Fano and Reflexive Polytopes from Feynman Integrals

TL;DR

This paper classifies Fano and reflexive polytopes arising from Feynman integrals, revealing their sparse occurrence and deep connections to Calabi--Yau geometries, thus linking quantum field theory to complex algebraic varieties.

Contribution

It provides a systematic classification of Fano and reflexive polytopes from Feynman integrals, highlighting their geometric structures and connections to Calabi--Yau varieties.

Findings

Sparse occurrence of reflexive and Fano polytopes in analyzed Feynman graphs.

Identification of reflexive polytopes encoding degenerate Calabi--Yau manifolds.

Explicit links between polytopes and algebraic surfaces like del Pezzo and K3 surfaces.

Abstract

We classify the Fano and reflexive polytopes that arise from quasi-finite Feynman integrals. These polytopes appear as scaled Minkowski sums of the Newton polytopes associated with the Symanzik graph polynomials. For one-loop graphs and multiloop sunset graphs, we identify the Fano and reflexive cases by computing the number of interior points from the associated bivariate Ehrhart polynomials. More generally, we utilize the properties of Symanzik polynomials and their symmetries to conduct a direct search over all Feynman graphs in generic kinematics with up to ten edges and nine loops. We find that such cases are remarkably sparse: for example, we find only two two-dimensional reflexive polytopes, three three-dimensional reflexive polytopes, and four three-dimensional Fano polytopes. We also reveal a surprising feature of one-loop $N$-gon integrals in higher dimensions: their associated reflexive polytopes encode degenerate Calabi--Yau $(N-2)$-folds. We further analyze the geometric structures encoded by these polytopes and exhibit explicit connections with del Pezzo surfaces, $K3$ surfaces, and Calabi--Yau threefolds. Since reflexive polytopes naturally correspond to Calabi--Yau varieties, our classification demonstrates that quasi-finite Feynman integrals, with reflexive polytopes, are intrinsically linked to Calabi--Yau period integrals.