[Paper Review] Dynamics of D-branes II. The standard action --- an analogue of the Polyakov action for (fundamental, stacked) D-branes

Chien‐Hao Liu, Shing‐Tung Yau|arXiv (Cornell University)|Apr 11, 2017

Black Holes and Theoretical Physics33 references3 citations

TL;DR

This paper introduces a new standard action for D-branes that generalizes the Polyakov action for fundamental strings, formulating D-brane dynamics as a non-Abelian gauged sigma model on Azumaya/matrix manifolds with a fundamental module and connection. The key contribution is the derivation of equations of motion via first and second variations, establishing a foundational framework for quantizing D-branes analogous to the Polyakov approach in string theory.

ABSTRACT

We introduce a new action $S_{standard}^{(ρ,h; Φ,g,B,C)}$ for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This `standard action' is abstractly a non-Abelian gauged sigma model --- based on maps $φ: (X^{\!A\!z},E; abla) ightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ with a fundamental module $E$ with a connection $ abla$ to $Y$ --- enhanced by the dilaton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples $(φ, abla)$ to Ramond-Ramond field. In a special situation, this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. With the analysis developed in D(13.1) (arXiv:1606.08529 [hep-th]) for such maps and an improved understanding of the hierarchy of various admissible conditions on the pairs $(φ, abla)$ beyond D(13.2.1) (arXiv:1611.09439 [hep-th]) and how they resolve the built-in obstruction to pull-push of covariant tensors under a map from a noncommutative manifold to a commutative manifold, we develop further in this note some covariant differential calculus needed and apply them to work out the first variation --- and hence the corresponding equations of motion for D-branes --- of the standard action and the second variation of the kinetic term for maps and the dilaton term in this action. Compared with the non-Abelian Dirac-Born-Infeld action constructed in D(13.1) along the same line, the current note brings the Nambu-Goto-string-to-Polyakov-string analogue to D-branes. The current bosonic setting is the first step toward the dynamics of fermionic D-branes (cf. D(11.2): arXiv:1412.0771 [hep-th]) and their quantization as fundamental dynamical objects, in parallel to what happened to the theory of fundamental strings during years 1976--1981.

Motivation & Objective

• To develop a dynamical framework for D-branes analogous to the Polyakov action for fundamental strings.
• To resolve obstructions in pulling back covariant tensors from noncommutative to commutative manifolds using improved admissible conditions on maps and connections.
• To formulate a standard action incorporating dilaton, gauge, and Chern-Simons terms, unifying harmonic map and gauge theory structures.
• To derive the first and second variations of the action to obtain equations of motion for D-brane dynamics.
• To lay the groundwork for quantizing fermionic D-branes and extending the theory beyond the bosonic case.

Proposed method

• Formulates D-brane dynamics on an Azumaya/matrix manifold $(X^{ ext{Az}}, ilde{E}; abla)$ with a fundamental module and connection, mapping to a target space-time $Y$.
• Constructs the standard action $S_{ ext{standard}}^{( ho,h; ho,g,B,C)}$ as an enhanced non-Abelian gauged sigma model with dilaton, gauge-theory, and Ramond-Ramond Chern-Simons terms.
• Applies covariant differential calculus on noncommutative manifolds to handle the pullback of tensors under maps from noncommutative to commutative spaces.
• Computes the first variation of the action to derive the equations of motion for the map $ ho$ and connection $ abla$.
• Analyzes the second variation of the kinetic and dilaton terms to study stability and linearized dynamics.
• Uses results from prior works (arXiv:1606.08529, arXiv:1611.09439) on admissible conditions and harmonic maps in noncommutative geometry.

Experimental results

Research questions

• RQ1 How can a Polyakov-like action be constructed for D-branes, given their richer internal structure compared to fundamental strings?
• RQ2 What are the appropriate geometric and analytic conditions (admissible conditions) that allow consistent pullback of tensors from a noncommutative D-brane worldvolume to a commutative spacetime?
• RQ3 How do the first and second variations of the standard action yield the equations of motion and linearized dynamics for D-brane configurations?
• RQ4 In what way does the standard action unify harmonic map theory and non-Abelian gauge theory in the context of D-branes?
• RQ5 What is the role of the Ramond-Ramond Chern-Simons term in coupling the D-brane connection and map to background fields?

Key findings

• The standard action is formulated as a non-Abelian gauged sigma model on an Azumaya/matrix manifold with a fundamental module and connection, incorporating dilaton, gauge, and Ramond-Ramond terms.
• The first variation of the action yields equations of motion that generalize the harmonic map and Yang-Mills equations in the presence of background fields.
• The second variation of the kinetic and dilaton terms is computed, showing that the integral of the curvature term vanishes when the equations of motion are imposed.
• The theory resolves the built-in obstruction to pullback of covariant tensors from noncommutative to commutative manifolds through refined admissible conditions.
• The framework provides a direct analogue of the Polyakov action for fundamental strings, now extended to D-branes with richer internal geometry.
• The construction sets the stage for quantizing fermionic D-branes and developing a full quantum theory of D-branes, mirroring the historical development of string theory from 1976–1981.