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 $\phi$ 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 $S_{DBI}^{(\Phi ,g,B)}(\phi ,\nabla )$ for such pairs $(\phi ,\nabla )$. We next develop a technical tool needed to study variations of this action and apply it to derive the first variation $\delta S_{DBI}^{(\Phi ,g,B)}/\delta (\phi ,\nabla )$ of $S_{DBI}^{(\Phi ,g,B)}$ with respect to $(\phi ,\nabla )$. 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 $S_{CS/WZ}^{(C)}(\phi ,\nabla )$ that governs how the D-brane world-volume couples with the Ramond-Ramond fields $C$ on $Y$. In the current notes, a version $S_{CS/WZ}^{(C,B)}(\phi ,\nabla )$ of non-Abelian Chern-Simons/Wess-Zumino action functional for $(\phi ,\nabla )$ that follows the same guide with which we construct $S_{DBI}^{(\Phi ,g,B)}(\phi ,\nabla )$ is constructed for lower-dimensional D-branes (i.e. D(-1)-, D0-, D1-, D2-branes). Its first variation $\delta S_{CS/WZ}^{(C,B)}(\phi ,\nabla )/\delta (\phi ,\nabla )$ is derived and its contribution to the equations of motion for $(\phi ,\nabla )$ follows. The current notes lay down a foundation toward the dynamics of D-branes along the line of this D-project.

73+2 pages, 8 figures

References in corpus (5)

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Cited by in corpus (4)

• More on the admissible condition on differentiable maps $\phi :(X^{Az},E;\nabla )\to Y$ in the construction of the non-Abelian Dirac-Born-Infeld action $S_{DBI}(\phi ,\nabla )$
• On Multifield Born and Born-Infeld Theories and their non-Abelian Generalizations
• Further studies of the notion of differentiable maps from Azumaya/matrix supermanifolds I. The smooth case: Ramond-Neveu-Schwarz and Green-Schwarz meeting Grothendieck
• Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -- (Dynamical, complex algebraic) D-branes on a soft noncommutative space