On an extension of the universal monodromy representation for $\mathbb{P}^1\backslash\{0,1,\infty\}$

TL;DR

This paper extends the universal monodromy representation for the thrice-punctured projective line to a 1-cocycle of the modular group, linking it to polylogarithms and the Riemann zeta function's properties.

Contribution

It introduces an injective 1-cocycle of PSL(2, Z) into power series with non-commuting variables, expanding the monodromy representation with an S_3 twist.

Findings

Provides a new framework for the monodromy representation involving the modular group.

Establishes a connection between the monodromy action and the analytic continuation of the zeta function.

Offers multiple proofs of the zeta function's functional equation using polylogarithm symmetries.

Abstract

The Chen series map giving the universal monodromy representation of $\mathbb{P}^1\backslash\{0,1,\infty\}$ is extended to an injective 1-cocycle of $PSL(2, \mathbb{Z})$ into power series with complex coefficients in two non-commuting variables, twisted by an action of $S_3.$ The definition of the 1-cocycle is effected by parallel transport of flat sections of the bundle, also with an $S_3$ twisting, along paths in $\mathbb{P}^1\backslash\{0,1,\infty\}$ which are explicitly associated to elements of $PSL(2, \mathbb{Z})$. The resulting action of the modular group on the polylogarithm generating function is shown to yield a family of proofs of the analytic continuation and functional equation of the Riemann zeta function.