Feynman integrals and critical modular $L$-values

Broadhurst [12] conjectured that the Feynman integral associated to the polynomial corresponding to $t = 1$ in the one-parameter family $(1 + x_1 + x_2 + x_3)(1 + x^{-1}_1 + x^{-1}_2 + x^{-1}_3) - t$ is expressible in terms of $L(f, 2)$, where $f$ is a cusp form of weight $3$ and level $15$. Bloch, Kerr and Vanhove [8] have recently proved that the conjecture holds up to a rational factor. In this paper, we prove that Broadhurst’s conjecture is true. Similar identities involving Feynman integrals associated to other polynomials in the same family are also established.

2010 Mathematics Subject Classification

Primary 11F67. Secondary 81Q30.

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Published 20 June 2016