This article reviews the generalization of field theory to space-time with noncommuting coordinates, starting basics and covering most active directions research. Such theories are now known emerge from limits M string describe quantum Hall states. In last few years they have been studied intensively, many qualitatively new phenomena discovered, on both classical level.

Effective field theories in type I and II superstring for D-branes located at points the orbifold C^2/Z_n are supersymmetric gauge whose content is conveniently summarized by a `quiver diagram,' Lagrangian includes non-metric couplings to moduli: particular, twisted sector moduli couple as Fayet-Iliopoulos terms theory. These describe on resolved ALE spaces. Their spaces of vacua smooth metrics Yang-Mills instantons, explicitly computable. For U(N) construction exactly reproduces results...

We show that in certain superstring compactifications, gauge theories on noncommutative tori will naturally appear as D-brane world-volume theories. This gives strong evidence they are well-defined quantum It also a physical derivation of the identification proposed by Connes, Douglas and Schwarz Matrix theory compactification torus with M constant background three-form tensor field.

We discuss how D = 5 maximally supersymmetric Yang-Mills theory (MSYM) might be used to study or even define the (2, 0) in six dimensions. It is known that compactification of on a circle leads MSYM. A variety arguments suggest relation can reversed, and all degrees freedom are already present If so, this should have consequences for SYM perturbation theory. explore whether it imply orders finiteness, else an unusual between cutoff gauge coupling. S-duality reduction 4 may provide...

We show that boundary conditions in topological open string theory on Calabi–Yau (CY) manifolds are objects the derived category of coherent sheaves, as foreseen homological mirror symmetry proposal Kontsevich. Together with conformal field considerations, this leads to a precise criterion determining supersymmetry preserving branes at any point CY moduli space, completing II-stability.

We explore the extent to which a local string theory dynamics in anti-de Sitter space can be determined from its proposed Conformal Field Theory (CFT) description. Free fields bulk are constructed CFT operators, but difficulties encountered when one attempts incorporate interactions. also discuss general features of black hole as seen perspective. In particular, we argue that singularity AdS_3 holes is resolved

We study topological properties of the D-brane resolution three-dimensional orbifold singularities, C3/Λ, for finite abelian groups Λ. The vacuum moduli space is shown to fill out backgroun space-time with Fayet-Iliopoulos parameters controlling size blow-ups. This can be classically described by a gauged linear sigma model, which non-generic in manner that projects non-geometric regions its phase diagram, as anticipated from number perspectives.