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In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion ofAzumaya $C^{\infty}$-manifolds with a fundamental module and morphismstherefrom to a complex manifold. In the current sequel, we use this notion togive 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 stringtheory -- as special Lagrangian morphisms from such objects with a unitary,minimally flat connection-with-singularities. This merges Donaldson's pictureof special Lagrangian submanifolds and the Polchinski-Grothendieck Ansatz forD-branes on a Calabi-Yau space. Basic phenomena of D-branes such asHiggsing/un-Higgsing and large- vs. small-brane wrapping can be realized viadeformations of such morphisms. Classical results of Alexander, Hilden, Lozano,Montesinos, and Thurston suggest then a genus-like expansion of thepath-integral of D3-branes. Similarly for D2-branes and M2-branes. In the lastsection, we use the technical results of Joyce on desingularizations of specialLagrangian submanifolds with conical singularities to explain how A-branes thusdefined can be driven and re-assemble under a split attractor flow, as studiedin an earlier work of Denef. This section is to be read witharXiv:hep-th/0107152 of Denef and arXiv:math.DG/0303272 of Joyce.

D-branes of A-type, their deformations, and Morse cobordism of A-branes  on Calabi-Yau 3-folds under a split attractor flow:  Donaldson/Alexander-Hilden-Lozano-Montesinos-Thurston/Hurwitz/Denef-Joyce  meeting Polchinski-Grothendieck