Thomas–Yau-type conjecture for stability conditions on the Fukaya category

Let $(X,\omega)$ be a Calabi–Yau manifold and let $\mathcal{D}^b\mathcal{F}(X,\omega)$ denote its Fukaya category. A stability condition assigns stability to isotopy classes of Lagrangians, while a partial differential equation on the space of Lagrangians specifies a geometric condition such as the special Lagrangian equation.

Fukaya-category stability–equation conjecture. There is a class of stability conditions on the Fukaya category such that for each stability condition in this class, there is an associated partial differential equation on the space of Lagrangians such that stability of an isotopy class of Lagrangians is equivalent to the existence of a Lagrangian in the isotopy class solving the equation, in such a way that the special Lagrangian equation is a special case.

This is a proposed mirror-symmetry extension of the Thomas–Yau picture, relating Bridgeland-type stability on the Fukaya category to geometric partial differential equations. The source leaves the conjecture deliberately vague and notes substantial subtleties, including possible restrictions near the large complex structure limit and the need to formalise the relationship between Bridgeland stability, special Lagrangians, and deformed Hermitian Yang–Mills connections.