Multiple zeta values and periods of moduli spaces 𝔐 ¯ 0 , n

Francis C. S. Brown

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 3, page 371-489
  • ISSN: 0012-9593

Abstract

top
We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces 𝔐 0 , n of Riemann spheres with n marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on 𝔐 0 , n and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces.

How to cite

top

Brown, Francis C. S.. "Multiple zeta values and periods of moduli spaces $\overline{\mathfrak {M}}_{0,n}$." Annales scientifiques de l'École Normale Supérieure 42.3 (2009): 371-489. <http://eudml.org/doc/272243>.

@article{Brown2009,
abstract = {We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces $\mathfrak \{M\}_\{0,n\}$ of Riemann spheres with $n$ marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on $\mathfrak \{M\}_\{0,n\}$ and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces.},
author = {Brown, Francis C. S.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {moduli spaces; multiple zeta values; iterated integrals; polylogarithms; associators; associahedra},
language = {eng},
number = {3},
pages = {371-489},
publisher = {Société mathématique de France},
title = {Multiple zeta values and periods of moduli spaces $\overline\{\mathfrak \{M\}\}_\{0,n\}$},
url = {http://eudml.org/doc/272243},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Brown, Francis C. S.
TI - Multiple zeta values and periods of moduli spaces $\overline{\mathfrak {M}}_{0,n}$
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 3
SP - 371
EP - 489
AB - We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces $\mathfrak {M}_{0,n}$ of Riemann spheres with $n$ marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on $\mathfrak {M}_{0,n}$ and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces.
LA - eng
KW - moduli spaces; multiple zeta values; iterated integrals; polylogarithms; associators; associahedra
UR - http://eudml.org/doc/272243
ER -

