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Si Li and author suggested in that, in some cases, the AdS/CFT correspondencecan be formulated in terms of the algebraic operation of Koszul duality. Inthis paper this suggestion is checked explicitly for $M2$ branes in an$\Omega$-background. The algebra of supersymmetric operators on a stack of $K$$M2$ branes is shown to be Koszul dual, in large $K$, to the algebra ofsupersymmetric operators of $11$-dimensional supergravity in an$\Omega$-background (using the formulation of supergravity in an$\Omega$-background presented in arXiv:1610.04144).  The twisted form of supergravity that is used here can be quantized to allorders in perturbation theory. We find that the Koszul duality result holds toall orders in perturbation theory, in both the gravitational theory and thetheory on the $M2$. (However, there is a certain non-linear identification ofthe coupling constants on each side which I was unable to determineexplicitly).  It is also shown that the algebra of operators on $K$ $M2$ branes, as $K \to\infty$, is a quantum double-loop algebra (a two-variable analog of theYangian). This algebra is also the Koszul dual of the algebra of operators onthe gravitational theory. An explicit presentation for this algebra ispresented, and it is shown that this algebra is the unique quantization of itsclassical limit. Some conjectural applications to enumerative geometry ofCalabi-Yau threefolds are also presented.

Holography and Koszul duality: the example of the $M2$ brane