Introduction to M(atrix) theory and noncommutative geometry, Part II

TL;DR

This paper reviews noncommutative ${\mathbb R}^{d}$ spaces, focusing on algebraic structures, soliton and instanton solutions, and the ADHM construction within noncommutative geometry, emphasizing differences between unital and nonunital algebras.

Contribution

It provides a detailed analysis of soliton and instanton solutions, including the noncommutative ADHM construction, in the context of noncommutative geometry over various function algebras.

Findings

Exact soliton solutions in Yang-Mills-Higgs systems

Approximate solitons in scalar theories at large \theta

Thorough description of noncommutative ADHM instantons

Abstract

This review paper is a continuation of hep-th/0012145 and it deals primarily with noncommutative ${\mathbb R}^{d}$ spaces. We start with a discussion of various algebras of smooth functions on noncommutative ${\mathbb R}^{d}$ that have different asymptotic behavior at infinity. We pay particular attention to the differences arising when working with nonunital algebras and the unitized ones obtained by adjoining the unit element. After introducing main objects of noncommutative geometry over those algebras such as inner products, modules, connections, etc., we continue with a study of soliton and instanton solutions in field theories defined on these spaces. The discussion of solitons includes the basic facts regarding the exact soliton solutions in the Yang-Mills-Higgs systems as well as an elementary discussion of approximate solitons in scalar theories in the $\theta \to \infty$ limit. We concentrate on the module structure and topological numbers characterizing the solitons. The section on instantons contains a thorough description of noncommutative ADHM construction, a discussion of gauge triviality conditions at infinity and the structure of a module underlying the ADHM instanton solution. Although some familiarity with general ideas of noncommutative geometry reviewed in the first part is expected from the reader, this part is largely independent from the first one.