References

top
  1. [1] K. Aomoto, Fonctions hyperlogarithmiques et groupes de monodromie unipotents, J. Fac. Sci. Univ. Tokyo Sect. IA Math.25 (1978), 149–156. Zbl0416.32020MR509582
  2. [2] K. Aomoto, Addition theorem of Abel type for hyper-logarithms, Nagoya Math. J.88 (1982), 55–71. Zbl0545.33014MR683242
  3. [3] K. Aomoto, Special values of hyperlogarithms and linear difference schemes, Illinois J. Math.34 (1990), 191–216. Zbl0684.33010MR1046562
  4. [4] V. I. Arnolʼd, The cohomology ring of the group of dyed braids, Mat. Zametki5 (1969), 227–231. Zbl0277.55002MR242196
  5. [5] A. A. Beilinson, A. B. Goncharov, V. V. Schechtman & A. N. Varchenko, Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of a pair of triangles in the plane, in Grothendieck Festschrift, 86, Birkhäuser, 1990, 135–171. Zbl0737.14003MR1086885
  6. [6] A. Borel & J-P. Serre, Corners and arithmetic groups, Comment. Math. Helv.48 (1973), 436–491. Zbl0274.22011MR387495
  7. [7] F. C. S. Brown, Polylogarithmes multiples uniformes en une variable, C. R. Math. Acad. Sci. Paris338 (2004), 527–532. Zbl1048.11053MR2057024
  8. [8] F. C. S. Brown, Single-valued hyperlogarithms and unipotent differential equations, preprint. 
  9. [9] K. T. Chen, Extension of C ∞ function algebra by integrals and Malcev completion of π 1 , Advances in Math.23 (1977), 181–210. Zbl0345.58003MR458461
  10. [10] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc.83 (1977), 831–879. Zbl0389.58001MR454968
  11. [11] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math., Vol. 163, Springer, 1970. Zbl0244.14004
  12. [12] P. Deligne, Théorie de Hodge. III, Publ. Math. I.H.É.S. 44 (1974), 5–77. Zbl0237.14003MR498552
  13. [13] P. Deligne, Le groupe fondamental de la droite projective moins trois points, in Galois groups over 𝐐 (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 16, Springer, 1989, 79–297. Zbl0742.14022MR1012168
  14. [14] P. Deligne & A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup.38 (2005), 1–56. Zbl1084.14024MR2136480
  15. [15] S. L. Devadoss, Tessellations of moduli spaces and the mosaic operad, in Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math. 239, Amer. Math. Soc., 1999, 91–114. Zbl0968.32009MR1718078
  16. [16] S. L. Devadoss, Combinatorial equivalence of real moduli spaces, Notices Amer. Math. Soc.51 (2004), 620–628. Zbl1093.14509MR2064149
  17. [17] A. C. Dixon, On a certain double integral, Proc. London Math. Soc.2 (1905), 8–15. Zbl35.0320.01MR1577292JFM35.0320.01
  18. [18] V. G. Drinfelʼd, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal ( 𝐐 ¯ / 𝐐 ) , Leningrad Math. Journal2 (1991), 829–860. Zbl0728.16021MR1080203
  19. [19] J. Écalle, Singularités non abordables par la géométrie, Ann. Inst. Fourier (Grenoble) 42 (1992), 73–164. Zbl0940.32013MR1162558
  20. [20] M. Falk & R. Randell, The lower central series of a fiber-type arrangement, Invent. Math.82 (1985), 77–88. Zbl0574.55010MR808110
  21. [21] S. Fischler, Groupes de Rhin-Viola et intégrales multiples, J. Théor. Nombres Bordeaux15 (2003), 479–534. Zbl1074.11040MR2140865
  22. [22] E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, in The moduli space of curves (Texel Island, 1994), Progr. Math. 129, Birkhäuser, 1995, 199–230. Zbl0851.18005MR1363058
  23. [23] A. B. Goncharov, The dihedral Lie algebras and Galois symmetries of π 1 ( l ) ( ℙ 1 - ( { 0 , ∞ } ∪ μ N ) ) , Duke Math. J.110 (2001), 397–487. Zbl1113.14020MR1869113
  24. [24] A. B. Goncharov, Multiple ζ -values, Galois groups, and geometry of modular varieties, in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, 2001, 361–392. Zbl1042.11042
  25. [25] A. B. Goncharov, Multiple polylogarithms and mixed Tate motives, preprint arXiv:math.AG/0103059. 
  26. [26] A. B. Goncharov, Periods and mixed motives, preprint arXiv:math.AG/0202154. 
  27. [27] A. B. Goncharov & Y. I. Manin, Multiple ζ -motives and moduli spaces ℳ ¯ 0 , n , Compos. Math.140 (2004), 1–14. Zbl1047.11063
  28. [28] J. Gonzalez-Lorca, Série de Drinfel’d, monodromie et algèbres de Hecke, Thèse de doctorat, École Normale Supérieure, 1998. 
  29. [29] P. Griffiths & W. Schmid, Recent developments in Hodge theory: a discussion of techniques and results, in Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, 1975, 31–127. Zbl0355.14003MR419850
  30. [30] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publ. Math. I.H.É.S. 29 (1966), 95–103. Zbl0145.17602MR199194
  31. [31] R. M. Hain, The geometry of the mixed Hodge structure on the fundamental group, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., 1987, 247–282. Zbl0654.14006MR927984
  32. [32] R. M. Hain, Classical polylogarithms, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., 1994, 3–42. Zbl0807.19003MR1265550
  33. [33] R. M. Hain & R. MacPherson, Higher logarithms, Illinois J. Math.34 (1990), 392–475. Zbl0737.14014MR1046570
  34. [34] M. Hoang Ngoc, M. Petitot & J. Van Den Hoeven, Shuffle algebra and polylogarithms, in Formal Power Series and Algebraic Combinatorics 1998, Toronto. Zbl0965.68129
  35. [35] M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin.11 (2000), 49–68. Zbl0959.16021MR1747062
  36. [36] M. M. Kapranov, The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation, J. Pure Appl. Algebra 85 (1993), 119–142. Zbl0812.18003MR1207505
  37. [37] V. G. Knizhnik & A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B247 (1984), 83–103. Zbl0661.17020MR853258
  38. [38] T. Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math.82 (1985), 57–75. Zbl0574.55009MR808109
  39. [39] E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, 1973, Pure and Applied Mathematics, Vol. 54. Zbl0264.12102MR568864
  40. [40] I. Lappo-Danilevskii, Mémoires sur la théorie des systèmes des équations différentielles linéaires, Chelsea, New York, 1953. Zbl0011.34903
  41. [41] A. R. Magid, Lectures on differential Galois theory, University Lecture Series 7, Amer. Math. Soc., 1994. Zbl0855.12001MR1301076
  42. [42] L. Boutet de Monvel, Polylogarithmes, http://www.math.jussieu.fr/~boutet. 
  43. [43] P. Orlik & H. Terao, Arrangements of hyperplanes, Grund. Math. Wiss. 300, Springer, 1992. Zbl0757.55001MR1217488
  44. [44] H. Poincaré, Sur les groupes d’équations linéaires, Acta Mathematica 4 (1884). JFM16.0252.01
  45. [45] G. Racinet, Doubles mélanges des polylogarithmes multiples aux racines de l’unité, Publ. Math. Inst. Hautes Études Sci.95 (2002), 185–231. Zbl1050.11066MR1953193
  46. [46] D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra58 (1979), 432–454. Zbl0409.16011MR540649
  47. [47] G. Rhin & C. Viola, On a permutation group related to ζ ( 2 ) , Acta Arith.77 (1996), 23–56. Zbl0864.11037MR1404975
  48. [48] J. D. Stasheff, Homotopy associativity of H -spaces. I, Trans. Amer. Math. Soc. 108 (1963), 275–292. Zbl0114.39402
  49. [49] T. Terasoma, Selberg integrals and multiple zeta values, Compositio Math.133 (2002), 1–24. Zbl1003.11042MR1918286
  50. [50] C. Voisin, Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics 77, Cambridge University Press, 2003. Zbl1032.14002MR1997577
  51. [51] M. Waldschmidt, Valeurs zêta multiples. Une introduction, J. Théor. Nombres Bordeaux 12 (2000), 581–595. Zbl0976.11037MR1823204
  52. [52] M. Yoshida, Fuchsian differential equations, Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, 1987. Zbl0618.35001MR986252
  53. [53] J. Zhao, Multiple polylogarithms: analytic continuation, monodromy, and variations of mixed Hodge structures, in Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000), Nankai Tracts Math. 5, World Sci. Publ., River Edge, NJ, 2002, 167–193. Zbl1058.32005MR1945360
  54. [54] S. A. Zlobin, Integrals that can be represented as linear forms of generalized polylogarithms, Mat. Zametki [Math. Notes] 71 (2002), 782–787 [711–716]. Zbl1049.11077MR1936201
  55. [55] W. Zudilin, Well-poised hypergeometric transformations of Euler-type multiple integrals, J. London Math. Soc.70 (2004), 215–230. Zbl1065.11054MR2064759

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.