Free Lie algebroids and the space of paths

• Mikhail Kapranov ^[1]
  1. [1] Yale University

     Yale University

     Town of New Haven, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 13, Nº. 2, 2007, págs. 277-319
• Idioma: inglés
• DOI: 10.1007/s00029-007-0041-9
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• Resumen

  • 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.

    Ma