Communications in Number Theory and Physics

Volume 15 (2021)

Number 4

Wrońskian algebra and Broadhurst–Roberts quadratic relations

Pages: 651 – 741



Yajun Zhou (Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, New Jersey, U.S.A.; and Academy of Advanced Interdisciplinary Studies (AAIS), Peking University, Beijing, China)


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