The Dynamics of D-branes with Dirac-Born-Infeld and Chern-Simons/Wess-Zumino Actions

We have explained, and shown by feature stringy examples, why a D-brane in superstring theory, when treated as a fundamental dynamical object, can be described by a map φ from an Azumaya/matrix manifold X with a fundamental module with a connection (E,∇) to the target spacetime Y . In this sequel, we construct a non-Abelian Dirac-Born-Infeld action functional S (Φ,g,B) DBI (φ,∇) for such pairs (φ,∇) when the target spacetime Y is equipped with a background (dilaton, metric, B)-field (Φ, g, B) from closed strings. We next develop a technical tool needed to study variations of this action and apply it to derive the first variation δS (Φ,g,B) DBI /δ(φ,∇) of S (Φ,g,B) DBI with respect to (φ,∇). The equations of motion that govern the dynamics of D-branes then follow. We introduce a new action S (ρ,h;Φ,g,B,C) standard for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This ‘standard action’ is abstractly a non-Abelian gauged sigma model based on maps φ : (X, E;∇) → Y from an Azumaya/matrix manifold X with a fundamental module E with a connection ∇ to Y enhanced by the dilaton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples (φ,∇) to RamondRamond field. In a special situation, this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. A complete action for a D-brane worldvolume must include also the Chern-Simons/Wess-Zumino term S (C) CS/WZ (φ,∇) that governs how the D-brane world-volume couples with the Ramond-Ramond fields C on Y . The current notes lay down a foundation toward the dynamics of D-branes along the line of this research project.