Higher genus polylogarithms on families of Riemann surfaces

Abstract

We construct polylogarithms on families of pointed Riemann surfaces of any genus which describe monodromies of nilpotent meromorphic connections with simple poles on these families. Furthermore, we show that the polylogarithms are computable as power series in deformation parameters and their logarithms associated with these families whose coefficients are essentially expressed by multiple zeta values.