Brackets in the Pontryagin Algebras of Manifolds

Abstract

Given a smooth oriented manifold $M$ with non-empty boundary, we study the Pontryagin algebra $A=H_*(\Omega)$ where $\Omega$ is the space of loops in $M$ based at a distinguished point of $\partial M$. Using the ideas of string topology of Chas-Sullivan, we define a linear map $\{\{-,-\}\} : A \otimes A \to A \otimes A$ which is a double bracket in the sense of Van den Bergh satisfying a version of the Jacobi identity. For $\dim(M)>2$, the double bracket $\{\{-,-\}\}$ induces Gerstenhaber brackets in the representation algebras associated with $A$. This extends our previous work on the case $\dim(M)=2$ where $A=H_0(\Omega)$ is the group algebra of $\pi_1(M)$ and $\{\{-,-\}\}$ induces the classical Poisson brackets on the moduli spaces of representations of $\pi_1(M)$.