Graded Lagrangians, exotic topological D-branes and enhanced triangulated categories

I point out that (BPS saturated) A-type D-branes in superstring compactification on Calabi-Yau threefolds correspond to graded special Lagrangian submanifolds, a particular case of the graded Lagrangian submanifolds considered by M. Kontsevich and P. Seidel. Combining this with the categorical formulation of cubic string field theory in the presence of D-branes, I consider a collection of topological D-branes wrapped over the same Lagrangian cycle and derive its string field action from first string-theoretic principles. The result is a Z-graded version of super-Chern-Simons field theory living on the Lagrangian cycle, whose relevant string field is a degree one superconnection in a Z-graded superbundle, in the sense previously considered in mathematical work of J. M. Bismutt and J. Lott. This gives a refined (and modified) version of a proposal previously made by C. Vafa. I analyze the vacuum deformations of this theory and relate them to topological D-brane composite formation, upon using the general formalism developed in a previous paper. This allows me to identify a large class of topological D-brane composites (generalized, or ‘exotic’ topological D-branes) which do not admit a traditional description. Among these are objects which correspond to the ‘covariantly constant sequences of flat bundles’ considered by Bismut and Lott, as well as more general structures, which are related to the enhanced triangulated categories of Bondal and Kapranov. I also give a rough sketch of the relation between this construction and the large radius limit of a certain version of the ‘derived category of Fukaya’s category’. This paper forms part of a joint project with Prof. S. Popescu, a brief announcement of which can be found in the second part of the note hep-th/0102183. The paralel B-model realization, as well as the relation with the enhanced triangulated categories of Bondal and Kapranov, was recently discussed by D. E. Diaconescu in the paper hep-th/0104200, upon using the observations contained in that announcement.