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

TL;DR

This paper develops a graded string field theory framework for topological D-branes on Calabi-Yau threefolds, analyzing the BV formalism, master equations, and condensation processes, revealing new insights into D-brane interactions and classifications.

Contribution

It introduces a $ ext{Z}$-graded BV formalism for topological D-branes, providing a detailed analysis of the extended string field action and new classifications of brane pairs.

Findings

The extended string field action satisfies the classical master equation.

Condensation of certain brane pairs results in BRST trivial states, akin to a closed string vacuum.

Six distinct types of brane pairs are identified, challenging previous assumptions.

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.