Harrison Cohomology and Abelian Deformation Quantization on Algebraic Varieties

TL;DR

This paper explores abelian deformation quantizations of algebraic varieties, emphasizing the role of Harrison cohomology in classifying such deformations, especially on singular varieties and specific examples like (anti-) de Sitter space.

Contribution

It demonstrates how Harrison cohomology can classify abelian deformations on algebraic varieties, including those with singularities, and provides explicit examples relevant to physics.

Findings

Harrison cohomology is trivial on smooth manifolds, making first-order abelian *-products trivial.

Deformations trivial at first order can be nontrivial as exact deformations.

The coordinate algebra of (anti-) de Sitter space is a nontrivial deformation of Minkowski space.

Abstract

Abelian deformations of ordinary algebras of functions are studied. The role of Harrison cohomology in classifying such deformations is illustrated in the context of simple examples chosen for their relevance to physics. It is well known that Harrison cohomology is trivial on smooth manifolds and that, consequently, abelian *-products on such manifolds are trivial to first order in the deformation parameter. The subject is nevertheless interesting; first because varieties with singularities appear in the physical context and secondly, because deformations that are trivial to first order are not always (indeed not usually) trivial as exact deformations. We investigate cones, to illustrate the situation on algebraic varieties, and we point out that the coordinate algebra on (anti-) de Sitter space is a nontrivial deformation of the coordinate algebra on Minkowski space -- although both spaces are smooth manifolds.