The algebraic index theorem and deformation quantization of Lagrange-Finsler and Einstein spaces

Abstract

Various types of Lagrange and Finsler geometries, Einstein gravity, and modifications, can be modelled by nonholonomic distributions on tangent bundles/manifolds when the fundamental geometric objects are adapted to nonlinear connection structures. We can convert such geometries and physical theories into almost Kähler/Poisson structures on (co)tangent bundles. This allows us to apply deformation quantization formalism to almost symplectic connections induced by Lagrange–Finsler and/or Einstein fundamental geometric objects. There are constructed respective nonholonomic versions of the trace density maps for the zeroth Hochschild homology of deformation quantization of distinguished algebras (in this work, adapted to nonlinear connection structure). Our main result consists in an algebraic index theorem for Lagrange–Finsler and Einstein spaces. Finally, we show how the Einstein field equations for gravity theories and geometric mechanics models can be imbedded into the formalism of deformation quantization and index theorem.