The Thomas-Yau conjecture is largely motivated by mirror symmetry, based on the analogy between special Lagrangian geometry on the $A$-side (‘symplectic’), and stability conditions in the derived category of coherent sheaves on the $B$-side (‘holomorphic’). It is worth emphasizing that mirror symmetry is quantum by nature, and only sometimes admits classical geometric interpretations. The mathematical predictions of open string mirror symmetry chiefly fall into the following three classes:

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  (Homological mirror symmetry, i.e. the categorical approach) Given a (compact) Calabi-Yau manifold $X$ viewed as a symplectic manifold, one expects to find a (compact) Calabi-Yau manifold $X^{\vee}$ viewed as a complex manifold (sometimes over the Novikov field), such that the (derived, idempotent closed) Fukaya category $D^{\pi}Fuk(X)$ is equivalent as a triangulated category to the derived category of coherent sheaves $Coh(X^{\vee})$. Frequently, there is a preferred $A_{\infty}$ enhancement on both sides, such that $D^{\pi}Fuk(X)\simeq D^{b}Coh(X^{\vee})$ holds as an $A_{\infty}$-equivalence. The concrete implication is that Lagrangian branes $L$ correspond to objects $E$ of $D^{b}Coh(X^{\vee})$, such that the Floer groups are identified with the $Ext$ groups, which explains the name ‘homological’:

  $$HF^{*}(L,L^{\prime})\simeq Ext^{*}(E,E^{\prime}).$$

  Homological mirror symmetry has many ramifications beyond Calabi-Yau manifolds, involving for instance Fano manifolds and Landau-Ginzburg models. As a general feature, homological mirror symmetry only uses pure symplectic topology (vis. a vis. complex geometry) on either side of the mirror. It is the best understood aspect of mirror symmetry, with many special cases proven.

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  (SYZ mirror symmetry, i.e. the moduli space approach) Taken in a somewhat generalised and very optimistic sense ¹⁵¹⁵ 15 The SYZ paper [78] focuses primarily on special Lagrangian torus fibrations, but hints at more general special Lagrangian branes. Some more discussions can be found in Fukaya’s work [34, section 8.5]., the Strominger-Yau-Zaslow philosophy says that after taking into account the quantum instanton corrections, then the moduli space of (semi)stable Lagrangian branes within fixed homology classes acquires a modified complex structure, and can be identified with certain moduli spaces of (semi)stable objects from the mirror side. The stable Lagrangian branes are believed to be related to special Lagrangians via the Thomas-Yau conjecture. Most of the literature concentrates on the restricted case of (special) Lagrangian torus fibrations arising from the large complex structure limit, in which case the moduli space of Lagrangian branes supported on the fibres inside $X$ should recover the moduli space of points on $X^{\vee}$, namely the mirror manifold $X^{\vee}$ itself ¹⁶¹⁶ 16 This SYZ mirror construction has seen the intense research effort on the symplectic side by Auroux [8], Abouzaid, Fukaya, among many others. The very recent progress [83] achieves a non-archimedean mirror from Fukaya categorical considerations, but the nature of singular fibres is still not adequately understood. The skeptics may reasonably doubt if special Lagrangians exist at all in the region with large curvature inside the Calabi-Yau manifold. Even if they do exist, there is insufficient evidence that the fibration structure persists. This is maybe the weakest point of the SYZ conjecture.. Beyond the torus fibration case, the moduli space of Lagrangian branes defined by the Mauer-Cartan equation typically has singularities, which is compatible with the fact that the moduli space of semistable objects in $D^{b}Coh(X^{\vee})$ typically is very singular.

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  (DT theory, i.e. the counting approach) In the special case of Calabi-Yau 3-folds, the moduli spaces/stacks of semistable objects have virtual dimension zero, so one can hope to extract enumerative invariants, which can be thought of as integrals over the moduli space, the most basic version being the Euler number. Donaldson-Thomas theory is a highly elaborate framework for extracting such numbers using virtual techniques, on the side of $D^{b}Coh(X^{\vee})$. It is suggested (e.g in [34, section 8.5]) that one can assign DT invariants to the moduli space of semistable objects in the Fukaya category¹⁷¹⁷ 17 This would presuppose suitable properness and algebraicity on the moduli space of special Lagrangian objects, which is a highly nontrivial claim related to the Thomas-Yau conjecture. For instance, in the SYZ special Lagrangian torus fibration case, compactifying the moduli space requires good understanding in the singular region, which is far beyond current knowledge., and these should be equivalent to the DT invariants on the mirror $X^{\vee}$. The role of Thomas-Yau conjecture is that in principle it transforms a problem about counting special Lagrangian objects, into a categorical framework involving Calabi-Yau $A_{\infty}$-categories and stability conditions, which then supposedly ¹⁸¹⁸ 18 modulo heroic efforts feeds into the grand machinery of Kontsevich and Soibelman [46][47][48][49]. This DT perspective may be regarded as the ultimate goal of Thomas-Yau conjecture ¹⁹¹⁹ 19 As an analogy, the practical calculation of Donaldson’s ASD instanton invariants depends largely on the solution of Hermitian-Yang-Mills equation on algebraic surfaces., and is close to the physical applications involving BPS states counting, although on the $A$-side it is the most removed from mathematical attempts.

It is worth emphasizing that taking stability questions into account requires one to go beyond homological mirror symmetry. While the Fukaya category (vis. a vis. $D^{b}Coh(X^{\vee})$) depends only on the symplectic (holomorphic) data, the stability condition remembers information about the holomorphic volume form (polarisation class), which now sees the $B$-side ($A$-side). As such, neither the above strong version of the SYZ mirror symmetry, nor the DT mirror symmetry are consequences of homological mirror symmetry alone, as indicated already in the SYZ paper [78].²⁰²⁰ 20 In the words of SYZ, they required the full equivalence of the two type II string theories, including the full BPS spectrum. A very general framework that encompasses all the three aspects of mirror symmetry is in Kontsevich and Soibelman [46], but how to fit the $A$-side into this picture is wildly conjectural.