Duke Mathematical Journal

A K3 in ϕ4

Francis Brown and Oliver Schnetz

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Abstract

Inspired by Feynman integral computations in quantum field theory, Kontsevich conjectured in 1997 that the number of points of graph hypersurfaces over a finite field Fq is a (quasi-) polynomial in q. Stembridge verified this for all graphs with at most twelve edges, but in 2003 Belkale and Brosnan showed that the counting functions are of general type for large graphs. In this paper we give a sufficient combinatorial criterion for a graph to have polynomial point-counts and construct some explicit counterexamples to Kontsevich’s conjecture which are in ϕ4 theory. Their counting functions are given modulo pq2 (q=pn) by a modular form arising from a certain singular K3 surface.

Article information

Source
Duke Math. J., Volume 161, Number 10 (2012), 1817-1862.

Dates
First available in Project Euclid: 27 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1340801625

Digital Object Identifier
doi:10.1215/00127094-1644201

Mathematical Reviews number (MathSciNet)
MR2954618

Zentralblatt MATH identifier
1253.14024

Subjects
Primary: 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture)
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 11G25: Varieties over finite and local fields [See also 14G15, 14G20] 14M12: Determinantal varieties [See also 13C40] 81T18: Feynman diagrams

Citation

Brown, Francis; Schnetz, Oliver. A K3 in $\phi^{4}$. Duke Math. J. 161 (2012), no. 10, 1817--1862. doi:10.1215/00127094-1644201. https://projecteuclid.org/euclid.dmj/1340801625


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References

  • [1] P. Aluffi and M. Marcolli, “Feynman motives and deletion-contraction relations” in Topology of Algebraic Varieties and Singularities, Contemp. Math. 538, Amer. Math. Soc., Providence, 2011, 21–64.
  • [2] P. Belkale and P. Brosnan, Matroids, motives, and a conjecture of Kontsevich, Duke Math. J. 116 (2003), 147–188.
  • [3] S. Bloch, “Motives associated to sums of graphs” in The Geometry of Algebraic Cycles, Clay Math. Proc. 9, Amer. Math. Soc., Providence, 2010, 137–143.
  • [4] S. Bloch, H. Esnault, and D. Kreimer, On motives associated to graph polynomials, Comm. Math. Phys. 267 (2006), 181–225.
  • [5] D. J. Broadhurst and D. Kreimer, Knots and numbers in $\phi^{4}$ theory to $7$ loops and beyond, Internat. J. Modern Phys. C 6 (1995), 519–524.
  • [6] F. Brown, Mixed Tate motives over $\mathbb{Z}$, Annals of Math. (2) 175 (2012), 949–976.
  • [7] F. Brown, The massless higher-loop two-point function, Comm. Math. Phys. 287 (2009), 925–958.
  • [8] F. Brown, On the periods of some Feynman graphs, preprint, arXiv:0910.0114v2 [math.AG]
  • [9] F. Brown and K. Yeats, Spanning forest polynomials and the transcendental weight of Feynman graphs, Comm. Math. Phys. 301 (2011), 357–382.
  • [10] F. Chung and C. Yang, On polynomials of spanning trees, Ann. Comb. 4 (2000), 13–25.
  • [11] D. Doryn, Cohomology of graph hypersurfaces associated to certain Feynman graphs, Commun. Number Theory Phys. 4 (2010), 365–415.
  • [12] D. Doryn, On one example and one counterexample in counting rational points on graph hypersurfaces, Lett. Math. Phys. 97 (2011), 303–315.
  • [13] H. Esnault and E. Viehweg, On a rationality question in the Grothendieck ring of varieties, Acta Math. Vietnam. 35 (2010), 31–41.
  • [14] G. Kirchhoff, Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme gehührt wird, Ann. Phys. Chem. 72 (1847), 497–508.
  • [15] R. Livné, Motivic orthogonal two-dimensional representations of $\operatorname{Gal}(\overline{\mathbf{Q}}/{\mathbf{Q}})$, Israel J. Math. 92 (1995), 149–156.
  • [16] OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A140686 (accessed 26 April 2012).
  • [17] W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math. 310 (1979), 151–170.
  • [18] O. Schnetz, Quantum periods: A census of $\phi^{4}$-transcendentals, Commun. Number Theory Phys. 4 (2010), 1–47.
  • [19] O. Schnetz, Quantum field theory over $F_{q}$, Electron. J. Combin. 18 (2011), no. 102.
  • [20] M. Schütt, CM newforms with rational coefficients, Ramanujan J. 19 (2009), 187–205.
  • [21] J.-P. Serre, Cours d’arithmétique, 2nd ed., Le Mathématicien 2, Presses Universitaires de France, Paris, 1977.
  • [22] T. Shioda and H. Inose, “On singular $K3$ surfaces” in Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, 119–136.
  • [23] R. P. Stanley, Spanning trees and a conjecture of Kontsevich, Ann. Comb. 2 (1998), 351–363.
  • [24] J. R. Stembridge, Counting points on varieties over finite fields related to a conjecture of Kontsevich, Ann. Comb. 2 (1998), 365–385.
  • [25] S. Weinberg, High-energy behavior in quantum field-theory, Phys. Rev. (2) 118 (1960), 838–849.