Parallel Transport on Higher Loop Spaces

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.