Towards a Noncommutative Geometric Approach to Matrix Compactification

TL;DR

This paper develops a noncommutative geometric framework for matrix compactifications in M(atrix) theory, providing a unified method to derive solutions and formulate gauge theories on quantum spaces.

Contribution

It introduces a systematic approach linking algebraic structures to matrix compactifications using noncommutative geometry, yielding new solutions and a novel gauge theory formulation.

Findings

Derived new solutions for orbifolds and orientifolds.

Unified algebraic method for matrix compactifications.

Formulated gauge theory on quantum spaces.

Abstract

In this paper we study generic M(atrix) theory compactifications that are specified by a set of quotient conditions. A procedure is proposed, which both associates an algebra to each compactification and leads deductively to general solutions for the matrix variables. The notion of noncommutative geometry on the dual space is central to this construction. As examples we apply this procedure to various orbifolds and orientifolds, including ALE spaces and quotients of tori. While the old solutions are derived in a uniform way, new solutions are obtained in several cases. Our study also leads to a new formulation of gauge theory on quantum spaces.