Nonabelian Phenomena on D-branes

TL;DR

This work surveys the nonabelian dynamics of multiple D-branes, highlighting how matrix-valued transverse scalars ${\Phi^i}$ on a $U(N)$ worldvolume induce noncommutative geometry and enable phenomena such as the dielectric effect and giant gravitons. It presents the nonabelian generalization of the Born-Infeld and Wess-Zumino actions, detailing features like the nonabelian pullback, interior product, and the $Q^i{}_j$ structure, while discussing the limits of the symmetric trace and necessary commutator corrections. The dielectric effect and giant gravitons illustrate how background fields stabilize higher-dimensional noncommutative configurations and how dual abelian pictures emerge in the large-$N$ limit, with connections to M-theory, AdS/CFT, and Matrix theory. Together, these insights reveal a cohesive framework in which nonabelian D-brane dynamics bridges gauge theories, noncommutative geometry, and spacetime geometry through brane polarization, bound states, and dual descriptions.

Abstract

A remarkable feature of D-branes is the appearance of a nonabelian gauge theory in the description of several (nearly) coincident branes. This nonabelian structure plays an important role in realizing various geometric effects with D-branes. In particular, the branes' transverse displacements are described by matrix-valued scalar fields and so noncommutative geometry naturally appears in this framework. I review the action governing this nonabelian theory, as well as various related physical phenomena such as the dielectric effect, giant gravitons and fuzzy funnels.