  We study the sunset Feynman integral defined as the scalar two-point
self-energy at two-loop order in a two dimensional space-time.
  We firstly compute the Feynman integral, for arbitrary internal masses, in
terms of the regulator of a class in the motivic cohomology of a 1-parameter
family of open 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.
