Hodge Theory and String Topology

Abstract

Let $M$ be an oriented smooth manifold of dimension $n$. The free loop space $\Lambda M$ of $M$ is the space of piecewise smooth maps from the circle to $M$. Chas and Sullivan defined the string product$$ CS : H_i (\Lambda M;\\mathbb{Z}) \otimes H_j (\Lambda M; \mathbb{Z}) \to H_{i + j - n} (\Lambda M; \mathbb{Z}). $$ Goresky and Hingston defined a string coproduct $$ GH: H_k (\Lambda M, M; \mathbb{Q}) \to \bigoplus_{i + j = k - n + 1} H_i (\Lambda M, M; \mathbb{Q}) \otimes H_j (\Lambda M, M; \mathbb{Q}). $$ When $M$ is a simply-connected complex algebraic manifold, the cohomology of $\Lambda M$ has a natural mixed Hodge structure with weights greater or equal to degree. In this case, we show that the string operations, $CS$ and $GH$, are morphisms of mixed Hodge structure after a suitable Tate twist. We prove this by giving a de Rham description of the string operations in terms of Chen's iterated integrals.