Local mirror symmetry and the sunset Feynman integral

Abstract

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.