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

TL;DR

The paper addresses the need for an off-shell, graded description of A-type D-branes on Calabi–Yau manifolds. It derives a ${f Z}$-graded version of super-Chern–Simons theory on a special Lagrangian cycle for a collection of graded D-branes, interpreting the open-string field as a degree-one superconnection on a ${f Z}$-graded superbundle. It shows that vacuum deformations generate a broad class of exotic A-type branes, including flat complexes and covariantly constant sequences, and links these to enhanced triangulated categories (Bondal–Kapranov) and to extended moduli spaces of open strings, with partial connections to the large-radius Fukaya category. The work provides a physically motivated framework for an extended A-model open-string sector, offering a bridge between topological string field theory, higher-categorical structures, and mirror symmetry, and it outlines avenues for incorporating disk instanton effects and connections to Fukaya’s category in future research.

Abstract

I point out that (BPS saturated) A-type D-branes in superstring compactifications on Calabi-Yau threefolds correspond to {\em 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 {\em topological} D-branes wrapped over the same Lagrangian cycle and {\em derive} its string field action from first principles. The result is a {\em $\Z$-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, by 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'.