Resumen de Free Lie algebroids and the space of paths

Mikhail Kapranov

• 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 PX 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 PX to a formal groupoid Π X which can be seen as the formal neighborhood of X inside the space of paths.

  We establish a relation between Π X and the stable map spaces of Kontsevich.

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