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A K3 in ϕ4\phi^4

Jun 21, 2010
28 pages
Published in:
  • Duke Math.J. 161 (2012) 10, 1817-1862
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Abstract: (arXiv)
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 \F_q is a (quasi-) polynomial in qq. Stembridge verified this for all graphs with ≤12\leq12 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 counter-examples to Kontsevich's conjecture which are in ϕ4\phi^4 theory. Their counting functions are given modulo pq2pq^2 (q=pnq=p^n) by a modular form arising from a certain singular K3 surface.
Note:
  • Final version
  • Feynman graph
  • modular
  • surface
  • phi**n model: 4
  • perturbation theory: higher-order
  • mathematical methods
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