Noncommutative Solitons and D-branes

TL;DR

This thesis reviews noncommutative BPS solitons, focusing on instantons and monopoles, and explores their applications to D-brane dynamics, proposing new extensions of integrable systems in lower dimensions.

Contribution

It provides a comprehensive review of noncommutative solitons, including exact solutions via ADHM and Nahm constructions, and introduces noncommutative extensions of integrable systems.

Findings

Analysis of noncommutative instantons and monopoles

Development of noncommutative extensions of integrable systems

Systematic review of ADHM/Nahm constructions on noncommutative spaces

Abstract

This thesis is designed for a comprehensive review of noncommutative (BPS) solitons with applications to D-brane dynamics including our works. We focus on noncommutative instantons and monopoles and study various aspects of the exact solutions by using Atiyah-Drinfeld-Hitchin-Manin (ADHM) and Nahm constructions. Finally we propose noncommutative extensions of integrable systems and soliton theories in lower dimensions in collaboration with Kouichi Toda, which would pioneer a new study area of integrable systems. Appendix is devoted to a brief and systematic review of formal aspects of ADHM/Nahm construction and Nahm transformation on commutative spaces. This article is also a step to a comprehensive review of ADHM/Nahm construction on both commutative and noncommutative spaces. Comments are welcome.