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

Abstract

We introduce a new action $S_{standard}^{(\rho ,h;\Phi ,g,B,C)}$ 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 $\phi :(X^{Az},E;\nabla )\to Y$ from an Azumaya/matrix manifold $X^{Az}$ with a fundamental module $E$ with a connection $\nabla$ to $Y$ --- enhanced by the dilaton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples $(\phi ,\nabla )$ 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 $(\phi ,\nabla )$ 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.

75+2 pages, 6 figures

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Cited by in corpus (3)

• 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
• 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