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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Hyperelliptic curves and planar 2-loop Feynman gra
phs - Andrew Harder (Lehigh University)
DTSTART;TZID=Europe/London:20220720T143000
DTEND;TZID=Europe/London:20220720T153000
UID:TALK175022AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/175022
DESCRIPTION:According to work of Bloch-Esnault-Kreimer and sub
sequent work of Brown\, Feynman integrals can be e
xpressed as relative periods of \;complements
of hypersurfaces in projective space called Feynma
n 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 Feynma
n graph hypersurfaces is rather poorly understood\
, even in basic examples. I will focus on the hype
rsurface defined by the vanishing of the second Sy
manzik polynomial. In this case\, it has been know
n 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 general
ized 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 gen
eralize this in a different direction by focusing
on Feynman graphs with first homology of rank 2 wh
ich have two trivalent vertices connected by an ed
ge. These are the so-called (n\,1\,m)-graphs. Rece
ntly\, Bloch has studied the case where n=m=3 and
has shown that in this case the "motive" is an ell
iptic curve. We generalize this to all n and m\, s
howing that if n+m is even then the corresponding
graph hypersurface has hyperelliptic motive\, and
that the genus of this curve depends on the dimens
ion D of the underlying physical theory. In the ca
se where n=m=3\, this recovers precisely Gram dete
rminants from quantum field theory. \;\nThis i
s joint work with C. Doran\, A. Novoseltsev\, and
P. Vanhove.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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