Noncommutative Geometry and Twisted Little-String Theories

TL;DR

This thesis explores the interplay between noncommutative geometry and twisted Little-String theories, establishing connections between D-brane dynamics, instantons, and moduli spaces in supersymmetric gauge theories.

Contribution

It demonstrates that moduli spaces of twisted compactifications correspond to instanton moduli spaces on noncommutative tori, linking string theory, noncommutative geometry, and supersymmetric gauge theories.

Findings

Moduli spaces of vacua are equivalent to instanton moduli spaces on noncommutative tori.

Large classes of supersymmetric gauge theories have moduli spaces described by noncommutative instantons.

Established a connection between D-brane dynamics and noncommutative geometry.

Abstract

In this thesis we will discuss various aspects of noncommutative geometry and compactified Little-String theories. First we will give an introduction to the use of noncommutative geometry in string theory. Thereafter we will present a proof of the connection between D-brane dynamics and noncommutative geometry. Then we will explain the concept of instantons in noncommutative gauge theories. The last chapters shift the focus to Little-String- and $(2,0)$-theories. We study compactifications of these theories on tori with twists. First we study the case of two coinciding branes in detail. Afterwards we study the case of an arbitrary number of coinciding branes. The main result here is that the moduli spaces of vacua for the twisted compactifications are equal to moduli spaces of instantons on a noncommutative torus. A special case of this is that a large class of gauge theories with $\SUSY{2}$ supersymmetry in D=4 or $\SUSY{4}$ in D=3 has moduli spaces which are moduli spaces of instantons on noncommutative tori.