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Résumé |
We study the Feynman integral for the sunset graph defined as the scalar two-point self-
energy at two-loop order. The Feynman integral is evaluated for all inequal internal masses in
two space-time dimensions. Two calculations are given for the Feynman integral; one based on
an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs
differential equation, and the other using arithmetic algebraic geometry, motivic cohomology,
and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a
family of regulator periods associated to a family of elliptic curves. Using an Hodge theoretic
(B-model) approach, we show that the integral is given by a sum of elliptic dilogarithms
evaluated at the divisors determined by the punctures. Secondly we associate to the sunset
elliptic curve a local non-compact Calabi-Yau 3-fold, obtained as a limit of elliptically fibered
compact Calabi-Yau 3-folds. By considering the limiting mixed Hodge structure of the Batyrev
dual A-model, we arrive at an expression for the sunset Feynman integral in terms of the local
Gromov-Witten prepotential of the del Pezzo surface of degree 6. This expression is obtained
by proving a strong form of local mirror symmetry which identifies this prepotential with the
second regulator period of the motivic cohomology class (Based on work done in collaboration
with Spencer Bloch and Matt Kerr.) |
