abstract: In this talk (joint work with M.Spera, to appear in "Quantum Topology") we deal with low-dimensional matrix representations of the pure braid group in three and four strands in a geometrical framework: such new and easy to handle invariants of braids are obtained via holonomy of nilpotent flat connections. The simple but crucial observation is that the pure braid group is the fundamental group of the configuration space of n distinct points on the complex plane: nilpotent connections are built by imposing that their curvature just involves Arnol'd's relations, so it vanishes. Then their parallel transport - the calculation of which requires Chen's iterated path integrals, hyperlogarithms and their monodromy- gives rise to many-parameter matrix representations of the pure braid group. In particular Heisenberg type representations are obtained as well. Furthermore, upon increasing the dimension of the representation matrices, one gets finer invariants..