Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space

TL;DR

This paper explores the embedding of Feynman integrals associated with Calabi-Yau geometries into weighted projective spaces, revealing new examples and properties relevant to quantum field theory calculations.

Contribution

It introduces a systematic approach to embedding Calabi-Yau hypersurfaces in weighted projective spaces and identifies new Feynman integral examples related to these geometries.

Findings

Identification of degree-$2k$ hypersurfaces in weighted projective spaces relevant to Feynman integrals

Description of properties of these hypersurfaces and their relation to quantum field theory

Additional examples at three and four loops provided as ancillary data

Abstract

It has recently been demonstrated that Feynman integrals relevant to a wide range of perturbative quantum field theories involve periods of Calabi-Yaus of arbitrarily large dimension. While the number of Calabi-Yau manifolds of dimension three or higher is considerable (if not infinite), those relevant to most known examples come from a very simple class: degree-$2k$ hypersurfaces in $k$-dimensional weighted projective space $\mathbb{WP}^{1,\ldots,1,k}$. In this work, we describe some of the basic properties of these spaces and identify additional examples of Feynman integrals that give rise to hypersurfaces of this type. Details of these examples at three and four loops are included as ancillary files to this work.