Despite substantial progress, many essential difficulties still need to be overcome before the dHYM equation can give rise to a Bridgeland stability condition on $D^{b}Coh(X^{\vee})$:

• •

  Can one relax the large phase assumption?

• •

  Is there a generalisation of dHYM equation to higher rank vector bundles? (Currently, there is no well established PDE, let alone how to solve it.)

• •

  Can this story be extended to complexes of vector bundles?

• •

  How can one compare the answer with the A-side of the mirror?

Time will tell how far one can push in this program, but we would like to momentarily play the skeptic’s advocate:

• •

  The differential geometric motivation ³⁰³⁰ 30 There is an independent physics motivation for dHYM from the Dirac-Born-Infeld action. The DBI action is however not an exact result, but depends on the assumption that certain derivative terms of the curvature can be ignored. Unlike the A-model side where the central charge $Z(L)=\int_{L}\Omega$ is believed to hold exactly, on the B-side the central charge formula $Z_{X^{\vee}}(E)=-\int_{X^{\vee}}e^{-\sqrt{-1}\omega_{X^{\vee}}}ch(E)$ is believed to be subject to worldsheet instanton corrections. of dHYM assumes semiflat ambient Kähler metrics. Such metrics only arise naturally if the manifold has toric symmetry, or as a good approximate description for degenerating Calabi-Yau metrics near the large complex structure limit/large volume limit. Near this limit, the dHYM equation may be an improvement on the HYM equation as the mirror version of special Lagrangians. But far away from such limits, the instanton corrections cannot be ignored, and indeed a general Kähler metric has no toric symmetry.

• •

  The reliance on the large phase assumption is a possible indication of a breakdown once we move too far from a large phase limit. In the closely related special Lagrangian graph equation, relaxing the phase condition is known to result in rather severe singularities for the viscosity solutions.

• •

  Being the section of a torus fibration is a serious assumption on the topology of a Lagrangian.

• •

  The difficulty with higher rank vector bundles is a possible indication that $Coh(X^{\vee})$ may be not the natural abelian subcategory of $D^{b}Coh(X^{\vee})$ suited for dHYM. Furthermore, there is no a priori guarantee for arbitrary stability conditions to correspond to PDEs,³¹³¹ 31 If a Bridgeland stability condition arises from a PDE, there is no a priori guarantee that its deformations also come from PDEs. or to have any classical geometrical interpretation at all. In particular, the $B$-side mirror to the hypothetical special Lagrangian stability condition may well be an abstract stability condition with no particular PDE interpretation.

• •

  Due to the lack of imagination, it is hard to see how PDEs can know about the degree shifts of a complex of vector bundles, and how it can detect homotopy equivalence of complexes.

If we take this skeptic view, then we do not expect an exact comparison between the hypothetical Bridgeland stabilities on both sides of the mirror, without adding very substantial assumptions such as toric symmetry. But when HYM shares the same qualitative features as dHYM, which admit a mirror interpretation, it lends more plausibility for these features to appear also on the special Lagrangians.