[Paper Review] (Dis)assembling Special Lagrangians

Frederik Denef|ArXiv.org|Jul 18, 2001

Black Holes and Theoretical Physics35 references17 citations

TL;DR

This paper establishes a microscopic connection between split attractor flows in 4D $σ=2$ supergravity and the geometric deformation of special Lagrangian submanifolds (SLGs) in Calabi-Yau manifolds. By analyzing how changes in complex structure induce controlled deformations of SLGs—leading to splitting into simpler components—it provides a classification scheme for SLGs without explicit construction, enabling determination of marginal stability walls and deformation moduli spaces via attractor flow trees.

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.

Motivation & Objective

• To bridge the gap between macroscopic attractor flow structures in supergravity and microscopic D-brane configurations on special Lagrangian submanifolds.
• To provide a geometric, deformation-based framework for understanding SLG splitting and decay without relying on homological algebra or explicit SLG constructions.
• To derive a classification scheme for arbitrary SLGs using attractor flow trees, encoding stability and moduli information.
• To offer a concrete method for computing marginal stability walls and deformation moduli spaces of SLGs through geometric flow structures.

Proposed method

• Uses infinitesimal deformations of compact special Lagrangian submanifolds under changes in complex structure to model physical decay processes.
• Applies a physical analogy to steady heat flow to interpret the evolution of SLGs along attractor flow paths.
• Constructs a geometric flow that preserves the special Lagrangian condition while deforming the submanifold, leading to topological splitting.
• Defines a decomposition procedure by following attractor flow trees backward to assemble complex SLGs from simpler building blocks.
• Employs orientation-preserving and orientation-reversing behavior of tangent bases across singularities to classify possible degenerations.
• Utilizes local models of SLGs as connected sums of $σ^3$-like manifolds to analyze transverse intersection products and topological invariants.

Experimental results

Research questions

• RQ1 How do split attractor flows in 4D $σ=2$ supergravity relate to the geometric deformation and splitting of special Lagrangian submanifolds in Calabi-Yau manifolds?
• RQ2 What determines the direction and topology of SLG decay under complex structure deformation, and how does this relate to attractor flow splits?
• RQ3 Can the moduli space and stability walls of arbitrary SLGs be determined without explicit construction, using only flow tree data?
• RQ4 How do orientation properties of tangent bases across singularities constrain possible SLG degenerations in odd- and even-dimensional cases?
• RQ5 To what extent can the $Π$-stability conjecture be understood or verified through this geometric flow-based decomposition framework?

Key findings

• The deformation of a special Lagrangian under complex structure flow preserves the SLG condition and can lead to topological splitting into simpler components.
• For odd-dimensional SLGs, asymptotic tangent bases from incoming and outgoing trajectories remain consistently oriented, while for even-dimensional ones, they reverse orientation.
• The direction of decay in SLG splitting is determined by the flow direction in the attractor flow tree, with distinct topological outcomes depending on the complex structure evolution.
• The attractor flow tree structure encodes the domain of stability and deformation moduli space of the original SLG, even without explicit construction.
• The method provides a systematic way to decompose any SLG into elementary building blocks via reverse flow, enabling classification without solving for the SLG explicitly.
• Intersection products of SLGs can be computed via local contributions in complex coordinate planes, with sign determined by orientation conventions and the factor $(-1)^{d(d-1)/2}$ for $d$-dimensional manifolds.