Graded Chern-Simons field theory and graded topological D-branes

TL;DR

The paper develops a comprehensive BV framework for graded topological D-branes in the A-model on Calabi-Yau threefolds, proving that the extended graded Chern-Simons action satisfies the classical master equation within a Z-graded graded supermanifold formalism. It provides both a covariant geometric BV treatment and a detailed component-level construction of master actions for graded D-brane pairs, revealing six inequivalent BV theories corresponding to relative brane grades. It analyzes brane condensation, showing acyclic composites can represent closed-string vacua under topological assumptions and linking composites to points in the extended moduli space. The work connects constructive and geometric approaches, outlines how extended moduli and off-shell open string dynamics relate to homological mirror symmetry, and lays groundwork for quantum BV analyses in graded open-string systems, including extensions to the B-model.

Abstract

We discuss graded D-brane systems of the topological A model on a Calabi-Yau threefold, by means of their string field theory. We give a detailed analysis of the extended string field action, showing that it satisfies the classical master equation, and construct the associated BV system. The analysis is entirely general and it applies to any collection of D-branes (of distinct grades) wrapping the same special Lagrangian cycle, being valid in arbitrary topology. Our discussion employs a $\Z$-graded version of the covariant BV formalism, whose formulation involves the concept of {\em graded supermanifolds}. We discuss this formalism in detail and explain why $\Z$-graded supermanifolds are necessary for a correct geometric understanding of BV systems. For the particular case of graded D-brane pairs, we also give a direct construction of the master action, finding complete agreement with the abstract formalism. We analyze formation of acyclic composites and show that, under certain topological assumptions,all states resulting from the condensation process of a pair of branes with grades differing by one unit are BRST trivial and thus the composite can be viewed as a closed string vacuum. We prove that there are {\em six} types of pairs which must be viewed as generally inequivalent. This contradicts the assumption that `brane-antibrane' systems exhaust the nontrivial dynamics of topological A-branes with the same geometric support.