Iterated integral and the loop product

In this article we discuss a relation between the string topology and differential forms based on the theory of Chen’s iterated integrals and the cyclic bar complex.

💡 Research Summary

The paper establishes a concrete bridge between string topology and the algebra of differential forms by exploiting Chen’s theory of iterated integrals together with the cyclic bar complex. The central object of study is the free loop space (LM) of a smooth manifold (M). In string topology, Chas and Sullivan introduced a loop product on the homology (H_*(LM)) that endows it with a graded‑commutative algebra structure and, together with a Batalin–Vilkovisky (BV) operator, a rich Gerstenhaber‑BV algebra. However, the original definition of the loop product is geometric and does not directly involve differential forms.

The author’s strategy is to replace the geometric chains on (LM) by a purely algebraic model built from the de Rham algebra (\Omega^(M)). The cyclic bar complex (B_(\Omega^*(M))) is a chain complex whose elements are tensors of differential forms arranged cyclically; it carries a natural degree‑shifting product (\odot) that reflects the concatenation of loops at the algebraic level.

A key technical construction is the iterated‑integral map
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