Communications in Number Theory and Physics
Volume 15 (2021)
Wrońskian algebra and Broadhurst–Roberts quadratic relations
Pages: 651 – 741
Through algebraic manipulations onWrońskian 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 Wrońskian 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.
Bessel moments, Feynman integrals, Wrońskian matrices, Bernoulli numbers
2010 Mathematics Subject Classification
Primary 11B68, 33C10, 34M35. Secondary 81T18, 81T40.
To the memory of Dr. W (1986–2020)
This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).
Received 2 January 2021
Accepted 13 May 2021
Published 6 October 2021