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

Abstract

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 $\varphi$ from an Azumaya/matrix manifold $X^{Az}$ (cf. D-brane world-volume) with a fundamental module with a connection $(E,\nabla)$ (cf. Chan-Paton bundle) to the target space-time $Y$. In this sequel, we construct a non-Abelian Dirac-Born-Infeld action functional $S_{DBI}^{(\Phi, g, B)}(\varphi,\nabla)$ for such pairs $(\varphi,\nabla)$. We next develop a technical tool needed to study variations of this action and apply it to derive the first variation $\delta S_{DBI}^{(\Phi,g,B)}/\delta(\varphi,\nabla)$ of $S_{DBI}^{(\Phi,g,B)}$ with respect to $(\varphi,\nabla)$. 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 $S_{CS/WZ}^{(C)}(\varphi,\nabla)$ that governs how the D-brane world-volume couples with the Ramond-Ramond fields $C$ on $Y$. In the current notes, a version $S^{(C,B)}_{CS/WZ}(\varphi,\nabla)$ of non-Abelian Chern-Simons/Wess-Zumino action functional for $(\varphi,\nabla)$ that follows the same guide with which we construct $S^{(\Phi,g,B)}_{DBI}(\varphi,\nabla)$ is constructed for lower-dimensional D-branes (i.e. D(-1)-, D0-, D1-, D2-branes). Its first variation $\delta S^{(C,B)}_{CS/WZ}(\varphi,\nabla)/\delta(\varphi,\nabla)$ is derived and its contribution to the equations of motion for $(\varphi, \nabla)$ follows. The current notes lay down a foundation toward the dynamics of D-branes along the line of this D-project.