D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifolds

Abstract

We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, $k$-times differentiable maps, and smooth maps from an Azumaya/matrix manifold with a fundamental module to a (commutative) real manifold $Y$ is developed. Such maps are meant to describe D-branes or matrix branes in string theory when these branes are light and soft with only small enough or even zero brane-tension. When $Y$ is a symplectic manifold (resp. a Calabi-Yau manifold; a $7$-manifold with $G_2$-holonomy; a manifold with an almost complex structure $J$), the corresponding notion of Lagrangian maps (resp. special Lagrangian maps; associative maps, coassociative maps; $J$-holomorphic maps) are introduced. Indicative examples linking to symplectic geometry and string theory are given. This provides us with a language and part of the foundation required to study themes, new or old, in symplectic geometry and string theory, including (1) $J$-holomorphic D-curves (with or without boundary), (2) quantization and dynamics of D-branes in string theory, (3) a definition of Fukaya category guided by Lagrangian maps from Azumaya manifolds with a fundamental module with a connection, (4) a theory of fundamental matrix strings or D-strings, and (5) the nature of Ramond-Ramond fields in a space-time. The current note D(11.1) is the symplectic/differential-geometric counterpart of the more algebraic-geometry-oriented first two notes D(1) ([L-Y1]) (arXiv:0709.1515 [math.AG]) and D(2) ([L-L-S-Y], with Si Li and Ruifang Song) (arXiv:0809.2121 [math.AG]) in this project.

arXiv comment100+2 pages, 12 figures

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Cited by in corpus (9)

• Thirty years of Erice on the brane
• D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theory
• Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes --- with remarks on the non-Abelian Chern-Simons/Wess-Zumino term
• Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case
• 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
• 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
• D-branes and synthetic/$C^{\infty }$-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifolds