Local mirror symmetry and the sunset Feynman integral

By Spencer Bloch, M. Kerr, Pierre Vanhove. 2016

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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.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/503294c0dca3f9e2574b98f2048f7efc90bd8fb9

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Citations in the corpus

Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism (2017, background): “Our paper is not the first to consider elliptic generalisations of MPLs [14, 17, 19, 20, 24, 31, 45, 49, 50] or iterated integrals over integration kernels that involve square roots of polynomials of degree greater than two [51].”
Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral (2016, methodology): “…knowledge of the mathematical structures that appear in multiloop Feynman integrals, thanks to the combined use of various computational techniques, such as to the method of differential equations [1–3], the introduction of a class of special functions (dubbed originally harmonic polylogarithms,…”