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.
Categories: math-ph hep-ph math.AG math.MP