Contents Online

# 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

DOI: https://dx.doi.org/10.4310/CNTP.2020.v14.n2.a2

#### Authors

#### Abstract

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.

#### Keywords

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