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

We consider D-branes in string theory and address the issue of how todescribe them mathematically as a fundamental object (as opposed to a solitonicobject) of string theory in the realm in differential and symplectic geometry.The notion of continuous maps, $k$-times differentiable maps, and smooth mapsfrom an Azumaya/matrix manifold with a fundamental module to a (commutative)real manifold $Y$ is developed. Such maps are meant to describe D-branes ormatrix branes in string theory when these branes are light and soft with onlysmall 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 manifoldwith an almost complex structure $J$), the corresponding notion of Lagrangianmaps (resp. special Lagrangian maps; associative maps, coassociative maps;$J$-holomorphic maps) are introduced. Indicative examples linking to symplecticgeometry and string theory are given. This provides us with a language and partof the foundation required to study themes, new or old, in symplectic geometryand string theory, including (1) $J$-holomorphic D-curves (with or withoutboundary), (2) quantization and dynamics of D-branes in string theory, (3) adefinition of Fukaya category guided by Lagrangian maps from Azumaya manifoldswith a fundamental module with a connection, (4) a theory of fundamental matrixstrings or D-strings, and (5) the nature of Ramond-Ramond fields in aspace-time. The current note D(11.1) is the symplectic/differential-geometriccounterpart 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 andRuifang Song) (arXiv:0809.2121 [math.AG]) in this project.