Introduction to M(atrix) theory and noncommutative geometry

TL;DR

This paper provides a comprehensive introduction to M(atrix) theory and noncommutative geometry, highlighting their mathematical foundations and applications in string theory and physics.

Contribution

It offers a self-contained review connecting noncommutative geometry with M(atrix) theory, including detailed discussions on noncommutative tori and dualities, for physicists and mathematicians.

Findings

Review of BFSS and IKKT matrix models

Explanation of noncommutative tori and Morita equivalence

Discussion of instantons and noncommutative orbifolds

Abstract

Noncommutative geometry is based on an idea that an associative algebra can be regarded as "an algebra of functions on a noncommutative space". The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang-Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang-Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics. In this paper we give a mostly self-contained review of some aspects of M(atrix) theory, of Connes' noncommutative geometry and of applications of noncommutative geometry to M(atrix) theory. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and $SO(d,d|{\mathbb Z})$-duality, an elementary discussion of instantons and noncommutative orbifolds. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and to mathematicians who would like to learn about some new problems arising in theoretical physics.