Twisted holography from the B-model on a 7-fold

I study a topological string construction of the holographic duality between Kodaira-Spencer gravity on the Calabi-Yau 7-fold $\mathcal{O}(-1)^4\to\mathbb{PT}$ in the presence of a stack of $N$ backreacted D5 branes wrapping twistor space, $\mathbb{PT}$. The theory on the stack of branes is the twistor uplift of self-dual $\mathcal{N}=4$ gauge theory. I show that turning on a bulk superpotential and twisting the brane theory by the dual supercharge reduces the duality to twisted holography which relates the B-model on AdS$_3\times S^3\cong SL(2,\mathbb{C})$ to the 2d chiral algebra subsector of $\mathcal{N}=4$. I do an analogous computation for the twistor uplift of self-dual $\mathcal{N}=2$ by working on the Calabi-Yau 7-fold $\mathcal{O}(-2,-2)\oplus \mathcal{O}(0,-1)^2\to\mathbb{CP}^1\times\mathbb{PT}$. I also connect twists of the twistor uplift of self-dual $\mathcal{N}=4$ with the matrix model found by supersymmetric localization on $S^4$ and the Dijkgraaf-Vafa matrix model construction.