On the deformation of path algebras

TL;DR

This paper investigates how certain cohomology elements induce deformations in the path algebra of manifolds, with some deformations being quantized, and explores examples like the torus and sphere for physical insights.

Contribution

It characterizes conditions under which second de Rham cohomology induces algebra deformations and examines their quantization, providing new insights into geometric and physical interpretations.

Findings

Deformations are induced by specific cohomology elements.

Some deformations are quantized when the image is non-zero.

Examples include the torus and 2-sphere with potential physical relevance.

Abstract

Those elements of the second de Rham cohomology group of a connected, oriented Riemannian manifold which map its second homotopy group to zero or to a discrete subgroup of the reals induce deformations of the path algebra of the manifold. If the image is not identically zero then the induced deformations are quantized. We examine the simplest examples, namely, the torus and the 2-sphere, and consider possible physical interpretations of the deformations of their path algebras.