Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals
Roman N. Lee, Andrei A. Pomeransky
TL;DR
The paper formulates a constructive criterion for when a system of differential equations on the Riemann sphere can be reduced to a global normalized Fuchsian form, of which the ε-form for multiloop integrals is a special case. It develops an algorithm to test and realize such a normalization by decomposing a transition matrix U into a product of X(x) and Y(1/x) factors, drawing a deep connection to holomorphic vector bundles via the Birkhoff-Grothendieck theorem. The authors show that reducibility hinges on the existence of this two-chart decomposition, and they illustrate both success and failure cases with explicit examples. They further discuss the role of variable changes x → p(y)/q(y) to aid normalization and outline directions for generalizing beyond ε-form and extending the transformation class. Overall, the work provides a rigorous framework to determine when ε-form (or its generalizations) is achievable and offers practical procedures for multiloop calculations."
Abstract
We consider the question of reducibility of the differential system to normalized Fuchsian form on the Riemann sphere. The differential equations for the multiloop integrals in $ε$-form constitute a particular example of the normalized Fuchsian form. We formulate the algorithmic criterion of reducibility. We also consider the question of the proper choice of variable in the differential system suitable for its reduction to $ε$-form.