Noncommutative Geometry and D-Branes

TL;DR

This paper explores how noncommutative geometry provides a natural framework for describing D-brane systems in string theory, linking geometric operators to physical actions and revealing deep connections between geometry and string dynamics.

Contribution

It demonstrates that the Dirac operator in noncommutative geometry corresponds to the supercharge in D-brane systems, reproducing super Yang-Mills actions from geometric principles.

Findings

Connes' Yang-Mills action matches D-brane effective action.

The Dirac operator acts as the supercharge for strings.

Geometric features have natural physical interpretations in string theory.

Abstract

We apply noncommutative geometry to a system of N parallel D-branes, which is interpreted as a quantum space. The Dirac operator defining the quantum differential calculus is identified to be the supercharge for strings connecting D-branes. As a result of the calculus, Connes' Yang-Mills action functional on the quantum space reproduces the dimensionally reduced U(N) super Yang-Mills action as the low energy effective action for D-brane dynamics. Several features that may look ad hoc in a noncommutative geometric construction are shown to have very natural physical or geometric origin in the D-brane picture in superstring theory.