The twisted holography Koszul duality conjecture

Consider a $D$-brane in type IIA or IIB string theory, a brane in the topological string, or a brane in $M$-theory, viewed as a defect in the string theory or supergravity theory. Let $N$ be the number of branes, and perform a twist of the supersymmetric gauge theory on the branes and of the dual supergravity theory. The relevant operator algebras are the algebra of operators of the twisted gauge theory on the stack of $N$ branes in the limit $N\to\infty$ and the algebra of operators in perturbative twisted supergravity on the defect. Twisted holography Koszul duality conjecture. These two algebras are Koszul dual. The statement is expected to require modification when the brane sources flux: in general one expects a deformation of the Koszul dual, while an exact match is expected when no flux is sourced. This gives a mathematical formulation of part of twisted AdS/CFT; the conjecture is supported by examples, including $M2$ branes in an $\Omega$-background, but the general statement remains conjectural.