D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck

In this continuation of (L-Y3) (arXiv:0709.1515 (math.AG)), (L-L-S-Y) (arXiv:0809.2121 (math.AG)), (L-Y4) (arXiv:0901.0342 (math.AG)), (L-Y5) (arXiv:0907.0268 (math.AG)), and (L-Y6) (arXiv:0909.2291 (math.AG)), we give an overview of the posted part of the project and then take it as background to introduce Azumaya noncommutative C ∞ -manifold and the four aspects of morphisms therefrom to a projective complex manifold. This gives us then a description of supersymmetric D-branes of A-type in a Calabi-Yau manifold along the line of the Polchinski-Grothendieck Ansatz. The notion of Kahler differentials and their tensors for an Azumaya noncommutative space are introduced. Donaldson's picture of Lagrangian and special Lagrangian submanifolds as selected from the zero-locus of a moment-map on a related space of maps can be merged into the setting of morphisms from Azumaya man- ifolds with a fundamental module. As a pedagogical toy model for illustration, we study D-branes of A-type in a Calabi-Yau torus. Simple as it is, it reveals already several features of D-branes of A-type, including their assembling/disassembling. The short-vs.-long string wrapping behavior of matrix-strings in the string-theory literature can be produced in this context as well. In addition to the previous comparison with stringy works made, the 4th theme (subtitled: "Gomez-Sharpe vs. Polchinski-Grothendieck") of Sec. 2.4 is to be read with the work (G-Sh) (arXiv:hep-th/0008150), while the 2nd theme (subtitled: "Donagi- Katz-Sharpe vs. Polchinski-Grothendieck") of Sec. 4.2 is to be read with the work (D-K-S) (arXiv:hep-th/0309270). Sec. 4.3, though not yet ready to be subtitled "Denef vs. Polchinski- Grothendieck", is to be read with the work (De) (arXiv:hep-th/0107152). Some directly related string-theory remarks are added to the end of each section.