An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by $X^{(n)}$ the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on $H^1(X^{(n)})$ to define an integrable connection on the trivial vector bundle on $X^{(n)}$ with fiber the universal algebra of the Lie algebra associated with the descending central series of ${\pi }_1$ of $X^{(n)}$. The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.