Wro\'nskian algebra and Broadhurst-Roberts quadratic relations

TL;DR

This paper provides a new algebraic proof of Broadhurst-Roberts quadratic relations using Wrońskians and explores their implications for Feynman diagrams, motivic L-functions, and algebraic geometry.

Contribution

It introduces a novel algebraic approach to prove quadratic relations and generalizes them, connecting Feynman integrals with motivic L-functions and intersection pairings.

Findings

New proof of Broadhurst-Roberts quadratic relations

Derivation of non-linear sum rules for Feynman diagrams

Infinite family of determinant identities compatible with Deligne's conjectures

Abstract

Through algebraic manipulations on Wro\'nskian matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts, along with some generalizations. In the Wro\'nskian framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove's differential operators, and compute the Betti intersection pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst--Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne's conjectures for critical values of motivic $L$-functions.