# Hyperelliptic curves and planar 2-loop Feynman graphs

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KA2W03 - Mathematical physics: algebraic cycles, strings and amplitudes

According to work of Bloch-Esnault-Kreimer and subsequent work of Brown, Feynman integrals can be expressed as relative periods of complements of hypersurfaces in projective space called Feynman graph hypersurfaces, which defined as vanishing loci of products of the first and second Symanzik polynomials. Despite having very straightforward combinatorial definitions, the geometry of Feynman graph hypersurfaces is rather poorly understood, even in basic examples. I will focus on the hypersurface defined by the vanishing of the second Symanzik polynomial. In this case, it has been known for quite some time that for a 2-vertex, 3-edge graph the corresponding graph hypersurface is an elliptic curve. Recently Klemm et al. have generalized this to graphs with 2-vertices and n-edges to show that the corresponding graph hypersurace is a Calabi-Yau (n-1)-fold. In this talk, I will generalize this in a different direction by focusing on Feynman graphs with first homology of rank 2 which have two trivalent vertices connected by an edge. These are the so-called (n,1,m)-graphs. Recently, Bloch has studied the case where n=m=3 and has shown that in this case the “motive” is an elliptic curve. We generalize this to all n and m, showing that if n+m is even then the corresponding graph hypersurface has hyperelliptic motive, and that the genus of this curve depends on the dimension D of the underlying physical theory. In the case where n=m=3, this recovers precisely Gram determinants from quantum field theory.  This is joint work with C. Doran, A. Novoseltsev, and P. Vanhove.

This talk is part of the Isaac Newton Institute Seminar Series series.

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