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More on the admissible condition on differentiable maps $φ: (X^{\!A\!z},E;\nabla)\rightarrow Y$ in the construction of the non-Abelian Dirac-Born-Infeld action $S_{DBI}(φ,\nabla)$

In D(13.1) (arXiv:1606.08529 [hep-th]), we introduced an admissible condition on differentiable maps $φ: (X^{\!A\!z}, E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ (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 $(φ,\nabla)$ 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 $(φ,\nabla)$ and the full action $S_{DBI}(φ,\nabla)+S_{CS/WZ}(φ,\nabla)$ 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 $φ(X^{\!A\!z})$ to the dynamics of $(φ, \nabla)$. We conclude with a refined definition of admissible $(φ,\nabla)$ and a remark on the anomaly factor in the integrand of the Chern-Simons/Wess-Zumino term $S_{CS/WZ}(φ,\nabla)$ for D-branes based on the current study.