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SUMMARY:Hyperelliptic curves and planar 2-loop Feynman graphs - Andrew Har
 der (Lehigh University)
DTSTART:20220720T133000Z
DTEND:20220720T143000Z
UID:TALK175022@talks.cam.ac.uk
DESCRIPTION:According to work of Bloch-Esnault-Kreimer and subsequent work
  of Brown\, Feynman integrals can be expressed as relative periods of&nbsp
 \;complements of hypersurfaces in projective space called Feynman graph hy
 persurfaces\, which defined as vanishing loci of products of the first and
  second Symanzik polynomials. Despite having very straightforward combinat
 orial definitions\, the geometry of Feynman graph hypersurfaces is rather 
 poorly understood\, even in basic examples. I will focus on the hypersurfa
 ce defined by the vanishing of the second Symanzik polynomial. In this cas
 e\, it has been known for quite some time that for a 2-vertex\, 3-edge gra
 ph the corresponding graph hypersurface is an elliptic curve. Recently Kle
 mm et al. have generalized this to graphs with 2-vertices and n-edges to s
 how that the corresponding graph hypersurace is a Calabi-Yau (n-1)-fold. I
 n 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 v
 ertices connected by an edge. These are the so-called (n\,1\,m)-graphs. Re
 cently\, 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 h
 as 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\, t
 his recovers precisely Gram determinants from quantum field theory.&nbsp\;
 \nThis is joint work with C. Doran\, A. Novoseltsev\, and P. Vanhove.
LOCATION:Seminar Room 1\, Newton Institute
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