Special Fano geometry from Feynman integrals

TL;DR

This paper explores the connection between Feynman integrals in quantum field theory and special Fano algebraic varieties, extending geometric interpretations beyond Calabi-Yau spaces to include Fano types.

Contribution

It introduces the study of special Fano varieties in the context of Feynman integrals, broadening the geometric framework for understanding these integrals.

Findings

Fano varieties are relevant in the geometric interpretation of Feynman integrals.

The Hodge structure of these varieties is characterized by their dimension and charge Q.

This approach extends the algebraic-geometric understanding of Feynman integrals beyond Calabi-Yau cases.

Abstract

One of the fundamental open questions in QFT is what kind of functions appear as Feynman integrals. In recent years this question has often been considered in a geometric context by interpreting the polynomials that appear in these integrals as defining algebraic varieties. One focal point of the past decade has in particular been the class of Calabi-Yau varieties that arise in some types of Feynman integrals. A class of manifolds that includes CYs as a special case are varieties of special Fano types. These varieties were originally introduced because the class of CY spaces is not closed under mirror symmetry. Their Hodge structure is of a more general type and the middle cohomology in particular is determined by two integers, the dimension of the manifold and a charge $Q$. In the present paper this class of manifolds is considered in the context of Feynman integrals.