Advertisement

Communications in Mathematical Physics

, Volume 267, Issue 1, pp 181–225 | Cite as

On Motives Associated to Graph Polynomials

  • Spencer BlochEmail author
  • Hélène Esnault
  • Dirk Kreimer
Article

Abstract

The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.

Keywords

Exact Sequence Span Tree Projective Space Hopf Algebra Exceptional Divisor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artin, M.: Théorème de finitude pour un morphisme propre; dimension cohomologique des schémas algébriques affines. In SGA 4, tome 3, XIV, Lect. Notes Math., Vol. 305, Berlin-Heidelberg-New York: Springer, 1973, pp. 145-168.Google Scholar
  2. 2.
    Borel A. (1977). Cohomologie de SL n et valeurs de fonctions zêta aux points entiers. Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) 4(4):613–636zbMATHMathSciNetGoogle Scholar
  3. 3.
    Belkale P., Brosnan P. (2003). Matroids, Motives, and a Conjecture of Kontsevich. Duke Math. J. 116(1):147–188CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Broadhurst D., Kreimer D. (1995). Knots and numbers in Φ4 theory to 7 loops and beyond. Int. J. Mod. Phys. C 6:519CrossRefzbMATHADSMathSciNetGoogle Scholar
  5. 5.
    Broadhurst D., Kreimer D. (1997). Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B 393(3-4):403–412CrossRefzbMATHADSMathSciNetGoogle Scholar
  6. 6.
    Deligne P., Goncharov A. (2005). Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Éc. Norm. Sup. (4) 38(1): 1–56zbMATHMathSciNetGoogle Scholar
  7. 7.
    Deligne, P.: Cohomologie étale. SGA 4 1/2, Springer Lecture Notes 569 Berlin-Heidelberg-New York: Springer, 1977Google Scholar
  8. 8.
    Deninger C. (1997). Deligne periods of mixed motives, K-theory, and the entropy of certain \(\mathbb{Z}^n\)-actions. JAMS 10(2):259–281CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dodgson C.L. (1866). Condensation of determinants. Proc. Roy. Soc. London 15:150–155CrossRefGoogle Scholar
  10. 10.
    Esnault, H., Schechtman, V., Viehweg, E.: Cohomology of local systems on the complement of hyperplanes. Invent. Math. 109, 557–561 (1992); Erratum: Invent. Math. 112, 447 (1993)Google Scholar
  11. 11.
    Goncharov A., Manin Y. (2004). Multiple zeta motives and moduli spaces \(\overline{M}_{0,n}\). Compos. Math. 140(1):1–14CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Itzykson J.-C., Zuber J.-B. (1980). Quantum Field Theory. Mc-Graw-Hill, New YorkGoogle Scholar
  13. 13.
    Stembridge J. (1998). Counting Points on Varieties over Finite Fields Related to a Conjecture of Kontsevich. Ann. Combin. 2:365–385CrossRefMathSciNetGoogle Scholar
  14. 14.
    Soulé C. (1986). Régulateurs, Seminar Bourbaki, Vol. 1984/85. Asterisque No. 133–134, 237–253Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Spencer Bloch
    • 1
    Email author
  • Hélène Esnault
    • 2
  • Dirk Kreimer
    • 3
    • 4
  1. 1.Dept. of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Mathematik, FB6, MathematikUniversität Duisburg-EssenEssenGermany
  3. 3.IHESBures sur YvetteFrance.
  4. 4.Boston UniversityBostonUSA

Personalised recommendations