Flat connections on configuration spaces and braid groups of surfaces

We construct an explicit bundle with flat connection on the configuration space of n points on a complex curve. This enables one to recover the ‘1-formality’ isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n   points on a surface and an explicitly presented Lie algebra, and to extend it to a morphism from the full braid group of the surface to the semidirect product of the associated group with the symmetric group SnSn.