Modelling $A$-branes with foliations

TL;DR

This paper develops a foliation-based framework to model A-branes in mirror Calabi-Yau threefolds, elucidating their moduli spaces, spectral networks, and connections to quiver representation theory, advancing understanding of mirror symmetry and BPS states.

Contribution

It introduces a novel foliation approach to describe A-branes and their moduli spaces, linking geometric degenerations with spectral networks and quiver invariants.

Findings

Explicit description of A-brane moduli spaces via cell decomposition

Decomposition of spectral networks into basic objects related to DT invariants

Establishment of a local map connecting brane moduli and quiver invariants

Abstract

A certain class of $A$-branes in mirrors of toric Calabi-Yau threefolds can be described through the framework of foliations. This allows to develop an explicit description of their moduli spaces based on a cell decomposition, with strata of various dimensions glued together in a way that is dictated by partial degenerations of the underlying special Lagrangian. Examples of $A$-branes associated with `wild' BPS states are considered in detail. The torus fixed points in their moduli spaces provide a decomposition of $m$-herds spectral networks into a number $|\Omega|$ of basic connected objects, where $\Omega$ is the the corresponding rank-zero Donaldson-Thomas (DT) invariant. A relation between the surgery parameters of the special Lagrangian and the baryonic semi-invariants of the representation theory of $m$-Kronecker quivers is also discussed, providing a local map between moduli spaces of branes related by homological mirror symmetry.