Higher genus polylogarithms associated with the D'Hoker-Hidding-Schlotterer connection

We study higher genus polylogarithms associated with the D'Hoker-Hidding-Schlotterer (DHS) connection defined on families of Riemann surfaces of any genus with marked points. Under separating degenerations of Riemann surfaces, we show that the DHS connection can be constructed by gluing the Brown-Levin (BL) connections on elliptic curves and the Knizhnik-Zamolodchikov (KZ) connections on pointed Riemann spheres. Therefore, the associated higher genus polylogarithms can be described explicitly as power series in deformation parameters and their logarithms for the families of Riemann surfaces whose coefficients are expressed by multiple zeta values and elliptic multiple zeta values.