Single-valued flat connections in several variables on higher genus Riemann surfaces

Authors: Eric D’Hoker, Oliver Schlotterer

Pre-print: UUITP–37/25

Abstract: Polylogarithms on Riemann surfaces may be constructed efficiently in terms of flat connections that can enjoy various algebraic and analytic properties. In this paper, we present a single-valued and modular invariant connection DHS on the configuration space Cfn(Σ) of an arbitrary number n of points on an arbitrary compact Riemann surface Σ with or without punctures. The connection DHS generalizes an earlier construction for a single variable and is built out of the same integration kernels. We show that DHS is flat on Cfn(Σ). For the case without punctures, we relate it to the meromorphic multiple-valued Enriquez connection E in n variables on the universal cover Σ̃ of Σ by the composition of a gauge transformation and an automorphism of the Lie algebra in which DHS and E take values. In a companion paper, we shall establish the equivalence between the flatness of these connections and the corresponding interchange and Fay identities, for arbitrary compact Riemann surfaces.