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
In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of Azumaya C 1 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 with a unitary, minimally flat connectionwith-singularity. This merges Donaldson’s picture of special Lagrangian submanifolds (1999) and the Polchinski-Grothendieck Ansatz for D-branes in a Calabi-Yau space (Sec. 2.1). The Higgsing/un-Higgsing and the large- vs. small-brane wrapping of A-branes in string theory can be achieved via deformations of such morphisms (Sec. 2.2 and Sec. 2.3). For the case of Calabi-Yau 3-folds, classical results of Alexander (1920), Hilden (1974) and Montesinos (1976), Thurston (1982), and Hilden-Lozano-Montesinos (1983) on 3-manifolds branchedcovering S 3 implies that any embedded special Lagrangian submanifold with a complex vector bundle with a unitary flat connection on a Calabi-Yau 3-fold is the image of a special Lagrangian morphism from an Azumaya 3-sphere with a fundamental module, with a unitary minimally flat connection. This suggests a genus-like expansion of the path-integral of D3-branes in type IIB string theory compactified on Calabi-Yau 3-folds that resembles the genus expansion of the path-integral of strings (Sec. 2.4.2). Similarly, for the path-integral of D2-branes and M2-branes respectively. In Sec. 3, we use the technical results of Joyce (2002-2003) on desingularizations of special Lagrangian submanifolds with conical singularities to explain how supersymmetric D3-branes thus defined can be driven and re-assemble under a reverse split attractor flow at a point on the wall of marginal stability in Type IIB superstring theory compactified on varying Calabi-Yau 3-folds, studied by Denef (2001). This last section is to be read alongside the works [De3] (arXiv:hep-th/0107152) of Denef and [Joy3: V] (arXiv:math.DG/0303272) of Joyce. To cover the basic type of deformations of morphisms from Azumaya spaces in this note and its sequel, we discuss in Sec. 1 Morse cobordisms of manifolds and their promotion to Morse cobordisms of Azumaya manifolds with a fundamental module, and of morphisms from Azumaya manifolds to complex manifolds. The notion of cone of special Lagrangian cycles in a Calabi-Yau manifold is brought out in Sec. 2.4.1 for further study. A summary of the needed results from Joyce is given in the appendix.