Hierarchies in relative Picard-Lefschetz theory

Abstract

We prove a relative version of the Picard-Lefschetz theorem, describing the variation of relative homology groups $H_d(Y_t\setminus A_t,B_t\setminus A_t)$ in the fibers of a smooth fiber bundle $Y\to T$ of complex manifolds with $A\cup B\subset Y$ transverse. From this we derive the vanishing of certain iterated variations, a system of constraints dubbed "hierarchy". As applications, we rederive the known analytic structure of Aomoto polylogarithms and massive one loop Feynman integrals. Moreover, we introduce the "simple type" to prove hierarchy constraints in degenerate cases where the Picard-Lefschetz formula does not apply, e.g. the massless triangle or the ice cream cone Feynman diagram. We compare our findings with a "classical" hierarchy of iterated variations (from 1960's $S$-matrix theory) and show how our setup not only explains, but also refines the latter. In order to do so, we need to further resolve the geometry of Feynman motives: We boldly blow up what no one has blown up before.

arXiv comment85 pages, 27 figures

References in corpus (8)

• On Motives Associated to Graph Polynomials
• Feynman graph polynomials
• Cluster adjacency properties of scattering amplitudes
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Landau Discriminants
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals