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

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KAH - K-theory, algebraic cycles and motivic homotopy theory

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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