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

Abstract

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.