Noncommutative field theories from a deformation point of view

TL;DR

This paper reviews the geometric structure of noncommutative field theories by examining their deformation from classical manifolds using star products and explores how matter and gauge fields are incorporated through deformation quantization.

Contribution

It provides a comprehensive framework for understanding noncommutative field theories via deformation quantization of manifolds, vector bundles, and principal fiber bundles.

Findings

Deformation of space-time manifolds using star products.

Quantization of vector bundles representing matter fields.

Deformation of principal fiber bundles for gauge theories.

Abstract

In this review we discuss the global geometry of noncommutative field theories from a deformation point of view: The space-times under consideration are deformations of classical space-time manifolds using star products. Then matter fields are encoded in deformation quantizations of vector bundles over the classical space-time. For gauge theories we establish a notion of deformation quantization of a principal fiber bundle and show how the deformation of associated vector bundles can be obtained.