Communications in Number Theory and Physics

Volume 14 (2020)

Number 2

Genus-zero and genus-one string amplitudes and special multiple zeta values

Pages: 413 – 452



Don Zagier (Max-Planck-Institut für Mathematik, Bonn, Germany)

Federico Zerbini (UFR de mathématique et d’informatique, Strasbourg, France)


In this paper we show that in perturbative string theory the genus-one contribution to formal $2$‑point amplitudes can be related to the genus-zero contribution to $4$‑point amplitudes. This is achieved by studying special linear combinations of multiple zeta values that appear as coefficients of the amplitudes. We also exploit our results to relate closed strings to open strings at genus one using Brown’s single-valued projection, proving a conjecture of J. Broedel, O. Schlotterer, and F. Zerbini [3].

Received 21 July 2019

Accepted 17 December 2019

Published 30 March 2020