ser. 303, Cambridge Univ. Press (2003) 177–201.Google Scholar
  26. 26.
    Meinardus G.: Über Partitionen mit Differenzenbedingungen. Math. Z. 1 (1954) 289–302.CrossRefGoogle Scholar
  27. 27.
    Merkur'ev, A. S. and Suslin, A.: The group K3 for a field. Izv. Akad. Nauk SSSR 54 (1990) 522–545; Math. USSR-Izv. 36 (1991) 541–565.zbMATHGoogle Scholar
  28. 28.
    Nahm, W.: Conformal field theory and the dilogarithm. In 11th International Conference on Mathematical Physics (ICMP-11) (Satellite colloquia: New Problems in the General Theory of Fields and Particles), Paris (1994) 662–667.Google Scholar
  29. 29.
    Nahm, W.: Conformal Field Theory, Dilogarithms, and Three Dimensional Manifold. In Interface between physics and mathematics (Proceedings, Conference in Hangzhou, P.R. China, September 1993), eds. W. Nahm and J.-M. Shen, World Scientific, Singapore (1994) 154–165.Google Scholar
  30. 30.
    Nahm, W.: Conformal field theory and torsion elements of the Bloch group. Frontiers in Number Theory, Physics, and Geometry II, 67–132.Google Scholar
  31. 31.
    Nahm, W., Recknagel, A. and Terhoeven, M., Dilogarithm identities in conformal field theory. Mod. Phys. Lett. A8 (1993) 1835–1847.Google Scholar
  32. 32.
    Neumann, W. and Zagier, D.: Volumes of hyperbolic three-manifolds. Topology 24 (1985) 307–332.zbMATHCrossRefGoogle Scholar
  33. 33.
    Neumann, W.: Extended Bloch group and the Cheeger-Chern-Simons class. Geom. Topol. 8 (2004) 413–474.zbMATHCrossRefGoogle Scholar
  34. 34.
    Rhin, G. and Viola, C.: On a permutation group related to ?(2). Acta Arith. 77 (1996) 23–56.zbMATHGoogle Scholar
  35. 35.
    Sachdev, S.: Polylogarithm identities in a conformal field theory in threedimensions. Phys. Lett. B309 (1993) 285–288.Google Scholar
  36. 36.
    Terhoeven, M.: Dilogarithm identities, fusion rules and structure constants of CFTs. Mod. Phys. Lett. A9 (1994) 133–142.Google Scholar
  37. 37.
    Terhoeven, M.: Rationale konforme Feldtheorien, der Dilogarithmus und Invarianten von 3-Mannigfaltigkeiten. Thesis, Bonn University, 1995. Scholar
  38. 38.
    Volkov, A. Y. and Faddeev, L. D.: Yang-Baxterization of a quantum dilogarithm. Zap. Nauchn. Sem. S.-Peterburg 224 (1995) 146–154; J. Math. Sci. 88 (1998) 202–207.Google Scholar
  39. 39.
    Voros, A.: Airy function—exact WKB results for potentials of odd degree. J. Phys. A 32 (1999) 1301–1311.zbMATHCrossRefGoogle Scholar
  40. 40.
    Wojtkowiak, Z.: Functional equations of iterated integrals with regular singularities. Nagoya Math. J. 142 (1996) 145–159.zbMATHGoogle Scholar
  41. 41.
    Wojtkowiak, Z.: A note on functional equations of the p-adic polylogarithms. Bull. Soc. Math. France 119 (1991), 343–370.zbMATHGoogle Scholar
  42. 42.
    Yoshida, T.: On ideal points of deformation curves of hyperbolic 3-manifolds with one cusp. Topology 30 (1991) 155–170.zbMATHCrossRefGoogle Scholar
  43. 43.
    Zagier, D.: The remarkable dilogarithm. J. Math. and Phys. Sciences 22 (1988) 131–145; also appeared as: “The dilogarithm function in geometry and number theory” in Number Theory and Related Topics, papers presented at the Ramanujan Colloquium, Bombay 1988, Tata and Oxford (1989) 231–249.zbMATHGoogle Scholar
  44. 44.
    Zagier, D.: The Bloch—Wigner—Ramakrishnan polylogarithm function. Math. Annalen 286 (1990) 613–624.zbMATHCrossRefGoogle Scholar
  45. 45.
    Zagier, D.: Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields. In Arithmetic Algebraic Geometry (G. v.d. Geer, F. Oort, J. Steenbrink, eds.), Prog. in Math. 89, Birkhäuser, Boston (1990) 391–430.Google Scholar
  46. 46.
    Zagier, D.: Special values and functional equations of polylogarithms. In The Structural Properties of Polylogarithms, ed. L. Lewin, Mathematical Surveys and Monographs 37, AMS, Providence (1991) 377–400.Google Scholar
  47. 47.
    Zagier, D and Gangl, H.: Classical and elliptic polylogarithms and special values of L-series. In The Arithmetic and Geometry of Algebraic Cycles, Proceedings, 1998 CRM Summer School, Nato Science Series C, Vol. 548, Kluwer, Dordrecht-Boston-London (2000) 561–615.Google Scholar
  48. 48.
    Zudilin, W.: Quantum dilogarithm. Preprint, Bonn and Moscow (2006), 8 pages.Google Scholar
  49. 49.
    Zwegers, S.: Mock theta functions. Thesis, Universiteit Utrecht, 2002.Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Don Zagier
    • 1
    • 2
  1. 1.Max-Planck-Institut für MathematikBonnGermany
  2. 2.Collège de FranceParisFrance

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