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Rothschild Lecture: Elliptic curves associated to two-loop graphs (Feynman diagrams)

Presented by: 
Spencer Bloch
Wednesday 29th January 2020 - 16:00 to 17:00
INI Seminar Room 1
Two loop Feynman diagrams give rise to interesting cubic hypersurfaces in n variables, where n is the number of edges. When n=3, the cubic is obviously an elliptic curve. (In fact, a family of elliptic curves parametrized by physical parameters like momentum and masses.) Remarkably, elliptic curves appear also for suitable graphs with n=5 and n=7, and conjecturally for an infinite sequence of graphs with n odd. I will describe the algebraic geometry involved in proving this. Physically, the amplitudes associated to one-loop graphs are known to be dilogarithms. Time permitting, I will speculate a bit about how the presence of elliptic curves might point toward relations between two-loop amplitudes and elliptic dilogarithms.   

This is joint work with C. Doran, P. Vanhove, and M. Kerr.   

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons