Free Lie algebroids and the space of paths
Abstract
Description
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 P 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 P to a formal groupoid G which can be seen as the formal neighborhood of X inside the space of paths. We establish a relation of G with the stable map spaces of Kontsevich.
42 pages, revised version, to appear in Selecta Math
42 pages, revised version, to appear in Selecta Math