HOLONOMIES FOR CONNECTIONS WITH VALUES IN L∞-ALGEBRAS

Given a flat connection alpha on a manifold M with values in a filtered L-infinity-algebra g, we construct a morphism hol(alpha)(infinity) : C center dot (M) -> B (U) over cap (infinity)(g), which generalizes the holonomy map associated to a flat connection with values in a Lie algebra. The construction is based on Gugenheim's A(infinity)-version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of con fi guration spaces of points in Euclidean space R-d, and to generalizations of the holonomy representations of braid groups.