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

Abstract

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 oftopological D-branes wrapped over the same lagrangian cycle and derive its string field action from first string-theoretic principles. The result is a -graded version of super-Chern-Simons field theory living on the lagrangian cycle, whose relevant string field is a degree one superconnection in a -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.