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

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.