Two Lectures on D-Geometry and Noncommutative Geometry

TL;DR

The paper surveys two interconnected threads for extending geometry in string/M-theory: D-geometry, which asks what metric D-branes perceive in a given background and how substringy physics modifies effective geometry, and noncommutative geometry, which arises naturally in D-brane/M-theory settings via deformation quantization and gauge theories on noncommutative spaces. It develops D-geometry through D$0$-brane probes, conformal field theory boundary data, orbifold and resolved-orbifold constructions, and the D$0$-D$6$ system, highlighting how moduli-space metrics can be computed from sigma-model beta functions and FI-terms, with explicit examples like Eguchi-Hanson and $\mathbb{C}^3/\mathbb{Z}_3$. It then introduces noncommutative geometry as a framework for brane coordinates and gauge theories, detailing deformation quantization, the noncommutative torus, fuzzy spaces, and gauge theory on NC spaces, including quantization, quotients, and dualities; it connects these developments to Matrix theory and M-theory via NC torus dualities and instanton moduli, suggesting a unified description of D-geometry in terms of noncommutative structures. The overarching message is that nonlocal stringy effects can be captured by deformations of classical geometry, with concrete computational tools (beta functions, FI-terms, star-products, Morita equivalence) providing a route to universal equations of motion and dualities in a finite-string-scale regime.

Abstract

This is a write-up of lectures given at the 1998 Spring School at the Abdus Salam ICTP. We give a conceptual introduction to D-geometry, the study of geometry as seen by D-branes in string theory, and to noncommutative geometry as it has appeared in D-brane and Matrix theory physics.

Figures (1)

• Figure 1: A string excitation on a D1-brane (the thick line) on $T^2$. It is a bilocal object and the resulting non-locality is precisely described by the NC gauge theory.