Our goal is to look for obstructions to the existence of special Lagrangians within given $D^{b}Fuk(X)$ classes, which is the ‘easy direction’ of the conjectural stability condition. Before specializing to a technically oversimplified setup, we first explain the features we expect from these obstructions, which may hold in much more general contexts. The mirror analogy (cf. our discussion on the $\mu$-stability in section 2.5) suggests:

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  The obstructions are associated to certain positivity of signs, which essentially depend on the integrability of Kähler geometry.

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  The quantity involved in the obstruction can be expressed as an integral over a moduli space of worldsheet instantons (i.e. holomorphic curves), and its sign comes from a pointwise positivity of the integrand on the moduli space.

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  The input from Floer theory is associated to a distinguished triangle in $D^{b}Fuk(X)$, or possible generalisations to several Lagrangians.

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  The role of the holomorphic volume form enters via cohomological integrals.

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  There is no need for the complex Monge-Ampère equation. Only the almost Calabi-Yau condition is needed.

Furthermore, out of the many moduli spaces that may arise in Floer theory, we will only make use of certain $(n-1)$-dimensional moduli spaces of holomorphic curves, whose associated $(n+1)$-dimensional universal family provides bordism currents between the $n$-dimensional Lagrangians. Here are some a priori reasons why we restrict attention to these:

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  The holomorphic volume form is naturally integrated over $n$-cycles. This explains the dimension.

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  We need bordism currents canonically associated to the distinguished triangles. In Floer theory, the $A_{\infty}$-structure only becomes an invariant when considered as a whole, and individual $A_{\infty}$ products are not invariants, so invariance constrains how moduli spaces can enter into stability conditions. As mentioned in section 3.1, the existence of the bordism current is the geometric manifestation of linear relations in the zeroth Hochschild homology $HH_{0}$ of the Fukaya category, which contains important invariant information.

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  From a variational viewpoint which will be discussed more fully in Chapter 5, it is desirable to extend Floer theory to Lagrangians with much weaker regularity, in the varifold and current sense. We shall explain there that most of Floer cohomologies and $A_{\infty}$ products cannot be expected to pass to the limit when the Lagrangians degenerate in such weak topologies, and we hope that the bordism currents we use are among the few pieces of Floer theory that may be well behaved under rather severe degenerations of Lagrangians.

These requirements are very stringent. We notice two other features:

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  We shall crucially rely on the almost calibrated condition.

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  When we test the stability of $L$ via the distinguished triangle $L_{1}\to L\to L_{2}\to L_{1}[1]$, we do not wish to assume $L_{1}$ or $L_{2}$ is special Lagrangian. In our view, stability conditions should be expressed in Floer theoretic terms, without a priori knowledge of what special Lagrangians there are inside a given almost Calabi-Yau manifold.