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

TL;DR

This paper explores the mathematical structure of topological D-branes in Calabi-Yau compactifications, introducing graded Lagrangian submanifolds and a $ ext{Z}$-graded super-Chern-Simons theory, revealing new exotic D-brane objects and their categorical relations.

Contribution

It develops a $ ext{Z}$-graded superconnection framework for topological D-branes, connecting graded Lagrangians with enhanced triangulated categories and exotic brane formations.

Findings

Derived a $ ext{Z}$-graded super-Chern-Simons theory for D-branes.

Identified a class of exotic topological D-branes beyond traditional descriptions.

Linked the construction to the derived Fukaya category and Bismut-Lott structures.

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'.