We study the Kaluza–Klein reduction of various 6d holomorphic theories. The KK reduction is analyzed in the BV formalism, resulting in theories that come from the holomorphic topological twist of 3d \({\mathcal {N}} = 2\) supersymmetric field theories. Effective interactions of the KK theories at the classical level can be obtained at all orders using homotopy transfer theorem. We also analyze a deformation of the theories that comes from deforming the spacetime geometry to \(SL_2(\mathbb {C})\) due to the brane back-reaction. We study the boundary chiral algebras for the various KK theories. Using Koszul duality, we argue that by properly choosing a boundary condition, the boundary chiral algebra coincides with the universal defect chiral algebra of the original theory. This perspective provides a unified framework for accessing the chiral algebras that arise from both twisted holography and celestial holography programs.
