Feynman integrals as flat bundles over the complement of Landau varieties

TL;DR

This paper shows that Feynman integrals can be viewed as flat vector bundles with a connection over the complement of Landau varieties, linking physics integrals to geometric and topological structures.

Contribution

It establishes that Feynman integrals form a flat bundle with a Gauss-Manin connection over the Landau variety complement, emphasizing geometric and Riemann-Hilbert perspectives.

Findings

Feynman integrals form a flat vector bundle over Landau variety complement.

The connection is described by a differential equation involving Landau polynomials.

Suggests a geometric approach to understanding Feynman integrals via Riemann-Hilbert data.

Abstract

We demonstrate that Feynman integrals of a fixed diagram form a flat vector bundle over the complement of Landau varieties that possesses a connection \begin{equation} \frac{\partial}{\partial p_{i,\mu}}f_\beta(p_{i,\mu})=\sum_{\beta'} \sum_k \sum_{I_1,...,I_k} \frac{A^{I_1,...,I_k}_{i,\mu,\beta,\beta'}(p)}{L_{I_1}(p)...L_{I_k}(p)} f_{\beta'}(p) \end{equation} where $L_I(p)$ are the Landau polynomials (multidiscriminants). This is the Gauss-Manin connection for the original integral. This result suggests a shift of focus from the integrals to the geometry of the complement of Landau varieties and Riemann-Hilbert data associated with these varieties.