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# Communications in Number Theory and Physics

## Volume 10 (2016)

### Number 1

### Feynman integrals and critical modular $L$-values

Pages: 133 – 156

DOI: http://dx.doi.org/10.4310/CNTP.2016.v10.n1.a5

#### Author

#### Abstract

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.

Published 20 June 2016