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Feynman Integrals, Toric Geometry and Mirror Symmetry

  • Pierre VanhoveEmail author
Chapter
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals. We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series. We explain how this allows to determine the minimal differential operator acting on the Feynman integrals. We illustrate the method on sunset integrals in two dimensions at various loop orders. The graph polynomials of the multi-loop sunset Feynman graphs lead to reflexive polytopes containing the origin and the associated variety are ambient spaces for Calabi-Yau hypersurfaces. Therefore the sunset family is a natural home for mirror symmetry techniques. We review the evaluation of the two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm. The equivalence between these two expressions is a consequence of (1) the local mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the sunset graph polynomial and (2) that the sunset Feynman integral is expressed in terms of the local Gromov-Witten prepotential of this del Pezzo surface.

Notes

Acknowledgements

It is a pleasure to thank Charles Doran and Albrecht Klemm for discussions. The research of P. Vanhove has received funding the ANR grant “Amplitudes” ANR-17- CE31-0001-01, and is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. N\(^\circ \) 14.641.31.0001.

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Authors and Affiliations

  1. 1.CEA, DSM, Institut de Physique ThéoriqueIPhT, CNRS, MPPU, URA2306SaclayFrance
  2. 2.National Research University Higher School of EconomicsMoscowRussian Federation

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