Generalized complexes and string field theory

TL;DR

The paper shows that to achieve a unitary description of D-brane dynamics in associative open string field theory, one must include all D-brane composites formed by condensation. It formulates the theory in terms of differential graded (dg) categories, introduces vacuum-shift data via Maurer–Cartan deformations, and identifies generalized complexes as the natural objects describing brane composites. A minimal, quasiunitary completion, the quasi-unitary cover c({\cal A}), is constructed as a dg category of degree-one generalized complexes and proven to be closed under D-brane formation. This provides a broad, algebraic framework for D-brane dynamics with potential connections to K-theory, homological mirror symmetry, and broader open-closed string theory classifications.

Abstract

I discuss the axiomatic framework of (tree-level) associative open string field theory in the presence of D-branes by considering the natural extension of the case of a single boundary sector. This leads to a formulation which is intimately connected with the mathematical theory of differential graded categories. I point out that a generic string field theory as formulated within this framework is not closed under formation of D-brane composites and as such does not allow for a unitary description of D-brane dynamics. This implies that the collection of boundary sectors of a generic string field theory with D-branes must be extended by inclusion of all possible D-brane composites. I give a precise formulation of a weak unitarity constraint and show that a minimal extension which is unitary in this sense can always be obtained by promoting the original D-brane category to an enlarged category constructed by using certain generalized complexes of D-branes. I give a detailed construction of this extension and prove its closure under formation of D-brane composites. These results amount to a completely general description of D-brane composite formation within the framework of associative string field theory.