  In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of
Azumaya $C^{\infty}$-manifolds with a fundamental module and morphisms
therefrom to a complex manifold. In the current sequel, we use this notion to
give a prototypical definition of supersymmetric D-branes of A-type (i.e.
A-branes) -- in an appropriate region of the Wilson's theory-space of string
theory -- as special Lagrangian morphisms from such objects with a unitary,
minimally flat connection-with-singularities. This merges Donaldson's picture
of special Lagrangian submanifolds and the Polchinski-Grothendieck Ansatz for
D-branes on a Calabi-Yau space. Basic phenomena of D-branes such as
Higgsing/un-Higgsing and large- vs. small-brane wrapping can be realized via
deformations of such morphisms. Classical results of Alexander, Hilden, Lozano,
Montesinos, and Thurston suggest then a genus-like expansion of the
path-integral of D3-branes. Similarly for D2-branes and M2-branes. In the last
section, we use the technical results of Joyce on desingularizations of special
Lagrangian submanifolds with conical singularities to explain how A-branes thus
defined can be driven and re-assemble under a split attractor flow, as studied
in an earlier work of Denef. This section is to be read with
arXiv:hep-th/0107152 of Denef and arXiv:math.DG/0303272 of Joyce.

Symplectic Geometry;High Energy Physics - Theory;Algebraic Geometry