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

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 $\mathbb{Z}$-graded} version of super-Chern-Simons field theory living on the Lagrangian cycle, whose relevant string field is a degree one superconnection in a $\mathbb{Z}$-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'.

31 pages, 4 figures, minor typos corrected; v3: changed to JHEP style

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• Holomorphic potentials for graded D-branes
• (Dis)assembling Special Lagrangians
• Linear Sigma Models for Open Strings
• Graded Chern-Simons field theory and graded topological D-branes
• On the non-commutative geometry of topological D-branes
• Gauge-fixing, semiclassical approximation and potentials for graded Chern-Simons theories
• An analytic torsion for graded D-branes
• Graded D-branes and skew-categories
• Derived categories and stacks in physics