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Feynman integrals, toric geometry and mirror symmetry

Pierre Vanhove

TL;DR

The paper develops a toric-GKZ framework to study Feynman integrals by analyzing maximal cuts, deriving minimal Picard-Fuchs operators, and applying this to the sunset integral. It demonstrates two equivalent analytic representations: an elliptic dilogarithm based on the sunset elliptic curve and a trilogarithm expansion tied to a local mirror-symmetry prepotential on a del Pezzo surface. The results connect the geometry of the graph polynomials to Calabi-Yau and Gromov-Witten theory, showing how mirror symmetry provides a natural language for evaluating and transforming sunset integrals. The approach generalizes to higher-loop cases and suggests efficient, geometry-driven pathways for analytic and numerical evaluations of complex Feynman integrals with multiple mass scales.

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.

Feynman integrals, toric geometry and mirror symmetry

TL;DR

The paper develops a toric-GKZ framework to study Feynman integrals by analyzing maximal cuts, deriving minimal Picard-Fuchs operators, and applying this to the sunset integral. It demonstrates two equivalent analytic representations: an elliptic dilogarithm based on the sunset elliptic curve and a trilogarithm expansion tied to a local mirror-symmetry prepotential on a del Pezzo surface. The results connect the geometry of the graph polynomials to Calabi-Yau and Gromov-Witten theory, showing how mirror symmetry provides a natural language for evaluating and transforming sunset integrals. The approach generalizes to higher-loop cases and suggests efficient, geometry-driven pathways for analytic and numerical evaluations of complex Feynman integrals with multiple mass scales.

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.

Paper Structure

This paper contains 30 sections, 163 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Feynman integrals
  3. The parametric representation
  4. Maximal cut
  5. The differential equations
  6. The logarithmic integral
  7. Elliptic curve
  8. Toric geometry and Feynman graphs
  9. Toric polynomials and Feynman graphs
  10. Some important special cases
  11. The GKZ approach : a review
  12. Hypergeometric functions and GKZ system
  13. The Gauß hypergeometric series
  14. The massive one-loop graph
  15. Maximal cut
  16. ...and 15 more sections

Figures (1)

  • Figure 1: The Newton polyhedron associated with the sunset second Symanzik polynomial. The coordinates $(a,b)$ of the vertices we the powers of $x^a y^b$ and we give the value of the coefficient in $\phi_\circleddash(p^2,\underline\xi^2,x,y,1)$.