Translation-Invariant Noncommutative Gauge Theories, Matrix Modeling and Noncommutative Geometry

TL;DR

This paper develops a matrix-based framework for translation-invariant noncommutative gauge theories, exploring their algebraic structures, quantization, and anomaly calculations within a noncommutative geometric setting.

Contribution

It introduces a novel matrix modeling approach using differential graded algebras and quantum groups for noncommutative gauge theories, linking star product cohomology to loop calculations.

Findings

Loop calculations depend solely on {eta}-cohomology class of star product.

Noncommutative geometric quantization is formulated.

Consistent anomalies and Schwinger terms are described in the noncommutative setting.

Abstract

A matrix modeling formulation for translation-invariant noncommutative gauge theories is given in the setting of differential graded algebras and quantum groups. Translation-invariant products are discussed in the setting of {\alpha}-cohomology and it is shown that loop calculations are entirely determined by {\alpha}-cohomology class of star product in all orders. Noncommutative version of geometric quantization and (anti-) BRST transformations is worked out which leads to a noncommutative description of consistent anomalies and Schwinger terms.