D-膜和Azumaya/矩阵非交换differential geometry，I：D-膜作为弦理论中的基本对象和从具有基本模的Azumaya/矩阵流形到实流形的可微映射
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
摘要：
我们考虑弦理论中的D-膜，并解决如何在数学上将它们描述为微分和辛几何领域中弦理论的基本对象（而不是孤立对象）的问题。给出了从具有基本模的Azumaya/矩阵流形到（交换）实流形$Y$的连续映射、$k$ -次可微映射和光滑映射的概念.这样的映射是用来描述弦理论中的D膜或矩阵膜，当这些膜轻而软，只有足够小甚至为零的膜张力时。当$Y$是一个辛流形时（分别为一个卡-丘流形;一个具有$G_2$ -完整性的$7$ -流形;一个具有殆复结构的流形$J$），相应的拉格朗日映射的概念（分别是.特殊拉格朗日映射;结合映射，余结合映射; $J$ -全纯映射）。指示性的例子连接到辛几何和弦理论。这为我们提供了一种语言和部分基础，需要研究辛几何和弦理论中的新的或旧的主题，包括（1）$J$ -全纯D-曲线（2）弦理论中D-膜的量子化和动力学，（3）由Azumaya流形的Lagrange映射引导的具有带联络的基本模的Komaya范畴的定义，（4）基本矩阵弦或D弦理论;（5）时空中Ramond-Ramond场的性质。当前的音符D（11.1）是该项目中更面向代数几何的前两个音符D（1）（[L-Y1]）（arXiv：0709.1515 [math.AG]）和D（2）（[L-L-S-Y]，与Si Li和Ruifang Song）（arXiv：0809.2121 [math.AG]）的辛/微分几何对应物。
英文摘要：
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.
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