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

Chien‐Hao Liu, Shing‐Tung Yau|arXiv (Cornell University)|Nov 29, 2016

Black Holes and Theoretical Physics12 references3 citations

TL;DR

This paper refines the admissible condition for differentiable maps $\varphi: (X^{\!A\!z},E;\nabla) \to Y$ in the construction of the non-Abelian Dirac-Born-Infeld (DBI) action, showing that Condition (1) alone ensures the masslessness of the connection $\nabla$ from the open-string perspective, while Condition (2) enforces decoupling of the nilpotent fuzzy cloud of $\varphi(X^{\!A\!z})$ from the dynamics. The authors propose a refined definition of admissibility and a modified anomaly factor in the Chern-Simons/Wess-Zumino (CS/WZ) term for D-branes.

ABSTRACT

In D(13.1) (arXiv:1606.08529 [hep-th]), we introduced an admissible condition on differentiable maps $φ: (X^{\!A\!z}, E; abla) ightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ (with the fundamental module $E$) with a connection $ abla$ 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 $(φ, abla)$ and to render $ abla$ 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 $(φ, abla)$ and the full action $S_{DBI}(φ, abla)+S_{CS/WZ}(φ, abla)$ more transparent. In particular, we show that Condition (1) alone already implies masslessness of $ abla$ 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 $(φ, abla)$. We conclude with a refined definition of admissible $(φ, abla)$ and a remark on the anomaly factor in the integrand of the Chern-Simons/Wess-Zumino term $S_{CS/WZ}(φ, abla)$ for D-branes based on the current study.

Motivation & Objective

• To clarify the geometric and physical implications of the two-part admissible condition on maps $\varphi: (X^{\!A\!z},E;\nabla) \to Y$ in non-Abelian DBI action construction.
• To demonstrate that Condition (1) alone ensures the masslessness of the connection $\nabla$ from the open-string field theory perspective.
• To show that Condition (2) leads to decoupling of the nilpotent fuzzy cloud of $\varphi(X^{\!A\!z})$ from the dynamics of $\varphi$ and $\nabla$.
• To refine the definition of admissible pairs $(\varphi, \nabla)$ based on geometric and physical consistency.
• To propose a modified anomaly factor in the integrand of the Chern-Simons/Wess-Zumino term $S_{\mathrm{CS/WZ}}(\varphi, \nabla)$ for D-branes based on the current analysis.

Proposed method

• Introduces a refined admissible condition on $\varphi$ and $\nabla$ by analyzing two components: Condition (1) and Condition (2), derived from prior work.
• Analyzes Condition (1) as a massless condition for $\nabla$ via open-string dynamics, showing it ensures the absence of mass terms in the effective action.
• Uses a canonical splitting of the pullback map $\varphi^\sharp$ and the concept of generic covariantly-invariant substructures to formalize Condition (1).
• Applies the $C^\infty$-scheme of uniform type and module of uniform type to characterize the geometric structure of $\varphi$ and $\nabla$.
• Introduces a refined diagram relating $\varphi$, its reduction $\varphi_{\mathrm{red}}$, and the lift $\breve{f}$ to $\breve{X}$, showing that under Condition (2), $S_{\mathrm{DBI}}$ and $S_{\mathrm{CS/WZ}}$ are preserved.
• Proposes a modified anomaly factor $\pi_{\breve{\varphi},*}(\sqrt{\hat{A}(T_*\breve{X})/\hat{A}(N_{\breve{f}})})$ for the CS/WZ term, which is identified with an $\mathrm{End}_\mathbb{C}(E)$-valued differential form on $X$.

Experimental results

Research questions

• RQ1 Does Condition (1) alone suffice to ensure the masslessness of the connection $\nabla$ in the open-string effective field theory?
• RQ2 What geometric role does Condition (2) play in decoupling the nilpotent fuzzy cloud of $\varphi(X^{\!A\!z})$ from the dynamics of $\varphi$ and $\nabla$?
• RQ3 How can the anomaly factor in the Chern-Simons/Wess-Zumino term be consistently extended to the noncommutative setting of $X^{\!A\!z}$?
• RQ4 Can the full action $S_{\mathrm{DBI}}(\varphi, \nabla) + S_{\mathrm{CS/WZ}}(\varphi, \nabla)$ be consistently reduced to the classical limit $\varphi_{\mathrm{red}}$ and $\breve{f}$ under Condition (2)?
• RQ5 What is the refined definition of admissible $(\varphi, \nabla)$ that ensures both physical consistency and geometric transparency?

Key findings

• Condition (1) alone is sufficient to ensure the masslessness of the connection $\nabla$ from the open-string field theory perspective, as it guarantees the absence of mass-generating terms in the effective action.
• Condition (2) ensures that the nilpotent fuzzy cloud of $\varphi(X^{\!A\!z})$ decouples from the dynamics of $\varphi$ and $\nabla$, allowing the action to be consistently reduced to the classical limit.
• Under Condition (2), the actions $S_{\mathrm{DBI}}(\varphi, \nabla)$ and $S_{\mathrm{CS/WZ}}(\varphi, \nabla)$ are equal to their counterparts on the reduced map $\varphi_{\mathrm{red}}$ and the lift $\breve{f}$, preserving physical consistency.
• The anomaly factor in the CS/WZ term is proposed to be extended as $\pi_{\breve{\varphi},*}(\sqrt{\hat{A}(T_*\breve{X})/\hat{A}(N_{\breve{f}})})$, which is canonically identified as an $\mathrm{End}_\mathbb{C}(E)$-valued differential form on $X$.
• The refined definition of admissible $(\varphi, \nabla)$ incorporates both geometric consistency and physical requirements, particularly masslessness and decoupling.
• The modified CS/WZ action is defined as $S_{\mathrm{CS/WZ}}^{(C,B)}(\varphi, \nabla) = T_{m-1} \int_X \mathrm{Re} \left( \mathrm{Tr} \left( \varphi^\diamond C \stackrel{\scriptstyle\mbox{\tiny$\odot$}}{\wedge} e^{2\pi\alpha' F_\nabla + \varphi^\diamond B} \stackrel{\scriptstyle\mbox{\tiny$\odot$}}{\wedge} \pi_{\breve{\varphi},*}\sqrt{\hat{A}(T_*\breve{X})/\hat{A}(N_{\breve{f}})} \right) \right)_{(m)}$, preserving equality with the reduced action under admissibility.