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

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.

References in corpus (6)

• Supersymmetric gauge theory and the Yangian
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• Twisted supergravity and its quantization
• Quantization of open-closed BCOV theory, I
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Cited by in corpus (19)

• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Cohomological Hall algebras, vertex algebras and instantons
• Twisted Supergravity and Koszul Duality: A case study in AdS${}_3$
• Twisted M2 brane holography and sphere correlation functions
• Feynman diagrams and $\Omega$-deformed M-theory
• Twisted eleven-dimensional supergravity
• Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
• Monopole Operators and Bulk-Boundary Relation in Holomorphic Topological Theories
• Twisted holography of defect fusions
• Homotopy Algebras in Higher Spin Theory
• Evidence for an Algebra of ${\mathbf{G}}_2$ Instantons
• Twisted holography on AdS${}_3\times S^3\times K3$ & the planar chiral algebra
• On even spin $W_{\infty }$
• R-matrices and Miura operators in 5d Chern-Simons theory
• A domain wall in twisted M-theory
• A String Dual for Partially Topological Chern-Simons-Matter Theories
• The elliptic Casimir connection of a simple Lie algebra
• Fermi-gas correlators of ADHM theory and triality symmetry
• The matrix-extended $W_{1+\infty }$ algebra