Wrońskian factorizations and Broadhurst-Mellit determinant formulae

Abstract

Drawing on Vanhove's contributions to mixed Hodge structures for Feynman integrals in two-di\-men\-sion\-al quantum field theory, we compute two families of determinants whose entries are Bessel moments. Via explicit factorizations of certain Wrońskian determinants, we verify two recent conjectures proposed by Broadhurst and Mellit, concerning determinants of arbitrary sizes. With some extensions to our methods, we also relate two more determinants of Broadhurst--Mellit to the logarithmic Mahler measures of certain polynomials.

i+32 pages. Proof of two conjectures proposed by Broadhurst-Mellit (arXiv:1604.03057), using Vanhove's theory (arXiv:1401.6438). Evaluation of vacuum determinants via Mahler measures, in a new Section 5

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