On Motives Associated to Graph Polynomials
Spencer Bloch, Hélène Esnault, Dirk Kreimer
TL;DR
This work surveys the emergence of motivic structures in graph polynomials and Feynman graph periods, focusing on primitive divergent graphs in four-dimensional scalar theories. It develops a framework combining graph-polynomial geometry, Schwinger parametrization, and blowups to define and analyze the graph motive, showing that wheel and spoke graphs yield Tate middle cohomology and de Rham generators. The authors connect graph motives to periods that relate to zeta-values, discuss correspondences with Feynman quadrics, and examine the extent to which graph hypersurfaces are mixed Tate. They also explore potential Hopf-algebraic structures linking graph motives to the algebra of multiple zeta values, illustrating both motivic richness and certain Tate-bottom cases. The results provide concrete motivic and cohomological descriptions for wheels and suggest directions for understanding more complex graphs and their periods.
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.