Communications in Number Theory and Physics

Volume 14 (2020)

Number 2

Absence of irreducible multiple zeta-values in melon modular graph functions

Pages: 315 – 324



Eric D’Hoker (Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California at Los Angeles)

Michael B. Green (Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom; and Centre for Research in String Theory, School of Physics, Queen Mary University of London, United Kingdom)


The expansion of a modular graph function on a torus of modulus $\tau$ near the cusp is given by a Laurent polynomial in $y = \pi \operatorname{Im} (\tau )$ with coefficients that are rational multiples of single-valued multiple zeta-values, apart from the leading term whose coefficient is rational and exponentially suppressed terms. We prove that the coefficients of the non-leading terms in the Laurent polynomial of the modular graph function $D_N (\tau)$ associated with a melon graph is free of irreducible multiple zeta-values and can be written as a polynomial in odd zeta-values with rational coefficients for arbitrary $N \geq 0$. The proof proceeds by expressing a generating function for $D_N (\tau)$ in terms of an integral over the Virasoro–Shapiro closed-string tree amplitude.


modular graph function, multiple zeta values

2010 Mathematics Subject Classification

Primary 11M32. Secondary 81T30.

The research of ED is supported in part by the National Science Foundation under research grant PHY-16-19926. MBG has been partially supported by STFC consolidated grant ST/L000385/1, by a Leverhulme Emeritus Fellowship, and by a Simons Visiting Professorship at the NBIA.

Received 3 May 2019

Accepted 30 October 2019

Published 30 March 2020