Feynman integrals as flat bundles over the complement of Landau varieties

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.