A symbol and coaction for higher-loop sunrise integrals

We construct a symbol and coaction for the l-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;i. 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.

Automated summary

We have constructed a symbol and coaction for the l-loop sunrise family of integrals, both for equal-mass and generic-mass cases. These examples are the first concrete instances of symbols and coactions for integrals involving Calabi-Yau threefolds and higher. We have recast the differential equations satisfied by these integrals into a unipotent form in order to obtain a symbol of finite length. We have also augmented the integrals by including ratios of maximal cuts and tau;i in a natural way. We discuss the connection of this construction to symbols and coactions for multiple polylogarithms, elliptic multiple polylogarithms, and notions of transcendental weight.