(Dis)assembling Special Lagrangians

TL;DR

This paper builds a microscopic D-brane approach to special Lagrangian submanifolds by linking their deformations to attractor-flow data in Calabi–Yau compactifications. By interpreting the flow of complex structure along attractor trajectories as Hamiltonian deformations, it provides a concrete procedure to disassemble any SLG into simple building blocks and reassemble them, thereby illuminating marginal stability walls and deformation moduli without requiring explicit SLG constructions. The framework yields a natural flow-tree classification of SLGs, connects to Joyce-type stability and Pi-stability concepts, and is illustrated with simple $T^6$ examples and the Quintic, offering a tool to study BPS spectra, moduli spaces, and potentially black-hole microstate counting in string theory. The approach foregrounds the interplay between geometry, stability, and D-brane dynamics, suggesting avenues for explicit computations and extensions to quantum corrections.

Abstract

We explain microscopically why split attractor flows, known to underlie certain stationary BPS solutions of four dimensional N=2 supergravity, are the relevant data to describe wrapped D-branes in Calabi-Yau compactifications of type II string theory. We work entirely in the context of the classical geometry of A-branes, i.e. special Lagrangian submanifolds, avoiding both the use of homological algebra and explicit constructions of special Lagrangians. Our results provide a way to disassemble and assemble arbitrary special Lagrangians to and from more simple building blocks, giving a concrete way to determine for example marginal stability walls and deformation moduli spaces.