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Free Lie algebroids and the space of paths

Mikhail Kapranov
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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 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.

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Mikhail Kapranov. 2007-07-17. Free Lie algebroids and the space of paths. https://arxiv.org/abs/math/0702584

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