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We study the Kaluza-Klein reduction of various $6d$ holomorphic theories. TheKK reduction is analyzed in the BV formalism, resulting in theories that comefrom the holomorphic topological twist of $3d$ $\mathcal{N} = 2$ supersymmetricfield theories. Effective interactions of the KK theories at the classicallevel can be obtained at all orders using homotopy transfer theorem. We alsoanalyze a deformation of the theories that comes from deforming the spacetimegeometry to $SL_2(\mathbb{C})$ due to the brane back-reaction. We study theboundary chiral algebras for the various KK theories. Using Koszul duality, weargue that by properly choosing a boundary condition, the boundary chiralalgebra coincides with the universal defect chiral algebra of the originaltheory. This perspective provides a unified framework for accessing the chiralalgebras that arise from both twisted holography and celestial holographyprograms.

Twisted Holography and Celestial Holography from Boundary Chiral Algebra