  We construct a parallel transport on higher loop spaces of a manifold in term
of a higher dimensional generalization of iterated path integrals. Under mild
assumptions, we define a de Rham complex on higher loop spaces and we recover a
known result of Hain of a de Rham structure on higher homotopy groups of a
manifold.
  The key ingredient is a new definition of iterated integrals on membranes,
which also have applications in number theory, algebraic geometry and
mathematical physics.
