Free Lie algebroids and the space of paths

By M. Kapranov. 2007

48 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cb18d5651055267f5e29886a9d355b6b4d4c188d

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the universal enveloping algebra of a Lie algebroid (2010, methodology): “Following [5], we write A⊗̄RA for the submodule of A⊗R A given by the kernel of the map θ,”