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 $S_{DBI}(\phi ,\nabla )$

Abstract

In D(13.1) (arXiv:1606.08529 [hep-th]), we introduced an admissible condition on differentiable maps $\phi :(X^{Az},E;\nabla )\to Y$ from an Azumaya/matrix manifold $X^{Az}$ (with the fundamental module $E$) with a connection $\nabla$ on $E$ to a manifold $Y$ in order to resolve a pull-push issue in the construction of a non-Abelian-Dirac-Infeld action $S_{DBI}$ for $(\phi ,\nabla )$ and to render $\nabla$ massless from the aspect of open strings. The admissible condition ibidem consists of two parts: Condition (1) and Condition (2). In this brief note, we examine these two conditions in more detail and bring their geometric implications on $(\phi ,\nabla )$ and the full action $S_{DBI}(\phi ,\nabla )+S_{CS/WZ}(\phi ,\nabla )$ more transparent. In particular, we show that Condition (1) alone already implies masslessness of $\nabla$ from open-string aspect; and that the additional Condition (2) implies a decoupling of the nilpotent fuzzy cloud of $\phi (X^{Az})$ to the dynamics of $(\phi ,\nabla )$. We conclude with a refined definition of admissible $(\phi ,\nabla )$ and a remark on the anomaly factor in the integrand of the Chern-Simons/Wess-Zumino term $S_{CS/WZ}(\phi ,\nabla )$ for D-branes based on the current study.

arXiv comment15+2 pp, 2+1 figures

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Cited by in corpus (3)

• Dynamics of D-branes II. The standard action --- an analogue of the Polyakov action for (fundamental, stacked) D-branes
• Further studies of the notion of differentiable maps from Azumaya/matrix supermanifolds I. The smooth case: Ramond-Neveu-Schwarz and Green-Schwarz meeting Grothendieck
• $N=1$ fermionic D3-branes in RNS formulation I. $C^{\infty }$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\hat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspace