Iterated integrals over higher dimensional loops (1203.3768v1)

Abstract: We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the membrane case.

• The paper introduces a novel framework for n-dimensional iterated integrals using n-forms to generalize classical integration.
• It demonstrates homotopy invariance under reparametrization, ensuring robust integration over complex manifold structures.
• It conjectures a higher-dimensional de Rham theorem that links iterated integrals with higher homotopy groups, opening new research directions.

Higher Dimensional Iterated Integrals and Their Conjectured de Rham Theorem

The paper "Iterated integrals over higher dimensional loops" by Anton Deitmar and Ivan Horozov presents an extension of classical iterated integrals to higher dimensions and formulates a conjecture regarding a higher-dimensional version of Chen’s de Rham theorem. This work advances the understanding of homotopy theory and its interplay with differential forms through the introduction of a general framework for iterated integrals over n-dimensional membranes.

Key Contributions

1. Higher Dimensional Iterated Integrals: The authors define n-dimensional iterated integrals using n-forms and constructions involving multiple observers, each witnessing events in potentially different orders. This generalization offers a novel characterization of integrations over manifolds, moving beyond the traditional one-dimensional framework.
2. Reparametrization and Homotopy Invariance: The paper explores the properties of these integrals, showing invariance under homotopic deformations that respect certain foliations. This section establishes robustness in the definition and practical applications to varying manifold conditions.
3. Shuffle Relations and Composition Rules: The research further explores algebraic properties such as shuffle relations, which dictate how these integrals behave under permutations of forms and composition laws for sequential integrals over membranes with shared base points. These foundational rules provide a structured algebraic understanding crucial for further theoretical exploration.
4. Conjectured de Rham Theorem: A bold conjecture is proposed extending Chen's theorem to higher dimensions. This posits an isomorphism between the iterated integrals over n-loops and a specific hom space related to higher homotopy groups. The paper proves this conjecture for the case s = 1, providing a concrete step toward validating the broader conjecture.

Implications and Future Work

The implications of this research are significant both theoretically and practically. By extending the notion of iterated integrals to higher dimensions, this work could illuminate new pathways in topology and geometry, impacting the paper of complex manifolds and differential topology. This approach also has potential applications in mathematical physics, where higher-dimensional integrals frequently appear.

Furthermore, the conjectured de Rham theorem invites further exploration. Confirmation of the conjecture might transform how certain algebraic topological invariants are considered within homotopy theory. Future work could focus on experimental verifications for cases beyond s = 1 or exploring connections to non-commutative geometry, influenced by Manin's work on modular symbols.

In summary, Deitmar and Horozov have set the groundwork for a substantial generalization in the paper of iterated integrals, with significant theoretical potential to influence multiple domains in mathematics. Their conjecture and partial proof open new lines of inquiry that are likely to prompt further research into the topological structures underpinning these integral constructs.