A symbol and coaction for higher-loop sunrise integrals

Abstract

We construct a symbol and coaction for the ll-loop sunrise family of integrals, both for equal-mass and generic-mass cases. These constitute the first concrete examples of symbols and coactions for integrals involving Calabi-Yau threefolds and higher. In order to achieve a symbol of finite length, we recast the differential equations satisfied by these integrals in a unipotent form. We augment the integrals in a natural way by including ratios of maximal cuts \tau_iτi. We discuss the relationship of this construction to constructions of symbols and coactions for multiple polylogarithms and elliptic multiple polylogarithms, and its connection to notions of transcendental weight.