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# Communications in Number Theory and Physics

## Volume 8 (2014)

### Number 3

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

Pages: 369 – 402

DOI: http://dx.doi.org/10.4310/CNTP.2014.v8.n3.a1

#### Author

#### Abstract

The ideas behind the Tsuchiya-Kanie representations of braid groups on spaces of $N$-point correlation functions are emulated to represent the modular group $PSL(2, \mathbb{Z})$ on a space of degenerate $3$-point correlation functions. This extends the Chen series map giving the universal monodromy representation of $\mathbb{P}^1 \backslash \{ 0, 1,\infty \}$ 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 with elements of $PSL(2, \mathbb{Z})$. Injectivity is proven using a De Rham-type theorem due to Chen. The resulting action of $PSL(2, \mathbb{Z})$ 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.