Parallel Transport on Higher Loop Spaces

TL;DR

This paper develops a higher-dimensional parallel transport framework on loop spaces using generalized iterated integrals, connecting de Rham complexes with higher homotopy groups and introducing membrane integrals with broad mathematical applications.

Contribution

It introduces a novel higher-dimensional iterated integral approach for parallel transport on higher loop spaces, extending de Rham structures to higher homotopy groups.

Findings

Established a de Rham complex on higher loop spaces.

Connected higher homotopy groups with de Rham structures.

Introduced membrane iterated integrals applicable in multiple mathematical fields.

Abstract

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.