The Algebraic Index Theorem and Fedosov Quantization of Lagrange-Finsler and Einstein Spaces

TL;DR

This paper develops an algebraic index theorem for Lagrange-Finsler and Einstein spaces by applying Fedosov quantization to their almost symplectic structures, linking geometric models with deformation quantization.

Contribution

It introduces a novel algebraic index theorem for these geometries and demonstrates how Einstein equations can be embedded into deformation quantization frameworks.

Findings

Constructed nonholonomic trace density maps for deformation quantization.

Established an algebraic index theorem for Lagrange-Finsler and Einstein spaces.

Embedded Einstein field equations into the deformation quantization formalism.

Abstract

Various types of Lagrange and Finsler geometries and the Einstein gravity theory, 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 Kahler/ Poisson structures on (co)tangent bundles. This allows us to apply the Fedosov 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.