Algebraic cycles and motivic iterated integrals II

TL;DR

This paper develops a natural framework for constructing and interpreting multiple zeta values, polylogarithms, and multiple logarithms within the Hopf algebra of framed mixed Tate motives, extending previous results and unifying divergent cases.

Contribution

It introduces a new framework for elements in the Hopf algebra of framed mixed Tate motives, enabling a unified interpretation of multiple zeta values and polylogarithms, including divergent cases.

Findings

All multiple zeta values, including divergent ones, are elements of the Hopf algebra.

The pro-unipotent completion of the torsor of paths is a mixed Tate motive.

Multiple logarithms are elements of the Hopf algebra under certain conditions.

Abstract

This is a sequel to our previous paper (joint with Furusho). It will give a more natural framework for constructing elements in the Hopf algebra of framed mixed Tate motives according to Bloch and Kriz. This framework allows us to extend our previous results to interpret all multiple zeta values (including the divergent ones) and the multiple polylogarithms in one variable as elements of this Hopf algebra. It implies that the pro-unipotent completion of the torsor of paths on projective line minus three points, is a mixed Tate motive in the sense of Bloch-Kriz. Also It allows us to interpret the multiple logarithm as an element of this Hopf algebra as long as the products of consecutive arguments are not 1.