Derived Categories and Zero-Brane Stability
Paul S. Aspinwall, Albion Lawrence
TL;DR
The paper establishes that the sum of a class of open-string topological field theories on a Calabi–Yau variety X is equivalent to the bounded derived category of coherent sheaves \\mathbf{D}(X), by introducing a graded B-model with marginal deformations and computing the Q-cohomology as hyperext groups. It then shows that physical D-brane data, encoded in objects and morphisms of these theories, reproduces \\mathbf{D}(X) up to the natural physical equivalences, with tachyon condensation realized as cone constructions that implement brane recombination. The work analyzes 0-brane stability under monodromy in the Kahler moduli space, using the conjectured Fourier–Mukai actions associated to the discriminant, and demonstrates that 0-branes can become unstable after encircling discriminant components, with implications for birational invariance. Finally, the paper discusses the relationship between topological and physical branes, linear sigma-model realizations of complexes, and the birational invariance of \\mathbf{D}(X) across flop-like transitions, highlighting deep connections between string theory, derived categories, and birational geometry.
Abstract
We define a particular class of topological field theories associated to open strings and prove the resulting D-branes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0-brane on any Calabi-Yau threefold must become unstable along some path in the Kahler moduli space. As a byproduct of this analysis we see how the derived category can be invariant under a birational transformation between Calabi-Yaus.