Modelling A-branes with foliations

A bstract 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 |Ω| of basic connected objects, where Ω is 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.