Kapustin's hyperkähler Fukaya–deformation-quantisation correspondence

Let $(\mathrm{X},\mathrm{I},\mathrm{J},\mathrm{K})$ be a hyperkähler manifold with symplectic form $\omega_\mathrm{J}$, and set

$$\sigma_\mathrm{I}=\omega_\mathrm{J}+i\omega_\mathrm{K}.$$

Let $\mathcal{DF}(\mathrm{X},\omega_\mathrm{J})$ be the Fukaya category and let $\mathbf{D}_{\mathrm{dg,h}}(\widehat{\mathscr{W}}_\mathrm{X})$ be the differential graded category of holonomic deformation-quantisation modules. Kapustin's correspondence conjecture. There is a quasi-equivalence

$$\mathcal{DF}(\mathrm{X},\omega_\mathrm{J})\simeq \mathrm{Ind}_{\mathrm{Nov}/\mathbf{C}((\hbar))}\left(\mathbf{D}_{\mathrm{dg,h}}(\widehat{\mathscr{W}}_\mathrm{X})\right),$$

between the Fukaya category and the category associated to the holomorphic symplectic manifold $(\mathrm{X},\mathrm{I},\sigma_\mathrm{I})$. This is a categorical form of the A/B-brane duality suggested by Kapustin; the paper presents it as a conjectural correspondence and does not prove it.