Renormalisation group for multiple zeta values

TL;DR

This paper develops a framework to compare different solutions to the renormalisation problem for multiple zeta values, especially at non-positive arguments, using algebraic group actions, thus clarifying their relationships.

Contribution

It introduces a novel algebraic framework that relates various solutions to the renormalisation problem via group actions, addressing an open question in the field.

Findings

Provides a transparent comparison of solutions at non-positive values

Establishes a free and transitive group action on the set of solutions

Clarifies the relationship between different renormalisation solutions

Abstract

Calculating multiple zeta values at arguments of any sign in a way that is compatible with both the quasi-shuffle product as well as meromorphic continuation, is commonly referred to as the renormalisation problem for multiple zeta values. We consider the set of all solutions to this problem and provide a framework for comparing its elements in terms of a free and transitive action of a particular subgroup of the group of characters of the quasi-shuffle Hopf algebra. In particular, this provides a transparent way of relating different solutions at non-positive values, which answers an open question in the recent literature.