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

TL;DR

This paper explicitly verifies the Koszul duality between the algebra of supersymmetric operators on M2 branes and 11-dimensional supergravity in an Omega-background, demonstrating the duality holds to all orders in perturbation theory.

Contribution

It provides a detailed check of the algebraic Koszul duality in the context of M2 branes and supergravity, including the explicit algebraic structures involved.

Findings

Koszul duality holds to all orders in perturbation theory

Algebra of operators on M2 branes becomes a quantum double-loop algebra at large K

Explicit presentation of the quantum algebra and its unique quantization

Abstract

Si Li and author suggested in that, in some cases, the AdS/CFT correspondence can be formulated in terms of the algebraic operation of Koszul duality. In this 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 of supersymmetric 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 all orders in perturbation theory. We find that the Koszul duality result holds to all orders in perturbation theory, in both the gravitational theory and the theory on the $M2$. (However, there is a certain non-linear identification of the coupling constants on each side which I was unable to determine explicitly). 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 the Yangian). This algebra is also the Koszul dual of the algebra of operators on the gravitational theory. An explicit presentation for this algebra is presented, and it is shown that this algebra is the unique quantization of its classical limit. Some conjectural applications to enumerative geometry of Calabi-Yau threefolds are also presented.