Free Lie algebroids and the space of paths

Abstract.

We construct algebraic and algebro-geometric models for the spaces of unparametrized paths. This is done by considering a path as a holonomy functional on indeterminate connections. For a manifold X, we construct a Lie algebroid \({\mathcal{P}}_X\) which serves as the tangent space to X (punctual paths) inside the space of all unparametrized paths. It serves as a natural receptacle of all “covariant derivatives of the curvature” for all bundles with connections on X.

If X is an algebraic variety, we integrate \({\mathcal{P}}_X\) to a formal groupoid \(\widehat{\Pi}_X\) which can be seen as the formal neighborhood of X inside the space of paths. We establish a relation between \(\widehat{\Pi}_X\) and the stable map spaces of Kontsevich.