Bar complex of the Orlik–Solomon algebra

Let A be an arrangement of complex hyperplanes and MA the complement of the union of hyperplanes in A. The Orlik–Solomon algebra of A determines a subcomplex of the de Rham complex of the loop space of MA, which is called the bar complex of the Orlik–Solomon algebra. The dual of this complex is isomorphic to the tensor algebra of the homology of MA equipped with a derivation arising from the product structure of the Orlik–Solomon algebra. Based on this construction we give an explicit description of Chenʹs iterated integrals of logarithmic forms depending only on the homotopy class of a loop.