Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes --- with remarks on the non-Abelian Chern-Simons/Wess-Zumino term

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In earlier works, D(1) (arXiv:0709.1515 [math.AG]), D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we have explained why a D-brane in string theory, when treated as a fundamental dynamical object, can be described by a map φ from an Azumaya/matrix manifold XAz (cf. D-brane world-volume) with a fundamental module with a connection (E,∇) (cf. Chan-Paton bundle) to the target space-time Y. In this sequel, we construct a non-Abelian Dirac-Born-Infeld action functional SDBI⁽Φ, ᵍ, ᵇ⁾(φ,∇) for such pairs (φ,∇). We next develop a technical tool needed to study variations of this action and apply it to derive the first variation δSDBI⁽Φ,ᵍ,ᵇ⁾/δ(φ,∇) of SDBI⁽Φ,ᵍ,ᵇ⁾ with respect to (φ,∇). The equations of motion that govern the dynamics of D-branes then follow. A complete action for a D-brane world-volume must include also the Chern-Simons/Wess-Zumino term SCS/WZ⁽ᶜ⁾(φ,∇) that governs how the D-brane world-volume couples with the Ramond-Ramond fields C on Y. In the current notes, a version S⁽ᶜ,ᵇ⁾CS/WZ(φ,∇) of non-Abelian Chern-Simons/Wess-Zumino action functional for (φ,∇) that follows the same guide with which we construct S⁽Φ,ᵍ,ᵇ⁾DBI(φ,∇) is constructed for lower-dimensional D-branes (i.e. D(-1)-, D0-, D1-, D2-branes). Its first variation δS⁽ᶜ,ᵇ⁾CS/WZ(φ,∇)/δ(φ,∇) is derived and its contribution to the equations of motion for (φ, ∇) follows. The current notes lay down a foundation toward the dynamics of D-branes along the line of this D-project.